Compactifications of String/M-Theory and
the Swampland
A Study of the AdS4 Mass Spectrum of Eleven-Dimensional
Supergravity on the Squashed Seven-Sphere
Master’s thesis in Physics
JOEL KARLSSON
DEPARTMENT OF PHYSICS
CHALMERS UNIVERSITY OF TECHNOLOGY
Gothenburg, Sweden 2021
www.chalmers.se

Master’s thesis 2021
Compactifications of String/M-Theory and
the Swampland
A Study of the AdS4 Mass Spectrum of Eleven-Dimensional
Supergravity on the Squashed Seven-Sphere
Joel Karlsson
Department of Physics
Division of Subatomic, High Energy and Plasma Physics
Group of Mathematical Physics
Chalmers University of Technology
Gothenburg, Sweden 2021
Compactifications of String/M-Theory and the Swampland:
A Study of the AdS4 Mass Spectrum of Eleven-Dimensional
Supergravity on the Squashed Seven-Sphere
Joel Karlsson
© Joel Karlsson, 2021.
Supervisor: Bengt E. W. Nilsson, Department of Physics
Master’s Thesis 2021
Department of Physics
Division of Subatomic, High Energy and Plasma Physics
Group of Mathematical Physics
Chalmers University of Technology
SE-412 96 Gothenburg
Telephone: +46 31 772 1000
Typeset in LATEX
Printed by Chalmers Digital Printing
Gothenburg, Sweden 2021
Compactifications of String/M-Theory and the Swampland:
A Study of the AdS4 Mass Spectrum of Eleven-Dimensional
Supergravity on the Squashed Seven-Sphere
Joel Karlsson
Department of Physics
Chalmers University of Technology
Abstract
The landscape of possible four-dimensional low-energy effective theories arising from
compactifications of string/M-theory seems vast, one estimate suggesting the number
10272 000. This might lead one to believe that any consistent-looking effective field
theory coupled to gravity can be obtained as a low-energy limit of string theory.
However, a set of “swampland” conjectures suggests that this is not true and that,
in fact, there is an even larger set of effective field theories that cannot be obtained
in this way. In particular, the AdS instability swampland conjecture asserts that
nonsupersymmetric anti-de Sitter vacua are unstable. These swampland criteria can
have implications for, for instance, low-energy physics and cosmology.
M-theory is a nonperturbative unification of all superstring theories. Its low-energy
limit, eleven-dimensional supergravity, admits two compactifications on the squashed
seven-sphere. One of the solutions has one unbroken supersymmetry (N = 1) while
the other has none (N = 0). Due to the AdS instability swampland conjecture, the
latter should be unstable. However, this has not been demonstrated explicitly. To
study the stability of the N = 0 vacuum, we investigate the mass spectrum of the
theory. The main advancement compared to previous attempts is the realisation that
all mass operators in the Freund–Rubin compactification are related to a universal
Laplacian. This allows us to make significant simplifications and relate Weyl tensor
terms to group invariants. One limitation of the group-theoretical method we employ
is that it can lead to false roots. This is remedied, at least in part, by demanding
that the fields form supermultiplets in the N = 1 case. Although we arrive at an
eigenvalue spectrum for all operators of interest, there is a hint that the results may
be incomplete. Thus, we do not reach a decisive conclusion regarding the investigated
type of instability.
Keywords: squashed seven-sphere, mass spectrum, flux compactification, M-theory,
string theory.
iii

Acknowledgements
I would like to thank my supervisor, Bengt E. W. Nilsson, for his support and for
suggesting the project. Bengt has always taken the time to answer and discuss the
questions I have had. I would also like to thank my fellow students for interesting
discussions on mathematical and physical subjects, in particular Rolf Andréasson,
Markus Klyver, Ludvig Svensson and Eric Nilsson. An additional thank you to
Ludvig for providing useful comments on my text. Lastly, a big thank you to all of
my friends and family for their support.
Joel Karlsson, Gothenburg, June 2021
v

Table of contents
List of tables ix
1 Introduction 1
1.1 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The swampland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 The weak gravity and AdS instability swampland conjectures . 5
1.2.2 The swampland, the Standard Model and cosmology . . . . . 6
1.2.3 Unstable nonsupersymmetric AdS vacua and GSMOs . . . . . 7
2 Supersymmetry and supergravity 9
2.1 Supersymmetry in Minkowski spacetime . . . . . . . . . . . . . . . . 10
2.1.1 The super-Poincaré algebra . . . . . . . . . . . . . . . . . . . 11
2.1.2 Supermultiplets . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 The Wess–Zumino model . . . . . . . . . . . . . . . . . . . . . 16
2.2 Superspace formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Revisiting the Wess–Zumino model . . . . . . . . . . . . . . . 20
2.2.3 Superdifferential forms . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Coordinate transformations in superspace . . . . . . . . . . . 27
2.3 Eleven-dimensional supergravity . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2 The spacetime theory . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.3 Component formulation with left-action . . . . . . . . . . . . 37
3 Supergravity compactifications 39
3.1 Freund–Rubin compactification . . . . . . . . . . . . . . . . . . . . . 40
3.2 Anti-de Sitter, mass operators and supersymmetry . . . . . . . . . . . 46
3.2.1 Defining mass in AdS . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.2 Masses from operators on the internal space . . . . . . . . . . 47
3.2.3 Spin(3, 2)-representations and supersymmetry . . . . . . . . . 50
3.2.4 Differential operators and a universal Laplacian . . . . . . . . 52
4 Homogeneous spaces 57
4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Harmonic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 The coset master equation . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Squashed sphere geometry 69
5.1 Squashed S7 as a principal bundle . . . . . . . . . . . . . . . . . . . . 69
5.2 Coset construction with arbitrary squashing . . . . . . . . . . . . . . 73
vii
Table of contents
6 Eigenvalue spectra of the squashed seven-sphere 81
6.1 0-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 1-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3 2-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.4 Symmetric rank-2 tensors . . . . . . . . . . . . . . . . . . . . . . . . 86
6.5 3-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.6 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.7 Vector-spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.8 Spectrum summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7 Mass spectrum and supermultiplets 99
7.1 Unbroken supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2 The left-squashed N = 1 vacuum . . . . . . . . . . . . . . . . . . . . 100
8 Conclusions 107
A Conventions and representations 109
A.1 Representations and index notation . . . . . . . . . . . . . . . . . . . 109
A.2 The Lorentz group and special orthogonal groups . . . . . . . . . . . 110
A.3 Quadratic Casimirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B Spinors 115
B.1 Spinors in arbitrary even dimension . . . . . . . . . . . . . . . . . . . 116
B.2 Spinors in arbitrary odd dimension . . . . . . . . . . . . . . . . . . . 119
B.3 Spinors in four dimensions . . . . . . . . . . . . . . . . . . . . . . . . 120
B.4 Spinors in eleven dimensions . . . . . . . . . . . . . . . . . . . . . . . 122
C Octonions 125
C.1 Spin(7), octonions and G2 . . . . . . . . . . . . . . . . . . . . . . . . 127
C.2 Structure constant identities . . . . . . . . . . . . . . . . . . . . . . . 129
D Differential forms 131
D.1 The Hodge dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
D.2 The exterior derivative and de Rham cohomology . . . . . . . . . . . 133
D.3 The codifferential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
D.4 The Hodge–de Rham operator and harmonic forms . . . . . . . . . . 135
E Bundles, gauge theory and gravity 137
E.1 Fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
E.2 Gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
E.3 Einstein–Cartan gravity . . . . . . . . . . . . . . . . . . . . . . . . . 141
F Grassmann numbers 145
G Solving the supergravity Bianchi identities 149
G.1 Solution to the supergravity Bianchi identities . . . . . . . . . . . . . 162
References 163
viii
List of tables
2.1 Massive supermultiplets in four-dimensional Minkowski spacetime. . . 15
2.2 Massless supermultiplets in four-dimensional Minkowski spacetime. . 16
3.1 Mass operators in Freund–Rubin compactification. . . . . . . . . . . . 48
3.2 Relation between energy and mass in AdS4. . . . . . . . . . . . . . . 51
3.3 The unitary N = 1 supermultiplets in AdS4. . . . . . . . . . . . . . . 52
6.1 Decomposition of Spin(7)-representations when restricted to G2. . . . 82
6.2 Summary of eigenvalues of differential operators on the squashed S7. 97
7.1 Possible particle masses in the left-squashed vacuum. . . . . . . . . . 101
7.2 Possible asymptotic values of E0 in the left-squashed vacuum. . . . . 102
7.3 Multiplicities of irreducible G-representations. . . . . . . . . . . . . . 103
7.4 Asymptotic massive higher spin supermultiplet families. . . . . . . . . 104
7.5 Asymptotic Wess–Zumino supermultiplet families. . . . . . . . . . . . 105
A.1 Group and Lie algebra transformation laws. . . . . . . . . . . . . . . 110
A.2 Conventions for quadratic Casimir operators. . . . . . . . . . . . . . . 113
B.1 Symmetric and antisymmetric matrices in the D = 11 Γ-basis. . . . . 123
ix
List of tables
x
1. Introduction
The perhaps greatest challenge in fundamental physics is quantum gravity. During
the twentieth century, two theories transformed the scientific worldview radically.
The first, Einstein’s theory of relativity, tells us that space and time are not absolute
but relative to the observer and that only spacetime, not space and time separately,
is physically meaningful (special relativity). Furthermore, spacetime is dynamical: it
curves as a result of the presence of matter and energy and gravity is a manifestation
of this curvature (general relativity). The second is quantum theory1, describing
matter by wavefunctions with nonclassical properties such as entanglement and only
giving probabilistic predictions for observable measurement outcomes.
General relativity and the Standard Model of particle physics, a (special-)relativistic
quantum field theory, are often considered the two most successful theories in all
of physics. Yet, they seem incompatible [1]. Due to the nonclassical properties of
reality, demonstrated for instance by experiments that violate the Bell inequality [2],
the consensus is that gravity has to be quantum or emergent from more fundamental
quantum degrees of freedom. Still, the quantum nature of gravity remains unclear.
There are, however, several approaches to quantum gravity, including string/M-theory
[1] and loop quantum gravity [3]. This thesis is concerned with the former.
String theory is a framework that generalises quantum mechanics to extended one-
dimensional objects, strings.2 There are five (critical) string theories with fermions,
known as Type I, Type IIA, Type IIB, heterotic SO(32) and heterotic E8×E8, which
all live in ten dimensions [1]. These are related by various dualities, for instance
T-duality (inverting the radius of a compactified dimension) and S-duality (inverting
the string coupling constant) [4]. Thus, they may be viewed as perturbative regimes
of an underlying nonperturbative theory known as M-theory [5]. Remarkably, the
low-energy limit of M-theory is eleven-dimensional supergravity, which, as the name
suggests, is formulated in eleven dimensions rather than the ten of the superstring
theories [4]. Eleven is special in the sense that it is the maximum number of
dimensions in which one can have supersymmetry without particles with spins
greater than two and the highest dimension that admits super p-branes [5].3
One may ask how string theory in ten dimensions and supergravity in eleven can
have any prospect of describing the four-dimensional universe we perceive, with
1Quantum theory is not a physical theory but a theoretical framework. Similarly, there is a
theoretical framework subsuming general relativity.
2String theory also contains higher-dimensional extended objects known as branes.
3This depends crucially on the assumption of Minkowski signature and can be avoided in twelve
dimensions with two timelike directions, a fact that is used in F-theory [5], [6].
1
1. Introduction
three dimensions of space and one of time. One possible answer is that some of
the dimensions are small and compact and, therefore, not observed in experiments.
This is known as Kaluza–Klein compactification [1]. A useful analogy is the surface
of a rope, which looks one-dimensional from afar but is really two-dimensional.
Another, not mutually exclusive, possibility is the brane-world scenario, in which
our four-dimensional spacetime corresponds to a defect created by branes [1].4
Although compact dimensions can solve the problem of why we do not observe ten
or eleven dimensions in experiments, it also gives rise to another problem, namely,
the problem of choosing a compact manifold. When compactifying M-theory to four
dimensions, one obtains a low-energy effective theory which depends on the specifics
of the compactification. One estimate suggests that there is at least 10272 000 such
vacua [7], although it is unclear whether they are all distinct or if some are related
by dualities [6]. It is then reasonable to ask whether any consistent-looking effective
field theory coupled to gravity can be obtained as a low-energy limit of an M-theory
vacuum. A set of “swampland” conjectures claims, to the contrary, that the space of
effective field theories that cannot be obtained in this way is even larger [6], [8]. In
contrast to the string landscape, these are said to belong to the swampland.
When compactifying eleven-dimensional supergravity to four dimensions, the internal
space, that is, the compact dimensions, is of course seven-dimensional. Thus, the
seven-sphere, S7, may be used as internal space. In the Freund–Rubin ansatz, a
special form of flux compactification, the compact manifold has to be an Einstein
space, Rmn ∝ gmn, to satisfy the supergravity field equations. The seven-sphere
admits two Einstein metrics, the usual maximally symmetric round one and a
squashed metric with fewer isometries. There are two vacua with the squashed
seven-sphere as internal space, related by “skew-whiffing”, that is, by reversing the
direction of the flux. One of these has one unbroken supersymmetry while the other
has none [9]. We will refer to the N = 1 solution as the left-squashed vacuum
and the N = 0 solution as right-squashed. Since the metric on the internal space
is a spacetime scalar, the four-dimensional theories obtained from the squashed
sphere can be viewed as spontaneously broken phases of the N = 8 maximally
supersymmetric theory obtained from the round S7 [9], [10].
One swampland conjecture, which we will refer to as the AdS instability swampland
conjecture, asserts that nonsupersymmetric anti-de Sitter (AdS) vacua are unstable
and, hence, belong to the swampland [11]. Accordingly, the right-squashed vacuum
described above, which is AdS, is expected to be unstable. However, no instability
has been explicitly demonstrated for this vacuum. For instance, due to the relation
with the left-squashed N = 1 vacuum, the Breitenlohner-Freedman bound [12], [13]
is not violated [14]. If present, the instability must, therefore, arise in some other
way. One possibility is that the vacuum is shifted significantly by an instability
indicated by a tadpole. For the shift to be significant, the field with the tadpole has
to correspond to a global singlet marginal operator (GSMO) in the conformal field
4In such scenarios, there may be noncompact extra dimensions. If the space is a warped product,
one can still get four-dimensional gravity with an inverse-square law [1].
2
1.1. Thesis outline
theory (CFT) dual to the supergravity theory [15]. Other possibilities for instabilities
include the formation of a “bubble of nothing” [16] and brane-jet instabilities [17].
In this thesis, we focus on tadpole instabilities related to GSMOs. As explained in
[18], such instabilities can occur not only for the elementary scalar fields in the theory
but also for composite fields. Hence, one needs detailed information of considerable
parts of the Kaluza–Klein mass spectrum to investigate whether any GSMO-related
instability occurs. Since the masses are determined by the eigenvalue spectra of
certain differential operators on the internal space [19], we aim to derive said spectra.
1.1 Thesis outline
The thesis is structured as follows. In the remainder of this chapter, we provide
some context for the problem and elaborate on the kind of instability we are going
to investigate. The reader is assumed to be familiar with quantum field theory and
general relativity. Also, familiarity with some aspects of string theory is assumed,
at least on a conceptual level. Detailed knowledge about supersymmetry and
supergravity is not assumed and we, therefore, present some relevant background,
including the superspace construction of eleven-dimensional supergravity, in chapter 2.
In chapter 3, we discuss compactifications of eleven-dimensional supergravity, in
particular the Freund–Rubin ansatz. We also discuss the concept of mass in AdS
and present the expressions for the mass operators, in terms of differential operators
on the internal space, and the N = 1 supermultiplets in AdS4. In the last section of
the chapter, section 3.2.4, we relate the mass operators to a universal Laplacian that
later proves to be of great use. This realisation is not present in the literature as far
as we know.
In chapter 4, we review aspects of the geometry of and harmonic analysis on
homogeneous spaces G/H. Some of the material is based on [20] but we generalise
the discussion to an arbitrary G-invariant metric. We also derive the equations,
in section 4.3, that are later used to find the eigenvalue spectra of the squashed
seven-sphere.
Two constructions of the squashed-seven sphere are presented in chapter 5. We use
the above to realise the squashed S7 with arbitrary squashing parameter as a coset
space G/H with G = Sp(2)× Sp(1) and H ' Sp(1)× Sp(1). The Einstein-squashed
seven-sphere, on which we compactify eleven-dimensional supergravity, is of course
of particular interest. However, we also see that the round S7 comes from a metric
of indefinite signature on G.
The eigenvalue spectra of all operators of interest on the (Einstein-)squashed seven-
sphere are derived in chapter 6, using the coset construction of the previous chapter
and the equations derived in section 4.3. We do not solve any differential equations
explicitly but rather use group theoretical techniques. A limitation of the method is
that it can lead to false roots.
3
1. Introduction
Recently, large parts of the eigenvalue spectra of the squashed seven-sphere were
uploaded to arXiv [21], although substantial parts were known before that as well.
Our calculations, which are completely independent, agree with all previous results
and extend them by providing eigenvalues for iD/ 3/2 independent of supersymmetry
requirements.
Having investigated the squashed seven-sphere in detail, we return to eleven-dimen-
sional supergravity and derive the mass spectrum and supermultiplet structure upon
compactification on the squashed S7 in chapter 7. We focus on the left-squashed
N = 1 vacuum and use the fact that the fields must fall into supermultiplets to
eliminate false roots. In this way, we arrive at eigenvalue spectra that are consistent
with supersymmetry. However, there seem to be degeneracies that we have not been
able to explain. This could be taken as an argument for our results being incomplete.
The analysis regarding whether GSMO-related instabilities can occur in the right-
squashed vacuum is not completed. We end with some concluding remarks and a
discussion on what is needed for this in chapter 8. Further results will hopefully be
presented in a future publication [22].
Conventions, notation and some mathematical preliminaries are presented in appen-
dices A to F. In appendix G we solve the Bianchi identities of eleven-dimensional
supergravity, a calculation too long to fit in the main text.
We do not attempt to always give complete lists of original references. In particular,
several textbooks are used for well-known results, especially when of mathematical
nature. Also, the review article [19] is used for large parts of the background material
regarding compactifications of eleven-dimensional supergravity.
1.2 The swampland
As already mentioned, the swampland program aims to demonstrate that not all
consistent-looking effective field theories (EFTs) coupled to gravity can be obtained
from M-theory compactifications. Theories that cannot be obtained in this way are
thought to not admit a finite UV-completion and are therefore considered inconsistent
[6]. Some swampland conjectures are concerned with quantum gravity more generally
while others are concerned with string/M-theory more specifically. In this section,
we provide some background regarding the swampland program to put the project in
a wider context and expand on some details concerning GSMO-related instabilities.
Before turning to specific conjectures and their consequences, we note that many
of the swampland conjectures have not been rigorously proven. Indeed, the lack
of a complete, nonperturbative definition of M-theory is a significant obstruction
to such proofs. Instead, the conjectures are motivated by, among other things,
realisations and examples from string theory and black hole physics [6]. To be
able to claim predictions from string theory based on swampland conjectures, one
must be confident enough of the validity of the relevant conjectures, within the
4
1.2. The swampland
theoretical framework, to believe that the fault lies in string theory itself, and not
the conjectures, in the event that repeated experiments would violate the prediction.
Otherwise, the experiments can only be said to test the swampland conjectures and
not string theory. This motivates further theoretical study of the subject to refine
and strengthen the conjectures.
1.2.1 The weak gravity and AdS instability swampland
conjectures
The weak gravity conjecture states, loosely, that gravity is the weakest force in any
consistent theory of quantum gravity [23]. More specifically, for a U(1) gauge field,
the conjecture asserts that there must be a state of mass M and charge Q such that
M ≤ |Q|, (1.1)
MPl
where MPl is the Planck mass [23]. Several arguments motivate the conjecture [6],
[11], [23]. Firstly, it agrees with what is observed in nature and known string theory
compactifications [23]. It can also be motivated by black hole physics as follows.
A Reissner–Nordström black hole, that is, a charged black hole in 4 dimensions, is
described by the metric
( r2 ) ( r2 )−1
ds2 = − 1− 2 rM + Q 22 dt + 1− 2
rM + Q dr22 + r
2dΩ2, (1.2)
r r r r
where rM = `Pl M/MPl = rS/2 is half of the Schwarzschild radius, rQ = `PlQ is the
length scale associated with the charge and dΩ2 is the usual metric on S2. We see
from the metric that there are horizons a√t
r = rM ± r2M − r2Q. (1.3)
If we imagine turning up the charge from Q = 0, corresponding to a Schwarzschild
black hole, the outer horizon shrinks while the inner one grows. When r2Q = r2M ,
corresponding to M/MPl = |Q|, the two coincide and the black hole is said to be
extremal. For larger charges, there is no horizon and we get a naked singularity,
violating the cosmic censorship hypothesis. Thus, for extremal black holes to be
able to evaporate via Hawking radiation without violating the cosmic censorship
hypothesis, there has to be a state satisfying (1.1). For macroscopic black holes
with M MPl, such evaporation is expected to be possible to avoid large numbers
of Planck scale black hole remnants and large numbers of exactly stable objects
not protected by symmetry [23]. This does not apply to Bogomol’nyi–Prasad–
Sommerfield (BPS) states, which saturate (1.1) and whose stability is protected by
supersymmetry [4], [11]. The weak gravity conjecture applies to charged branes in
string theory as well; the gravitational attraction is conjectured to be weaker or
equally strong as the electric repulsion [11].
There is a sharpened version of this conjecture that states that equality only occurs for
BPS states in supersymmetric theories [11]. This can be motivated by the argument
5
1. Introduction
presented above since the phase space of the emission of particles saturating (1.1)
vanishes [11]. Note that not all states must satisfy (1.1); it is only required that
there exists some state satisfying it.
The sharpened weak gravity conjecture has a consequence related to nonsupersym-
metric AdS and holography. To see this, we consider Maldacena’s [24] original
construction of the AdS/CFT correspondence. The conformal field theory lives on a
stack of N coincident branes and decouples from the bulk theory in the low-energy
limit. The geometry is described by a black brane supergravity solution, valid
for large N , whose near-horizon limit typically reduces to a product of AdS and a
sphere. In the nonsupersymmetric setting, the electric repulsion between the branes is
stronger than the gravitational attraction, by the sharpened weak gravity conjecture
[11]. This renders the system unstable. Furthermore, the lifetime approaches zero in
the near-horizon limit due to the gravitational time dilation [11].
Although the above depends somewhat on the specific construction, [11] conjectures
that nonsupersymmetric AdS holography with a low-energy description in terms of
finitely many matter fields coupled to Einstein gravity belongs to the swampland. If
true, this means that all nonsupersymmetric AdS vacua of M-theory are unstable.
We refer to this as the AdS instability swampland conjecture5.
1.2.2 The swampland, the Standard Model and cosmology
Some swampland conjectures can be related to the Standard Model of particle
physics or cosmology. Here, we describe a couple of examples of this. The first
example comes from considering compactifications of the Standard Model to three
or two dimensions. At first, this might seem peculiar but if the Standard Model
can be obtained from M-theory so can its compactifications. Thus, if stable AdS
vacua can be obtained from compactifications of the Standard Model, the Standard
Model itself belongs to the swampland according to the AdS instability swampland
conjecture. Aspects of this are investigated in [29]–[31]. In particular, [29], [30] find
that the Standard Model augmented with Majorana neutrino masses gives rise to
AdS vacua after compactification with current values of the neutrino masses and the
cosmological constant. However, [30] also finds that this can be avoided by adding
a light beyond-the-Standard-Model particle or if the neutrinos are Dirac fermions
and the lightest neutrino is sufficiently light.6 As pointed out in [30], the above
argument can be reversed to provide a lower bound on the cosmological constant
based on the neutrino masses. Note that the above constraints only apply if the AdS
vacuum obtained from compactification of the Standard Model is stable, as discussed
in [30], [31].
5Not to be confused with the (gravitational) AdS instability conjecture, which asserts that AdS
is unstable to black hole formation under arbitrary small perturbations [25], [26] and has been
proven for some gravity-matter systems [27], [28].
6Note that the smallest neutrino mass is not bounded from below by current neutrino oscillation
experiments.
6
1.2. The swampland
The implications of swampland conjectures have also been studied in the context of
cosmology. This is based, for instance, on the swampland conjecture known as the
refined de Sitter conjecture which states that the effective low-energy potential V (φ)
for scalar fields φi must satisfy
‖∇ ‖ ≥ c
′
V V or min∇2 cV ≤ − 2 V, (1.4)MPl MPl
for universal positive constants c, c′ ∼ O(1), in any consistent theory of quantum
gravity [32], [33]. In these expressions, ‖∇V ‖2 = gij(φ)∇iV∇jV where gij(φ) is
the field-space metric from the kinetic term in the Lagrangian, that is, Lkin. =
−gij(φ)/2 ∂ µµφi∂ φj, and the minimum refers to the minimum eigenvalue of the
Hessian ∇2V in an orthonormal field-frame. The name of the conjecture comes from
the fact that it excludes (meta-)stable de Sitter vacua [33]. If true, this would imply
that the state of the universe is unstable. One possibility is a quintessence model
where the cosmological “constant” asymptotically goes to zero with cosmic time,
which would have consequences for the dark energy equation of state [6]. Notably,
recent studies based on observations suggest an evolving equation of state for dark
energy [34]. However, in [35], it is argued that a metastable de Sitter vacuum
produced by the KKLT (Kachru–Kallosh–Linde–Trivedi) mechanism [36] avoids the
problems motivating the swampland conjecture and KKLT could, therefore, still be
a viable mechanism for producing de Sitter vacua in string theory. They further
note that many proposed quintessence models, including those presented in [32], are
excluded at high significance by cosmological data.
In [37], the refined de Sitter conjecture is applied to single-field inflation models.
They consider, in particular, the ratio between scalar and tensor modes in primordial
fluctuations, r, and the scalar spectral index, ns, parameterising the scale dependence
of the scalar fluctuations. For consistency between observational data and the single-
field slow-roll inflation model, they find that c . O(0.1) or c′ . O(0.01) depending
on which inequality in (1.4) applies to the inflaton potential. Depending on the
precision of the statement that c, c′ ∼ O(1) in the swampland conjecture, this result
is in considerable tension with the conjecture. As stated above, the refined de Sitter
conjecture should apply to this model. The validity of this application has however
been questioned in [35].
1.2.3 Unstable nonsupersymmetric AdS vacua and GSMOs
As already mentioned, the AdS instability swampland conjecture asserts that non-
supersymmetric AdS belongs to the swampland. Here, we elaborate on the kind of
instability we aim to investigate for the squashed seven-sphere compactification of
eleven-dimensional supergravity, namely, instabilities related to tadpoles and global
singlet marginal operators (GSMOs).
Consider an M-theory vacuum dual to a conformal field theory (CFT) in the N →
∞ limit. At large N < ∞, there can be tadpoles in the supergravity theory,
corresponding to 1/N corrections of the β-functions of the dual CFT, that signifies
7
1. Introduction
a shift of the true vacuum [18]. This does not happen in supersymmetric cases but
can lead to an instability for nonsupersymmetric theories [18]. The question is then
whether there is a 1/N perturbed vacuum close by in parameter space or not, the
latter signalling instability. In [15], it is argued that only fields dual to operators
that are marginal in the N → ∞ limit can give rise to such instabilities. This is
based on the 1/N expansion of the β-functions, which is of the form
a
β(g) ∼ (∆− d)(g − g∗) + + . . . , (1.5)
N
for a coupling constant g ∼ g∗ of some operator O, where g = g∗ at the N → ∞
conformal fixed point, d is the spacetime dimension of the CFT and ∆ is the scaling
dimension of the operator O [18]. From this, we see that the correction to the
fixed point is of order 1/N and, thus, goes to zero in the large N limit, as long as
∆− d 6= 0. For a marginal operator, ∆ = d and the 1/N correction of the β-function
may eliminate the conformal fixed point completely. This implies that the limit
N →∞ cannot be taken smoothly since the theory would flow to a point far from
g∗ for all finite N [18].
If the β-function of a marginal operator has a saddle point at g∗ or the 1/N correction
is in the right direction in the case of a local extremum, the fixed point would only
receive a small 1/N correction [15]. Thus, the presence of a marginal operator does
not imply that the vacuum is unstable [18].
A tadpole instability can only develop for fields that are neutral with respect to the
gauge symmetries in the supergravity theory since it would, otherwise, explicitly
break gauge invariance [15]. Thus, the corresponding operator need not only be
marginal but also invariant under the global symmetries of the CFT, that is, it
has to be a GSMO [15], [18]. However, as emphasised in [18], the GSMO can be a
multi-trace operator, corresponding to a composite field in the supergravity theory.
To be able to examine the presence of GSMO-related instabilities from the super-
gravity side, we need the relation between the scaling dimension ∆ and properties of
the fields in AdS. For this, the picture of the AdS/CFT correspondence presented in
[38], [39], where it was realised that the CFT lives on the conformal boundary of
AdS, is useful.7 The scaling dimension of an operator is then seen to be related to
the asymptotic behaviour of the dual field [38] and coincides with the dimensionless
energy E0 of the field [18]. Thus, in the case of an AdS4 vacuum, GSMOs corresponds
to (possibly composite) fields with E0 = 3. In [18], an argument that such GSMOs are
always present in nonsupersymmetric vacua related to N ≥ 2 vacua by skew-whiffing
is presented. However, the situation for the N = 0 vacuum of eleven-dimensional
supergravity compactified on the squashed seven-sphere, whose skew-whiffed partner
has N = 1, remains unclear.
7Note that the Lie algebras of the isometry group SO(d− 1, 2) of AdSd and the conformal group
Conf(d− 2, 1) are isomorphic.
8
2. Supersymmetry and
supergravity
Supersymmetry is a symmetry that relates bosons and fermions, that is, commuting
and anticommuting fields. It has been proposed as a possible solution to a number
of current problems, for instance, dark matter, and is needed in string theories
with fermions [1]. Also, supersymmetry can ensure stability and finiteness of a
theory. For example, N = 4 super-Yang–Mills, a supersymmetric quantum field
theory, is finite to all orders in perturbation theory [40]. Although some argued
that superpartners, particles predicted by supersymmetry, would be experimentally
discovered at the Large Hadron Collider [1], there is, as of today, no experimental
evidence for supersymmetry [41], [42]. Still, supersymmetry remains a large area of
interest for theories beyond the Standard Model.
Supergravity combines supersymmetry with ideas from general relativity. These
theories can be formulated on supermanifolds, a generalisation of ordinary manifolds
that uses both ordinary (commuting) and fermionic (anticommuting) coordinates,
or as field theories with gauged supersymmetry on an ordinary spacetime manifold.
Although there are renormalisable and, as noted above, even finite supersymmetric
quantum field theories, supergravity theories are in general nonrenormalisable and,
therefore, considered as effective field theories [6]. Some of these, in the landscape,
arise as effective low-energy descriptions of (possibly compactified) M-theory while
others, in the swampland, are thought to not admit a finite UV-completion and are,
thus, deemed inconsistent as quantum theories [6]. Specifically, five supergravity
theories in D = 10 are the massless tree-level approximations of the five consistent
superstring theories [1]. Apart from these ten-dimensional supergravity theories, there
is an eleven-dimensional supergravity theory which is an effective low-energy limit of
M-theory. M-theory is a quantum theory, first conjectured by Witten, that unifies
the string theories and whose quantum structure is inherently nonperturbative and
remains largely unknown [5]. The eleven-dimensional supergravity theory is related to
the ten-dimensional supergravity theories via duality transformations. The simplest
such relation is that type IIA supergravity can be obtained by dimensional reduction
of D = 11 supergravity [1] and the relations to other supergravity theories can be
understood via string dualities [1], [4], [6]. The eleven-dimensional supergravity
theory is the theory with which this thesis is concerned.
This chapter gives an introduction to supersymmetry and eleven-dimensional super-
gravity. In this thesis, we are interested in supersymmetric theories in an anti-de
Sitter (AdS) spacetime (after compactification). Still, we start by considering the
simpler case of supersymmetry in Minkowski spacetime. Due to the limited scope
9
2. Supersymmetry and supergravity
of the thesis, the presentation here is incomplete in many ways, although relatively
self-contained. For more thorough introductions to supersymmetry, see for instance
[43], [44].
2.1 Supersymmetry in Minkowski spacetime
In this section, we give an introduction to supersymmetry. For simplicity, we do
this in the setting of a four-dimensional Minkowski spacetime. From a theoretical
perspective, the interest in supersymmetry is motivated by the Coleman–Mandula
theorem [45]. This theorem states, under quite general assumptions, that the Lie
algebra of a connected symmetry group of the S-matrix containing the Poincaré
algebra is locally isomorphic to the direct product of the Poincaré algebra and a
Lie algebra of internal symmetries.8 Essentially, this means that the spacetime
symmetries can only be extended with internal symmetries in a trivial way. One
way around this is to not consider a Lie algebra but a Lie superalgebra. A Lie
superalgebra is a generalisation of a Lie algebra that allows for “anticommuting”
generators as well as ordinary “commuting” generators.9 Formally, it is a Z2-graded
vector space equipped with a bilinear Lie superbracket [·, ·} satisfying [47]
[X, Y } = −(−1)|X||Y |[X, Y }, (2.1a)
(−1)|X||Z|[X, [Y, Z}}+ (−1)|Z||Y |[Z, [X, Y }}+ (−1)|Y ||X|[Y, [Z,X}} = 0, (2.1b)
where X, Y and Z are elements of the Lie superalgebra which are pure in the grading
and |X| denotes the degree (0 or 1) of X. (2.1) are the natural graded generalisations
of the anticommutative property and the Jacobi identity of the ordinary Lie bracket.
Given an associative superalgebra, a Lie superalgebra can be constructed by defining
[47]
[X, Y } = XY − (−1)|X||Y |Y X. (2.2)
This means that, if a Lie superalgebra is represented by linear operators on a vector
space, the superbracket corresponds to the anticommutator if both elements are of
odd degree and the commutator otherwise.10
The generalisation of the Coleman–Mandula theorem to the Lie superalgebra setting
is the Haag–Łopuszański–Sohnius theorem [48]. This theorem gives a classification
of possible Lie superalgebras generating symmetries of the S-matrix for a theory in
Minkowski spacetime. To give an introduction to supersymmetry, we consider, in
particular, the super-Poincaré algebra.
8Note that, since the theorem is concerned with symmetries of the S-matrix, it does not apply to
spontaneously broken symmetries [45], see [46] for a counterexample.
9We have put quotes around (anti-)commuting since the generators may fail to (anti-)commute, as
measured by the superbracket.
10This is similar to how bosonic creation and annihilation operators commute with fermionic
creation and annihilation operators.
10
2.1. Supersymmetry in Minkowski spacetime
2.1.1 The super-Poincaré algebra
The super-Poincaré algebra (in four dimensions) is a Lie superalgebra with neither
central charges nor internal symmetries that can be used to describe supersymmetry
in a four-dimensional Minkowski spacetime. The generators of this algebra are the
translations Pa, Lorentz generators Lab and supercharges Qiα and their conjugates
Q̄α̇i = (Qi †α) . Here, α is a Weyl-spinor index while i = 1, . . . , N where N is the
number of supersymmetries. The nonvanishing independent superbrackets are [43]
[L cdab, L ] = −2(L )[c |e|d]ab eL , (2.3a)
[L cab, P ] = −(L )c dab dP , (2.3b)
[L ,Qiab α} = −(L β iab)α Qβ, (2.3c)
{Qiα, Q̄ i aβ̇j} = 2δjσαβ̇Pa, (2.3d)
where σa are the Pauli matrices (see appendix B.3 for conventions and some identi-
ties).11 From this, it is easy to see that, if the super-Poincaré algebra is realised as
operators on a Hilbert space, the energy is bounded from below. To calculate the
energy, we must fix a direction of time. This introduces δαβ̇ as an invariant under
spatial rotations. Contracting (2.3d) with δαβ̇ gives
1
δαβ̇{Qi , Q̄ } = 2δi(−2)(−H) =⇒ H = δαβ̇{Qiα β̇j j 4 α, Q̄β̇i}. (2.4)N
From this,
〈ψ|H|ψ〉 = 14 δ
αβ̇〈ψ|QiαQ̄β̇i + Q̄ iβ̇iQα|ψ〉 ≥ 0, (2.5)N
since ∑ ∑
〈ψ|Qi δαβ̇α Q̄ 2β̇i|ψ〉 = ‖Q̄α̇iψ‖ , 〈ψ|Q̄ αβ̇ iβ̇iδ Qα|ψ〉 = ‖Qi ψ‖2α . (2.6)
i,α i,α
Note that a global minimum of H is attained if Qiα|ψ〉 = 0 = Q̄α̇i|ψ〉, in which case
|ψ〉 is a supersymmetric vacuum. Such a state |ψ〉 might, however, not exist in which
case the supersymmetry is said to be spontaneously broken [43].12
Since Qiα and Q̄α̇i are spinor generators, they raise or lower the spin of a state by
1/2. To see this explicitly, consider the commutator with Sz ∝ L12, the spin operator
in the z-direction.13 By (2.3c)
[S ,Qi i 12 β i i 12 β̇z α] = −2(σ )α Qβ, [Sz, Qα̇i] = −2(σ̄ )α̇ Qβ̇i. (2.7)
Since σ12 = −iσz and σ̄12 = iσz, Qi1 and Q̄2̇i lower while Qi2 and Q̄1̇i raise Sz by a
half unit. By the spin-statistics theorem, this implies that the fermionic Q operators
interchange bosons and fermions.
11Here, we have written the superbrackets in an almost convention-independent way although (2.3d)
reveals that Pa is Hermitian and, hence, identified with −i∂a.
12Note the difference between this and, for instance, the spontaneously broken symmetry in the
Standard Model. In the latter, an invariant state exists but does not minimise the energy.
13With the normalisation of the Lorentz algebra from appendix A.2 and the conventional [Si, Sj ] =
i kijkS , the exact relation is Si = −i jkijkL , which explains the factors in (2.7).
11
2. Supersymmetry and supergravity
2.1.2 Supermultiplets
We now turn to the possible particle contents of supersymmetric theories, with super-
symmetry implemented by the super-Poincaré algebra, referred to as supermultiplets.
Particles are identified with the nontrivial irreducible unitary representations of the
Poincaré algebra. Physically, this means that, given a particle, it will not change into
another particle under translations, rotations or boosts. That the representations
are required to be irreducible corresponds to the fact that we are interested in
elementary particles. To investigate the particle content, we will therefore consider
irreducible representations not of the complete super-Poincaré algebra but, rather, of
a subalgebra k which leaves the momentum pa invariant, that is, we consider particles
in a specific frame. The subalgebra k includes, P a, Qiα, Qα̇i and the subalgebra of
the Lorentz algebra leaving pa invariant.
As explained in [44], the above procedure induces a unique representation of the
complete algebra. We will only consider supermultiplets consisting of a finite number
of particles and will assume that pa 6= 0.14 Note that P aPa is a Casimir of the
superalgebra, whence all particles in a supermultiplet have the same mass.15
Before investigating the massive and massless supermultiplets in more detail, we
give a final remark. Consider the fermion parity operator (−1)Nf acting by +1
(−1) on bosonic (fermionic) one-particle states in an irreducible finite-dimensional
representation of k [43]. From the above remarks, it follows that (−1)NfQiα =
−Qi Nfα(−1) and similarly for Q̄α̇i. Using this and cyclicity of the trace16, we find
that [ ] [ ]
2δiσaj αβ̇pa tr(−1)
Nf = tr (−1)Nf{Qiα, Q̄ Nf i iβ̇j} = tr (−1) (QαQ̄β̇j + Q̄β̇jQα) = 0.
(2.8)
Since p 6= 0, this implies that the number of bosonic and fermionic states in the
representation are equal.
Massive supermultiplets
To get the particle content in the massive case, we follow [43] and consider the rest
frame in which pa = (m, 0, 0, 0)a. This corresponds to a choice of a timelike vector,
whence Spin(3, 1) is broken to Spin(3) ' SU(2) and δαβ̇ becomes invariant. Thus,
the subalgebra of the Lorentz algebra contained in k is isomorphic to so(3). By
defining
aiα := √
1 i 1Qα, a
†
α̇i = √ Q̄2 2 α̇i
(2.9)
m m
we obtain, from (2.3d), the canonical anticommutation relations
{aiα, a
† i i j † †
α̇j} = δjδαα̇, {aα, aβ} = 0, {aα̇i, a } = 0. (2.10)β̇j
14The case p = 0 corresponds to the Minkowski vacuum since the only finite-dimensional unitary
irreducible representation, in this case, is the trivial representation.
15Since we do not observe superpartners with identical mass in nature, this means that supersym-
metry must be spontaneously broken if implemented in nature.
16This is valid since the trace is over the Hilbert space and matrix elements are complex (not
Grassmann) numbers.
12
2.1. Supersymmetry in Minkowski spacetime
Since we consider finite-dimensional representations, there exists some state |Ω〉,
known as a Clifford vacuum, which is quenched by all annihilation operators, aiα|Ω〉 =
0. From (2.3c), we see that, given one such state, there must be a complete so(3)-
representation of such states, describing the same particle but with different Sz-
eigenvalues. To get an irreducible representation, this so(3)-representation must be
irreducible [43].
We begin by considering the case when the Clifford vacuum has spin 0, that is, the
case in which there is precisely one state |Ω〉 such that aiα|Ω〉 = 0. This is known as
the fundamental supermultiplet [49]. Note that it contains 22N states since there are
2N anticommuting creation operators.
(2.10) is manifestly (SU(2)×U(N ))-invariant. There is, however, a larger invariance
group that can be found by defining
Γi = ai + a† , ΓN+i1 1̇ = a
i †
i 2 + a2̇ ,i
Γ2N+
(2.11)
i = i(ai1 − a
† ), Γ3N+i = i(ai − a†1̇i 2 2̇ ).i
These are Hermitian and satisfy {Γr,Γs} = 2δrs, where r, s = 1, . . . , 4N , which we
recognise as the generators of a Clifford algebra with invariance group SO(4N ). The
22N states transform under the Dirac spinor representation of this SO(4N ), with the
irreducible Weyl spinor representations corresponding to bosons and fermions [49].
To understand the physical content of the representation, we want to label the
particles by their spin. Thus, we wish to keep the original SU(2) manifest. It is,
however, convenient to consider a larger subgroup of SO(4N ) than the originally
manifest SU(2)× U(N ), namely SU(2)× Sp(N ) [43].17 To this end, define
qiα = aiα, qN+iα =  βγ̇
†
αβδ aγ̇i. (2.12)
These satisfy the (SU(2)× Sp(N ))-invariant anticom(mutation) relationsmn
{qm nα , qβ} = − mn mn
0 1
αβΛ , Λ = − 0 , (2.13)1
where m,n = 1, . . . , 2N . When breaking SO(4N ) to SU(2)×Sp(N ) the Dirac spinor
representation decomposes as [49]
⊕N ( ( ))22N → N + 1− Nk, dk , (2.14)
k=0
where the first label, N + 1− k = 2s+ 1, is the dimension of the irreducible spin-s
representation of SU(2) and the seco(nd la)bel,( )
(N ) = 2Ndk −
2N
k k − 2 , (2.15)
17Sp(N ) is sometimes, for instance in [43], [49], denoted USp(2N ).
13
2. Supersymmetry and supergravity
is the dimension of the irreducible representation of Sp(N ) consisting of traceless18
completely antisymmetric tensors Tm1...m . Hence, the number of spin-s particlesk
is (N )dN−2s.
The number of particles with each spin can also be calculated directly with combina-
torics by analysing the spin in the z-direction. Due to (2.7) it is clear that, since we
start from an s = 0 Clifford vacuum, the highest Sz eigenvalue in the supermultiplet
is N /2 which occurs with multiplicity 1. This implies that there is exactly one
spin-N /2 representation with dimension N + 1 corresponding to the k = 0 term in
(2.14). Similarly, there are 2N states with sz = (N − 1)/2 since we can either use
all raising operators, a†1̇ , and then one of the lowering operators, a
†
2̇ , (N choices) ori i
all but one raising operator (N choices). Generalising this to sz = n/2 we see, by
Vandermonde’s identity, that there are
N∑− ( )( ) ( )n N N = 2N+ (2.16)k=0 n k k N − n
states. The terms in this sum are interpreted as raising the sz eigenvalue n + k
times and then lowering it k times. Since there is one sz state for every irreducible
representation with s ≥ sz, the multiplicity of the spin-s representation is the number
of sz = s states minus the number of sz = s + 1 states. Hence, the multiplicity of
the representation with s(pin s =)n/2(is )
2N − 2N = (N )d
N − n N − n− 2 N−n, (2.17)
which is consistent with the above result from the group-theoretic approach.
Now consider the general case of a spin-s Clifford vacuum |Ωs〉. The sz eigenvalues
can be obtained as all possible sums of one sz value from the vacuum and one from the
creation operators, whence the supermultiplet is given by angular momentum addition
with the fundamental supermultiplet. The dimension of the general supermultiplet is
given by 22N (2s+ 1). In table 2.1, massive N = 1 and N = 4 supermultiplets with
spin at most 2 are presented; a complete list can be found in [43]. Note that N = 4
is the maximal number of supersymmetries if we require that the spin is at most 2.
18The trace is of course taken with the antisymmetric invariant Λmn.
14
2.1. Supersymmetry in Minkowski spacetime
Table 2.1: Multiplicities of the spin representations, that is, the number
of particle types of each spin, for the irreducible massive N = 1 and N =
4 supermultiplets. Ωs specifies the spin, s, of the Clifford vacuum of the
representation.
N = 1 N = 4
Spin Ω0 Ω1/2 Ω1 Ω3/2 Ω0
0 2 1 42
1/2 1 2 1 48
1 1 2 1 27
3/2 1 2 8
2 1 1
Massless supermultiplets
In the massless case, p2 = 0, the particles have no rest frame. Instead, again
following [43], we analyse the situation in the frame in which pa = (E, 0, 0, E)a. This
corresponds to a choice of a lightlike vector, which breaks Spin(3, 1) to the double
cover of SE(2) [50].19 Since we only consider finite-dimensional representations, they
are labelled by their helicity, which coincides with the spin in the z-direction due to
our choice of coordinates. From (2.3d) we se(e that )
{Qi 2E 0α, Q̄β̇j} = 2δij 0 0 . (2.18)
αβ̇
Since the α = 2 operators anticommute with everything they act by 0 on any
representation [43] whence no new states are obtained by applying them. Thus, we
need only introduce N creation and annihilation operators
i = √1 1a Qi1, a
†
i = √ Q , (2.19)2 E 2 1̇iE
where ai lowers and a†i raises the helicity by a half unit. Introducin(g )a Clifford
vacuum |Ωλ〉 with helicity λ, we see that we get a total of 2N states and N states ofk
helicity λ+ k/2. These supermultiplets are, in general, not CPT-invariant since they
are not symmetric around helicity 0 [43]. To create CPT-invariant supermultiplets,
two supermultiplets with opposite helicities can be added. In table 2.2, the CPT-
invariant N = 1 and N = 8 supermultiplets with spin at most 2 are presented. Note
that N = 8 is the maximal number of supersymmetries if we require the spin to be
at most 2. This assumption is often, but not universally, employed due to no-go
theorems for higher spins [51].
19E(n) denotes the isometry group of Euclidean space.
15
2. Supersymmetry and supergravity
Table 2.2: Multiplicities of the helicities for the massless CPT-invariant N = 1
and N = 8 supermultiplets. Ωλ specifies the helicity, λ, of the lowest-helicity
Clifford vacuum of the representation. The N = 8 supermultiplet is irreducible
while the N = 1 supermultiplets contain two irreducible parts to make them
CPT-invariant.
N = 1 N = 8
Helicity Ω−2 Ω−3/2 Ω−1 Ω−1/2 Ω−2
2 1 1
3/2 1 1 8
1 1 1 28
1/2 1 1 56
0 2 70
−1/2 1 1 56
−1 1 1 28
−3/2 1 1 8
−2 1 1
2.1.3 The Wess–Zumino model
Here, we give an example of a supersymmetric theory in Minkowski spacetime
known as the Wess–Zumino model [52]. The theory consists of a massless N = 1
supermultiplet with two scalar and two spinor real on-shell degrees of freedom, see
table 2.2. We formulate the theory using a Weyl spinor ψα and a complex scalar
φ. Note that the spinor has four real off-shell degrees of freedom but only two real
on-shell degrees of freedom. The Lagrangian of the free theory is
L = −∂aφ∗∂aφ− iψ̄σ̄a∂aψ, (2.20)
which is real since( )
iψ̄ σ̄aα̇β
∗
α̇ ∂aψβ = iψ σaαβ̇∂ aβ̇αα aψ̄β̇ ' iψβ̇σ̄ ∂aψα, (2.21)
where, in the step indicated by ', we have used integration by parts and disregarded
boundary terms.
The supersymmetry transformation can be written as
√ √
δξφ = 2ξαψα, δ a β̇ξψα = 2iσαβ̇ ξ̄ ∂aφ, (2.22)
√
where the factor 2 is purely conventional. Note that the transformation parameter
ξα is fermionic since supersymmetry interchanges bosons and fermions. The above
implies that
δ ∗
√ √
α̇ α̇ aα̇β ∗
ξφ = 2ξ̄α̇ψ̄ , δξψ̄ = 2iσ̄ ξβ∂aφ . (2.23)
16
2.2. Superspace formalism
From this, we see that the Lagrangian is, indeed, supersymmetric since
√ √ √ √
δξL = − 2ξ̄∂aψ̄∂aφ− 2∂aφ∗ξ∂aψ − 2∂ φ∗b ξσbσ̄a∂aψ + 2ψ̄σ̄aσbξ̄∂a∂bφ '√ √ √ √
' − 2ξ̄∂aψ̄∂aφ− 2∂ ∗ a ∗aφ ξ∂ ψ + 2∂aφ ξ∂aψ + 2∂ aaψ̄ξ̄∂ φ = 0, (2.24)
where we have integrated by parts and used properties of the Pauli matrices.
To see the relation to the super-Poincaré algebra, consider the commutator [δξ, δε] of
two supersymmetry transformations. From δξδεφ = 2iεσaξ̄∂aφ, we immediately find
[δ , δ ]φ = (ξαε̄β̇ − εαξ ε ξ̄β̇)2σaαβ̇(−i∂a)φ. (2.25)
Similarly, δξδεψα = 2iσa ε̄β̇ξγ∂aψγ whenceαβ̇
[δ , δ ]ψ = 2iσa δγδβ̇ ̇ δξ ε α αβ̇ δ ̇ (ε̄ ξ − ξ̄
̇εδ)∂aψγ. (2.26)
Writing the deltas using σb σ̄β̇γδ̇ b = −2δ
γ
δ δ
β̇
̇ and then using σaσ̄b = −2ηab − σbσ̄a this
becomes
[δ , δ ]ψ = (ξαε̄β̇ − εαξ ε ξ̄β̇)2σaαβ̇(−i∂a)ψ − i(ξσ
bε̄− εσbξ̄)σ abσ̄ ∂aψ. (2.27)
By writing a supersymmetry transformation as δξ = ξQ + ξ̄Q̄ [43], we find, from
(2.3d),
[δξ, δε] = ξα{Qα, Q̄β̇}ε̄β̇ − εα{Q , Q̄ β̇ α β̇ α β̇α β̇}ξ̄ = (ξ ε̄ − ε ξ̄ )2σaαβ̇Pa. (2.28)
Identifying Pa with (−i∂a), we see that this agrees with (2.25) and (2.27) except for
the last term in (2.27). Note that this term vanishes on-shell since, then, ψ satisfies
the Weyl equation σ̄a∂aψ = 0. Hence, the supersymmetry is said to close on-shell.
This is expected from the representation theory of the super-Poincaré algebra since
there is always an equal number of bosonic and fermionic states in a representation.
In the Wess–Zumino model, as presented here, their numbers are equal on-shell but
not off-shell. Thus, we expect that we can make the algebra close off-shell by adding
an auxiliary complex scalar field with no on-shell degrees of freedom. In section 2.2.2,
we will see that this is indeed the case.
2.2 Superspace formalism
In this section, we introduce the formalism of superspace and superfields. With
this formalism, one can construct manifestly supersymmetric theories. For instance,
one can add interaction terms to the Wess–Zumino model without having to check
that the Lagrangian is supersymmetric by hand, as we did in section 2.1.3. Fur-
thermore, it provides insight into the quantum theory by simplifying calculations
and explaining seemingly miraculous cancellations in component calculations by
keeping the supersymmetry manifest [40], [44], [53]. It is also the language that we
will use to construct eleven-dimensional supergravity. We will not attempt to give
17
2. Supersymmetry and supergravity
a mathematically rigorous presentation of supermanifolds20 but provide the tools
necessary to formulate supergravity theories.
Superspace21 is parameterised by supercoordinates zM = (xm, θµ, θ̄ )Mµ̇ or zM =
(xm, θµ)M , where xm are real and Grassmann-even (bosonic) while θµ and θ̄µ̇ are
Grassmann-odd (fermionic), see appendix F. In dimensions and signatures in which
the irreducible spinor(s) are Majorana, θµ can be taken as real and, then, the
supercoordinates are (xm, θµ). For extended supersymmetry, that is, with N > 1,
one adds additional anticommuting coordinates [44].
2.2.1 Superfields
Here, we introduce the concept of superfields. We consider the case of flat superspace
with 4 bosonic dimensions and supercoordinates zA = (xa, θα, θ̄ Aα̇) , although several
aspects naturally generalise to arbitrary dimension. Similar to how translations
and rotations can be implemented as differential operators on ordinary space, we
wish to realise the supercharges, Qα and Q̄α̇, as differential operators on superspace.
Following [43], we define
Qα = ∂α − iσa β̇αβ̇ θ̄ ∂a, Q̄α̇ = −∂ + iθ
βσaα̇ βα̇∂a, (2.29)
from which we find the single nonvanishing anticommutator
{Qα, Q̄ } = 2iσaβ̇ αβ̇∂a = −2σ
a
αβ̇Pa. (2.30)
Note the difference in sign compared to (2.3d). However, by changing coordinates
xa →7 −xa, which implies Pa 7→ −Pa, we recover the super-Poincaré algebra as
previously defined.
The partial derivate ∂̄α̇ does not anticommute with Qα. Hence, we introduce
supercovariant derivatives
D = ∂ + iσa β̇α α αβ̇ θ̄ ∂a, D̄α̇ = −∂̄α̇ − iθ
βσaβα̇∂a, (2.31)
satisfying
{Dα, D̄β̇} = −2iσaαβ̇∂a, {Dα, Dβ} = 0, (2.32a)
{Dα, Qβ} = 0, {Dα, Q̄β̇} = 0. (2.32b)
We raise and lower the indices on Dα and D̄α̇ using the usual convention for spinors,
in contrast to how the indices on ∂ α̇α and ∂̄ are raised and lowered, see appendix F.
Now that we have defined the supercharges as differential operators and introduced
supercovariant derivatives, we are ready to introduce superfields. A superfield
F (x, θ, θ̄) is a Grassmann-even function on superspace. Since α can only take two
20For such a treatment, see for instance [54].
21With “superspace”, we refer to any supermanifold, not necessarily flat superspace.
18
2.2. Superspace formalism
values, we can form θ2 = θαθ but θ2θαα = 0 and similarly for dotted indices.22 Hence,
we may expand F in powers of θ and θ̄ and only get a finite number of terms
F (x, θ, θ̄) = f(x) + θψ(x) + θ̄χ̄(x) + θ2m(x) + θσaθ̄va(x) + θ̄2n(x)
+ θ2θ̄λ̄(x) + θ̄2θη(x) + θ2θ̄2h(x). (2.33)
Here, the x-dependent expansion-coefficients are referred to as component fields.
The supersymmetry transformation of a superfield is defined as [43]
δξF = (ξQ+ ξ̄Q̄)F, (2.34)
while the supersymmetry transformations of the component fields are defined by
δξF = δξf + θ δξψ + θ̄ δξχ̄+ θ2 δξm+ θσaθ̄ δξv + θ̄2a δξn
+ θ2θ̄ δξλ̄+ θ̄2θ δξη + θ2θ̄2 δξh. (2.35)
(2.34) is so important that we do not call functions of x, θ and θ̄ superfields if they
do not obey it (similarly to how the word tensor is used in physics). Thus, ∂αF is,
for instance, not a superfield since {Q̄β̇, ∂α} =6 0. Given two superfields F and G,
a linear combination of them is, however, a superfield since Qα and Q̄α̇ are linear
operators. Similarly, the product FG is a superfield since the supercharges are graded
derivations, that is, they satisfy the super-Leibniz’s rule. Also, the supercovariant
derivative of a superfield is again a superfield since Dα and D̄α̇ anticommute with
Qα and Q̄α̇.
Since the supersymmetry transformation is linear, the space of superfields transform
under a representation of the superalgebra. However, as is clear from (2.33), an
unconstrained superfield contains quite many spacetime fields and the representation
is not irreducible [43]. To get an irreducible representation, the superfield must be
constrained. To not break Lorentz invariance or supersymmetry, the constraining
equations should be Lorentz invariant and respect supersymmetry in the sense that
the variation δξF also satisfies the constraints [43]. If the theory should be kept off-
shell, the constraining equations should, furthermore, not imply differential equations
for the remaining component fields. Two possibilities are
D̄Φ = 0, V ∗ = V, (2.36)
where Φ is called a chiral superfield while V is called a vector superfield. It turns
out that every supersymmetric renormalisable Lagrangian can be written in terms of
chiral and vector superfields [43].
22Note that θαθβ = kαβθ2 since there is only one antisymmetric combination. By contracting with
αβ , one finds k = −1/2.
19
2. Supersymmetry and supergravity
2.2.2 Revisiting the Wess–Zumino model
Now that we have introduced superfields, we demonstrate the formalism by studying
chiral superfields in more detail. As it will turn out, we will find a superspace
formulation of the Wess–Zumino model. As stated above, a chiral superfield satisfies
D̄Φ = 0. Naturally, this forces the θ̄-dependence of the field. To see this explicitly,
define
T := exp(−i 1θσaθ̄∂a) = 1− iθσaθ̄∂ 2 2a + 4θ θ̄ , (2.37)
which satisfies TD̄α̇ = −∂̄α̇T . With a short calculation, one can verify that T−1 =
exp(iθσaθ̄∂a), as expected. Thus, since T is invertible, D̄α̇Φ = 0 is equivalent to
∂̄α̇TΦ = 0. Hence, TΦ can be expanded, in terms of component fields, as
√
TΦ(x, θ, θ̄) = φ(x) + 2θψ(x) + θ2F (x), (2.38)
√
where the 2 might seem arbitrary at this point. Acting with T−1 gives, in agreement
with [43],
√
Φ = φ+ 2θψ + θ2F + iθσaθ̄∂aφ+ √
i 1
θ2θ̄σ̄a∂ ψ + θ2a 4 θ̄
2φ. (2.39)
2
To derive the supersymmetry transformation of the component fields, note that they
are obtained as the analogous components in the θ-expansion of δξF , see (2.35). Since
φ = Φ|θ=0=θ̄, this means that δξφ = δξΦ|θ=0=θ̄. Henceforth, we will omit θ = 0 = θ̄ at
the evaluation bar and simply write, for instance, Φ| to save ink. Note that, acting
with (ξQ+ ξ̄Q̄) and (ξD + ξ̄D̄) gives the same result when evaluated at θ = 0 = θ̄
since only the ∂α and ∂̄α̇ terms su∣∣rvive. Using thi∣∣s, and th∣∣at Φ is chiral, we find√
δξφ = (ξQ+ ξ̄Q̄)Φ∣ = (ξD + ξ̄D̄)Φ∣ = ξDΦ∣ = 2ξψ. (2.40)
W√e recognise this from (2.22), which expla√ins why we normalised ψ with a factor of
2 in (2.38). Next, we note that∣ DαΦ| = 2ψα whence
= √1 ∣ 1 ∣∣δξψα Dα(ξQ+ ξ̄Q̄)Φ∣ = √ (ξD∣+ ξ̄D̄)DαΦ∣ =2 2
= √1 ( β ∣ 1ξ DβDα − ξ̄β̇{D̄β̇, Dα})Φ∣ = √ (2ξβαβF + 2iσa β̇2 2 αβ̇ ξ̄ ∂aφ) =√ √
= 2ξαF + 2iσaαα̇ξ̄α̇∂aφ. (2.41)
Here, it is important that we move Dα to the right of the supercharges before we
replace them with covariant derivatives. This is reminiscent of (2.22) but there is
an extra term involving the field F . More on this later. In the above, we used that
DαD
2
βθ = 2βα. This implies that F = −1/4D2Φ|, which we now use to compute
the supersymmetry transformation∣ of F , ∣ ∣
δ F = −1D2( + 1ξ 4 ξQ ξ̄Q̄)Φ∣∣ = −4(ξD ∣+ ξ̄D̄)D
2Φ∣∣ = −14 ξ̄D̄D2Φ∣∣ =
√
= 1 ξ̄α̇(D2D̄ − 2{ ∣D̄ ,D }Dβ)Φ∣ = 2iξ̄α̇σa β4 α̇ α̇ β βα̇∂aψ =√
= 2iξ̄σ̄a∂aψ. (2.42)
20
2.2. Superspace formalism
Having found the supersymmetry transformations of the component fields we would
like to construct a supersymmetric Lagrangian. To obtain an ordinary spacetime
Lagrangian, L, that does not depend on the Grassmann variables, we integrate over
θ and θ̄. For this, we use the Berezin integral, see appendix F. Therefore, we want to
know what quantities∫can be integrated to a s∫upersymmetric Lagrangian. Note that
d2θ d2θ̄ ∂α = 0, d2θ d2θ̄ ∂̄α̇ = 0, (2.43)
since the integral is only nonzero for θ2θ̄2-terms. Hence, any superfield integrated over
θ and θ̄ transforms into a total x-derivative under a supersymmetry transformation.
Thus, the integral of any superfield gives a supersymmetric Lagrangian, disregarding
boundary terms. We can now write a∫ Lagrangian for a chiral superfield Φ as
L = d2θ d2θ̄Φ∗Φ. (2.44)
As we will soon see, this is a kinetic Lagrangian, even though it does not contain
any explicit derivatives. Instead, the derivatives will come from (2.39). Note that Φ∗
is an antichiral superfield, D ∗αΦ = 0, whence Φ∗Φ is neither chiral nor antichiral.
However, it is a superfield, which is all that matters for L to be supersymmetric. To
calculate the integrals, we use ∫that ∣
d2θ d2 ' 1θ̄ D2 2∣16 D̄ ∣, (2.45)
where the difference, which comes from the θ-term in D̄, is a total x-derivative. Using
that∫Φ is chiral ( )∣
d2θ d2θ̄Φ∗Φ ' 1 (D2D̄2 ∗16 Φ )Φ− 2(DαD̄
2Φ∗)(DαΦ) + (D̄2Φ∗)(D2Φ) ∣∣ =
= φ∗φ+ i∂ ψ̄β̇σ̄a ψα + F ∗a β̇α F '
' −∂aφ∂aφ− iψ̄σ̄a∂aψ + F ∗F, (2.46)
where, in the second step, we have used
D 2αD̄ Φ∗ = (D̄2Dα + 2{Dα, D̄β̇}D̄β̇)Φ∗ = −4iσa β̇ ∗αβ̇∂aD̄ Φ , (2.47a)
D2D̄2Φ∗ = −4iσa ∂ DαD̄β̇Φ∗ = −8σa σ̄bβ̇ααβ̇ a αβ̇ ∂a∂ Φ
∗
b = 16Φ∗. (2.47b)
Since no derivatives of F enter in L, we say that F is an auxiliary field. Note that
if we impose the Euler–Lagrange equation F = 0, we recover the Lagrangian and
supersymmetry transformations of the Wess–Zumino model in section 2.1.3. Without
imposing F = 0, the supersymmetry transformations close off-shell due to (2.30).23
Thus, we have realised the supersymmetry off-shell by introducing an auxiliary field
F , as alluded to in section 2.1.3.
23Due to how we define supersymmetry transformations of component fields, δξδεΦ| = δεδξφ and
similarly for ψ and F [43]. Thus, [δξ, δε] gives the same result as in section 2.1.3 due to the sign
difference between (2.3d) and (2.30).
21
2. Supersymmetry and supergravity
Interactions
At this point, it might seem like we have not gained much by introducing the
superspace formalism. To illustrate part of the power of the formalism, we consider
supersymmetric interactions. As noted above, the superspace formalism also provides
insight into supersymmetric quantum field theories.
To only get renormalisable interactions, we require that the coupling constants have
positive momentum dimensions. For this analysis, we need to define [θ]. As usual,
in momentum dimensions, [x] = −1, [φ] = 1, which implies [Φ] = 1. From this,
and [ψ] = 3/2, we get [θ] = −1/2 which is consistent with the dimensions in (2.29)
and (2.31). Hence, if we integrate with respect to both θ2 and θ̄2, the integrand must
have dimension 2 since [L] = 4.24 Such a term can, hence, only have two powers
of Φ. However, from (2.45), we see that any chiral integrand will only contribute
with boundary terms. Similarly, any antichiral integrand will only contribute with
boundary terms since the integration over θ and θ̄ can be carried out in any order.
Thus, the only possible term of this kind is the kinetic term.
∫
To construct interaction terms, we instead use d2θ + c.c. Such an integral will, in
general, not produce a θ̄-independent result. However, if the integrand is chiral, any
term containing θ̄ will be a total x-derivative, as seen from (2.39), that we disregard.
A term of this kind in the Lagrangian is supersymmetric due to (2.43) and the fact
that δξΦ can be written without ∂̄α̇-terms since D̄α̇Φ = 0. Since we only integrate
over θ or θ̄ in the interaction terms, the integrand must have dimension 3. Hence,
we write ∫ ( ) ∫ ( )
L = d2 mint. θ λΦ + 2 Φ
2 + g 33Φ + d
2θ̄ λΦ∗ + m(Φ∗)2 + g (Φ∗)3 , (2.48)
∫ 2 3
where [λ] = 2, [m] = 1 and [g] = 0. Using that d2θ and −1/4D2| differs by a total
x-deriva∫tive, we compute the component field interactions
2 1 ∣2 ∣
∫ d θΦ ' −4D Φ∣ = F, ∣ (2.49a)
∫ d
2θΦ2 ' −1(Dα2 ΦDαΦ + ΦD
2Φ)∣∣ = ∣−ψ
αψα + 2φF, (2.49b)
d2θΦ3 ' −3(2ΦDα4 ΦDαΦ + Φ
2D2Φ)∣∣ = −3φψαψα + 3φ2F. (2.49c)
Generalising this to multiple chiral superfields Φi, we can write the most general
renormal∫isable supersymm[e∫tric Lagrangian as( ) ]
L = d2 1 1θ d2θ̄Φ∗ i 2 i i j i j kiΦ + d θ λiΦ + 2mijΦ Φ + 3gijkΦ Φ Φ + c.c. =
= −∂[aφ∗∂ φi − iψ̄ σ̄a∂ ψi + F ∗ ii a i a i F + ]
+ 1λiF i − 2m
i
ijψ ψ
j +m φiF j − g φiψjψkij ijk + g iijkφ φjF k + c.c. , (2.50)
24 ∫Note that d2θ contributes +1 to the dimension due to how the integral is defined.
22
2.2. Superspace formalism
where mij and gijk are completely symmetric [43].25 Note that even after adding
interactions, F i are still auxiliary fields without dynamics. The Euler–Lagrange
equation for F i reads
F ∗i + λ j ji +mijφ + gijkφ φk = 0. (2.51)
We can now eliminate the auxiliary fields from the theory by inserting the solution
into (2.50).26 The Lagrangian then becomes
L = −∂aφ∗∂ φii a − i
1 1
ψ̄iσ̄
a∂aψ
i − m ψiψj2 ij − 2m
∗ijψ̄iψ̄j+
− g φiψjψk − g∗ijk ∗ijk φi ψ̄jψ̄k − F ∗i (φ)F i(φ). (2.52)
Here, the last term is a potential term containing powers of φ from order zero to four.
We recognise the kinetic and mass terms for the scalars and spinors, the Yukawa
interactions and the scalar potential from quantum field theory. Note that, due to
the supersymmetry, there are very particular relations between the masses, Yukawa
couplings and parameters in the potential.
The last term is the only term where a vacuum expectation value can enter. From the
Lagrangian, we can see explicitly that the conclusions regarding nonnegative energy
and spontaneously broken supersymmetry in section 2.1.1 are valid. If 〈F i〉 = 0
we get a vacuum with 0 energy which is supersymmetric since a supersymmetry
transformation leaves all fields unchanged. If, on the other hand, 〈F i〉 =6 0, the
ground state energy is positive and the supersymmetry is spontaneously broken since
ψi is not invariant under a supersymmetry transformation. Spontaneously broken
supersymmetry can only be guaranteed if λi, mij and gijk are such that there is no
solution to (2.51) with F i = 0 but can also be obtained as a metastable vacuum
where the energy is only minimised locally [55].
2.2.3 Superdifferential forms
Having introduced the concepts of superspace and superfields, we turn to superdif-
ferential forms, or superforms, which we will use to formulate eleven-dimensional
supergravity. Supersymmetry transformations form a subgroup of the diffeomorphism
group of a supermanifold [43]. Thus far, we have only considered flat superspace
and global, or rigid, supersymmetry transformations. To be able to formulate super-
gravity theories that are manifestly invariant under general diffeomorphisms, that is,
coordinate transformations, we introduce superdifferential forms.27 Superforms are
not only useful for formulating supergravity, but supersymmetric Yang–Mills theories
as well. These formulations are super-analogous of Cartan’s formulation of general
relativity and Yang–Mills theory formulated with differential forms, see appendix E.
25An equivalent way of constructing supersymmetric Lagrangians is by picking out a component of
a superfield that transforms into a total x-derivative under supersymmetry, see [43].
26The dynamics remain the same when eliminating the auxiliary field, as can be proven in general.
27Note that a diffeomorphism is an active coordinate transformation. Any theory can be formulated
in a way invariant under passive coordinate transformations, that is, changes of coordinates.
23
2. Supersymmetry and supergravity
In the following, we will no longer separate commuting and anticommuting coordi-
nates. Instead, we work directly with supercoordinates zM = (xm, θµ, θ̄µ̇)M , where θ̄
may be omitted if we impose a Majorana condition on θ,28 satisfying
zMzN = (−1)|M ||N |zNzM , (2.53)
where |M | is 0 for M = m and 1 for M = µ, µ̇. Hence, the coordinates are
said to be graded-commutative. Similarly, we write dzM = (dxm, dθµ, dθ̄ Mµ̇) and
∂M = (∂m, ∂µ, ∂̄µ̇)M and introduce a “graded-anticommutative” wedge product
dzM ∧ dzN = −(−1)|M ||N |dzN ∧ dzM . (2.54)
A general superdifferential p-form can now be written as
Ω = 1!dz
M1 ∧ . . . ∧ dzMpΩMp...M1(z) = dzMIΩM (z), (2.55)p I
where MI is a multi-index, dzMI = dzM1 ∧ . . . dzMp and ΩM = ΩMp...M1/p!. Note theI
order of indices and placement of Ω. This is purely conventional but will turn out to
be practical. For Grassmann-even superforms, ΩM is Grassmann-odd if the numberI
of spinor indices is odd and Grassmann-even otherwise.29 The wedge product is, of
course, extended bilinearly to arbitrary superforms. It is a straightforward exercise
to show that the wedge product is associative, Λ∧ (Ω∧Ξ) = (Λ∧Ω)∧Ξ, and satisfies
Ω ∧ Λ = (−1)pqΛ ∧ Ω for a super p-form Ω and super q-form Λ [43], like the wedge
product of ordinary differential forms.
The exterior derivative of a superform Ω is defined as
dΩ = dzMI ∧ dzN∂NΩM (2.56)I
and is a super (p+ 1)-form where p is the form-degree of Ω. Note that this differs
from the conventional definition in ordinary space since dzN∂N is to the right of
dzMI . This implies that
d(Ω ∧ Λ) = Ω ∧ dΛ + (−1)|Λ|dΩ ∧ Λ, (2.57)
where |Λ| is the form-degree of Λ. As usual, d2 = 0 which, together with (2.57) and
dF = dzM∂MF for super 0-forms F , provides an alternative definition of the exterior
derivative [43].
Connection form, covariant derivative and field strength tensor
In gauge theory, one considers transformations under a gauged structure group
G. The structure group is a compact Lie group in case of Yang–Mills theory and
the Lorentz group in Cartan’s formulation of gravity. In the superspace setting,
we use right-action, so a tensorial super p-form Ωi transforms under some right-
representation ρ of the group, that is, Ω′i = Ωjρ(g) ij where g is a group element.
28This comment naturally applies throughout the rest of this section.
29When considering, for instance, a vector superform VM = dzNIV MN , this is complementedI
according to whether M is bosonic or fermionic.
24
2.2. Superspace formalism
The use of a right-action is motivated by our convention for the exterior derivative.
Henceforth, we will not write the representation explicitly but instead simply write
Ω′i = Ωjg ij or, dropping the indices as well, Ω′ = Ωg. Note that dΩ′ is not a tensor,
since d(Ωg) = Ω ∧ dg + dΩ g. To remedy this, we introduce the Lie algebra-valued
(local) connection 1-form φ and a covariant exterior derivative
D = d + φ, (2.58)
which acts on a tensor as DΩ = dΩ + Ω ∧ φ.30 To make DΩ a tensor, that is,
D′Ω′ = (d + φ′)(Ωg) = dΩ g + Ω ∧ dg + Ωg ∧ φ′ = (dΩ + Ω ∧ φ)g, (2.59)
we need
φ′ = g−1φg − g−1dg. (2.60)
In this equation, which defines how the connection form transforms, g−1φg is the
adjoint right action of a Lie group element on a Lie algebra element and, hence, a
well-defined element in the Lie algebra. The second term is also an element in the
Lie algebra.
With Tr generators of g = Lie(G), the connection 1-form may be written as φ = φriTr
where φr = dzNφr .31N The action on Ω is then
Ω ∧ φ = dzMI ∧ dzNφrNΩM iTr. (2.61)I
Next, we define the g-valued field strength, or curvature 2-form,
F = r i = 1F T dzM ∧ dzNF rr 2 NM iTr
:= dφ+ φ ∧ φ. (2.62)
Note that we define the wedge product between Lie algebra-valued forms using the
associative product32 of the universal enveloping algebra U(g) whence it is, in general,
U(g)-valued, not g-valued. F is, however, Lie algebra-valued as is seen from
∧ = ri ∧ si = 1φ φ φ T r sr φ Ts 2φ ∧ φ [iTr, iTs] =
: 1
2[φ ∧ φ], (2.63)
where [· ∧ ·] is a wedge product defined using the Lie bracket. Using the definition of
F , the transformation law of φ and 0 = d(g−1g) = g−1dg+dg−1g, it is straightforward
to show that F ′ = g−1Fg, whence F is a tensor carrying the adjoint representation.
Hence, φ acts on F as
F ∧ad φ = [F ∧ φ] = F r ∧ φs[iTr, iTs] = F ∧ φ− φ ∧ F. (2.64)
30In the super Yang–Mills context, the connection could be denoted by A in analogy with the
conventional notation in ordinary Yang–Mills theory.
31Here, we use the convention that a group element is g = exp(iT ) but keep the i close to the
generator to be able to switch conventions without effort.
32Given a representation, this corresponds to matrix multiplication.
25
2. Supersymmetry and supergravity
Using the definition of F to calculate dF , this implies
DF = d(dφ+ φ ∧ φ) + [F ∧ φ] = φ ∧ dφ− dφ ∧ φ+ [F ∧ φ] =
= 0, (2.65)
which is known as the Bianchi identity of the second type [43]. The Bianchi identity
of the first type is
D2Ω = D(dΩ + Ω ∧ φ) = (Ω ∧ dφ− dΩ ∧ φ) + (dΩ ∧ φ+ Ω ∧ φ ∧ φ) =
= Ω ∧ F. (2.66)
If we define F as the operator acting as FΩ = Ω∧F , the first Bianchi identity reads
D2 = F while the second Bianchi identity reads [D,F ] = 0 and follows immediately
from the first.33 Also, from the definition of the covariant derivate, it is clear that
D′ = gDg−1 where the juxtaposition denotes operator composition (D is not acting
on g−1) from which it follows that F ′ = gFg−1. This is consistent with the above
F ′ = g−1Fg since F acts from the right, F ′Ω = Ω ∧ g−1Fg.
Spin connection, vielbeins and torsion
Thus far, we have used the coordinate frame as a basis for tangent vectors. Now, we
consider another frame, related to the coordinate frame by a local change of basis
that is, in general, not induced by a change of coordinates,
E = E M∂ , EA = dzME AA A M M , (2.67)
where E MA E B BM = δA and E A NM EA = δNM . The vielbein EA generalises the concept
of vierbeins from four-dimensional spacetime. We define the vielbeins to transform
covariantly under local Lorentz transformations. Going forward, we use A,B,C, . . .
for Lorentz indices (flat) and M,N,P, . . . for “Einstein” indices (curved). As in the
case of ordinary manifolds, globally defined vielbeins EA do not exist in general [54]
but only for parallelisable supermanifolds, for instance, flat superspace. Instead,
the vielbeins are defined locally and related by local Lorentz transformations on
intersections of patches.
Since we are interested in fields carrying a spin-representation, we need a spin
connection ω. The curvature 2-form of the spin connection is denoted by R. In
gravity, ω is dynamical and the only connection we are concerned with, while in
super Yang–Mills, in a fixed background, there is a dynamical Yang–Mills connection
while ω is fixed. Note that a Lorentz transformation Λ BA does not mix the bosonic
and fermionic parts of tensors, that is, Λ BA is only nonzero when both indices are of
the same type. Splitting the index A = (a, α, α̇) each part transforms under its usual
vector or spinor representation. Hence, the spin connection ω B C de BA = E ωCde(L )A ,
where Lde are the Lorentz generators,34 and the curvature 2-form R BA are only
33Here, we distinguish the 2-form F from the operator F . Note, however, that the situation is
similar for the connection, where we use φ for both the operator and 1-form.
34We use the geometrical convention that a group element is Λ = exp(L) for the Lorentz group.
26
2.2. Superspace formalism
nonzero when both indices A B are either bosonic or fermionic. Explicitly, the
nonzero components of (Lde) BA are
(Lde) b = δ[dηe]b, (Lde) β 1 de βa a α = 4(Γ )α , (2.68)
whence all Lorentz algebra-valued quantities have this relation between their compo-
nents.
Given a spin connection and vielbeins, one may define the torsion 2-form as
TA = DEA = dEA + EB ∧ ω AB . (2.69)
The torsion transforms covariantly under local Lorentz transformations. From the
Bianchi identity of the first type (2.66), we see that
DTA = EB ∧R AB , (2.70)
which is known as the Ricci identity (with torsion).
2.2.4 Coordinate transformations in superspace
To conclude this section, we review the transformation laws for various fields under
an infinitesimal coordinate transformation35
zM 7→ z′M = zM + ξM . (2.71)
The transformation of a scalar field φ(z) is, as usual, defined by φ′(z′) = φ(z). This
is interpreted as moving the field: the value of the moved field, φ′, at the moved
point, z′, is the same as the value of the original field, φ, at the original point, z. For
the infinitesimal transformation in (2.71),
δξφ = −ξM∂Mφ, (2.72)
where δ φ := (φ′ξ (z)− φ(z))|O(ξ) (that is, only terms up to first order in ξ are kept
when computing δξφ).
Similarly, the transformation of a vector field is defined by V ′M(z′)∂′M = V M(z)∂M
and that of a covector by dz′MU ′M(z′) = dzMUM(z). Using the chain rules
∂z′M ∂z′Ndz′M = dzN , ∂ = ∂′
N M∂z ∂zM N
, (2.73)
one finds
∂ V M = −ξNξ ∂ MNV + V N∂NξM , ∂ NξUM = −ξ ∂NUM − ∂ NMξ UN . (2.74)
35We consider active transformations, that is, diffeomorphisms in superspace. Thus, z′ and z
describe different points in superspace and we only consider a single coordinate system.
27
2. Supersymmetry and supergravity
These equations, which agree with [43], can be generalised to arbitrary tensors. Then,
one picks up signs when moving factors through dzM and ∂M , for instance
δξΩMN = −ξP∂PΩ PMN − ∂Mξ ΩPN − (−1)|M |(|N |+|P |)∂ PNξ ΩMP =
= −ξP∂PΩMN − 2∂ P[Mξ Ω|P |N), (2.75)
where the last equality only holds in general for graded-antisymmetric ΩMN , corre-
sponding to a 2-form Ω. Here [MN) denotes graded antisymmetrisation of M N ,
that is, ordinary antisymmetrisation but picking up an extra sign when fermionic
indices pass through each other.36
Now, we consider fields that carry some representation of the Lorentz group.37 When
transforming a Lorentz vector field V A, we must combine the above with a local
Lorentz transformation to ensure that the result is Lorentz covariant. Therefore, we
write
δ V A = −ξMξ ∂MV A + V BL AB = −ξMD V A + (−1)|M ||B|ξMV Bω A + V BM MB L AB .
(2.76)
Demanding Lorentz covariance, that the expression is linear in ξ (there cannot be a
constant term since V ′ = V for ξ = 0) and linearity in V , that is, δξ(V A + UA) =
δ V A + δ UA, we find L A = −ξMξ ξ B ω AMB . We now require that all Lorentz tensors
transform with this local Lorentz transformation and find that, for a covector UA,
δ U = −ξM∂ U + ξMξ A M A ω BMA UB = −ξMDMUA. (2.77)
This generalises to multiple indices.
Since ξM is a Lorentz scalar, the covariant derivative acts on it by a partial derivative.
Hence, the above rules for variations of fields with only curved indices generalise
to tensors with Lorentz indices by replacing all partial derivatives with covariant
derivatives. Thus, the vielbein transforms as
δξE
A
M = −ξNDNE AM −D ξNM E AN . (2.78)
By noting that
DMξA = DM(ξNE AN ) = D ξNE A + (−1)|M ||N |M N ξNDME AN (2.79a)
T AMN = 2D A A[MEN) = DMEN − (−1)|M ||N |D ANEM (2.79b)
we see that (2.78) can be written as
δ E Aξ M = −DMξA − ξNT ANM . (2.80)
36Similarly, (MN ] denotes graded symmetrisation. The idea behind this notation is that the left
(right) symbol indicates what to do with bosonic (fermionic) indices.
37Parts of the following naturally generalise to the case of an arbitrary structure group.
28
2.3. Eleven-dimensional supergravity
Lastly, we turn to the transformation of the connection. Remembering the inhomo-
geneous term in the Lorentz transformation of the connection,
δ ω Bξ MA = −ξN∂Nω BMA − ∂ ξNω BM NA +
+ ω CMA( (−ξNω BNC )− (−ξNω C B N BNA ))ωMC − ∂M(−ξ ωNA ) =
= −2ξN ∂ B C B[NωM)A − ω[N |A| ωM)C =
= −ξNR BNMA . (2.81)
2.3 Eleven-dimensional supergravity
Eleven-dimensional supergravity, the theory this thesis is mainly concerned with,
was first formulated as a spacetime theory [56] and then later in superspace [57].
Here, we give a superspace formulation. The theory is invariant under the dif-
feomorphism group of a curved supermanifold, similar to general relativity but
in the setting of supermanifolds. We will construct the theory as a completely
geometrical theory in superspace with supercoordinates zM = (xm, θµ)M , where x
has D = 11 real Grassmann-even components and θ has 32 real Grassmann-odd
components. In section 2.3.3, we summarise the theory in component form with
left-action conventions.
Similar to what we did when formulating the Wess–Zumino model in superspace, see
section 2.2.2, we wish to reduce the number of component fields by placing constraints
on the superfields. This time, however, the constraints will imply equations of motion
for the remaining component fields and put the theory on-shell. Having constrained
the fields, there are Bianchi identities that are no longer automatically satisfied.
These give relations between the remaining components, including the equations of
motions.
After solving the constraints and Bianchi identities, we want the spacetime metric
gmn to remain as a physical field.38 The metric has
1
2(D − 1)(D − 2)− 1 = 44, (2.82)
independent on-shell degrees of freedom, since they sit in the traceless symmetric
transverse part [1]. Since spinors in D = 11 have 32 components and the graviton is
massless, we expect a total of 232/4 = 256 on-shell degrees of freedom, by an argument
analogous to that in section 2.1.1.39 Hence, there should be 128 fermionic degrees
of freedom and another 84 bosonic ones. The bosonic ones can be obtained from a
3-form B Abelian gauge potential. Gauge invariance then implies that there are
1
3!(D − 2)(D − 3)(D − 4) = 84 (2.83)
38We will work the vielbein, from which the metric is constructed as g a bmn = ηabem en .
39256 degrees of freedom is also the number we expect if maximal N = 8 supergravity in D = 4
should be obtainable via dimensional reduction of eleven-dimensional supergravity.
29
2. Supersymmetry and supergravity
degrees of freedom since only the transverse directions contribute. Lastly, we need
the fermionic degrees of freedom. 128 is precisely the number of on-shell degrees
of freedom of a massless spin-3/2 field in D = 11 [56], since the tensor product
of a transverse vector and spinor is 9 ⊗ 16 ' 16 ⊕ 128 [1]. Hence, we expect a
spin-3/2 gravitino ψ αm associated to some gauge invariance corresponding to local
supersymmetry [1].
In superspace, we have a dynamical vielbein E AM . The metric will then be obtained
form the θ = 0 component of E am and we can hope to similarly obtain the gravitino
from E αm . It is, however, not immediately clear how the 3-form B should be obtained
from the geometrical quantities in superspace. Therefore, we introduce a Lorentz
scalar 3-form B in superspace. This might seem to contradict the above claim that
the theory is entirely geometrical but, as we will see, this is not the case.
Starting from the vielbein, spin connection and 3-form, we can construct the curvature
2-form R BA , the torsion TA and the Abelian field strength H = DB = dB. These
satisfy the Bianchi identities
DTA = EB ∧R A, DR BB A = 0, DH = 0, (2.84)
where the middle equation is the Bianchi identity of the second type, (2.65), while
the other two are Bianchi identities of the first type, (2.66). According to a theorem
due to Dragon [58], these are not independent equations. Specifically, R BA can be
expressed in terms of the torsion by using the first identity and the second identity
is then automatically satisfied.40
2.3.1 Constraints
As mentioned above, we constrain the superfields to reduce the number of component
fields. This will put the theory on-shell. We arrive at the constraints motivated
by the field content of the theory and dimensional analysis, similar to [60]. Due to
Dragon’s theorem, we do not constrain the curvature 2-form, only the torsion and
the 3-form. Note that, since we expect the gravitino ψ αm to be related to the θ = 0
component of E α, the corresponding field strength S αm mn should be related to T αmn .
For the dimensional analysis, we use mass dimensions, so [dzm] = −1. Since, in the
superalgebra, the commutator of two supercharges is a translation in spacetime, we
need [dzµ] = −1/2. Starting from a superform Ω with dimension [Ω] = n, we can
deduce the dimensions of the components. For each bosonic index, the dimension of
the component is raised by 1 unit to balance the dimension of dzm. Similarly, the
dimension is raised by a half unit for every fermionic index. To be able to contract
upper with lower indices, the dimension of upper indices must contribute in the
opposite way.
40EB ∧ DR AB = 0 follows directly from Bianchi identities of the first type. One then has to check,
using that R BA is Lie algebra-valued, that this implies DR BA = 0. The theorem holds in D > 3
but fails in three dimensions, see for instance [59].
30
2.3. Eleven-dimensional supergravity
As usual, the derivatives have dimensions opposite to those of the coordinates, whence
the exterior derivative is dimensionless, [D] = 0. From their definitions, it is clear
that the dimension of the curvature and torsion components come solely from their
indices, which we may write as [R] = [T ] = 0. Since the 3-form, B, and its field
strength, H, are nongeometrical, we cannot derive their dimensions from the above.
However, it will turn out to be reasonable to set [Hmnpq] = 1, which implies [H] = −3.
We will constrain the components with flat indices. Still, we only keep components of
T and H corresponding to the field strengths S αmn and Hmnpq as dynamical. Apart
from that, we allow nondynamical components expressed in terms of Γ-matrices.
To not introduce dimensionful constants, these components must be dimensionless.
Thus, for H, we have the nonzero components
Habcd, Habγδ = 2i(Γab)γδ, (2.85)
where the second equation determines the normalisation of H.41,42 For T , we can
also form a nonvanishing component using the dynamical component of H, which
leads to the nonzero ( )
T γab , T
c = 2i(Γcαβ )αβ, T
γ
aβ = Hbcde k [b cde]
γ
1δa (Γ )β + k2(Γ bcde
γ
a )β , (2.86)
where k1 and k2 are, as of yet, undetermined dimensionless constants that will be
fixed by the Bianchi identities. Here, T cαβ = 2i(Γc)αβ is motivated by that flat
superspace should be a solution to the theory [61].43
Since T γaβ is expressed in terms of the only dynamical component field in H, the
theory could have been formulated without H, something we alluded to above. At
this point, it is not clear whether we, in that case, would have to implement additional
constraints on T or if it follows from the Bianchi identities that T γaβ can be written
in this way. By a more careful analysis, one can show that the 3-form, B, and its
field strength, H, do not have to be introduced by hand without adding additional
constraints [61]. In fact, the theory can be derived, without introducing the 3-form
by hand, from the single constraint T cαβ ∼ (Γc)αβ [62].
41Note that the notation here differs from chapter 3 and appendix B.4. Here Γa denotes the
eleven-dimensional Γ-matrices.
42H being a real 4-form implies that Habγδ is imaginary due to how Grassmann-odd quantities are
complex conjugated.
43Flat superspace has nonvanishing torsion, see for instance [43], [44].
31
2. Supersymmetry and supergravity
Solution to the Bianchi identities
To solve the Bianchi identities (2.84) subject to the constraints, we first write them
as tensor equations. For the torsion,
D D = 1
( )
T D EC ∧ EB2 T
D
BC =
= 1EC ∧ EB2 ∧
1
EADAT D + ECBC 2 ∧ T
B 1T DBC − 2T
C ∧ EBT DBC =
= 1EC ∧ EB ∧ EAD T D + EC2 A BC ∧ T
BT DBC =
= 1 C2E ∧ E
B ∧ 1EAD T D C B AA BC + 2E ∧ E ∧ E T
ET DAB EC . (2.87)
By completely analogous calculations for the curvature 2-form and the 4-form, we
find that the Bianchi identities on tensor form are
D T D E D D[A BC) + T[AB T|E|C) = R[ABC) , (2.88a)
D[AR E F EBC)D + T[AB R|F |C)D = 0, (2.88b)
D[AHBCDE) + 2T F[AB H|F |CDE) = 0. (2.88c)
As noted above, the second of these follows from the first due to Dragon’s theorem.
To proceed, one splits the equations into all different index types. Here, we present
the results of this analysis, a detailed derivation can be found in appendix G (see also
[60]). One finds that the only dynamical independent component fields of R, H and
T are the θ = 0 components of R γabcd, Habcd and Tab . The partially undetermined
components of the torsion, see (2.86), are found to be
1 ( )
T γaβ = −288H
[b
bcde 8δa (Γcde])
γ
β + (Γ bcde
γ
a )β . (2.89)
The field strengths satisfy Bianchi identities
R d[abc] = 0, (2.90a)
D[aHbcde] = 0, (2.90b)
D T δ[a bc] − T ε δ[ab Tc]ε = 0, (2.90c)
and equations of motion
R − 1 1η R = H H cde − 1 2ab 2 ab 12 acde b 96ηabH , (2.91a)
Dd = − 1H d1d2d3d4e1e2e3e4dabc 1152abc Hd1d2d3d4He1e2e3e4 , (2.91b)
T β(Γabc) αab β = 0. (2.91c)
32
2.3. Eleven-dimensional supergravity
2.3.2 The spacetime theory
Now that we have discussed eleven-dimensional supergravity in superspace, we turn
to the component formulation in spacetime. To distinguish between superfields and
spacetime fields, we put a hat on all superfields. The spacetime fields are defined
as the θ = 0 components of the corresponding superfields with curved form-indices
[60]. This is motivated geometrically since we, at least locally, can embed spacetime
in superspace by xm 7→ (xm, 0)M and the definition then ensures that spacetime
fields are tangent to the spacetime in this embedding. Lie algebra-indices, like
AB on ω BMA , and the flat indices on EA and TA are, however, kept flat. Local
supersymmetry corresponds to local translations in the θ-directions in superspace
whence invariance under gauged supersymmetry corresponds to invariance under the
choice of local embedding.
The first step in converting the superspace theory into a spacetime theory is to
impose gauge conditions. Then, we compute the supersymmetry transformations
and, lastly, derive the equations of motions for the spacetime fields. Note that this
section uses right-action conventions even in spacetime. In section 2.3.3, we switch
to left-action conventions and summarise the component formulation of the theory.
Gauge fixing
Consider an infinitesimal superspace coordinate transformation ξ̂M (z) combined with
an infinitesimal local Lorentz transformation L̂ BA (z). From (2.81), we find
δω̂ B N B C B C B BMA = −ξ̂ R̂NMA + ω̂MA L̂C − L̂A ω̂MC − ∂̂M L̂A . (2.92)
Since the theory in spacetime will contain gravity, we keep manifest invariance under
arbitrary spacetime coordinate transformations ξm(x) and local Lorentz transforma-
tions L BA (x). For M = µ, we can, however, use the θ-component of L̂ BA to set the
last term to a tensor C BµA (x) which is arbitrary apart from being Lie algebra-valued
in its last two indices. Hence, we can use that term to gauge away ω̂ BµA | and then
maintain that condition for transformations with arbitrary θ = 0 components of ξ̂M
and L̂ B.44A
Similarly, from (2.80),
δÊ A = −D̂ ξ̂A − ξ̂N T̂ A + Ê B AM M NM M L̂B . (2.93)
Here, we can use the θ-component of ξ̂A to set Ê aµ | = 0 and Ê α| = δαµ µ . It is
not trivial that this is possible. For an infinitesimal transformation, we can indeed
transform Ê Aµ | in any desired direction. This does, however, not imply that we can
set the corresponding components to whatever we like. For instance, we cannot set
Ê Aµ | = 0 since that would render Ê AM noninvertible. With the above gauge choice,
there is, however, no such singularity and we therefore expect it to be viable. In the
following, we assume this to be the case. Similar remarks apply to the other gauge
conditions.
44All evaluation bars denote evaluation at θ = 0.
33
2. Supersymmetry and supergravity
Lastly, we turn to B̂MNP . Since only the field strength Ĥ = d̂B̂ enters the theory,
we have an Abelian gauge symmetry
δλ̂B̂ = d̂λ̂, ⇐⇒ δλ̂B̂MNP = 3∂̂[M λ̂NP ), (2.94)
that leaves Ĥ invariant. Combining this with an infinitesimal coordinate transforma-
tion, see (2.75), we get
δB̂ = −ξ̂Q∂̂ B̂ − 3∂̂ ξ̂QMNP Q MNP [M B̂|Q|NP ) + 3∂̂[M λ̂NP ) =
= −ξ̂QĤQMNP − 3∂̂ Q[M(ξ B̂|Q|NP )) + 3∂̂[M λ̂NP ). (2.95)
In this transformation, we use the θ-component of λ̂mn to gauge away B̂µnp|.
The gauge w(e have arrived at)can be summarised as∣
A∣∣ a= em (x) Aψ αm (x)ÊM 0 α , Ê MA ∣
∣∣ ( )e m= a ( Mx) −ψ µa (x)0 µ , (2.96a)∣ δ∣ µ M ∣∣
δα A
ω̂ BµA ∣ = 0, B̂µnp∣ = 0. (2.96b)
Similar gauges are used in, for instance, [43], [63], [64]. Note that, on spacetime
fields, we convert between curved and flat indices using e am and δαµ , for example,
ψ αm = e am δαµψ µa . In superspace, on the other hand, curved and flat indices are
converted using Ê AM , which illustrates the importance of keeping spacetime fields
and θ = 0 components of superfields apart.
Since we have only used the θ-components of ξ̂M , L̂ BA and λ̂mn, we can still make
transformations with θ = 0 while maintaining the gauge with θ-components. From the
above, it is clear that ξm(x) corresponds to coordinate transformations in spacetime,
L BA (x) local Lorentz transformations and λmn(x) Abelian gauge transformations
of B µmnp. Since ξ (x) is fermionic, it must correspond to a gauged supersymmetry
transformation.
There are still some transformations that we have not considered, including higher
θ-components and λ̂MN with one or two fermionic indices. However, we are only
interested in e a am , ψm , Bmnp and ωm and these are invariant under such transforma-
tions.
Supersymmetry transformations
The (gauged) supersymmetry transformation is generated by ξµ(x). Note that
ξ̂α| = δαξµ = ξα and ξ̂a| = 0, since ξ̂A = ξ̂M Ê Aµ M . ∣From (2.93), we find
δ e a = −ξµ a∣ξ m T̂µm ∣. (2.97)
At this point, we need a torsion component with curved indices. As already mentioned,
flat indices in superspace are converted to curved ones using Ê AM , with sign factors
34
2.3. Eleven-dimensional supergravity
similar to those in (2.75). For the torsion
1d̂ẑN2 ∧ d̂
1 1
ẑM T̂ C B AMN = 2Ê ∧ Ê T̂
C
AB = 2d̂ẑ
N Ê BN ∧ d̂ẑM Ê A CM T̂AB
=⇒ T̂ C = (−1)|M |(|N |+|B|)MN Ê BN Ê AM T̂ CAB . (2.98)
Thus, the torsion componen∣∣t of interest is, see (2.86),
T̂ a∣ = −ψ βδαT̂ a = −2i(Γaµm m µ αβ )µβψ βm (2.99)
and the supersymmetry transformation of the vielbein
δξe
a
m = 2iξα(Γa)αβψ βm . (2.100)
For the spin-3/2 field, again using (2.93), ∣∣
δξψ
α = −D α ν αm mξ − ξ T̂νm ∣. (2.101)
By (2.89∣ ) and (2.98), ∣the relevan(t torsion component is
α∣T̂ ∣ ) ∣= e aδβT̂ α∣∣= 1 a [b1 b2b3b4] α b1b2b3b4 α ∣νm m ν βa 288em 8δa (Γ )ν + (Γa )ν Ĥb1b2b3b4∣. (2.102)
To express this in terms of the spacetime fields, we first relate the components of Ĥ
with flat and curved indices, similar to (2.98) but for Ĥ. We find
Ĥ = (−1)(|N |+|B|)|M |+(|P |+|C|)(|M |+|N |)+(|Q|+|D|)(|M |+|N |+|P |)MNPQ ·
· Ê DQ Ê C B AP ÊN ÊM ĤABCD, (2.103)
whence the spacetime field Hmnpq is given by, se∣∣e (2.85),H = Ê DÊ Cmnpq q p Ê Bn Ê Am ∣ĤABCD∣ = ( )
∣ = e d∣ q e
c b
p en e
a
m Ĥ ∣∣+ 6ψ δ γabcd [q ψp e b an em] 2i(Γab)γδ
=⇒ Ĥabcd∣ = H + 12iψ γabcd [a (Γbc) δ|γδ|ψd] =: H̃abcd. (2.104)
This equation illustrates, explicitly, the importance of distinguishing between θ = 0
components of superfields (Ĥabcd|) and spacetime fields (Habcd). Putting the above
together, we find the supersymmet(ry transformation )
δ ψ α = −D ξα − 1 ξβe a 8δ[b1(Γb2b3b4] α b1b2b3b4 αξ m m 288 m a )β + (Γa )β H̃b1b2b3b4 =
=: −D̃ αmξ . (2.105)
Lastly, we turn to the supersymmetry transformatio∣∣ n of Bmnp. From (2.95),
δξB
σ
mnp = −ξ Ĥσmnp∣. (2.106)
The relevant compone∣∣nt of Ĥ is, by (2.85) and∣∣ (2.103),
Ĥσmnp∣ = −3e c[p e b αn ψm] δδ ασĤδαbc∣ = −6i(Γ[mn)|σα|ψp] , (2.107)
whence the supersymmetry transformation is
δξB
α β
mnp = 6iξ (Γ[mn)|αβ|ψp] . (2.108)
35
2. Supersymmetry and supergravity
Equations of motion
Now that we have found the supersymmetry transformations, we derive the equations
of motion for the spacetime fields. We begin by examining the torsion. From (2.86)
and (2.98), ∣
T a = ψ βψ αT̂ a∣ α a βmn n m αβ∣∣ ∣ = −2iψm (Γ )α∣ βψn (2.109a)
T α = e be aT̂ α∣+ 2ψ βe aT̂ α∣mn n m ab [n m] aβ ∣. (2.109b)
By the definition of the torsion, T α α αmn = 2D̂[mÊn] | = 2D[mψn] , whence (2.89)
and (∣2.109b) giv[es∣
T̂ αab ∣ ( ) ]= 2e me n D ψ α 1 β [pa b [m n] + 288ψ[n| 8δ 1 (Γp2p3p4]) γ|m] β + (Γ p1p2p3p4 γ|m] )β H̃p1p2p3p4 =
= 2e ma e nb D̃[mψ αn] , (2.110)
with D̃m as defined in (2.105). Thus, the equation of motion for T αab , (2.91c), gives
the equation of motion for ψ αm
D̃ β mnp αmψn (Γ )β = 0. (2.111)
Writing the spin connection as ω bma = ω̊ b+K bma ma , where ω̊ is the unique torsion-free
spin connection and K bma the contorsion tensor, the definition of the torsion gives
T a = 2K amn [mn] . Since the contorsion is Lie algebra-valued, Kabc = K[ab]c −K[ac]b −
K[bc]a. Hence, by (2.109a),
K αabc = −iψa (Γc)αβψ
β
b + iψ αa (Γb)αβψ βc + iψ α βb (Γa)αβψc (2.112)
Turning to the equation of motion for H, we need to convert between curved and flat
indices on the derivative. The covariant derivative in spacetime is Dm = D̂m|, where
the result is evaluated at θ = 0 after D̂m has acted. In superspace, D̂M = Ê AM D̂A,
whence D̂m| = e am D̂a|+ ψ α∣ m D̂α|. Thu∣∣ ∣
s, using (2(.110) and (G.10),∣∣ )ηabD̂ Ĥ = ηabD Ĥ − ηabψ α −12iT̂ β(Γ ) ∣∣a bcde a bcde a [bc de] βα ∣ =
= DbH̃bcde + 24iψbα(Γ[bc) m n β|αβ|ed ee] D̃mψn . (2.113)
To write the equation of motion with curved indices, we have to move vielbeins
through the Da derivative. As explained in appendix E.3, we need to replace the
Lorentz connection with an affine connection to be able to do this. Thus, the equation
of motion (2.91b) becomes
∇mH̃ + 24iψmαmnpq (Γ[mn)
β
|αβ|D̃pψq] = −
1 r1r2r3r4s1s2s3s4
1152npq H̃r1r2r3r4H̃s1s2s3s4 .
(2.114)
Lastly, we turn to the equation of motion for R, given in (2.91a). From the above,
it is clear what happens to the right-hand side when switching from superspace to
36
2.3. Eleven-dimensional supergravity
spacetime. For the left-hand side, we need to relate Rmnpq to R̂abcd|. For arbitrary
indices
R̂ D = (−1)(|N |+|B|)|M |MNC Ê BÊ AR̂ DN M ABC , (2.115)
whence
∣∣ ∣ ∣ ∣
R̂ ∣ = e be a ∣mncd n m R̂abcd∣+ 2 ∣ ∣e bψ α β α[n m] R̂αbcd∣+ ψn ψm R̂αβcd∣. (2.116)
Using (2.110) a∣ nd (G.24) we fi(nd that the middle term is )
2e bψ α ∣[n m] R̂αbcd∣ = 4ie pe qψ α (Γ ) D̃ ψ βc d [m| [p| αβ |q] |n] −(Γ[p|) βαβD̃|n]ψ|q] −(Γ|n])αβD̃[pψ βq] ,
(2.117)
where, in all terms, the antisymmetrisations are [mn] and [p q]. By (2.104) and (G.11),
the third term ∣is
β α ∣∣ = 4i ( )ψ ψ R̂ ψ α 24δa1a2(Γa3a4) + (Γ a1a2a3a4n m αβcd 288 m c d αβ cd ) ψ βαβ n H̃a1a2a3a4 . (2.118)
Defining R̃abcd = R̂abcd| and collecting the above terms, we find( )
R̃ = R − 4i( ψ α 24δr1r2(Γr3r4) + (Γ r1r2r3r4 βmnpq mnpq 288 m p q αβ pq )αβ ψn H̃r1r2r3r)4+
− 4iψ α (Γ ) D̃ β β β[m| [p| αβ |q]ψ|n] − (Γ[p|)αβD̃|n]ψ|q] − (Γ|n])αβD̃[pψq] . (2.119)
With this definition, the equation of motion reads
R̃mn −
1
2gmnR̃ =
1 pqr 1 2
12H̃mpqr H̃n − 96gmnH̃ , (2.120)
where R̃mn := R̃ ppmn and R̃ = R̃ m 45m .
2.3.3 Component formulation with left-action
In this section, we give the most important equations from section 2.3.2 but in
left-action conventions. The crucial insight to convert a right-action to a left-action
is that (g g )−1 = g−1g−11 2 2 1 . Thus, given a right-action, we can define a left-action by
acting with g−1 from the right and, given a right-representation (v · g)a = vbg ab , we
get a left-representation (g · v)a = vb(g−1) ab . Hence, we need to replace all group
elements by their inverses and all Lie algebra elements by their negatives when
switching conventions.
We also switch conventions for the index order on differential forms. Here, we employ
the usual convention
Ω = 1 dxm1! ∧ . . . ∧ dx
mpΩm1...mp . (2.121)p
45Recall that we are working in right-action conventions, whence this definition agrees with R < 0
for AdS.
37
2. Supersymmetry and supergravity
We demand that the 3-form, B, is the same when switching convention. This implies
that we must replace B 46mnp with −Bmnp and Hmnpq with −Hmnpq. Since the spin
connection is Lie algebra-valued, it should be replaced with its negative. This implies
that the curvature 2-form is unchanged while its components R bmna change sign.47 In
these conventions, we define the Ricci tensor Rmn = R pmpn . Due to how we defined
the Ricci tensor in the other set of conventions, it does not change sign.
In this section we will, moreover, not write out spinor indices explicitly. Spinors
have an implicit subscript index (χ means χα), Dirac conjugated spinors have an
implicit superscript index (χ̄ means χ̄α) and Γ-matrices have their indices in the
usual positions (Γa means (Γa) βα ).48
From (2.112) we find that the spin( connection, with these con)ventions, is
ωabc = ω̊abc − i ψ̄aΓcψb − ψ̄aΓbψc − ψ̄bΓaψc , (2.122)
where ω̊mab is the torsion-free spin connection. The Lorentz covariant derivative is
Dm = ∂m + ωm and ∇m denotes the associated affine connection, see appendix E.3.
Rmnpq denotes the curvature tensor of the spin connection ωm.
The supersymmetry transformations are, by (2.100), (2.105) and (2.108),
δ e aξ m = −2iξ̄Γaψm, (2.123a)
δξψm = −D̃mξ, (2.123b)
δξBmnp = 6iξ̄Γ[mnψp], (2.123c)
and equations of motion, by (2.111), (2.114) and (2.120),
ΓmnpD̃nψp = 0, (2.124a)
∇mH̃ + 24i mΓ D̃ 1ψ ψ =  r1r2r3r4s1s2s3s4mnpq [mn p q] 1152 npq H̃r1r2r3r4H̃s1s2s3s4 (2.124b)
R̃mn −
1 1 1
2g
pqr 2
mnR̃ = 12H̃mpqr H̃n − 96gmnH̃ (2.124c)
where, by (2.104), (2.105) and (2.119(), )
D̃mξ = Dmξ −
1
H̃ [n1288 n1n2n3n4 8δm Γ
n2n3n4] − Γ n1n2n3n4m ξ, (2.125a)
H̃mnpq = Hmnpq + 12iψ̄[mΓnpψq], ( ) (2.125b)4i
R̃mnpq = Rmnpq −(288H̃r1r2r3r4ψ̄m 24δr1r2Γr3r4 + Γ r1r2r3r4p q pq ) ψn+
+ 4iψ̄[m| Γ[pD̃q]ψ|n] − Γ[p|D̃|n]ψ|q] − Γ|n]D̃[pψq] , (2.125c)
where, in the last line, the antisymmetrisations are [mn] and [p q].
46Note that Hmnpq = 4∂[mBnpq] in both conventions due to the difference in the definition of the
exterior derivative.
47This is consistent with Lie algebra elements being replaced by their negatives due to the additional
change of conventions for differential forms.
48Due to how we raise and lower spinor indices in D = 11, the Majorana condition χ = δβ̇χ†α α canβ̇
be written as χ̄α = −χα (see appendix B.4).
38
3. Supergravity compactifications
If eleven-dimensional supergravity describes reality (at low energies) an obvious
question that arises is why we perceive reality as having only four dimensions.
Historically, this question arose as early as 1921 when Kaluza [65] proposed a
unification of gravity and electromagnetism by introducing a fifth dimension.49 To
obtain Einstein’s and Maxwell’s field equations, Kaluza assumed, ad hoc, that all
fields are independent of the fifth dimension. This would also explain why we cannot
see the fifth dimension since there can be no dynamics in a direction in which
everything is constant. Still, it seems like an unmotivated assumption; if nature is
truly five-dimensional, why should all fields be constant in a specific direction?
A more satisfactory explanation was put forward by Klein [67] in 1926. Klein assumed
that the fifth dimension is periodic, that is, that the topology of spacetime is that of
R4×S1. Then, all fields can be expanded in Fourier series in the periodic coordinate
and ordinary gravity and electromagnetism correspond to the zero-modes in the
expansion [66]. Klein’s idea also explains the quantisation of electric charge, which
corresponds to momentum in the periodic dimension and is naturally quantised due
to the periodicity [68]. Assuming that the smallest unit of charge is that of the
electron, Klein derived the period of the compact dimension to be of order 10−30 m.
This also explains why we do not observe five dimensions in experiments since physics
at much larger scales would be averaged over the compact dimension. However,
momentum in the periodic dimension also gives the fields masses. With the above
period, these are of the same order as the Planck mass [66], that is, about 1022 times
the electron mass.
The situation for string and supergravity theories is similar. If some of the dimensions
form a compact manifold, this explains why we only observe four dimensions, provided
that the extra dimensions are sufficiently small. Similar to how electromagnetism
arises in Kaluza–Klein theory, isometries of the compact manifold, or internal space,
give rise to, possibly non-Abelian, gauge fields in spacetime [19]. As explained in
more detail in section 3.2.2 and also analogous to Kaluza–Klein theory, momentum
in the internal directions contribute to the mass of spacetime fields.
In this chapter, and the remainder of the thesis, we use slightly different notation and
conventions than in section 2.3.3. Uppercase indices are used as eleven-dimensional
spacetime indices, not superspace indices. In compactifications, Greek indices
(α, β, γ, . . . and µ, ν, ρ, . . .) are used for the resulting spacetime and Latin lowercase
(a, b, c, . . . and m,n, p, . . .) for the internal manifold, that is, the extra dimensions.
49The theory also contains a dilaton. At the time, this was, however, inconsistently set to zero [66].
39
3. Supergravity compactifications
Letters from the beginning of the alphabets are used for flat indices and letters from
the middle for curved ones. Spinor indices are not written out.
To distinguish between the components of the eleven-dimensional Γ-matrices in
the internal directions and the Γ-matrices on the internal manifold, we denote the
eleven-dimensional Γ-matrices by Γ̂A, similar to appendix B.4. Furthermore, we
denote the 3-form by A instead of√B and its field strength by F instead of H. We a√lso
rescale the gravitino ψM →7 ψM/ 2 and the supersymmetry parameter ξ 7→ −ξ/ 2.
Lastly, x denotes coordinates on the spacetime and y on the internal manifold.
With these conventions, the bosonic equations of motion, resulting from setting
ψM = 0 in (2.122) and (2.124), are
1 1
R − g R = F F PQR − 1 g F 2MN 2 MN 12 MPQR N 96 MN , (3.1a)
∇M 1FMNPQ = 1152
R1R2R3R4S1S2S3S4
NPQ FR1R2R3R4 FS1S2S3S4 , (3.1b)
where the curvature and covariant derivative are those of the torsion-free connection,
and the supersymmetry transformations are, by (2.123),
δ e A = iξ̄Γ̂Aξ M ψM , (3.2a)
δξAMNP = 3iξ̄Γ̂[MNψP ], ( ) (3.2b)
δξψM = D̃ ξ ' D ξ −
1 [N PQR] NPQR
M M 288FNPQR 8δM Γ̂ − Γ̂M ξ, (3.2c)
where, in the step indicated by ', we have dropped terms containing ψM . These
conventions (apart from how the Dirac conjugate is defined), equations of motion
and supersymmetry transformations agree with [1].
3.1 Freund–Rubin compactification
In theories with extra dimensions, we wish to achieve what is known as spontaneous
compactification. In contrast to ad hoc compactification, we do not simply postulate
that some dimensions are compact but instead look for stable ground state, or
vacuum, solutions to the field equations that describe, at least locally, a product
spaceMd×Mk [19]. Here, D = d+ k is the dimension of the complete reality in the
theory, d the dimension of spacetime (after compactification) and k the dimension of
the internal manifold. We will work towards the Freund–Rubin ansatz [69], which
is a way of achieving spontaneous compactification, but make some more general
comments before arriving at the full set of assumptions in the ansatz.
The first assumption we will employ is to assume that the vacuum spacetimeMd is
maximally symmetric.50 This is motivated experimentally and is a generally accepted
50We may add, as a zeroth assumption, that we assumeMd andMk to be spin manifolds so that
spinors can be defined globally. See [70] for an introduction to spin geometry.
40
3.1. Freund–Rubin compactification
assumption.51 The assumption implies that the vacuum expectation values of the
various fields can only be constructed from scalars, the metric and the Levi-Civita
tensor in the external space. We do not consider the possibility of topological phases
with vanishing vacuum expectation value of the metric.
The D = 11 spinor decomposes into the tensor product of the spinor in spacetime
and the spinor on the internal space when Spin(D−1, 1) is broken to Spin(d−1, 1)×
Spin(k), see appendix B.4. Hence, a spinor in D = 11 can be written as a sum of
terms on the form ε⊗ η, where ε is an anticommuting spinor in spacetime and η a
commuting spinor on the internal manifold.52 Since a nonzero spinor or vector-spinor
in spacetime would break maximal symmetry [19], we set ψ̊M = 0, where the overset
circle denotes that it is a vacuum value. Since ψM is the only fermion field in the
theory, this clearly solves its equation of motion. Note that this does not imply
that the vacuum expectation value of fermion bilinears vanish, that is, there can be
fermion condensates [72].53 In the following, we assume that all fermion bilinears
vanish as well, which implies that the relevant equations of motion are those in (3.1)
and, by (2.122), that the spin connection is torsion-free in the background.
Maximal symmetry forces the the x-dependence of the spacetime vacuum metric g̊µν
to be either that of Minkowski, de Sitter (dS), or anti-de Sitter (AdS) spacetime,
corresponding to R = 0, R > 0 and R < 0, respectively [19].54 In general, there may
also be a y-dependence, whence we write g̊µν = f(y)gm.s.µν (x) where f > 0 is known
as the warp factor and gm.s.µν (x) is maximally symmetric.55,56 Since a nonvanishing
spacetime vector field would break maximal symmetry, the mixed components of
the metric vanish, g̊µn = 0. Also, since the internal components g̊mn are spacetime
scalars, they must be x-independent to not break maximal symmetry.
Consider now the 4-form FMNPQ = (Fµνρσ, Fµνρq, Fµνpq, Fµnpq, Fmnpq). Here, maximal
symmetry forces F̊mnpq to be x-independent. Furthermore, any of the other compo-
nents can only be nonvanishing if it is a product of the completely antisymmetric
-tensor in spacetime and a tensor on the internal manifold. Thus, with d = 2, 3, 4
we may have nonzero F̊µνpq, F̊µνρq and F̊µνρσ, respectively. For other values of d, all
three of these vanish.
At this point, we make the additional assumption that F̊µνρq = 0 = F̊µνpq. Thus, we
can set F̊µνρσ = −6m̊µνρσ, with m = 0 in d =6 4 where ̊µνρσ does not make sense.
51A cosmology with a Big Bang singularity is clearly not maximally symmetric. However, this is
due to the matter content, not the vacuum.
52See [71] for a general discussion on the spinor bundle of product manifolds.
53Non-vanishing vacuum expectation values of fermion bilinears have consequences for the cosmo-
logical constant [72].
54Note that these are local considerations, that is, the spacetime is locally isometric to Minkowski,
dS or AdS.
55Here, we require that the spacetime metric is of constant signature (d− 1, 1). Signature changing
metrics have been discussed in the context of cosmology and quantum gravity, see [73]–[75].
56Intuitively, we glue together copies ofMd with different sizes overMk. For instance, S2 without
the poles is a warped product of a circle (the equator) and a line (a meridian).
41
3. Supergravity compactifications
Note that, for dimensional reasons, m has dimension mass and it is independent
of y due to the Bianchi identity ∂[MFNPQR] = 0. The Bianchi identity also implies
F̊mnpq = F̊mnpq(y), which, as noted above, also follows from maximal symmetry, and
∂[mFnpqr] = 0.
We now turn to the Einstein equation (3.1a). It is convenient to write the equation
for the vacuum values as
1
R̊ PQR
1 PQRS
MN = 12 F̊MPQR F̊N − 144 g̊MN F̊PQRS F̊ (3.3)
Using that F̊µνρσ = −6m̊µνρσ and thatMd is Lorentzian, we find
F̊ F̊ ρσλµρσλ ν = −6(6m)2g̊ , F̊ µρσλµν µρσλ F̊ = −24(6m)2. (3.4)
Since the vacuum metric is blo(ck diagonal, (3.3) spli)ts into
R̊ = − 1 g̊ 123m2µν 144 µν + F̊
mnpq
mnpq
1 1 (
F̊ , ) (3.5a)
R̊mn = 12 F̊
pqr pqrs 2
mpqr F̊n − 144 g̊mn F̊pqrs F̊ − 864m , (3.5b)
R̊µn = 0. (3.5c)
By choosing f(y) appropriately, we can make Rm.s. ∈ {−1, 0, 1 }, where Rm.s.(d) (d) is the
Ricci scalar of the maximally symmetric metric gm.s. µνµν . Contracting (3.5a) with g̊ ,
we find, since R̊µν is independent of f(y),
Rm.s.
( ) = (
( )
d) d
R̊ y 3 2 mnpq(d) ( ) = −144 12 m + F̊mnpq (y)F̊ (y) . (3.6)f y
Since f > 0, R̊(d)(y) is of constant sign. Furthermore, R̊(d)(y) is only zero at a
point y if both m = 0 and F̊mnpq(y) = 0 and, then, F̊mnpq = 0 at all points due to
R̊(d)(y) having constant sign. Thus, the only Minkowski solution, under the above
assumptions, is the zero flux case F̊MNPQ = 0 with a Ricci flat internal manifold,
R̊mn = 0. In all other cases, R̊(d) < 0, the spacetime is AdS and the internal manifold
has everywhere positive scalar curvature R̊(k) > 0.
Now, we assume that m 6= 0 and that there is no internal flux, that is, F̊mnpq = 0.
These are the last assumptions in the Freund–Rubin ansatz. As noted above, the
former forces d = 4 and k = 7 while the latter implies that f is independent of y by
(3.6). Under these assumptions, (3.5) immediately gives
R̊µν = −12m2g̊ 2µν , R̊mn = 6m g̊mn. (3.7)
Thus, the spacetime is AdS4 and the internal manifold an Einstein manifold with
positive scalar curvature. Hence, assuming that the internal manifold is complete
implies, by the Bonnet–Myers theorem [76], that it is compact and of finite diameter57.
57The diameter of a Riemannian manifold (that is, pseudo-Riemannian with Euclidean signature)
is the supremum of all distances.
42
3.1. Freund–Rubin compactification
The last part of (3.5), R̊µn = 0, is trivially satisfied since the only nonzero Christoffel
symbols of the Levi-Civita connection are Γ̊ νµ ρ (x) and Γ̊ nm p (y). Note that there not
being any other nonzero Christoffel symbols implies that the affine connection splits
as ∇̊M = ∇̊µ ⊕ ∇̊m, where ∇̊µ and ∇̊m are the Levi-Civita connections of AdS4 and
M7, respectively. For nontrivial warp factors, this is not generally true.
Note that the equation of motion for F , (3.1b), is satisfied since the right-hand side
immediately vanishes and ∇̊µ̊µνρσ = 0. With F̊mnpq 6= 0, there would have been a
nontrivial equation
∇̊mF̊mnpq =
1
̊ rstu4 npq F̊rstu. (3.8)
Unbroken supersymmetries
Thus far, we have not paid much attention to the gravitino, ψM , as its background
value is 0. For supersymmetry, it is, however, crucial. Since the supersymmetry
parameter is fermionic, each term in the transformations of the bosonic fields must
contain the gravitino, as is also evident from (3.2). Thus, all bosonic fields are
invariant under supersymmetry transformations in the vacuum. That the gravitino
is also invariant under a supersymmetry transformation in the vacuum is, therefore,
equivalent to the corresponding generator being unbroken.
As noted above, the spinor in eleven dimensions decomposes into a tensor product
of a four-component spinor in spacetime and an eight-component spinor on M7.
Explicitly, we write the eleven-dimensional Γ-matrices as
Γ̂α = γα ⊗ 1, Γ̂a = −γ5 ⊗ Γa, (3.9)
where, as in appendix B.4 but with slightly different notation, γα are the four-
dimensional γ-matrices, γ5 = −i γαβγδαβγδ /24 and Γa are the seven-dimensional
Γ-matrices.
In the Freund–Rubin vacuum, the supersymmetry transformation of the gravitino is
δξψM = D̃Mξ. Here, and in the following, we drop the overset circle to reduce clutter;
all quantities refer to vacuum values. Putting δξψM = 0, we obtain the (generalised)
Killing spinor equation
6m ( [ )D̃ µ νσρ] µνσρMξ = DMξ + 288µνσρ 8δM Γ̂ − Γ̂M ξ = 0 (3.10)
Since M7 is compact, ξ(x, y) can be expanded as ξ(x, y) = χI(x) ⊗ λI(y) where
λI(y) is a complete (infinite but countable) set of linearly independent spinors on
M7. Using  νρσµνρσΓ̂ = −6iγ5γµ ⊗ 1 and that the spin connection in D = 11 only
has nonzero components ω(µαβ and ωmab, the M) = µ part of (3.10) gives
D χIµ − imγ5γ χIµ ⊗ λI = 0, (3.11)
where, by abuse of notation, we use Dµ to denote the covariant derivative in AdS4.
Since λI are linearly independent, the parenthesis must vanish for every I. In
43
3. Supergravity compactifications
AdS4, this equation admits four linearly independent solutions εi(x) for χ, which
is the maximal number in d = 4 [19]. Due to the linear independence of these, we
may reorder the terms in the expansion and write ξ(x, y) = εi(x) ⊗ ηi(y), which
is now a sum of just four terms. The M = m part of (3.10) then reads, using
 Γ̂ µνρσµνρσ m = −24i1⊗ Γm, ( )
εi D mmηi + i 2 Γmηi = 0. (3.12)
Again the parenthesis must vanish for every i, now due to the linear independence of
εi. Hence, the most general solution ξ to δξψM = 0 is a sum of terms ε(x) ⊗ η(y)
where
D̃ 5µε := Dµε− imγ γµε = 0, (3.13a)
D̃ mmη := Dmη + i 2 Γmη = 0. (3.13b)
Since, as noted above, AdS4 admits four Killing spinors, we get four linearly indepen-
dent supercharges forming a spinorial supersymmetry generator Q for each linearly
independent solution to (3.13b). Accordingly, the number of supersymmetries, N , is
the number of linearly independent solutions to the Killing spinor equation (3.13b)
on the internal spaceM7.
Consider now the curvature of the connection D̃m on the spinor bundle ofM7. By
extending D̃m to ∇̃m, which acts on spinors as D̃m and on vectors as ∇m, and using
that ∇m is torsion-free, the curvature of D̃m may be computed as
( 2 )
[∇̃m, ∇̃n]η = [(∇m,∇n] + i
m
2 ∇m)(Γn)− i
m
2 ∇n(Γm)−
m
4 [Γm,Γn] η =
= 1 R pq4 mn − 2m
2δp qmn Γpqη, (3.14)
where we have also used that ∇m(Γn) = 0 and that Γpq/4 are the Lorentz generators
in the spinor representation. Recall that the Weyl tensor, in arbitrary dimension
d > 2, is
W pqmn = R pqmn −
4 [p q] 2
2R[m δn] + ( 1)( 2)Rδ
p q
mn. (3.15)d− d− d−
Thus, for our seven-dimensional Einstein manifold with Rmn = 6m2gmn,
W pqmn = R pqmn − 2m2δp qmn, (3.16)
which we recognise from (3.14). Hence, any solution η to the Killing spinor equation
(3.13b) also satisfies the integrability condition
W pqmn Γpqη = 0. (3.17)
The holonomy of a connection ∇̂ on a vector bundle E, with fibre F , at a point
p ∈ M, Holp(∇̂), is defined as the subgroup of GL(Ep) obtained by all possible
44
3.1. Freund–Rubin compactification
parallel transports58 around closed loops inM starting at p. For a proper introduction
to holonomy, see [78].59 For an introduction to bundles, see appendix E.1. If M
is connected, which we assume for our M7, the holonomy group is independent
of the base point p (up to conjugation) and we simply write Hol(∇̂) ⊆ GL(F ).
The restricted holonomy group Hol0(∇̂), defined as Hol(∇̂) but restricted to null-
homotopic loops, is a connected Lie subgroup of GL(F ) [78]; it is the identity
component of Hol(∇̂). For a principal connection, the holonomy is a subgroup of
the structure group [78]. In our case with D̃, the holonomy group Hol(D̃) is the
subgroup of invertible linear transformations on the space of 8-component spinors
obtained by the parallel transport maps defined by D̃.
The Lie algebra of the restricted holonomy group, hol(∇̂), is related to the curvature
of the connection. From pseudo-Riemannian geometry, this seems plausible since
the curvature gives the change of a vector when parallel transported around an
infinitesimally small parallelogram [79]. More precisely, the Lie algebra-valued
curvature 2-form take values in holp(∇̂) ⊆ gl(Ep) [78] and, by the Ambrose–Singer
theorem [80], holp(∇̂) is spanned by the curvature 2-form at all points connected to
p by piecewise smooth curves, parallel transported to p.60
Putting the above together, the integrability condition [∇̃m, ∇̃n]η = 0 implies that
the number of unbroken supersymmetries is at most the number of singlets in the
decomposition of the spinor when restricting so(7) to hol(D̃). Furthermore, by (3.14)
and (3.16), hol(D̃) is spanned by W abmn Σab, where Σab are the generators of so(7)
[19]. Note that D̃mη = 0 is stronger than [∇̃m, ∇̃n]η = 0, so there might be fewer
supersymmetries than singlets in the decomposition. Also, there can be at most
eight supersymmetries since spinors onM7 have eight components and the value
of η at a point determines its differential at the same point by the Killing spinor
equation (3.13b) [19].
For each vacuum with m =6 0, there is another vacuum obtained by skew-whiffing,
that is, reversing the direction of the flux, m 7→ −m. One can show that, except
for S7 with its usual round metric, at most one of the two solutions related by
skew-whiffing can admit Killing spinors and, hence, at most one of the solutions can
have unbroken supersymmetries [19]. This is known as the skew-whiffing theorem.
The above considerations are local. If the spaceM7 is not simply connected61, there
may, in addition, be global obstructions to the existence of Killing spinors [19].
58Parallel transport is defined by demanding that the covariant derivative along the curve vanishes.
See for instance [77], [78].
59As an example, the holonomy of the Levi-Civita connection on the sphere S2 is SO(2), that
is, parallel transport can rotate vectors arbitrarily but not change their lengths nor turn a
right-handed pair of vectors into a left-handed pair.
60It is, of course, important to consider the curvature not only at p since one can have a flat region
on a generally curved manifold with nontrivial holonomy.
61A connected manifold is simply connected if all closed loops are null-homotopic. The circle, S1,
is not simply connected.
45
3. Supergravity compactifications
3.2 Anti-de Sitter, mass operators and
supersymmetry
A general feature of Kaluza–Klein compactifications is that there are infinite towers
of fields obtained by expanding the D = 11 fields in modes on the internal space
[19]. The masses of these fields are related to certain differential operators on the
compactification manifoldM7. Before we turn to the specific expressions for these,
we should define what we mean by mass in AdS4. In Minkowski spacetime, the
mass is simply defined as (the nonnegative root of) M2 = −pµpµ. For perturbative
stability, that is, stability against small field fluctuations, M2 ≥ 0 is needed.62 In
AdS4, the situation is more complicated since the isometry group is SO(3, 2), which
does not contain momentum operators pµ. There are two ways forward, one can
study the field equations in AdS to try to come up with reasonable definitions of the
masses or one can investigate the unitary irreducible representations of Spin(3, 2)
to characterise the particles. Below, we review aspects of both approaches. As in
section 3.1, we use conventions in which the cosmological constant is Λ = −12m2,
that is, the curvature radius is 1/(2m) and R 2µν = −12m gµν .
3.2.1 Defining mass in AdS
Following [19], we define the masses for different spin s through the linear, free field
equations
s = 0: ∆0φ− 8m2φ+M2φ = 0, (3.18a)
= 1s : iγµ∇ χ−Mγ52 µ χ = 0, (3.18b)
s = 1: ∆ A +∇ ∇νA +M21 µ µ ν Aµ = 0, (3.18c)
s = 3 : iγµνρ2 ∇̃νψρ −Mγ
5γµνψν = 0, (3.18d)
s = 2: ∆ h ρ ρ 2 2L µν + 2∇(µ∇ hν)ρ −∇(µ∇ν)hρ + 24m hµν +M hµν = 0, (3.18e)
where ∆p is the Hodge–de Rham operator, see appendix D.4,
∇̃ 5µ = ∇µ − imγ γµ, (3.19)
as in (3.13a), and ∆L is the Lichnerowicz operator
∆Lhµν := −h − 2R ρ σµν µ ν hρσ + 2R
ρ
(µ hν)ρ. (3.20)
For s = 0, 1/2, these equations with M = 0 are Weyl invariant63 (if φ and χ given
proper Weyl weights) and they can be generalised Weyl-invariantly to arbitrary
dimension [82].64 As explained in [81], Weyl invariance together with diffeomorphism
62M2 ≥ 0 does not imply perturbative stability since the potential could have nonzero slope.
63A Weyl transformation is a local rescaling of the metric g 2µν →7 Ω (x)gµν [81].
64Without transforming m, (3.18a) with M = 0 should be written as ∆0φ+R/6φ = 0 to be Weyl
invariant.
46
3.2. Anti-de Sitter, mass operators and supersymmetry
invariance implies conformal invariance (although conformal invariance and diffeomor-
phism invariance do not imply Weyl invariance [81], [83] as sometimes claimed [84]).
One can show that (3.18a) and (3.18b) with M = 0 imply that φ and χ propagate on
the local light cones in AdS by using that ds2 = 0 is Weyl invariant and that AdS4
is related to four-dimensional Minkowski spacetime via a Weyl transformation [82].
While for s < 1, we may define masslessness by requiring propagation on the local
light cones, the situation for s ≥ 1 is different. Here, we instead define masslessness
by requiring gauge invariance. In d = 4 Minkowski space, p2 = 0 implies, via the
group theory of the Poincaré group, that the particle only has two states (helicity),
see section 2.1.2. However, to reduce the number of propagating degrees of freedom
to two for s ≥ 1, gauge invariance is needed, whence propagation on the light cone
and gauge invariance coincide [82]. This is not the case in arbitrary spacetimes.
Maxwell theory, that is, s = 1, is Weyl invariant in precisely d = 4 and photons,
therefore, propagate on the local light cones in AdS4, by the above argument [82].
The gauge-invariant s = 3/2, 2 theories are, however, not Weyl invariant in d = 4
and the fields propagate not only on but also in the interior of the light cone [82].
Another peculiarity regarding masses in AdS is the bound for perturbative stability.
For s > 0, the requirement is M2 ≥ 0, just as in Minkowski spacetime [19]. Remark-
ably, for s = 0, only M2 ≥ −m2 is required to avoid exponentially growing modes
for small field fluctuations [12], [13]. This is known as the Breitenlohner–Freedman
bound. A theory in which all scalar fields satisfy the Breitenlohner–Freedman bound
and all s > 0 fields satisfy M2 ≥ 0 is said to be BF stable. BF stability does not
imply perturbative stability since the slope of the potential can be nonvanishing, as
signalled by a tadpole.
3.2.2 Masses from operators on the internal space
As stated above, the masses of the fields in AdS4 are related to differential operators
on the internal space. We will not give a derivation of the mass operators for the
Freund–Rubin ansatz but make some comments on the derivation. For details, see
[19]. To derive the mass operators, one writes the D = 11 fields as their background
values plus a fluctuation (for instance, gMN(x, y) = g̊MN(x, y) + hMN(x, y) for the
metric) and derives the linearised field equations from D = 11 supergravity using the
background values of the Freund–Rubin ansatz. Then, gauge conditions are imposed
and the fluctuations are expanded in modes on the internal space, similar to what
we did when discussing supersymmetry in section 3.1. Finally, the mass operators
are derived by analysing the equations resulting from inserting the expansions in
the linearised field equations and comparing with (3.18). The result is presented in
table 3.1.
To make the below results plausible, note that, for instance, hµν is a scalar on the
internal space and will, hence, be expanded in scalar modes. Since s = 2 corresponds
to the transverse and traceless part of hµν , the s = 2 mass operator must act on
scalar fields onM7.
47
3. Supergravity compactifications
The operators appearing in table 3.1 are the Hodge–de Rham operator ∆p acting on
transverse p-forms with p = 0, 1, 2; the Lichnerowicz operator ∆L acting on transverse
traceless symmetric rank-2 tensors; the Dirac operator iD/ acting on Majorana spinors
and transverse Γ-traceless Majorana vector-spinors and the operator Q acting on
transverse 3-forms (defined in (3.35)).65 At this point, we switch to consider fields
with flat indices on the internal space. In particular, iD/ 3/2 acts both on the vector
and spinor index of the vector-spinor. All operators should be interpreted in terms
of their eigenvalues. Note that all of the operators are self-adjoint and respect the
transversality and tracelessness conditions, whence there are bases of eigenmodes
with real eigenvalues spanning the corresponding function spaces. We discuss this
further in section 3.2.4.
In the derivation of table 3.1, some special cases arise. In the table, the subscripts
label various towers of fields with the first (second) subscript referring to the top
(bottom) sign or, for iD/ and Q, the positive (negative) part of the spectrum of the
operator.66 The eigenvalues 7m2 of ∆0 in 0+1 and −7m/2 of iD/ 1/2 in 1/21 correspond
to singletons [19]. Singletons have no Poincaré analogue and are topological in the
sense that the fields have no degrees of freedom in the bulk, only on the boundary [85].
Still, as explained in [10], they must be kept in the theory. The last exception is the
eigenvalue 0 of ∆0 in 0+1 , which should be omitted from the physical spectrum [19].
Table 3.1: Mass operators in Freund–Rubin compactification of D = 11
supergravity. Here, s denotes spin, p parity and t labels the tower. The
operators are understood in terms of their eigenvalues. Singletons correspond
to 7m2 eigenvalues of ∆0 in 0+1 and −7m/2 of iD/ 1/2 in 1/21. The eigenvalue 0
of ∆0 should be omitted from 0+1 .
spt Mass operator
2+ ∆0
3 7m
/
2 −iD1/2 +1,2 2 √
1−2,1 ∆1 + 12m2 ± 6m ∆1 + 4m2
1+ ∆2
1 −iD 9m/2 1/2 −4,1 2
1 iD 3m/2 3/2 +3,2 2 √
0+3,1 ∆0 + 44m2 ± 12m ∆0 + 9m2
0+2 ∆L − 4m2
0− 22,1 Q + 6mQ+ 8m2
65Transversality means that ∇mYmn = 0 et cetera and a vector-spinor is Γ-traceless if Γmψm = 0.
66Note that the labels of the towers agree with [19] although other conventions differ.
48
3.2. Anti-de Sitter, mass operators and supersymmetry
Note that there is precisely one massless spin-2 field, that is, graviton, since ∆0 has
precisely one zero-mode. This is reassuring. From (3.13b), we see that each unbroken
symmetry gives a mode of iD/ 1/2 with eigenvalue 7m/2. As is clear from table 3.1, this
eigenvalue corresponds to massless spin-3/2 fields, that is, gravitinos, the gauge fields
of gauged supersymmetry [19].67 Similarly, unbroken gauge symmetries correspond
to 12m2 eigenvalues of ∆1 in 1−1 (∆1 has no zero-mode on compact Einstein spaces
with positive curvature [19]). There can be additional unbroken gauge symmetries
from zero-modes of ∆2 in 1+. These come from the Abelian gauge invariance in
D = 11 and all fields are, thus, neutral under them [19].68
Note that the masses of spins 3/2, 1/2 and 0− are sensitive to skew-whiffing, m 7→
−m. Also, by the skew-whiffing theorem, there are no spin-1/2 singletons for
supersymmetric vacua except for the round S7. By the same argument, a vacuum
with N = 0 related to a supersymmetric vacuum by skew-whiffing has exactly as
many spin-1/2 singletons as there are supersymmetries in the other vacuum. As we
will see below, the only supermultiplet containing singletons is the Dirac singleton
supermultiplet. Thus, there are no spin-0 singletons in supersymmetric vacua, except
for the round S7, nor their skew-whiffed partners since the 0+ spectrum is insensitive
to skew-whiffing. In fact, one can prove that the first nonzero eigenvalue of ∆0 is at
least 7m2 with equality only for the round S7 [19], whence there are never spin-0
singletons in any other cases.
From table 3.1, one can also draw some conclusions regarding BF stability. Firstly,
there cannot be any negative M2 values in 2+, 3/21,2, 1−, 1+2 , 1/21,2,3,4 nor 0+3 since
∆p is nonnegative and iD/ is Hermitian. One can prove that ∆1 ≥ 12m2 [14] which
is precisely what is needed to ensure M2 ≥ 0 for 1−1 . Since Q has real eigenvalues
and ∆0 ≥ 0, the 0+ and 0−1 1,2 towers satisfy, but might saturate, the Breitenlohner–
Freedman bound M2 ≥ −m2. The only remaining tower is 0+2 , for which the BF
stability criterion reads ∆L ≥ 3m2. This criterion is not satisfied by all compact
Einstein spaces with positive curvature [14]. If the vacuum is supersymmetric, it
is perturbatively stable (see below) [19]. Since the 0+2 spectrum is insensitive to
skew-whiffing, N = 0 skew-whiffed counterparts of supersymmetric vacua are BF
stable. As remarked above, this does not imply that they are perturbatively stable.
One class of unstable solutions are Riemannian products. If M7 = M(1) ×M(2)
with the product metric, that is, gmn is block diagonal over the two factors, then
there is a mode of ∆L with eigenvalue 0 < 3m2 corresponding to one of the factors
expanding while the other contracts [14].
67From (3.2c), we see that the supersymmetry transformation of the gravitino is δεψµ = D̃µε,
analogous to the transformation of the gauge potential in Yang–Mills theory.
68This is true of the supergravity fields; the M2 and M5-branes of M-theory, and the corresponding
supergravity solutions, are electrically and magnetically charged under this symmetry, respectively.
49
3. Supergravity compactifications
3.2.3 Spin(3, 2)-representations and supersymmetry
Similar to what we described in section 2.1.2, elementary particles in AdS4 correspond
to nontrivial irreducible unitary representations of Spin(3, 2), the double cover of
the identity component of the isometry group SO(3, 2) of AdS4. We denote the
generators of Spin(3, 2) by Mα̂β̂ where α̂, β̂, . . . are 5-dimensional indices. AdS4 is
the universal cover of the connected hyperboloid η xα̂xβ̂α̂β̂ = −1/(2m)2 [86].69
We split the 5-dimensional index as α̂ = (α, 5) in the basis in which ηα̂β̂ is block-
diagonal with blocks ηαβ and −1. By considering the Poincaré limit m→ 0, one sees
that Mαβ and Pα ∝ mMα5 are the AdS analogues of the Lorentz and momentum
generators of the Poincaré algebra, respectively [82]. Hence, M05 is identified as a
dimensionless energy operator. We consider representations with energy bounded
from below, that is, for which there is a smallest eigenvalue E0 of M05. The algebra
so(3, 2) is of rank 2 and irreducible representations can, hence, be specified by the
eigenvalues of two Casimir operators [88]. Equivalently, and more physically relevant,
the irreducible representations can be denoted by D(E0, s) where E0 is the lowest
energy eigenvalue in the representation and s the spin70 of the particle [86]. A
representation is said to be unitary if there exists an invariant, positive definite
scalar product on it. One can start from 2s+ 1 lowest energy states and construct
an invariant scalar product by declaring that these are orthonormal and that the
generators are (anti-)Hermitian (depending on conventions). Demanding that the
scalar product is positive definite then leads to unitarity bounds on E0 [86].71 The
result of this analysis is that
1
s < 1: E0 ≥ s+ 2 , s ≥ 1
: E0 ≥ s+ 1. (3.21)
There are some special representations. These are the massless D(2, 0) and, for all s,
D(s+ 1, s) and the Dirac singleton representations D(1/2, 0) and D(1, 1/2) [86].
The lowest energy eigenvalue, E0, characterising a representation can be related to
the AdS mass M discussed above. Here, we do not give a derivation but simply
quote the result in table 3.2 [19]. The sign ambiguities for s < 1 arise from a
quadratic equation and are eliminated for s ≥ 1 by the unitarity bounds.72 For
unitarity and E0 ∈ R, we see that M2s=0 ≥ −m2 is required for the plus sign, again
the Breitenlohner–Freedman bound, and −m2 ≤ M2s=0 ≤ 3m2 for the minus sign.
For the spinors, all real Ms=1/2 are allowed for the plus sign and |Ms=1/2| ≤ m for the
minus sign. One can prove that the absolute values of the eigenvalues of iD/ 1/2 onM7
are greater than or equal to 7|m|/2 [19]. Thus, Ms=3/2 as given in table 3.1 cannot
violate the unitarity bound in the table below. Lastly, for s = 1, 2, unitarity requires
M2 ≥ 0 in agreement with the above stability criteria. Note that the unitarity
69For example in [87], AdS is used for the hyperboloid and CAdS for the cover.
70The spin is, as usual, defined via the Casimir of the so(3) corresponding to spatial rotations.
71Actually, it is sufficient to demand positive semi-definiteness and then factor out the zero norm
states. This leads to multiplet shortening and corresponds to saturating the inequalities in (3.21).
See [86] for details.
72The different signs correspond to different boundary conditions [13], [38], [89].
50
3.2. Anti-de Sitter, mass operators and supersymmetry
bounds for s > 1 correspond to masslessness while for s ≤ 1/2 they correspond to
the singleton representations.
Table 3.2: Relation between energy, E0, and mass, M , for various spins, s, in
AdS4. The corresponding Spin(3, 2)-representations, D(E0, s), are unitary for
E0 ≥ Emin0 .
s E0 √ Emin0
2
0 3 ± 1 M 12 2 ∣∣ m1 3 1 ∣∣ ∣
2 + 1 2
± M
∣∣
2 2 2√ 1m ∣
2
1 3 + 1 M2 2 ∣ + 1 2∣ m
2
3 3 + 1 ∣2 2 2 ∣
∣
M ∣− 2∣ 5√m ∣ 2
2 3 + 1 M
2
2 2 m2 + 9 3
The above representations D(E0, s) can be combined into supermultiplets. The
superalgebra in AdS4 is not the super-Poincaré algebra of section 2.1.1 but the
orthosymplectic Lie superalgebra osp(N|4). This algebra is the graded extension
of so(N ) and sp(4,R) ' so(3, 2) where N denotes the number of supersymmetries
as usual. Accordingly, there are, apart from the 10 generators Mα̂β̂ of so(3, 2),
N (N−1)/2 generators T ij of so(N ) and 4N superchargesQi (with a suppressed Dirac
spinor index). The supersymmetry generators are Majorana and the nonvanishing
superbrackets are [86]
{Qi 1, Qj} = − ij2δ γ̂
α̂β̂C−1Mα̂β̂ + iC
−1T ij, (3.22a)
[M ,M γ̂δ̂α̂β̂ ] = 4i
[γ̂ δ̂]
η[α̂M ] , (3.22b)β̂
[M ,Qi] = iα̂β̂ 2 γ̂α̂β̂Q
i, (3.22c)
[T ij, Tkl] = 4i [i j]δ[kT l] , (3.22d)
[Tij, Qk] = 2iδk[iQj], (3.22e)
where γ̂α̂ are the γ-matrices of so(3, 2), that is, γ̂α = γα and γ̂5 = iγ5, and C is the
so(3, 2) charge conjugation matrix, Cγ̂α̂C−1 = +γ̂α̂T.73 By tracing with Cγ̂05 and
contracting with δij one finds
1
M = i j05 2 δN ijQ Cγ̂05Q ≥ 0, (3.23)
73Here, we use the convention with an i in the exponent when exponentiating to a group element.
51
3. Supergravity compactifications
since Cγ̂05 is symmetric and positive definite.74 Taken as a relation for the quantum
operators, this implies perturbative stability for supersymmetric vacua, that is, vacua
annihilated by the supersymmetry generators Qi.
By a method similar to that for the representations D(E0, s) one can determine the
possible unitary irreducible supermultiplets with energy bounded from below, see
[86]. The results for N = 1 were first obtained by Heidenreich [90] and for N = 8 by
Freedman and Nicolai [91]. We present the results for N = 1 in table 3.3.
Table 3.3: The unitary N = 1 supermultiplets in AdS4 with Spin(3, 2)-
representations ordered decreasingly by spin.
Class Multiplet name and unitary Spin(3, 2)-representations
1 Di(rac si)ngleto(n
1 1 ⊕ 1
)
D , 2 D 2 , 0
2 W(ess–Zu)mino s(upermultip)let for1 1 (E0 > 1 )
D E0, 2 ⊕D E0 + 2 , 0 ⊕D E0 −
1
2 , 0
3 Massless higher(spin supermu)ltiplets for s ≥ 1
D(s+ 1 1 1, s)⊕D s+ 2 , s− 2
4 Massive highe(r spin supermu)ltiplet1 1 (s for s ≥ 1 and)E0 > s+ 1
D( 1 1E0, s)⊕D E0 + 2 , s− 2 ⊕D E0 − 2 , s− 2 ⊕D(E0, s− 1)
3.2.4 Differential operators and a universal Laplacian
Here, we discuss some properties of the operators appearing in table 3.1 and relate
them to a universal Laplacian. We assume that the compact manifoldM7 is without
boundary or that the boundary conditions are such that all boundary integrals vanish.
To see where the universal Laplacian comes from, consider first the Hodge–de Rham
operator, or Hodge Laplacian,
∆p = δd + dδ, (3.24)
acting on p-forms. Here, d is the exterior derivative and δ the codifferential, see
appendix D. A p-form α is transverse if δα = 0, which follows from the definition
Dbaba1...ap−1 = 0 of transversality and the component formula for δα. Thus, ∆p maps
transverse p-forms to transverse p-forms since δ∆pα = δdδα. Also, ∆p is manifestly
self-adjoint and nonnegative since δ is the adjoint of d.
74Note that C is only well-defined up to a sign. The other sign gives M05 ≤ 0 and we must then
associate the energy with −M05 if we demand it to be bounded from below rather than above.
52
3.2. Anti-de Sitter, mass operators and supersymmetry
By using the definitions of the exterior derivative d and the codifferential δ we
immediately find
∆pα ba1...ap = −αa1...ap − p[D[a1 ,D ]α|b|a2...ap], (3.25)
where D is the torsion-free spin connection. Using the Ricci identity [Da,Db] =
RabcdΣcd, this can be written as
∆pα b1 b2 ba1...ap = −αa1...ap − p(p− 1)R[a a α1 2 |b1b2|a3...ap] + pR[a1 α|b|a2...ap], (3.26)
which is known as a Weitzenböck identity. There is another way of writing ∆p. For
this, note that the second term in (3.25) can be written as
p[D[a1 ,Db]α|b|a2...ap] = [Dc1 ,Dc2 ]pδ c1 c2αb[a |b| a2...a ] = [Dc1 ,Dc2 ]Σ
c1c2α
1 p a1...ap
. (3.27)
Again using the Ricci identity, we find
∆pαa1...ap = −αa1...ap − [D ,D ]Σb1b2b1 b2 αa1...ap =
= −α −R Σc1c2Σb1b2a1...ap b1b2c1c2 αa1...ap . (3.28)
This form of the Laplacian can be generalised to a field carrying any representation
of Spin(7). Thus, we define
∆ := −− [Da1 ,D ]Σa1a2a2 = −−R Σa1a2Σb1b2a1a2b1b2 . (3.29)
We refer to this as the universal Laplacian since it can act on a field carrying any
representation of Spin(7) and, as we will see, is related to all the other Laplacians
we are interested in.
Let us show that ∆ is self-adjoint. To this end, let YA be a field carrying any
finite-dimensional real representation of Spin(7).75 Since Spin(7) is compact, the
representation is unitary and there is an invariant symmetric nondegenerate δAB
with Euclidean signature. We have an∫L2 inner product defined by
〈Y,X〉 = volYAδABXB. (3.30)
To see that ∆ is self∫-adjoint with respect t∫o this inner product, note that
〈Y,−X〉 = − ∫ volYAδABXB = + volD Y δabδABa A DbXB =
= − vol∫Y δABA XB = 〈−Y,X〉, (3.31a)
〈Y,−DaD ab ab ABbΣ X〉 = − ∫ volYA(Σ ) DaDbXB =
= + ∫ volD Y (Σab)ABa A DbXB =
= − volDbDaYA(Σab)ABXB = 〈−DaDbΣabY,X〉, (3.31b)
75We restrict to the real case since that is what we are interested in and for notational convenience.
What follows is easily generalised to the complex case by replacing the symmetric invariant δAB
with a Hermitian invariant δĀB .
53
3. Supergravity compactifications
where we have used δAB to raise an index on (Σab) BA , that δAB and (Σab)AB are
invariant tensors and, in the last step, that (Σab)AB is antisymmetric both in a b and
AB. The last statement follows from δAB being a symmetric invariant. Thus, glossing
over some mathematical subtleties regarding the distinction between symmetric and
self-adjoint unbounded operators on infinite-dimensional Hilbert spaces [92], ∆ is
self-adjoint and there is a basis of eigenmodes of ∆ with real eigenvalues.
We would also like to show that ∆ respects the various conditions (transversality,
tracelessness, et cetera) placed on the fields. We do this separately in the cases of
interest. Since ∆ = ∆p when acting on p-forms and ∆p respects transversality, ∆
can be restricted to transverse p-forms.
Turning to symmetric rank-2 tensors, a short calculation shows that
∆hab = ∆Lhab := −h c d cab − 2Ra b hcd + 2R(a hb)c, (3.32)
where ∆L is the Lichnerowicz Laplacian as defined in (3.20) but here for the internal
space. By contracting a and b one immediately sees that ∆L maps traceless tensors
into traceless tensors. To show that ∆L respects transversality, that is, that Dahab = 0
implies Da∆Lhab =( 0, first )note that ( )
Da R c(a hb)c = 0, Da R c dh = Rac da b cd b Dahcd. (3.33)
Here, we have used that the manifold is Einstein, R 2ab = 6m δab, which implies that
DaRabcd = 0 by contracting the Bianchi identity D[aRbc]de = 0. When computing
Da∆Lhab, the first term in (3.32) gives
Dahab = Dc(DaDch +Rac dD h) +Rac dab c d ab a Dch ac ddb +R b Dchad =
= Dc Ra dca hdb +Ra dh ac d ac dcb ad +R b Dchad = 2R b Dchad. (3.34)
Since the right-hand sides of (3.33) and (3.34) cancel when combined as in (3.32),
∆L can indeed be restricted to the space of transverse hab.
Next, we turn to the 3-form operator Q, defined as
Q := ?d, (3.35)
which maps 3-forms into 3-forms. If α is a transverse 3-form,
∆3α = δdα = ?d ?dα = Q2α, (3.36)
since δ is acting on the 4-form dα. This gives the relation between Q and the
Laplacian ∆. In seven dimensions and Euclidean signature, ?2 = 1, ? is self-adjoint
and, thus,
Q∗ = δ ? = ?d ?2 = Q, (3.37)
that is, Q is self-adjoint. Also, Q maps into transverse 3-forms since δQ = − ?d2 = 0.
Turning now to half-integer spins, the relevant operator is the Dirac operator
iD/ = iΓaDa. (3.38)
54
3.2. Anti-de Sitter, mass operators and supersymmetry
The Dirac operator iD/ maps Majorana spinors to Majorana spinors. This is easily
seen in the basis from appendix C.1 since the Majorana condition then reduces to a
reality condition for each component and Γa are purely imaginary. It also maps the
space of transverse Γ-traceless vector-spinors, that is, vector-spinors ψa satisfying
Daψ = 0 = Γaa ψa, to itself since
ΓaiD/ψ = −iD/Γaψ + 2iDaa a ψa = 0, (3.39a)
DaiD/ψa = iD/Da +
i
ψ Ra ΓbΓcdψ + iΓbRa ca 4 bcd a ba ψc = 0, (3.39b)
where, in the last step, we used that Ra[bcd] = 0 and thatM7 is Einstein. Furthermore,
iD/ is self-adjoint. To see this, let ψA and χA be Majorana tensor-spinors, where A is
an arbitrary (flat) tensor index and we have suppressed the spinor indices, and note
that76 ∫ ∫
〈ψ, iD/χ〉 = ∫ volψTCδABiD/χ = vol iDaψTΓTCδABA B A a χB =
= vol (iD/ψ )TCδABA χB = 〈iD/ψ, χ〉. (3.40)
Here, the sign from the integration by parts cancels the sign from CΓa = −ΓTaC.
Lastly, we investigate the relation between iD/ and ∆. The square of the Dirac
operator is
(iD/ )2 = −ΓaΓbD D = − 1− Γaba b 2 [Da,Db]. (3.41)
Acting on ψA, this gives ( )
(iD/ )2 1ψA = −ψA − 2Γ
a1a2R b1b
1
2 B
a1a2 4Γb1b2ψA + (Σb1b2)A ψB =
= − 1 1ψ b1b2 a1a2 BA + 4RψA − 2Ra1a2 Γ (Σb1b2)A ψB. (3.42)
Acting instead with the Laplacian ∆ from (3.29) on ψA, we find, after a short
calculation,
∆ 1 1ψA = −ψA + 8Rψ − R
b1b2 a1a2 B
A 2 a1a2 Γ (Σb1b2)A ψB+
−R b1b2a1a2 (Σ
a1a2) BA (Σ Cb1b2)B ψC =
= (iD)2 − 1/ ψ Rψ −R b1b2 a1a2 B CA 8 A a1a2 (Σ )A (Σb1b2)B ψC . (3.43)
Thus, for spinors and vector-spinors, respectively,
(iD/ )2ψ = ∆ 21ψ + m24 ψ, (3.44a)
(iD/ )2 3ψ 2a = ∆ψa − 4m ψa, (3.44b)
76The Dirac operator can also be considered in the context of Dirac spinors. In this case, one
should use the Dirac conjugate instead of the Majorana conjugate in the L2 inner product. The
eigenvalues are the same since the Majorana eigenbasis provides an eigenbasis for the space of
Dirac spinors as well.
55
3. Supergravity compactifications
where we have used that Rab = 6m2δab. Note that, for Einstein spaces, the eigenvalues
of ∆ and (iD/ )2 on spinors and vector-spinors are related by a constant. This seems
to break down for higher tensor-spinors since the difference between ∆ and (iD/ )2 will
contain contributions from the Weyl tensor from the last term in (3.43). However,
we are only interested in spinors and vector-spinors, whence this is not a problem.
To conclude, we have seen that all of the operators ∆p, ∆L, Q and iD/ are self-adjoint
and can be restricted to the relevant functions spaces which, thus, have bases of
eigenmodes with real eigenvalues. Moreover, we have found that all of the operators
are related to a universal Laplacian ∆ (3.29)
∆ = −−R Σa1a2 b1b2a1a2b1b2 Σ , (3.45)
which differs from (iD/ )2 only by a constant when acting on spinors and vector-spinors
and coincides with ∆p, ∆L and Q2 = ∆3.
56
4. Homogeneous spaces
Homogeneous pseudo-Riemannian77 manifolds are a rich, yet particularly simple, set
of manifolds. Due to this, they are often used as the internal space in string and
supergravity compactifications. Examples of homogeneous spaces include Euclidean
spaces, spheres, flat tori and hyperbolic spaces in Euclidean signature; Minkowski,
de Sitter (dS) and anti-de Sitter (AdS) spaces in Lorentzian signature and super-
Minkowski and super-AdS superspaces.
A homogeneous space is, intuitively, a space in which all points are equivalent, or
“look the same”, in some sense appropriate to the setting. Technically, it is a space
M on which a group G of automorphisms acts transitively and effectively, that is,
for every x, y ∈ M there is a g ∈ G such that gx = y and there is no g ∈ G which
acts trivially on M. That G acts by automorphisms means that it preserves the
structure ofM and we require G to act effectively since, otherwise, it is really G/N ,
where N is the kernel of the G-action, that acts onM.78
Since we are interested in connected oriented pseudo-Riemannian manifolds, the
automorphisms are orientation-preserving isometries ofM, that is, G ⊆ Iso+(M). So,
a homogeneous (pseudo-Riemannian) manifold is a manifold on which the isometry
group acts transitively. The isometry group is a Lie group [93], [94], whence, in the
following, we consider Lie groups G.
The stabiliser Hy of y ∈M, that is, the subgroup of G fixing y, is called the isotropy
group of y. It is easy to see that the isotropy groups of different points inM are
conjugate subgroups in G. Thus, we often need not distinguish between them and
simply write H for the isotropy subgroup, which is a closed subgroup of G [77].79 Note
that the isotropy group Hy is a subgroup of SOg(TyM), where g is the G-invariant
metric onM.80 This follows from the fact that an isometry of a connected manifold
is determined by its value and differential at a single point [77]. Thus, since an
element of Hy fixes y, it is completely determined by its differential at y. Since Hy
preserves the metric and orientation onM, and particularly on the tangent space at
the fixed point, TyM, there is a natural homomorphism Hy → SOg(TyM) and we
may view Hy as a subgroup of SOg(TyM).
G is a principal H-bundle overM [95]. For an introduction to fibre bundles, see
77We include all signatures of the metric in “pseudo-Riemannian”.
78When considering spinor fields, we will have reason to loosen the latter requirement slightly.
79A closed subgroup of a Lie group is a Lie subgroup by Cartan’s theorem.
80With SOg(V ) we mean the orientation-preserving subgroup of GL(V ) that leaves the metric g on
V invariant.
57
4. Homogeneous spaces
appendix E.1. Similar to the frame bundle, we may view G as the bundle of H-frames
overM. To see this, pick an orthonormal frame at a point o and use the pushforward
by every element in g to get a set of frames. This set is naturally in one-to-one
correspondence with G since G acts effectively and isometries are determined by
their value and differential at a single point. Due to G preserving the metric of
M, the set will only contain orthonormal frames. Thus, G is a subbundle of the
orthonormal frame bundle and we have a reduction of the structure group from
SOg(dimM) to H.
By choosing an origin o ∈M we get a natural map φ : G/Ho →M by φ(gHo) = go.
It is easy to see that φ is a well-defined bijection and it is, in fact, a diffeomorphism
[77]. The G-action onM is realised on G/H by left-multiplication. By pulling back
the G-invariant metric onM to G/H, the two spaces become isometric whence we
do not distinguish them in the following.
Instead of starting from a manifoldM and then realising it as a coset space (provided
that the isometry group acts transitively), one can start from the groups G and H
and construct a homogeneous manifold as G/H. We do this in section 4.1. Note,
however, that there might exist a proper subgroup of Iso+(M) which acts transitively
onM [96], in which caseM can be described by different cosets G/H. Also, if we
loosen the requirement that G acts effectively, it may be possible to describeM by
additional cosets [19]. In section 4.2, we discuss harmonic analysis on coset spaces.
4.1 Geometry
Let G be a Lie group and H a closed subgroup with Lie algebras g and h, respectively.
We will assume that the algebras are reductive, that is, that there exists an Ad(H)-
invariant subspace m of g such that g = h ⊕ m [77]. Here, Ad is the adjoint
representation of G. Thus, reductivity means that the adjoint representation of
G splits, when restricted to H, into a direct sum of the adjoint representation
of H and another representation. The latter is a H-representation on m ' ToM
called the isotropy representation [96]. This implies that [h,m] ⊆ m while the
converse implication holds for connected H [96]. Reductivity is not a very restrictive
assumption: every homogeneous space admitting a G-invariant metric with Euclidean
signature is reductive [96]. Since we are, in this thesis, interested in compact manifolds
with Euclidean signature to be used in a Kaluza–Klein compactification, we find this
assumption acceptable.
In the above, we assumed that the G-action onM is effective. When constructing a
coset space, it is natural to ask what this means for G and H. Since any element
g ∈ G which acts trivially onM in particular fixes o, it follows that g ∈ Ho. Since
an element of Ho is determined by its differential at o, that is, its action on ToM,
this implies that the G-action is effective if and only if the isotropy representation of
H is faithful.
We wish to put a G-invariant metric on G/H. By a theorem [77], G-invariant tensor
58
4.1. Geometry
fields on G/H are in one-to-one correspondence with H-invariant tensors of the same
type on m. Thus, for a G-invariant metric on G/H we need a H-invariant symmetric
nondegenerate rank-2 tensor. Note that, if there is only one such tensor (up to a
constant factor), the coset space G/H, equipped with the G-invariant metric, is an
Einstein space, since the Ricci-tensor will also be proportional to the invariant [19].
This happens if the isotropy representation is irreducible [97].
In this section, we use indices A,B,C, . . . for g, a, b, c, . . . for m and i, j, k . . . for
h. With T CA generators and fAB structure constants of g, [h,m] ⊆ m and the
fact that h is a subalgebra of g implies that f jia = 0 and f aij = 0. We use gab to
denote the H-invariant tensor that defines the G-invariant metric on G/H and raise
and lower indices a, b, c . . . using gab and its inverse gab. Since, adg splits into adh
and the isotropy representation when restricted to h, the latter representation is,
explicitly, (T ) bi a = −f bia . Note that gab being h-invariant is equivalent to fiab being
antisymmetric in ab. Thus, the isotropy algebra is the subalgebra of sog(m) given by
Ti = −f abiabΣ , (4.1)
where Σab are the generators of sog(m),81 since the isotropy representation is faithful.
Following [19], [20], we can write a group element close to the identity as
g = exp(y · T(m)) exp(h · T(h)) (4.2)
where (T(h))i = Ti and (T(m))a = Ta are the generators of g in h and m, respectively,
and y and h are coordinates on G. Note that y · T(m) and h · T(h) are simple sums,
there are no vielbeins to convert the curved indices of the coordinates to flat indices,
like the ones on the generators.82 Still, the coordinates are “curved” due to the
noncommutativity of the generators. By writing a group element as in (4.2), we get
a natural representative of each coset gH
Ly = exp(y · T(m)). (4.3)
Since L−1y dLy is a g-valued 1-form [19], we can define 1-forms ea and Ωi by
ω̃ := L−1y dLy =: eaT ia + Ω Ti. (4.4)
Note that ea are left-invariant by construction. The G-invariant metric on G/H is,
in this local coordinate patch, given by
g a bmn = gabem en , (4.5)
where e am are the components of ea, that is, ea = dyme am . Here, the left-invariance
of ea ensures that G acts by isometries on G/H. We also see that the metric is
invariant under right-multiplication of Ly by h ∈ H due to gab being H-invariant.
This means that the metric is independent of which representative we choose for a
coset, which is needed to globally extend the metric on m ' ToM to G/H [96].
81See appendix A.2 for conventions regarding the normalisation of the ge(nerators of sog(m).82 )A familiar example is a SU(2)-group element close to the identity, exp iα2 σ1 + iβ2σ2 + iγ 32σ .
59
4. Homogeneous spaces
Note the similarity between ω̃ and the Maurer–Cartan form. In fact, ω̃ satisfies the
Maurer–Cartan equation
dω̃ + ω̃ ∧ ω̃ = 0, (4.6)
which follows from differentiating (4.4) by using
0 = d(L−1L ) = dL−1 L + L−1y y y y y dLy. (4.7)
By using ω̃ = eaT ia + Ω Ti and reductivity the Maurer–Cartan equation splits into
d a = −1e eb ∧ ecf a b i a i 1 a b i 1 j k i2 bc − e ∧ Ω fbi , dΩ = −2e ∧ e fab − 2Ω ∧ Ω fjk . (4.8)
From this, we can find an expression for the Levi-Civita spin connection ω, that is,
the unique torsion-free spin connection. Since it is torsion-free, 0 = dea + ωab ∧ eb.
From this and (4.8), one can read off that
ω a
1 a
[cb] = −2fcb − Ω
i a
[cf|i|b] . (4.9)
Since ωabc is antisymmetric in its last two indices ωabc = ω[ab]c − ω[ac]b − ω[bc]a, where
we have lowered the last index using gab, whence
1
ωabc = −2F − Ω
i
abc afibc, Fabc := 2fa[bc] − fbca. (4.10)
Note that the Jacobi identity and reductivity implies that all nonvanishing parts of
f CAB (f c k c kab , fab , fib and fij ) are h-invariant tensors. Thus, Fabc is also h-invariant
since gab is.
The curvature 2-form is, per definition, R ba = dω b + ω ca a ∧ ω bc .83 By direct compu-
tation, we find (from (4.8) and (4.10) )
dω b = ed ∧ ee 1 c b 1f i b 1 d i c b 1 j k i ba 4 de Fca + 2fde fia − 2e ∧ Ω fid Fca + 2Ω ∧ Ω fjk fia ,
( ) (4.11a)
ω c ∧ ω ba c =
1
ed ∧ eeF cF b + 1 1ed ∧ Ωi F cf b − f cF b + Ωi j c b4 da ec 2 da ic 2 ia dc ∧ Ω fia fjc .
(4.11b)
When adding these, the mixed terms ed ∧ Ωi cancel due to Fabc being h-invariant
and the Ωi ∧ Ωj terms cancel since (Ti) b = −f ba ia in the isotropy representation and
T[iTj] = f kij Tk/2. Thus, (
b = d ∧ e 1
)
R e e f i b
1 c b
a 2 de fia + 4fde Fca +
1
F c4 da F
b
ec . (4.12)
The curvature 2-form is related to the Riemann tensor by
R b = 1a 2R
b
cda dxc ∧ dxd, (4.13)
83Note that we use Rab to denote both the curvature 2-form and the Ricci tensor; it should be clear
from the context which is being referred to.
60
4.2. Harmonic analysis
whence
1 1
R bcda = f if bcd ia + 2f
eF b e bcd ea + 2F[c|a| Fd]e . (4.14)
Lastly, we mention that there is a special class of G-invariant metrics which are
particularly simple. Consider a positive definite G-invariant tensor gAB. One can then
take m as the orthogonal complement of h. This makes gAB block-diagonal on h⊕m
and gab, the restriction to m, H-invariant. Such a metric gab is said to be a normal
homogeneous metric on G/H [96]. Since gAB is G-invariant and block-diagonal, fabc
is completely antisymmetric and Fabc = fabc. Then, (4.14) simplifies to
R bcda = f icd f bia +
1
f e
1
2 cd f
b + e bea 2f[c|a| fd]e , (4.15)
which agrees with [20].
Spin geometry
To be able to globally define spinors onM = G/H, it must admit a spin structure,
that is, a lift of the structure group from SOg(m) to Sping(m).84 Not all coset
spaces G/H admit a spin structure, for instance CP2 ' SU(3)/U(2) does not [95].
Therefore, like [95], we assume that there is a covering group Ḡ of G such that the
embedding of H in SOg(m) lifts to an embedding of the corresponding cover H̄ of H
in Sping(m). This implies thatM' Ḡ/H̄ where H̄ is a subgroup of Sping(m). The
Ḡ-action onM is not effective since the lift is nontrivial. However, it is infinitesimally
effective in the sense that the isotropy representation of h̄ = h is faithful.
Analogous to what we saw above, Ḡ is a principal H̄-bundle overM and we have
a reduction of the structure group from Sping(m) to H̄ [95]. Correspondingly, the
associated vector bundles of the principal bundle of spin frames split into direct sums
of vector bundles carrying irreducible H̄-representations. This means that tensor
and spinor fields onM can be decomposed (globally) into pieces transforming under
some representation of H̄.
4.2 Harmonic analysis
In this section, we discuss harmonic analysis on coset spaces, a generalisation
of Fourier series and spherical harmonics. Harmonic analysis on coset spaces is
important in Kaluza–Klein compactifications since, if the (4 + k)-dimensional theory
is compactified on a manifold which is locally isometric toM4 ×Mk, the fields can
be expanded onMk using harmonics with spacetime fields as coefficients, yielding a
4-dimensional theory, see for instance [19], [95].
We assume that the manifold is spin and that the group of effective isometries lifts
as described in section 4.1. Since we will only be concerned with the lifted groups,
84For a proper introduction to spin structures and spin geometry, see [70].
61
4. Homogeneous spaces
we denote the lifted isometry group by G and the lifted isotropy group, which is a
subgroup of Sping(m), by H. Also, since the isometry group of a compact manifold
is compact [98] and the representation theory of compact Lie groups is particularly
well behaved, we restrict our attention to compact G. The relevant theory is based
on the Peter–Weyl theorem and the fact that G is a principal H-bundle over G/H.
The Peter–Weyl theorem
Recall that every representation of a compact Lie group G is unitary in the sense that
there exists a G-invariant positive definite scalar product, that is, δP̄Q is invariant.
This can be seen by Weyl’s unitarian trick [99]. The Peter–Weyl theorem [100] for
compact groups states that an orthogonal basis for L2(G), that is, complex square-
integrable85 functions on G, is provided by the matrix elements of all irreducible
representations. The o∫rthogonality relation is
〈ρ(σ)Q, ρ(τ) S〉 := dg ρ̄(σ)(g) Q̄ ρ(τ)( VGg) S = δ(σ)(τ)δ Q̄SP R P̄ R dim ρ(σ) P̄Rδ , (4.16)
G
where ρ(σ) and ρ(τ) are irreducible representations of G; σ and τ label86 all inequiva-
lent irreducible representations of G;87 P,Q,R, S are indices for the corresponding
representations; bars denote complex∫conjugation and
dg = VG (4.17)
G
is the volume of G. Thus, (ρ(σ)QP )σ,P,Q is an orthogonal basis for L2(G) and we may
expand a function X : G→ C as [10∑2]
X(g) = ρ(σ)(g) Q (σ)PP X Q . (4.18)
σ
Using that ρ(σ)(g) Qρ̄(σ)P (g) S̄δQS̄ = δPR̄, that is, that the representations are unitary,R̄
one finds the coefficients88
dim ρ(σ) ∫
X(σ)Q = dg ρ(σ)P (g−1)
Q
V P
X(g). (4.19)
G
G
Coset harmonics
Consider now a tensor (or spinor) field on a coset space G/H, that is, a section of a
vector bundle overM carrying a particular representation of Sping(m). As explained
above, these bundles split into direct sums of vector bundles carrying irreducible
H-representations due to the reduction of the structure group from Sping(m) to the
85The integration measure on G is known as the Haar measure. See [101] for details.
86We think of σ and τ as labels and do not employ the Einstein summation convention on them.
87Note that there are infinitely many inequivalent irreducible representations of (nonfinite) compact
Lie groups.
88With G = U(1) this is ordinary Fourier series.
62
4.2. Harmonic analysis
subgroup H. Thus, we wish to find a basis for vector bundles with H as structure
group. Such bundles are constructed from a vector space V , which H acts on
by a representation ρH , and the principal H-bundle G via the associated bundle
construction, see appendix E.1, and will be denoted G×ρ V [95]. A basis can thenH
be constructed by noting that sections of G×ρ V are in one-to-one correspondenceH
with V -valued functions on G satisfying the equivariance condition [95]
Xp(gh) = ρH(h−1) qp Xq(g). (4.20)
Here p, q are indices for the H-representation ρH . By the Peter–Weyl theorem, each
component Xp(g) can be expanded∑as
89
X (g) = ρ(σ)(g−1) QX(σ)Pp P p Q . (4.21)
σ
Imposing the equivariance condition and using that all functions ρ(σ)(g−1) QP are
independent, we find that
X(σ)Pp Q = ρH(h) q (σ)R (σ) −1p Xq Q ρ (h ) PR . (4.22)
Hence, for each fixed σ and Q, X(σ)Pp Q is an intertwiner between ρH and the restriction
ρ(σ)|H of the G-representation ρ(σ) to H. By Shur’s lemma, the only such intertwin-
ers are linear combinations of projections from the restricted G-representation to
subrepresentations equivalent to ρH . Thus, we can write (σ)P = (σ)ξ (σ)X Pp Q XQ Pξ p , where
(σ)
Pξ is the projection onto the ξ’th subrepresentation of ρ(σ)|H that is equivalent to
ρH , and ∑
X (g) = ρ(σ)(g−1) Q (σ)ξp pξ XQ . (4.23)
σ
In this expansion, we refer to the basis functions ρ(σ)(g−1) Qpξ as harmonics on the
coset.
There is a left G-action onXp, defined by g̃·Xp(g) = X (g̃−1p g). This G-representation
is said to be induced from the H-representation ρH [95], [101]. By identifying
Xp(g) with the coefficients (σ)ξXQ via the above expansion, we see that the induced
representation splits into a direct sum of irreducible G-representations, each (σ)ξXQ ,
for fixed σ and ξ, transforming under ρ(σ). That the multiplicity of ρ(σ) in the induced
representation coincides with the multiplicity of ρ in ρ(σ)H |H is known as Frobenius
reciprocity [101].
Lastly, we expand on the link between sections of G×ρ V , that is, fields carrying theH
H-representation ρH , and H-equivariant V -valued functions on G. This is explained
in more detail in appendix E.1. Given a local trivialisation of G, considered as a
principal H-bundle overM, we get local embeddings ψα : Uα → G, where {Uα }α
are coordinate charts onM [95]. Locally, a section of G ×ρ V is equivalent to aH
V -valued function onM and the section corresponding to Xp(g) is simply given by
Y αp (y) = X αp(ψ (y)). (4.24)
89Here, we apply the above expansion to X (g−1p ) for later convenience.
63
4. Homogeneous spaces
The H-equivariance of Xp is needed to ensure that Yp defines a global section of
G×ρ V . In the above chart, with local embedding given by y 7→ Ly, the expansionH
(4.23) reads ∑
Y (y) = ρ(σ)(L−1) Q (σ)ξp y pξ XQ , (4.25)
σ
which agrees with [102] apart from irrelevant normalisation of the coefficients.90
4.3 The coset master equation
In this section, we discuss what we will refer to as the coset master equation, which
we will use to compute the eigenvalue spectrum of the squashed S7 in chapter 6. The
equation is based on the fact that the tensor and spinor fields can be expanded in
terms of harmonics that come from the irreducible representations of G, as described
in the preceding section. For reasons explained below, we restrict our attention to
normal homogeneous metrics, so that Fabc = fabc is completely antisymmetric, and
compact Euclidean manifoldsM.
As explained above, G is a principal H-bundle overM. There is a natural principal
H-connection on this bundle induced by the splitting g = h⊕m [95]. In our local
coordinates, the H-connection is given by [20]
Ď im = ∂m + ΩmTi. (4.26)
Let ρ be any representation of G. Then, by (4.4),
Ď (Q) (Q)aρ(L−1y )P = −(T R −1a)P ρ(Ly )R , (4.27)
where Ď only acts on the first index of ρ(L−1y ), which we indicate by the parentheses
around Q. This is what we will refer to as the coset master equation. From (4.1)
and (4.10), we see that the torsion-free spin connection Da is related to the principal
H-connection Ďa in (4.26) by
Ď 1 bca = Da + 2fabcΣ , (4.28)
since Fabc = fabc in the normal homogeneous case. Note, however, that this relation
only is valid when Ďa acts on a tensor carrying a Sping(m)-representation since,
otherwise, the right-hand side is not defined. In particular, it cannot be used directly
in (4.27).
Since Ta are the generators of m, which are not block-diagonal over the irreducible
H-representations in ρ|H , Ta cannot act as matrices on the harmonics. However, if
multiple generators are combined to an element in the universal enveloping algebra,
U(g), which is block-diagonal over the H-representations, the corresponding analogue
90A well-known example is the expansion of a scalar field on S2 ' SO(3)/SO(2) ' Spin(3)/Spin(2)
in terms of spherical harmonics.
64
4.3. The coset master equation
of (4.27) can be restricted to any particular H-representation and then applies to
the harmonics. A short calculation shows that91
T −1 i −1aTbρ(Ly ) = −Ta(∂b + ΩbTi)ρ(Ly ) =
= −(∂ + ΩiT )T ρ(L−1)− Ωif c −1b b i a y b ai Tcρ(Ly ) =
= (∂ + ΩiT )Ď ρ(L−1)− Ωif cĎ ρ(L−1b b i a y b ia c y ) =
= ĎbĎaρ(L−1y ). (4.29)
Now we will make use of the assumption that the metric is normal homogeneous,
that is, that gab comes from the restriction of a G-invariant gAB. Then,
gabT T = −C + C , C := −gABT T , C := −gija b g h g A B h TiTj, (4.30)
where Cg and Ch are quadratic Casimir invariants of g and h, respectively.92,93 Here,
Cg acts by a constant on any particular G-representation ρ and Ch acts by a constant
on every irreducible part of ρ|H . Thus,
(Cg − Ch)Y = −̌Y, (4.31)
where Y is a field (with suppressed index) carrying a representation of Sping(m)
and ̌ := gabĎaĎb. We will refer to this equation as the quadratic master equation.
As noted above, the field Y splits into irreducible H-components each carrying an
induced G-representation. Ch acts by a constant on each irreducible H-component
and can thus be implemented as a matrix acting on the spin-index of Y while Cg
acts by a constant on each irreducible G-representation in the decompositions of the
induced representations and cannot be implemented as a matrix.
As we saw in section 3.2.4, the mass spectrum of a Freund–Rubin compactification
is related to the eigenvalue spectrum of a universal Laplacian (3.45)
∆Y = κ2Y, ∆ := −−RabcdΣabΣcd. (4.32)
SinceM is compact and Euclidean, both Sping(M) and G are compact, the finite-
dimensional representations of Sping(M) are unitary and the fields carrying such
representations form a unitary G-representation, with respect to the appropriate L2
inner product, which is, thus, completely reducible. Hence, the eigenmodes of ∆ fall
into irreducible representations of G since ∆ is manifestly invariant under isometries.
To be able to use (4.31) to compute the eigenvalues of ∆, we wish to relate  and ̌.
Using (4.28), that gab and f cab are H-invariant and that ρ(Σab) is an SOg(m)-invariant
for any representati(on ρ, we find )
̌ = Ďa D + 1f Σbc = + 1 ade 1 bc aa 2 abc 2f ΣdeDa + 2fabcΣ Ď =
= + f ΣbcĎaabc −
1
4f
ade bc
abcf Σ Σde. (4.33)
91In (4.29), Ďb acts not only on the first index of ρ(L−1y ) but also on the a-index.
92For semisimple g, Cg is some linear combination of the quadratic Casimirs of the simple constituent
Lie algebras, see appendix A.3.
93The normalisation here might not be conventional for concrete cases.
65
4. Homogeneous spaces
Combining this with (4.31) and (4.32) gives
f bcabcΣ Ďa = ∆− C + C +
1
R ΣabΣcd + f fadeg h abcd 4 abc Σ
bcΣde. (4.34)
We can simplify (4.34) a bit further. For this, note that, by (4.15),
R ΣabΣcd = f f icd ab 1 a bc de 1 a bc deabcd iab Σ Σcd + 2fabc f deΣ Σ − 2fabd f ceΣ Σ , (4.35)
whence, by (4.1),94
f ΣbcĎa = ∆− C + 3f fa Σbcabc g 4 abc de Σ
de − 12fabd f
a ΣbcΣdece . (4.36)
Remarkably, Ch from ̌ was cancelled by the first term in (4.15). As mentioned above,
fabc is an h-invariant and, hence, H0-invariant, where H0 is the identity component
of H. There may, however, be a larger group H̃ leaving fabc and gab invariant, such
that H0 ⊆ H̃ ⊆ Sping(m). This can lead to significant simplifications, as we will see
explicitly for the squashed seven-sphere in chapter 6. At this point, it is, however,
not clear that Ch will not re-enter in the calculation from ̌ or [Ďa, Ďb]. Note that
the last two terms in (4.36) can, for any Sping(m)-representation ρ, be expressed in
terms of projection operators that project onto the H̃-irreducible parts of ρ|H̃ , by
Shur’s lemma, since fabc is H̃-invariant.
For symmetric spaces95, which have f cab = 0 [19], this reduces the problem of finding
the eigenvalues of ∆ to the problem of decomposing the induced G-representation
into irreducible G-representations on which Cg is just a number. However, the case
we are ultimately interested in, the squashed seven-sphere, is not a symmetric space.
Curvature and torsion of Ď
Lastly, we give some properties of Ď. We have already seen, in (4.28), that
Ď = d + ω̌, ω̌ = ω + 1κ̌, κ̌abc = 2fabc, (4.37)
where κ̌ is the Lie algebra-valued contorsion 1-form of Ď. The torsion of this spin
connection is, as usual, defined by Ť a = Ďea = dea + ω̌a bb∧ e . Since D is torsion-free,
Ť a = κ̌ab ∧ eb, Ť cab = −2κ̌ c[ab] = −fabc. (4.38)
The Lie algebra-valued curvature 2-form is, per definition Ř = dω̌ + ω̌ ∧ ω̌. This is
related to the curvature 2-form R of the torsion-free spin connection by (E.21)
Ř = R + Ďκ̌− κ̌ ∧ κ̌, Ř b = R b b e b e bcda cda + 2Ď[cκ̌d]a + Ťcd κ̌ea − 2κ̌[c|a κ̌|d]e , (4.39)
94We also use the fact that it does not matter whether Σab acts on Σcd, as in ρ(Σ cdab · Σ ) =
2 [c|e (Σ |d]δ ρ )+ρ(Σ )ρ(Σcd), or is U(so (m))-multiplied by Σ , as in ρ(Σ ◦Σcd) = ρ(Σ )ρ(Σcda b e ab g cd ab ab ),
as long {ab} {cd} are symmetrised (as in, for instance, Rabcd = Rcdab).
95A space is said to be (locally) symmetric if there exists, for each y ∈M, a (local) isometry that
fixes y and reverses all geodesics through y [96]. By a theorem due to Cartan, a space is locally
symmetric if and only if DaRbcde = 0 [96].
66
4.3. The coset master equation
where the index expression follows from Ď(ecκ̌ b) = Ť cκ̌ bca ca −ec∧Ďκ̌ bca . Using (4.15)
and that Ďafbcd = 0 since fabc is h-invariant, this simplifies to
Ř i iabcd = fab ficd = (T )ab(Ti)cd. (4.40)
The Ricci identity Ď2Ω = Ř ∧ Ω can thus be written as
[Ďa, Ď cdb] = Řab Σcd − Ť c iab Ďc = (T )abTi + f cab Ďc, (4.41)
since Ď2Ω = D(eaD a a baΩ) = T ∧ DaΩ− e ∧ e ĎbĎaΩ. Note that we raise and lower
h-indices using gij.
67
4. Homogeneous spaces
68
5. Squashed sphere geometry
In this chapter, we study the geometry of the manifold on which we will compactify
eleven-dimensional supergravity: the squashed seven-sphere. With squashing, we
mean a smooth deformation of a homogeneous manifold, that is, a deformation of the
metric (the topology is unchanged), that keeps the manifold homogeneous. Although
homogeneity should be preserved, the isometry group may change when squashing.
For instance, when we squash the round S7, part of the SO(8) isometry is broken.
Note that the existence of a squashing deformation is nontrivial. For instance, one
cannot squash S2 [19].
Below, we present two constructions of the squashed seven-sphere. First, it is realised
as a nontrivial principal SU(2)-bundle over S4 and, second, as a coset space (the
subscripts are explained below)
Sp(2)× Sp(1)C
Sp(1) . (5.1)A × Sp(1)B+C
We also discuss the relation between these constructions and an isometric embedding
in the quaternionic projective space HP2.
5.1 Squashed S7 as a principal bundle
This construction is based on the fact that S7 can be realised as a principal SU(2)-
bundle over S4 and starts from the fact that the group of unit quaternions, which is
isomorphic to SU(2), has the topology of S3. Let U be a unit quaternion, parametrised
by Euler angles as [19]
U = ekφ/2eiθ/2ekψ/2, (5.2)
where i, j and k are the imaginary units of H. Consider the Lie algebra-valued 1-form
σ := 2U−1dU = iσ1 + jσ2 + kσ3 = hiσi, (5.3)
which is proportional to the Maurer–Cartan form. Here, σi are left-invariant 1-forms
and hi = (i, j, k)i.96 Using 0 = d(U−1U) = dU−1U +U−1dU and h kihj = −δij + ij hk
we immediately find
d 1 1σ = −2σ ∧ σ = − h 
i σj k2 i jk ∧ σ , (5.4)
which is essentially the Maurer–Cartan equation. Because of (5.4), we say that the
1-forms σi satisfy the su(2) algebra.
96The explicit expressions for σi in terms of the Euler angles can be found in [19].
69
5. Squashed sphere geometry
The quaternionic left-invariant 1-form σ and the invariant δij allows us to construct
a metric on S3 as97
ds2(S3) = ‖σ‖2 = σ iiσ . (5.5)
In fact, this is the standard metric on S3 up to a constant conformal factor [19]. That
σ is left-invariant, that is, invariant under U 7→ aU where a ∈ SU(2), is immediate
from (5.3). The metric ‖σ‖2 is, however, right-invariant as well, that is, invariant
under U 7→ Ua, since then σ 7→ a−1σa. Thus, the metric is said to be bi-invariant.
From the above metric on S3, the metric on S4 can be written as [19]
ds2(S4) = dµ2 + 14 sin
2 µ ‖Σ̃‖2, (5.6)
where 0 < µ < π and Σ̃ is a quaternionic left-invariant 1-form satisfying the
su(2) algebra. This construction uses that S4 without the north and south pole
is diffeomorphic to (0, π) × S3 and the sin2 µ factor gives the three-spheres their
correct sizes, smaller close to the poles and larger closer to the “equator”.98 The
coordinate patch described by these coordinates is a warped product space (compare
to section 3.1).
Now that we have briefly discussed the metrics on SU(2) ' S3 and S4 we turn to
the real case of interest, that is, an SU(2)-bundle over S4. We now have two S3
manifolds and use one real coordinate, µ, and two unit quaternions Ũ, Ṽ . Hence,
there are two independent su(2) algebras and we need two sets of imaginary units,
hŨ and hṼi i . Let σ̃ = hŨ σ̃ii and Σ̃ = hṼ ii Σ̃ be the su(2)-forms, constructed as in (5.3),
corresponding to the two unit quaternion coordinates. A metric can then be written
as
d 1s2 = dµ2 + 24 sin µ ‖Σ̃‖
2 + λ2‖σ̃ − A‖2, (5.7)
where A = hŨi Ai is a Yang–Mills SU(2) gauge potential. With A = 0 this would just
be S4 × S3, with λ determining the relative size of the factors, but if the potential
describes a topologically nontrivial instanton, the topology of the bundle is affected
[19]. In particular, with
Ai = cos2 µ i2 Σ̃ , (5.8)
the topology is that of S7 [19]. Note, however, that the topology of the chart covered
by our coordinates is still that of (0, π)× S3 × S3. The parameter λ in (5.7) will be
referred to as the squashing parameter.
Rewriting the metric
Before computing the Riemann tensor, we rewrite the metric in (5.7) as [19]
ds2 = dµ2 + 1 sin2 µ ‖$‖2 + 1λ2‖ν + cosµ $‖24 4 , (5.9)
97Note that, by writing the metric like this, we have chosen a length unit. Thus, we work in a
dimensionless unit system.
98The construction is analogous to glueing together circles of various sizes along a semicircle to
make a sphere.
70
5.1. Squashed S7 as a principal bundle
where
νi := σi + Σi, $i := σi − Σi, (5.10)
where σ and Σ are quaternionic left-invariant 1-forms satisfying the su(2) algebra
related to two unit quaternions U and V , respectively, as in (5.3). This form of the
metric comes from an isometric embedding of the squashed S7 in the quaternionic
projective space HP2 [19]. In the isometric embedding, only 0 < λ2 ≤ 1 is possible
[19], although there seems to be no such upper bound on λ2 in (5.7).
Note that we could not have written (5.10) as is without indices since σ contains hUi
while Σ contains hVi . To remedy this, we indicate which set of imaginary units is
being used with a superscript as σU = hUσi, σV = hV σi, $V = σVi i − ΣV and so on.
Since ‖σ‖ = σ iiσ regardless of which set of unit quaternions is being used, we need
not worry about this in the metrics (5.7) and (5.9). The relation between the two
constructions is
σ̃V = −V ΣV V −1, Σ̃V = V $V V −1. (5.11)
To see this, first note that the first and middle terms of the metrics (5.7) and (5.9)
are equal since V is a unit quaternion. That the last terms are also equal follows
from ∥∥∥ µ ∥∥∥2 ∥∥∥ µ ∥σ̃ − cos2 Σ̃ = Σ + cos2 $∥∥2 ∥∥2 2 = ∥
∥2
Σ + 12(σ − Σ) +
1 cosµ $∥2 ∥ =
= 14‖ν + cosµ $‖
2. (5.12)
Furthermore, that σ̃ from (5.11) satisfies the su(2) algebra is seen from
dσ̃ = −dV ∧ ΣV −1 − V dΣV −1 + V Σ ∧ dV −1 =
= −12V Σ ∧ ΣV
−1 + 12V Σ ∧ Σ
1 1
V −1 − 2V Σ ∧ ΣV
−1 = −2 σ̃ ∧ σ̃,
where we have dropped the superscript but hVi are the only unit quaternions appearing.
Similarly,
dΣ̃ = dV ∧$V −1 + V d$V −1 − V $ ∧ dV −1 =
= 12V Σ ∧ (
1 1
σ − Σ)V −1 − 2V (σ ∧ σ − Σ ∧ Σ)V
−1 + 2V (σ − Σ) ∧ ΣV
−1 =
= −12V (σ − Σ) ∧ (σ − Σ)V
−1 = −12Σ̃ ∧ Σ̃.
The spin connection and curvature
Now, we derive expressions for the spin connection, Riemann tensor, Ricci tensor
and curvature scalar, starting from the metric (5.9)
ds2 = d 1µ2 + 4 sin
2 µ ‖$‖2 + 1λ24 ‖ν + cosµ $‖
2. (5.13)
From this metric, we(see that an or)thonormal frame is provided by
eı̂ = 1 i2λ ν + cosµ $
i , e0 = d i = 1µ, e 2 sinµ $
i, (5.14)
71
5. Squashed sphere geometry
where we have split the seven-dimensional index as a = (̂ı, 0, i). Note that, from a
covariant perspective, the index on the first e should be i due to the right-hand side.
However, in the index split, we need to distinguish between i and ı̂. Therefore, the
notation is not completely covariant and, to avoid confusion, we will only use indices
ı̂, ̂, k̂, . . . on seven-dimensional objects and not on the SU(2)-invariants ijk and δij.
The torsion-free spin connection can be determined from
0 = T a := Dea = dea + ωab ∧ eb. (5.15)
To determine ω, we first have to compute dea. To this end, we compute dνi and d$i
using ( )
νi ∧ νj +$i ∧$j = 2(σi ∧ σj + Σi ∧ Σj), (5.16a)
νi ∧$j +$i ∧ νj = 2 σi ∧ σj − Σi ∧ Σj , (5.16b)
and express the results in terms of ea by inverting (5.14),
i = 2$ ei, νisin = −2 cot
2
µ ei + eı̂. (5.17)
µ λ
Using also that σ (and Σ satisfy the s)u(2) algebr(a, we find )
dνi = −1i σj ∧ σk + Σj ∧ Σk = −1i2 jk 4 jk ν
j ∧ νk +$j ∧$k =
= −1 + cos
2 µ i j k 2 i j k̂ 1 i ̂ k̂
sin2  jke ∧ e + cotµ  jke ∧ e − 2  jke ∧ e (5.18a)(µ λ) λ
d$i = −1i 1σj ∧ σk + Σj ∧ Σk = − i νj ∧$k2 jk 2 jk =
= 2 cosµ i j k 2sin2  jke ∧ e − sin 
i
jke
j ∧ ek̂. (5.18b)
µ λ µ
Thus, since dµ = e0,
de0 = 0, (5.19a)
d i = cot 0 ∧ i − 1e µ e e i ej ∧ ek̂jk + cotµ ijkej ∧ ek, (5.19b)λ
deı̂ = − λ 1λe0 ∧ ei − i ej ∧ ek − i ̂2 jk 2 jke ∧ e
k̂. (5.19c)
λ
Reading off ω[ab]c from (5.15) and (5.19), using the standard trick ωabc = ω[ab]c −
ω[ac]b − ω[bc]a, we find
ω i = − cot λµ ei + eı̂0 2 , (5.20a)
ω ı̂ = λei0 2 , ( ) (5.20b)
ωij = cot
λ 1
µ ijke
k + 2 − 
k̂
ijke , (5.20c)λ
1
ω k̂ı̂̂ = −2 ijke , (5.20d)λ
= −λ 0 − λω δ e  ki̂ 2 ij 2 ijke . (5.20e)
72
5.2. Coset construction with arbitrary squashing
The curvature 2-form and Riemann tensor are defined by
R b
1 b c
a = 2Rcda e ∧ e
d = dω ba + ω ca ∧ ω bc . (5.21)
Using (5.19) an(d (5.20), )we find
R i0 = 1−
3 2 0 ∧ i + 14λ e e 4(1− λ
2)ijke̂ ∧ ek̂, (5.22a)
λ2
R ı̂ 0 ı̂
1 2 i j k̂
0 = (4 e ∧ e )− 4(1− λ ) jke ∧ e , (5.22b)
Rij = 1− 3λ2 ei4 ∧
1
ej + 2(1− λ
2)eı̂ ∧ e̂, (5.22c)
ı̂̂ = 1 2 ij 0 k 1 1R 2(1− λ ) ke ∧ e + 2(1− λ
2)ei ∧ ej + ı̂ ̂4 2 e ∧ e , (5.22d)λ
2
Ri̂ = −1(1− λ2)ij e0 ∧ ek̂ + 1(1− λ2) λeı̂ ∧ ej + ei ∧ eĵ4 k 4 4 +
+14(1− λ
2)δij ˆ̀δk`ek ∧ e . (5.22e)
From this, it follows that the nonzero components of the Ricci tensor, R = R cab acb ,
are ( 2) ( ) ( )λ 2
R00 = 3 1− 2 , Rij = 3 1−
λ
δ , R = λ2 12 ij ı̂̂ + 2 2 δij. (5.23)λ
Finally, the curvature scalar is
= 3
( )
R 2 8− 2
1
λ2 +
λ2
. (5.24)
Note that, for sufficiently large λ2, the curvature scalar is negative.
The Ricci tensor is diagonal in the basis we have chosen. In particular, we see that
the manifold is Einstein if and only if λ2 = 1 or λ2 = 1/5. The λ2 = 1 solution
corresponds to the ordinary round S7 [19] while λ2 = 1/5 corresponds to what we
will call the Einstein-squashed or simply the squashed seven-sphere.
5.2 Coset construction with arbitrary squashing
As mentioned above, the squashed seven-sphere can be isometrically embedded in
HP2. More precisely, it can be realised as a distance-sphere, that is, as all points at a
fixed distance from an origin, inHP2 for squashing parameters in the range 0 < λ2 ≤ 1
[19]. This realisation provides insight into the isometry group of the squashed sphere.
In suitable inhomogeneous coordinates on HP2, one finds that left-multiplication
by quaternionic unitary 2× 2 matrices and right-multiplication by unit quaternions
leave both the metric of HP2 and the embedding equation invariant [19]. Thus, the
isometry group of the squashed sphere contains Sp(2) · Sp(1) as a subgroup, where
Sp(n) ' U(n,H) is the compact real form of Sp(2n,C), isomorphic to the quaternionic
73
5. Squashed sphere geometry
unitary group.99 Note that, by well-known exceptional isomorphisms Sp(2) ' Spin(5)
and Sp(1) ' Spin(3). The group Sp(2)·Sp(1) ⊂ SO(8) acts transitively and effectively
on the squashed sphere, whence the latter is a homogeneous space.
In this section, we use the theory from section 4.1 to construct the squashed S7, with
arbitrary squashing parameter, as a coset. As prescribed in section 4.1, we work with
spin groups, that is, Sp(2)× Sp(1) ⊂ Spin(8), since the manifold is spin, although
we will almost exclusively be concerned with the Lie algebras.
If we denote the (lifted) group of isometries by G = Sp(2)× Sp(1)C and break Sp(2)
to Sp(1)A × Sp(1)B ' Spin(4) (corresponding to fixing a SO(5)-vector), the isotropy
subgroup of G is H = Sp(1)A × Sp(1)B+C , where Sp(1)B+C denotes the diagonal
subgroup of Sp(1)B × Sp(1)C . Again, this is seen from the embedding in HP2 [19].
Thus, the squashed seven-sphere, with any squashing parameter 0 < λ2 ≤ 1, is
isometric to the coset space
G = Sp(2)× Sp(1)CSp(1) Sp(1) , (5.25)H A × B+C
with an appropriate metric. We will now demonstrate this in detail and find that
this is the case even for λ2 > 1. The construction is similar to that of [20] but we
generalise it to an arbitrary squashing parameter.
The metric
As explained in section 4.1, we need an h-invariant symmetric tensor gab to construct
the metric on the coset (gab is the metric with flat indices). By using, for instance,
[103], [104] one finds that the so(7)-representation 72 ' 1 ⊕ 27 contains two h-
singlets.100 Two is also the number of simple factors in g whence there are two
-invariants (1) and (2)g gAB gAB corresponding to the two quadratic Casimirs of g. Hence,
all G-invariant metrics on the coset are of normal homogeneous form.101 To get a
metric on the coset, we, therefore, start by finding the invariants (n)gAB. To this end,
we compute all commutators and the Cartan–Killing metric of g and then relate the
latter to the invariants via the Casimirs of sp(2) and sp(1)C .
To make everything explicit but not lose generality, we work in a faithful repre-
sentation. Recall that γ-matrices of so(5) can be constructed by joining γ5 to the
γ-matrices of so(4). We use the tensor product of the spinor representation of
so(5) ' sp(2) and the 2-dimensional spinor representation of so(3) ' sp(1)C . Thus,
99Here Sp(2) · Sp(1) = Sp(2) × Sp(1)/Z2 where Z2 is the diagonal subgroup of the centre. This
comes from the fact that left-multiplication by diag(−1,−1) and right-multiplication by −1 are
equivalent.
100The decomposition of the relevant so(7)-representations can also be found in [10].
101This is not entirely true in the strict sense of section 4.1 since the g-invariant may not be of
Euclidean signature. Also, there can be exceptions in degenerate cases.
74
5.2. Coset construction with arbitrary squashing
the generators c(an be)written as ( )
(A) = − i σi 0 ⊗ (B) i 0 0 (C) iTi 2 0 0 12, Ti = −2 0 ⊗ 12, T = − 1( ) σ ( i) 2 4
⊗ σi
i
(Q) = 1 0 −1T 20 2 0 ⊗
(Q)
12, Ti = −
i 0 σi
12 2 0
⊗ 12, (5.26)σi
where the labels A,B,C indicate which sp(1) algebra the generators belong to, T (Q)
are the remaining generators, 1n is the n × n unit matrix and σi are the Pauli
matrices. To see this, note that the γ-matrices of so(4) can be obtained from those
of so(3, 1) by multiplying γ0 by i. The generators of so(3, 1), which are proportional
to γαβ, are block-diagonal in the Weyl-basis of appendix B.3. The blocks in these
six generators are iσi and by appropriate linear combinations, (A)Ti and
(B)
Ti can be
obtained.102 This explicitly demonstrates the well-known exceptional isomorphisms
so(4) ' su(2) ⊕ su(2). The last generators of so(5) are proportional to γα5 and
are the ones denoted T (Q) above. The normalisations of the three commuting sp(1)
algebras are such that [Ti, Tj] =  kij Tk.
Apart from the sp(1) commutation relations, there are nonvanishing Lie brackets
between T (Q) and T (A),(B),(Q). By straightforward computation
[ (A) ( ] [Q) = −1 (Q) (A) (Q)] = 1 (Q) 1 (Q)[Ti , T0 ] 2Ti , [Ti , Tj ] 2δijT0 + 2
k
ij Tk , (5.27a)
(B) (Q) = 1 (Q) (B) (Q)[Ti , T0 ] 2Ti , [Ti , Tj ] = −
1 (Q) 1 k (Q)
2(δijT0 + 2i)j Tk , (5.27b)(Q) (Q) (A) (B) (Q) (Q) (A) (B)
Ti , T0 = Ti − Ti , Ti , Tj =  kij Tk + Tk . (5.27c)
From these commutation relations, it is easy to see that the Cartan–Killing metric
κ = f Df CAB AC BD , where A,B,C, . . . are g-indices, is block diagonal in our basis.
For instance, the (A), (Q)-block vanishes since T (A) only has nonvanishing brackets
[T (A), T (A)] ∼ T (A) and [T (A), T (Q)] ∼ T (Q) while [T (Q), T (A)] has no T (A) part and
[T (Q), T (Q)] has no T (Q) part. A short calculation gives,
κAB = diag(−3 · 13,−3 · 13,−2 · 13,−6 · 14)AB, (5.28)
where we have ordered the generators as (T (A), T (B), T (C), T (Q)).
We define the quadratic Casimir of g by
C ABg = 6κ TATB. (5.29)
Since g = sp(2) ⊕ sp(1)C , Cg is a linear combination of Csp(2) and Csp(1) . To findC
the coefficients, we compute adg(Cg). From the definition, we immediately find that
adg(Cg) is 6 · 13 on the (C)-block, that is, the sp(1)C part. After a short calculation,
we find that adg(Cg) is 6 ·13 on the (A)-block as well, whence it is 6 ·110 on the sp(2)
102Note that we use the convention in which a group element is g = exp(T ).
75
5. Squashed sphere geometry
part by Shur’s lemma. With the normalisation of the Casimirs from appendix A.3,
we conclude that
Cg = 2 Csp(2) + 3 Csp(1) , (5.30)C
since Csp(2)(adsp(2)) = 3 and Csp(1)(adsp(1)) = 2 and adg = ad 103sp(2)⊕ adsp(1) .C
The Casimirs Csp(2) and Csp(1) can be written asC
C = −gABsp(1) (1) TATB, Csp(2) = −gAB(2) TATB, (5.31)C
where gAB AB AB AB(1) = diag(0, 0,13, 0) and g(2) = diag(13,13, 0,14/2) are g-invariant
tensors. From these, we can form the invariants
(1)
gAB = diag(0, 0
(2)
, 13, 0)AB, gAB = diag(13, 13, 0, 2 · 14)AB. (5.32)
Note that (n)gAB is not the inverse of gAB. Rather, P (n) C =
(n)
g BC(n) A ABg(n) , for n = 1, 2, are
the projection operators onto sp(1)C and sp(2), respectively. Since there are two
simple factors in g, these span the space of g-invariants gAB.
We now write the g-invariant that we(will use to define the)metric on the coset as
= 1 (1)gAB 2 sin( ) cos θ gAB + sin
(2)
θ gAB , (5.33)θ
where θ is referred to as the squashing angle for reasons that will soon become
apparent. Apart from an overall constant factor, which is irrelevant for the geometry,
this is an arbitrary g-invariant except that we have to exclude θ = 0 due to the
prefactor. We need only consider half a revolution for the squashing angle since
θ 7→ θ + π leaves gAB invariant. At this point, we thus have two relevant regions
0 < θ < π/2 and π/2 < θ < π in which gAB is of signature (13, 0), that is, Euclidean,
and (10, 3), respectively, as well as the midpoint θ = π/2 in which gAB is degenerate.
As we will see, this will change when we go over to the metric on the coset.
Following section 4.1, we now wish to split g into a direct sum h ⊕ m such that
[h,m] ⊆ m and gAB is block-diagonal over the terms. At the same time, we will
switch to a basis in which gab = δab, where gab is the restriction of gAB to m (we use
a, b, c . . . for m-indices). To this end, we write the generators of g as
: (A) (B+C) = (B) + (C)h Ti , Ti Ti Ti , (5.34a)
m : T (Q) (T )α , Ti = f( )
(B)
θ Ti − tan
(C)
θ f(θ)Ti , (5.34b)
where √
f(θ) = 21 + tan . (5.35)θ
Note that (B+C)Ti generate sp(1)B+C , the diagonal subalgebra of sp(1)B ⊕ sp(1)C .
Here, we need to exclude the region in which tan θ ≤ −1 and θ = π/2 where tan θ
diverges. In the new basis for g,
gAB = diag(13/2, (1 + cot θ)13/2, 17)AB, (5.36)
103We use C to denote the Casimir operators and C for its eigenvalues.
76
5.2. Coset construction with arbitrary squashing
where the index A is split as A = (R, a) where R is an h-index and a an m-
index. The index R is further split as R = (r, ṙ) corresponding to the terms in
h = sp(1)A ⊕ sp(1)B+C . The metric in (5.36) is computed from (5.32) and (5.33)
by matrix multiplication, g′ C D BAB = MA MB gCD, where MA is the matrix relating
the two bases, T ′A = M BA TB. Note that the split h⊕m is reductive since gAB is g-
invariant which implies that f DAB gDC is completely antisymmetric whence f TSa = 0
follows from gAB being invertible and h being a subalgebra of g.
Before we turn to the Riemann and Ricci tensors, some comments about the range
of values of θ. Due to tan θ < −1 being excluded and θ ∼ θ + π, it is convenient to
consider θ in the interval −π/4 < θ < π/2. The quotient between the coefficients
of T (B) and T (C) in T (T ) are determined by requiring gAB to be diagonal in the
new basis while the factor f(θ) and the prefactor in (5.33) ensures gab = δab. Since
there are two -invariants (1),(2)h gab , we conclude that −π/4 < θ < π/2 corresponds
to linear combinations of these such that the result has signature (7, 0), that is,
Euclidean, while π/2 < θ < 3π/4 corresponds to signature (4, 3). Similarly, the points
θ = −π/4, π/2 corresponds to degenerate linear combinations. This situation is
precisely what one expects from there being two -invariants (1),(2)h gab . Recall, however,
that we also had to exclude θ = 0. Without the diverging prefactor in (5.33), gAB
would only have rank 3 for θ = 0. Clearly, such a gAB cannot be restricted to a
nondegenerate metric on h. However, it seems like θ = 0 will not be problematic
when we forget about gAB and only consider the coset since the divergences sit in
gRS, not gab, and there is no singularity in the basis in (5.34) at θ = 0. This agrees
with the above remark that the h-metric is degenerate for θ = −π/4, π/2, not θ = 0.
Note that the coset is Euclidean even when gAB is non-Euclidean (but nondegenerate)
as long as (B+C)Ti are time-like. The problematic θ = −π/4 corresponds to light-like
(B+C)
Ti .
The Riemann and Ricci tensors
To compute the Riemann tensor, we use (4.15). Thus, we first need to compute the
structure constants in the basis in (5.34). We use the index split A = (R, a) = (r, ṙ, a)
described above and further split a as a = (̂ı, 0, i), where (T ) (Q)Tı̂ = Tı̂ , T0 = T0 and
Ti = (Q)Ti .
Since (C)Ti commute with everything else, all commutators are easily computed
using the sp(1) commutation relations and (5.27). We find that the nonvanishing
components are
f t =  trs rs , f ṫṙṡ =  ṫṙṡ , (5.37a)
k = −1 1 1fr0 2δ
k
r , f
0 k k
rj = 2δrj, frj = 2rj ,
1 1 1
f k = k 0ṙ0 2δṙ , fṙj = −2δṙj, f
k =  k (5.37b)ṙj 2 ṙj ,
f k̂ṙ̂ =  k̂ṙ̂ ,
77
5. Squashed sphere geometry
f ti0 = δt, f ṫi i0 = −
1 ṫ k̂ = −f(θ)δ , f δk̂1 + cot i i0 2 i ,θ
t = t ṫ = 1 ṫ k̂ = f(θ)f k̂ij ij , fij 1 + cot ij , fij 2 ij ,θ (5.37c)
k = f(θ) k 0 = −f(θ) f(θ)fı̂0 2 δı̂ , fı̂j 2 δı̂j, f
k k
ı̂j = 2 ı̂j ,
2
f ṫı̂̂ = 1 + cot 
ṫ
ı̂̂ , f
k̂
ı̂̂ = f(θ)(1− tan θ) k̂,θ ı̂̂
where we have grouped the components based on whether they come from [h, h],
[h,m] or [m,m], in that order. Here, some of the θ-dependence comes from
−1 −1
(B) = tan θ (B+C) + f(θ) (T ) (C) 1 (B+C) f(θ) (T )Ti 1 + tan Tθ i 1 + tan T , T = T − T .θ i i 1 + tan θ i 1 + tan θ i
(5.38)
Computing the Riemann tensor using (4.15), we find
i 5 + 8 tan θ 0 i 1 + 2 tan θR i ̂ k̂0 = 8(1 + tan )e ∧ e + 8(1 + tan ) jke ∧ e , (5.39a)θ θ
R ı̂ = 1 e0 ∧ eı̂ − 1 + 2 tan θ i ̂ k0 8(1 + tan ) 8(1 + tan ) jke ∧ e , (5.39b)θ θ
ij = 5 + 8 tan θ i j 1 + 2 tan θR ı̂ ̂8(1 + tan θ)e ∧ e + 4(1 + tan θ)e ∧ e , (5.39c)
ı̂̂ = 1 + 2 tan θ ij 0 k 1 + 2 tan θ 1 + tan θR 4(1 + tan ) ke ∧ e + 4(1 + tan )e
i ∧ ej + eı̂ ∧ e̂2 , (5.39d)θ θ
i̂ = − 1 + 2 tan θ ij 0 k̂ 1 + 2 tan θR ı̂ j 1 i ̂8(1 + tan ) ke ∧ e + 8(1 + tan )e ∧ e + 8(1 + tan )e ∧ e +θ θ θ
+ 1 + 2 tan θ ij k8(1 + tan )δ δk ˆ̀e ∧
ˆ̀
e , (5.39e)
θ
where we, as in section 5.1, have dropped hats on indices on the su(2)-invariants and
it is understood that, for instance, i and ı̂ take the same value when they appear
in the same equation. This result agrees with (5.22) after identifying the relation
between the squashing parameter λ and the squashing angle θ as
1
λ2 = 2(1 + tan ) . (5.40)θ
The squashing parameter λ2 takes all values in the interval (0,∞) and decreases
monotonically for θ ∈ (−π/4, π/2). The values λ2 = 1, 1/5 for which the coset
manifold is Einstein, correspond to tan θ = −1/2 and tan θ = 3/2, respectively.
Hence, the round metric is not normal homogeneous in the strict sense since gAB has
indefinite signature for tan θ = −1/2. The Einstein-squashed sphere, on the other
hand, is not only normal homogeneous but standard homogeneous, that is, gAB ∝ κAB
for tan θ = 3/2. More specifically, gAB = −κAB/6 in the Einstein-squashed case.
Note that the two Einstein metrics are separated by θ = 0, corresponding to the
78
5.2. Coset construction with arbitrary squashing
only G-invariant metric on the coset that cannot be obtained from an invariant gAB.
Lastly, note that, for the Einstein-squashed S7,
c 1f cab = −√ aab , (5.41)5
where aabc are the octonion structure constants from appendix C. That fabc is
proportional to aabc only happens for the Einstein-squashed sphere since it depends
crucially on (1− tan θ) = −1/2.
79
5. Squashed sphere geometry
80
6. Eigenvalue spectra of the
squashed seven-sphere
In this chapter, we derive the eigenvalue spectra of the universal Laplacian of the
squashed seven-sphere. We will only consider the Einstein-squashed sphere and,
henceforth, refer to it simply as the squashed S7. We will consider eigenmodes of the
Laplacian, ∆, from section 3.2.4 and use (4.36) to determine the possible eigenvalues.
On a high level, the derivation goes as follows. First, assume that we have an
eigenmode of ∆ with some eigenvalue κ2. Since, as explained in section 4.3, the
eigenmodes of ∆ fall into irreducible representations of G = Sp(2)× Sp(1)C , Cg can
be replaced by its corresponding eigenvalue Cg on the relevant representation. We
will not investigate which irreducible G-representations occur in the G-representation
induced by the relevant irreducible H-representations. It has, however, been done
using Young tableaux techniques [10]. Replacing ∆ by κ2 and Cg by Cg, we are left
with a linear map that acts on the spin-index of the mode in the right-hand side
of (4.36). In the left-hand side, however, we have a first-order differential operator.
To get rid of this, we will use various techniques, such as squaring it, acting with
projection operators and combinations thereof. This will, eventually, result in a
polynomial equation for κ2. In general, there can be false roots, that is, solutions to
the polynomial equation that are not actual eigenvalues of ∆. We will not deal with
this in the current chapter. Note, however, that, as long as we do not introduce any
assumptions in the derivation, as we will not, κ2 being an eigenvalue of ∆ implies
that it is a root to the polynomial. Thus, although we can get false roots, we cannot
miss any eigenvalues.
In chapter 5, we used a dimensionless unit system such that, for Einstein-squashing,
(5.23)
= 27 = 189Rab 10δab, R 10 . (6.1)
To connect this to chapter 3, in which the internal manifold has Rab = 6m2δab, we
see that the dimensionless system results from setting
m2 = 920 . (6.2)
Here, we will mostly continue to use the dimensionless system for convenience. Note,
however, that there is a sign choice in (6.2) which is relevant for skew-whiffing and the
number of Killing spinors. In this chapter, we only concern ourselves with operators
on the squashed seven-sphere. The sign choice will become important in chapter 7,
where we investigate the number of unbroken supersymmetries and masses in the
compactifications of eleven-dimensional supergravity on the squashed seven-sphere.
81
6. Eigenvalue spectra of the squashed seven-sphere
We begin by concretising some details from section 4.3 for the case of interest. Note
that the g = sp(2)⊕ sp(1)C invariant is gAB = −κAB/6 for the Einstein-squashed S7.
The metric with flat indices is δab in the basis from section 5.2. Thus, comparing the
normalisations of the quadratic Casimirs Cg and Ch in section 4.3 and appendix A.3
gives the quadratic master equation (4.31)
− ̌Z = (Cg − Ch)Z. (6.3)
Next, the relevant structure constants are, as remarked in section 5.2,
1
fabc = −√ aabc, (6.4)5
where aabc are the octonion structure constants from appendix C. Hence, the covariant
H-derivative is (4.28)
Ď = D − √1a a aabcΣbc. (6.5)2 5
As explained in appendix C, the largest group that leave aabc and δab invariant is
G2 =: H̃ ⊃ H. Therefore, G2 will play an important role in the derivation of the
spectrum. In table 6.1, the decompositions of the relevant Spin(7)-representations
restricted to G2 are given. These can be found by using [103] or [104] or by looking
in [10]. Note that, since H is a subgroup of G2,
Ďaabcd = 0. (6.6)
For this reason, it will be convenient to work with Ďa instead of Da.
Table 6.1: Decomposition of irreducible Spin(7)-representations when restricted
to G2. 1, 7, 21 and 35 are p-forms for p = 0, 1, 2, 3; 27 is traceless symmetric
rank-2 tensors; 8 spinors and 48 vector-spinors. Each irreducible representation
is specified both using its dimension (in bold) and its Dynkin labels.
Spin(7) irrep. → G2 rep.
1 = (0, 0, 0) 1 = (0, 0)
7 = (1, 0, 0) 7 = (1, 0)
8 = (0, 0, 1) 1⊕ 7 = (0, 0)⊕ (1, 0)
21 = (0, 1, 0) 7⊕ 14 = (1, 0)⊕ (0, 1)
27 = (2, 0, 0) 27 = (2, 0)
35 = (0, 0, 2) 1⊕ 7⊕ 27 = (0, 0)⊕ (1, 0)⊕ (2, 0)
48 = (1, 0, 1) 7⊕ 14⊕ 27 = (1, 0)⊕ (0, 1)⊕ (2, 0)
In the coming sections, the projection operators onto the various irreducible G2-
representations in the Spin(7)-representations will appear. We denote these by Pn,
where n is the dimension of the G2-representation and the Spin(7)-representation
is understood from the context or index structure. Of immediate interest are the
projection operators P ∧27 and P14 from 21 = 7 , the adjoint representation of Spin(7),
82
6. Eigenvalue spectra of the squashed seven-sphere
to 7 and 14, respectively. Since aabc is a G2-invariant, it can be viewed as an
intertwiner between 21 and 7. Thus, the projection operator P7 : 21 → 21 that
projects onto the 7 ⊂ 21 is proportional to a acb1b2a1a2c . Working out the normalisation
through P 27 = P7 gives
(P ) b1b27 a1a2 =
1
a c b1b
1
2 b1b2
6 a1a2 ac = 3δa1a2 +
1
c b1b26 a1a2 , (6.7)
where c = ?a, see appendix C.104 Since P7 + P14 = 121, it immediately follows that
(P ) b1b 22 = δb b 11 2 b1b214 a1a2 3 a1a2 − 6ca1a2 . (6.8)
The Weyl tensor of the squashed sphere is, by (3.16), (4.1) and (4.15)
9
W cd = R cd − δcd = (T i) (T )cd + 1 a ea cd 1 ce d 9 cdab ab 10 ab ab i 10 ab e + 10a[a ab]e − 10δab =
= (T i) cd 6ab(Ti) − 5(P14)
cd
ab . (6.9)
This implies that, using the Casimirs from appendix A.3,
W ΣabΣcd 6abcd = 5 Cg2 − Ch. (6.10)
Since H ⊂ G2, a bcabc(Ti) = 0. This is seen by noting that aabc can be interpreted as
an intertwiner from 21 to 7 while (Ti)bc can be interpreted as an intertwiner from 21
to adh ⊂ adg2 = 14. Similarly, aabc(P14) debc = 0, whence also aabcW debc = 0.
The Ricci identity (4.41) reads ( )
[Ďa, Ďb] = (T i)abTi − √
1
a cĎ = W cd + 6ab c ab 5(P14)
cd Σ − √1ab cd a cab Ďc. (6.11)5 5
An important special case of this, which follows immediately from the above remark,
is
a bca ĎbĎc = −√
3 Ďa. (6.12)5
Lastly, since a da[b aa e de dec] = −3δbc + 6(P14)bc , (4.36) becomes
− √1 ΣabĎc = ∆− C − 6 3aabc g 5 Cso(7) + 2 Cg2 . (6.13)5
This is the equation we will use to derive the operator spectrum of the squashed
seven-sphere.
104In this chapter, we make heavy use of the octonion structure constant identities from appendix C.2.
We will, for the most part, not give references to these equations when using them.
83
6. Eigenvalue spectra of the squashed seven-sphere
6.1 0-forms
The scalars, or 0-forms, are trivial since all Casimirs vanish and ρ (Σab1 ) = 0. For a
0-form satisfying ∆Y = κ2Y and CgY = CgY , (6.13) immediately gives
κ2 = Cg. (6.14)
The G-representation induced by the scalar H-representation, that is, the G-repre-
sentation carried by scalar fields on the squashed S7, contains precisely one copy of
each irreducible G-representation (p, q; r) with p = r [10]. Thus, we know exactly
which values of Cg are possible. In this case, there are no false roots.
6.2 1-forms
We now turn to transverse 1-forms, Ya. Transversality means that DaYa = 0 which
is equivalent to ĎaYa = 0. Since Cso(7)(7) = 3 and Cg2(7) = 2, a 1-form eigenmode
of ∆ satisfies, by (6.13),
√ ( 3)
D1Ya := a bc 2a ĎcYb = − 5 κ − Cg − 5 Ya. (6.15)
Squaring the operator (D1 gives )
D 2Y = a( b1b21 a a Ďb2 a c1c2b1 Ďc2Yc1 )= −̌Y ba + Ď Ď b1b2b3aYb + ca Ďb1Ďb2Yb3 =
= 24 1Cg − Ch + − √5 D1 Ya + Ď
b
aĎ Yb, (6.16)5
where we have used that the(Ricci identity (6.11)) for 1-forms,
[Ďa, Ďb]Y d
6 d 1 d
c = Wabc + (P14)abc Yd − √5 a5 abd
Ď Yc, (6.17)
implies
[Ďb Ď ] = 12, a Yb Ya + √
1
5 D1Ya, (6.18a)5
c b1b2b3a Ď Ď
12 2
b1 b2Yb3 = 5 Ya −
√ D1Ya, (6.18b)5
since the Weyl tensor is traceless and W[abc]d = 0. Note that Ch re-entered the
calculation. However, (6.10) applied to 1(-forms sh)ows that
b c = 12Wa b Yc 5 − Ch Ya. (6.19)
Since the Weyl tensor is traceless, ChYa = 12/5Ya. Using this, (6.15), ĎaYa = 0 and
Ya 6= 0, (6.16) gives ( ) ( )2
Cg +
12 − − 2 + 35 Cg κ 5 = 5 −
3
Cg κ
2 + 5 , (6.20)
84
6.3. 2-forms
with solutions √
κ2 = 7 1 49Cg + √10 ± C5 g
+ 20 . (6.21)
Note that, from this calculation, we cannot determine whether both solutions occur
as eigenvalues of ∆1.
6.3 2-forms
Let Yab be a transverse eigenmode of the Hodge–de Rham operator, satisfying
∆Yab = κ2Yab, CgYab = CgYab and DaYab = 0. Since Cso(7)(21) = 5, Cg2(7) = 2 and
Cg2(14) = 4, the rewritten quadratic master equation (6.13) becomes105
2 ( )
D bc[2]Ya1a2 := a[a1| ĎcYb|a2] = √ Cg − κ
2 + 3P Y
5 7 a1a2
. (6.22)
The transversality condition DaYab = 0 can be written as
ĎbYba = √
1
a bca Ybc. (6.23)2 5
Define another differential operator D̃[2] by
D̃ b1b2[2]Ya1a2 := a[a Ď Y . (6.24)1 a2] b1b2
By using the definitions, the transversality condition and the projection operator in
(6.7), we find
c b1b2a1a2 D[2]Yb1b2 = √
3
P7Ya1a2 + 2D5 [2]
Ya1a2 − 2D̃[2]Ya1a2 . (6.25)
Taking the appropriate linear combination with D[2]Ya1a2 to get P7D[2]Ya1a2 in the
left-hand side and rearranging gives
3
D̃[2]Y = 2D[2]Y − 3P7D[2]Y + √ P7Y. (6.26)2 5
Note that, if Y is an eigenmode of ∆ with vanishing 7-part, D[2]Y = 0 since the
other terms in this equation trivially vanish. This depends on Y being transverse
since we used that in the derivation. We will not assume that Y has vanishing 7
part. However, this remark will prove useful later.
Using (6.22) to write the right-hand side of (6.26) in terms of Y and P7Y gives
√ [ ( ) ( ) ]
D̃[2]Y =
5 4
2 2 Cg − κ
2 − 3 C − κ2g + 5 P7 Y. (6.27)
105We put brackets around the 2 in D[2] to indicate the antisymmetrisation. D(2) will be defined
similarly but with symmetrisation.
85
6. Eigenvalue spectra of the squashed seven-sphere
Define Ya = a bca Ybc. We then immediately see that a b1b2a D̃[2]Yb1b2 = D1Ya, with D1
as defined in (6.15). (6.27) thus implies
√
5( )
D Y = − C − κ21 a 2 g +
12
5 Ya. (6.28)
The situation is now very similar to that in section 6.2. The only differences are that
Ya might not be transverse and can be 0. If Ya = 0, then D̃[2]Yab = ĎbYa = 0 and,
by (6.27), κ2 = Cg, since Yab 6= 0. Going forward, we hence assume Ya 6= 0.
To handle that Ya might not b(e transverse, w)e contract (6.28) with Ď
a and use (6.12)
to find
C − κ2 + 18 ag 5 Ď Ya = 0. (6.29)
Hence, either κ2 = C + 18/5 or Ďag Ya = 0. If ĎaYa = 0, the calculation in section 6.2
can be reused and gives (6.30c) below. Thus, the possibilities are
κ2 = Cg, (6.30a)
κ2 = 18Cg + 5 , √ (6.30b)
2 = + 11 ± √2 49κ Cg 5 C +5 g 20
, (6.30c)
where the first one applies to modes with P7Y = 0, the second applies to modes with
ĎaYa 6= 0 and the third possibility applies to modes with Ya 6= 0 but ĎaYa = 0. As
in section 6.2, we cannot, at this point, say whether all of these occur as eigenvalues
of ∆2.
6.4 Symmetric rank-2 tensors
Now we turn to the eigenvalues of the Lichnerowicz operator. Let Xa1a2 be a
transverse symmetric traceless rank-2 eigenmode of ∆, satisfying ∆X = κ2X and
DaXab = 0. Since Cso(7)(27) = 7 and Cg2(27) = 14/3, (6.13) becomes
√ ( )
D(2)Xa1a2 := a b1b2(a | Ďb2Xb1|a2) =
5 7
2 Cg − κ
2 + 5 Xa1a2 =
: kX
1 a1a2
. (6.31)
The transversality condition in terms of Ďa reads
ĎaXab = 0. (6.32)
Now define, for any rank-2 tensor Za1a2 ,
D Z b1b22 a1a2 := aa1 Ďb2Zb1a2 . (6.33)
An arbitrary rank-2 tensor consists of a trace, Z = Z aa , a traceless symmetric
part, Z(ab) − δabZ/7, and an antisymmetric part, Z[ab]. The corresponding Spin(7)-
representations are 1, 27 and 21, respectively, where the last one splits into 7⊕ 14
86
6.4. Symmetric rank-2 tensors
when restricted to G2. Consider D2Xab. The scalar part of this, that is, the trace,
vanishes since Xab is symmetric and aabc antisymmetric. By contracting D2Xb1b2
with aab1b2 , one sees immediately that the 7 part vanishes as well due to X being
traceless, symmetric and transverse. Define
Ya1a2 := D[2]Xa1a2 = D2Xa1a2 − kXa1a2 , (6.34)
which is a 2-form with vanishing 7-part by the above remarks and (6.31). Using
(6.12), we see that
ĎaD c1c2 a2Xab = aa Ď Ďc2Xc1b = 0, (6.35)
by transversality of Xab. Thus, Yab is a transverse 2-form.
We will now compute D 22 X. To this end, note that (6.10), applied to a traceless
symmetric rank-2 tensor, gives ( )
2W b1 b2a1 a2 Xb1b2 =
28
5 − Ch Xa1a2 . (6.36)
Using this, the symmetry properties of the Weyl tensor, the projection operators
(6.7) and (6.8) and the Ricci identity (6.11), we find
[Ďb, Ďa1 ]
1
Xba2 = 2C
1
hXa1a2 + √ D2Xa1a2 , (6.37a)5
c b1b b
1 2
2 3
a1 Ďb1Ďb2Xb3a2 = 2ChXa1a2 −
√ D
5 2
Xa1a2 . (6.37b)
Squaring D2, using the above, properties of Xab and the quadratic master equation
(6.3), we get
D 2X = −̌X + ĎbĎ X + c b1b2b32 a1a2 a1a2 a1 ba2 a1 Ďb1Ďb2Xb3a2 =
= − √1CgXa1a2 D2Xa1a2 . (6.38)5
Remarkably, the Ch from the quadratic master equation was cancelled by the two
halves in (6.37).
Combining (6.34) and (6.38), we see that ( )
k2
k 1
Xa1a2 + kYa1a2 + D2Ya1a2 = Cg − √ Xa1a2 − √ Ya1a2 . (6.39)5 5
The antisymmetric part of this is ( ) √ ( )
D Y = a b1b2[2] a1a2 [a | Ď = − +√
1
b2Yb1|a2] k Ya1a2 = −
5
C −κ22 g +
9
1 5 5
Ya1a2 . (6.40)
Since Yab is a transverse 2-form with vanishing 7-part, we see from (6.26) that
D[2]Y = 0. Hence, either Yab = 0 or κ2 = Cg + 9/5. If Yab = 0, the symmetric part
of (6.39) is ( )
k2 + √k − C X = 0. (6.41)
5 g a1a2
87
6. Eigenvalue spectra of the squashed seven-sphere
Note that this is not true if Ya1a2 =6 0 since D2Ya1a2 can have a symmetric part
even though Ya1a2 is antisymmetric. Inserting k from (6.31) and solving for κ2 gives
(6.42b) below, since Xab 6= 0. Thus, we have arrived at
κ2 = Cg +
9
5 , √ (6.42a)
κ2 = 8Cg + 5 ±
√2 1Cg + 20 , (6.42b)5
where the top row applies when P14D[2]Xab 6= 0 and the bottom one otherwise.
6.5 3-forms
A 3-form Yabc belongs to the irreducible representation 35 of Spin(7), which splits
into 1⊕ 7⊕ 27 when restricted to G2. Computing the Casimir eigenvalues on the
relevant representations using table A.2, the rewritten quadratic master equation
(6.13) becomes
3 ( )√ a b1b2 2 36[a | Ďb2Yb1|a2a3] = Cg − κ + 5 − 3P7 − 7P27 Ya1a2a3 . (6.43)5 1
for a transverse mode Yabc of ∆ with eigenvalue κ2. In section 6.3, we found a trick
that made the calculation very short. Here, we will use essentially the same method
but be a bit more systematic. We begin by analysing the irreducible parts of Yabc and
deriving expressions for the projection operators. Then, we analyse what implications
the transversality of Yabc has for the irreducible components. Lastly, we compute the
possible eigenvalues.
G2-components and projection operators
The G2-scalar in Yabc is, of course,
Y := ab1b2b3Yb1b2b3 , (6.44)
whence P1 is proportional to a b1b2b3a1a2a3a . Using P 21 = P1 to determine the constant
of proportionality gives
(P ) b1b2b 1 b b b 13 1 2 31 a1a2a3 = 42aa1a2a3a , P1Ya1a2a3 = 42aa1a2a3Y. (6.45)
Similarly, the 7-part is
Y := −c b1b2b3a a Yb1b2b3 . (6.46)
By P 27 = P7 we find that
( ) b 1P 1b2b37 a1a2a3 = −24c
cc b1b2b3a1a2a3 c =
= 1δb1 b2 b3 − 1 b b b + 31 2 3 [b1b2 b3]4 a1a2a3 24aa1a2a3a 8c[a a δa ], (6.47a)1 2 3
1
P Y = c c7 a1a2a3 24 a1a2a Yc.
(6.47b)
3
88
6.5. 3-forms
Lastly, the 27-part is
X := a b1b2 1a1a2 (a Ya2)b1b2 − 7δa1a2Y. (6.48)1
For this to be true,
(P ) b1b2b3 := (1− P − P ) b1b2b327 a1a2a3 1 7 a1a2a3 =
= 3δb1 b 1 32 b3 b1b2b3 [b1b2 b3]4 a1a2a3 + 56aa1a2a3a − 8c[a δ (6.49)1a2 a3]
has to be proportional to
( ( 1 )( [ 1 )b
δ 1
c
[a a
b2) b1b2 1 c2c3] c1c2c3
1 a2a3] − 7δ aa1a2a3 δ(b a1 b2) − 7δb1b2a =
= δc1 c 12 c3 − a ac1c2c 13 + [cc 1c2 c3] 1 [c c c ]a1a2a3 7 a1a2a3 2 [a a δ + a
1a 2 3
1 2 a3] 2 [a1a2 a3] . (6.50)
Indeed, by (C.48), we see that the latter is 4/3P27, whence
3
P b27Ya1a2a3 = 4a[a1a2 Xa3]b , (6.51)
since Xab is traceless.
Since 1 = P1 + P7 + P27, we can write Yabc in terms of the irreducible components as
= 1 1 3Y ca1a2a3 42aa1a2a3Y + 24ca1a2a3 Y + a
b
c 4 [a1a2 Xa3]b . (6.52)
We have seen that 1⊕ 27 sits in the symmetric part of
Za1a2 := a b1b2a1 Ya2b1b2 . (6.53)
Since there is no 14 in 7∧3 = 35, we can immediately say that P14Za1a2 = 0. This
can easily be verified by a direct computation as well. The 7-part of Za1a2 has to
vanish or be proportional to Ya by the representation theory. A direct calculation
immediately shows
Y = a b1b2a a Zb1b2 . (6.54)
Thus, Za1a2 is a rank-2 tensor with vanishing 14-components, containing all irreducible
components of Ya1a2a3 . Its irreducible components are
Y = Zbb , Xab = Z(ab) −
1
7δabY, Y
b1b2
a = aa Zb1b2 . (6.55)
Note that, since Zab has no 14-part,
1
Z[a1a2] = 6a
b
a1a2 Yb. (6.56)
89
6. Eigenvalue spectra of the squashed seven-sphere
Transversality
The transversality condition DaYabc can be written using Ďa as
Ďb 1Y b1b2ba1a2 = √ a[a Y5 1 a2]b1b2
. (6.57)
In terms of the irreducible components, this reads
√1 a b 1 1a1a2 Y = a
b b1b2
6 5 b 42 a1a2
ĎbY + 24ca1a2 Ďb1Yb2+
+ 1a b 11 b2 b1b24 a1a2 Ď Xb1b2 − 2a[a1| Ďb1Xb2|a2], (6.58)
where we have used (6.52) and (6.56). This equation contains both a 7-part and a
14-part. The 7-part can be obtained by contracting with aabc and the 14-part by
projecting with P14. They are
√1 (Y = Ď
b
a Xba +
1
7ĎaY +
1 b1b2
6aa ) Ďb1Yb2 , (6.59a)5
P14 Ď Y + 6a b1b2a1 a2 [a1| Ďb1Xb2|a2] = 0. (6.59b)
Computing the eigenvalues
(6.43) is a 3-form equation and hence contains a 1, 7 and 27 part. We begin by
analysing the former two and then turn to the last one. The scalar part of (6.43),
obtained by contracting with ab1b2b3 , is( )
√3 ĎbY = C − κ2 + 36b g 5 Y, (6.60)5
while the vector part, obtained by contracting with c b1b2b3a , is
4 1 ( )Ď Y − b 211b2 b 27 a 6aa Ďb1Yb2 − 3Ď Xba = − Cg − κ + 5 Ya. (6.61)
Using (6.59a) to eliminate ĎbXba from the latter gives
√ ( )
a b1b2Ď Y − 3Ď Y = 5 C − κ2 + 12a b2 b1 a g 5 Ya. (6.62)
Contracting this with Ďa and(using (6.60) t)o(eliminate Ď
aYa, we find
9 9 36)
5CgY = Cg − κ
2 + 25 Cg − κ + 5 Y. (6.63)
Hence, either Y = 0 or √
9 3 81
κ2 = Cg + 2 ±
√ Cg +5 20
. (6.64)
90
6.5. 3-forms
If Y = 0, (6.60) and (6.62) implies that Y(a is a transver)se 1-form satisfying√
a b1b2a Ďb2Yb1 = 5 Cg − κ2 +
12
5 Ya, (6.65)
Apart from numerical constants, the situation is identical to that in section 6.2.
Reusing [th(at calculation, we find
5 − 2 + 12
)2 ( ) ( )]
C κ + C − κ2 12 12g 5 g + 5 − Cg + 5 Ya = 0. (6.66)
Thus, either Ya = 0 as well or √
κ2 = 5 1 49Cg + 2 ±
√ Cg + 20 . (6.67)5
The only case remaining is when both Y = 0 and Ya = 0, that is, when Yabc only has
a nonvanishing 27-part. From (6.52), (6.53) and (6.55), we see that
3
X = a b1b2a1a2 a1 Ya2b1b2 , Ya1a2a3 = a
b
4 [a1a2 Xa3]b. (6.68)
The 7 and 14 parts of the transversality condition, (6.59), become
ĎaXab = 0, P14D[2]Xab = 0. (6.69)
The rewritten quadratic master equation (6.43), which now only has a nontrivial
27-part by (6.60) and (6.62), becomes
√ ( )
3a b1b2 1[a | Ďb2Y
2
1 b1|a2a3] = 5 Cg − κ + 5 Ya1a2a3 . (6.70)
Contract(ing this as in)the first half of ((6.68) gives√ 1 )5 C − κ2 + X = a b1b2 a c1c2Ď Y + 2a c1c2g 5 a1a2 a1 a2 c2 c1b1b2 b1 Ďc2Yc1b2a2 =
= a c1c2a2 Ďc2Xc1a1 + 2c
c2c1b2
a1 Ďc2Yc1b2a2 =
= −a c1c2a1 Ďc2Xc1a2 = −D2Xa1a2 , (6.71)
where we, in the second to last step, have used
2 bc c = 3c 1 2Y c bc1c2a dX = a bd bd da1 c1c2a2 2 a1 [c1c2 a2]d a1 Xda2 + aa2 Xda1 + aa1a2 X
b
d . (6.72)
From (6.69) and (6.71), we see that the situation is the same as that in section 6.4,
apart from numerical constants and the extra piece of information P14D[2]Xab = 0.
Hence, we can reuse that calculation but only get the two eigenvalues corresponding
to (6.42b). This gives the eigenvalues in (6.73c) below. To conclude, we have arrived
at the eigenvalues √
9 3 81
κ2 = Cg + ± √2 √Cg + 20 , (6.73a)5
2 = + 5 ± √1 49κ Cg 2 √C +5 g 20 , (6.73b)
1
κ2 = Cg + ± √
1 1
10 Cg + 20 , (6.73c)5
91
6. Eigenvalue spectra of the squashed seven-sphere
where the first line applies to modes with a nonzero 1-part, the second one to modes
with vanishing 1-part but nonzero 7-part and the last line applies to modes with only
a nonvanishing 27-part. Again, the list exhausts all possibilities but may contain
false roots.
6.6 Spinors
Now that we have dealt with all tensorial representations, we turn to the spinorial
ones, starting with the spinors. Thus, consider a spinor satisfying ∆ψ = κ2ψ.106 The
spinor representation 8 of Spin(7) splits into 1⊕ 7 when restricted to G2. Here, the
1 is the G2-invariant spinor η from appendix C. Since h ⊂ g2, Ďaη = 0. With the
normalisation η̄η = 1, the projection operators are
P1 = ηη̄, P7 = Γaηη̄Γa, (6.74)
where P1 + P7 = 1 by the Fierz identity (C.42). Hence, we define
Y := η̄ψ, Ya = −iη̄Γaψ, (6.75)
so that
ψ = Y η + iY Γaa η. (6.76)
Since Cso(7)(8) = 21/8, the rewritten qu(adratic master equa)tion (6.13) becomes
√1 a ab c 2 63
4 5 abc
Γ Ď ψ = Cg − κ + 20 − 3P7 ψ. (6.77)
Recall from appendix C that
aabc = iη̄Γabcη, cabcd = −η̄Γabcdη, (6.78)
while η̄Γaη = 0 = η̄Γabη since Γa and Γab are antisymmetric. Using this, we find that
the scalar and vector parts o(f (6.77) are
√3
)
ĎaYa = Cg − κ2 +
63
Y,
2 5 20
3 1 ( )
(6.79a)
√ ĎaY + √ a b1b2 2a Ďb2Yb1 = − Cg − κ +
3
20 Ya, (6.79b)2 5 2 5
respectively. Contracting the latter with Ďa gives, by using (6.12) and then eliminat-
ing ĎaYa using th(e scalar equat)ion,9 3 9 ( 9 )( )
C Y = √ C 2 a 220 g g − κ + 20 Ď Ya = Cg − κ + 20 Cg −
2 + 63κ 20 Y. (6.80)2 5
106Recall that, for Einstein manifolds, (iD/)2 only differs from ∆ by a constant when acting on
spinors and vector-spinor.
92
6.7. Vector-spinors
Thus, either κ2 is given by (6.82a) below or Y = 0. In the latter case, ĎaYa = 0, that
is, Ya is a transverse 1-form, by the scala(r equation and)the vector equation becomes√ 2 3D1Ya = −2 5 Cg − κ + 20 Ya. (6.81)
This situation is identical to that in section 6.2 apart from numerical constants.
Reusing that calculation gives (6.82b). Thus,√
2 = + 9 ± √3 81κ Cg 5 √Cg + 20 , (6.82a)2 5
κ2 = 1 1 49Cg + √10 ± C2 5 g
+ 20 , (6.82b)
where the top row applies to modes with nonzero scalar part, η̄ψ =6 0, and the second
otherwise.
6.7 Vector-spinors
Lastly, we turn to transverse Γ-traceless vector-spinors. These carry the represen-
tation 48 of Spin(7) which splits into 7 ⊕ 14 ⊕ 27 when restricted to G2. We
consider an eigenmode ψa of ∆ with eigenvalue κ2. The transversality and Γ-
tracelessness conditions are Daψa = 0 and Γaψa = 0, respectively. The so(7)-Casimir
is Cso(7)(48) = 49/8, whence the re(written quadratic master equation, (6.)13), becomes√
bcĎ 1 147aa cψb + 4a
c de 2
de Γ Ďcψa = 5 Cg − κ + 20 − 3P7 − 6P14 − 7P27 ψa. (6.83)
Before attempting to find κ2 from this, we analyse the irreducible components and
the transversality condition.
Irreducible G2-components
Note that 7 ⊕ 14 ⊕ 27 fits in 7⊗2, that is, a rank-2 tensor. Since we are used to
working with rank-2 tensors from sections 6.3 to 6.5, we will, here too, translate the
problem into one involving a rank-2 tensor. The 7-part of ψa is
Ya = iη̄ψa, (6.84)
where, as in section 6.6, η is the G2-invariant spinor. By the Fierz identity (C.42),
ψa = −iYaη + ΓbZbaη, Zab := η̄Γaψb. (6.85)
From Γaψa = 0, it immediately follows that Z aa = 0, that is, the 1-part of Zab is 0.
The 14 and 27 parts of ψa must be the corresponding parts in Zab since they clearly
do not sit in Ya. By the same representation theory, the 7 part of Zab is either 0 or
proportional to Ya. Using the Fierz identity (C.42), Γaψa = 0 and aabc = iη̄Γabcη, we
find
aabcZbc = iη̄Γabcηη̄Γbψc = −iη̄Γacψ ac = Y . (6.86)
Hence, Zab contains all three irreducible components of ψa. We define
Xab := Z(ab), Yab := Z 7[ab], Yab := P7Yab, Y 14ab := P14Yab. (6.87)
93
6. Eigenvalue spectra of the squashed seven-sphere
Transversality
In terms of Ďa, the transversality condition is
Ďa 1ψ = − √ aa Γbca bc ψa. (6.88)8 5
Picking out the 1 and 7 parts of this, by contracting with η̄ and η̄Γa, respectively,
ĎaY = 0, Ďba Zab = Ďb b
1
Xba − Ď Yba = √ Ya, (6.89)5
where we have used
η̄Γ Γcdψ = −a cdY − c cdeZ + δcZd − δdZca b a b a eb a b a b , (6.90)
which follows from (6.85).
Computing the eigenvalues
To convert (6.83) into an equation for Zab, we contract with η̄Γa. Again using (6.90),
we find ( )
− √3 Ď 1 1 7aY +√ a c1c2b b Ďc2Z c1c2 22 5 5 ac1
− √ a Ď Z
2 5 a c2 c1b
= Cg−κ + 20 +P14 +4P7 Zab.
(6.91)
Note that
ĎbY 7 = 1a cbba 6 a Ď
1
bYc = 6D1Ya, (6.92)
and that (6.89) relates the divergences of Xab and Yab to Ya. We can get two more
relations involving divergences by contracting (6.91) with Ďa and Ďb. This should
suffice to relate D1Ya ∝ ĎbY 7ba to Ya and we will then be in a situation similar to
that in section 6.2 since Ya is transverse by (6.89). First, contracting (6.91) with Ďb
and using the transversality condition, ĎbX b 14b(a and Ď Yba ca)n be expressed as√
ĎbY 14 = −3Ďb 7 5 2 81ba Yba + 3 (Cg − κ + 20)Ya, (6.93a)√
Ďb 5 93Xba = −2ĎbY 7ba + 3 C
2
g − κ + 20 Ya, (6.93b)
where we have used
ĎbĎ = 12 6aYb 5 Ya +
√ ĎbY 7ba, (6.94a)5
c1c2ĎbĎ = 3ab c2Zac1 5Ya, ( ) (6.94b)
a c1c2a ĎbĎ =
6 + √1Z Y Ďbc2 c1b 5 a Xba + 3Ď
bY 14 bba + 3Ď Y 7ba . (6.94c)5
Now contract (6.91) with Ďa. The first and last terms in the left-hand of (6.91) side
give
− √3
( )
ĎaĎ 3 12 1 cd a 3Y c
2 5 a b
= √ C − Y
2 5 g 5 b
, − √ aa Ď ĎdZcb = −2 5 10
Ď Zcb, (6.95)
94
6.7. Vector-spinors
respectively. Note that ĎcZcb can be expressed in terms of Y a 7b and Ď Yab by (6.93).
The right-hand side of (6.91) contracted with Ďa can similarly be expressed in terms
of Yb and ĎaY 7ab. Lastly, the middle term in the left-hand side of (6.91) is proportional
to
a cdb ĎaĎ Z cd a cd ad ac = ab ĎdĎ Zac + ab [Ď , Ďd]Zac. (6.96)
Here, the commutator term is
a cdb [Ďa, Ďd]Zac = −Yb + √
4 Ďc 2Y 7 c 14
5 cb
− √ Ď Ycb , (6.97)5
while the first term is, by (6.92) and (6.93),
√ ( 87)
a cd a cd a 7 2 a 7b ĎdĎ Zac = −4ab ĎdĎ Yac + 4 5 Cg − κ + 20 Ď Yab. (6.98)
In this expression, the first term in the right(-hand sid)e is
a cdĎ a 7b dĎ Yac =
1
a cda ef6 b c Ď
1 12 1
dĎfYe = 6 Cg + Yb −
√ Ďc5 Y
7
5 cb
. (6.99)
Putting the above together, we find
( √109) 5[( 57)2 1( 12)]
Cg − κ2 + D 240 1Ya = 2 Cg − κ + 20 − 4 Cg + 5 Ya. (6.100)
The situation is similar to that in section 6.2. Using (6.16), we get a fourth-order
equation in κ2 with solut√ions √
2 = + 14 ± √1 + 49 2 = + 31 ± √5 49κ Cg 5 C2 5 g 20
, κ Cg 10 C + . (6.101)2 5 g 20
If Ya = 0, both X 14ab and Yab are divergence-free by (6.93). The symmetric and
antisymmetric parts of (6.91) become ( )
√1 D(2)Xab − √
3
D 2
7
(2)Yab = (Cg − κ + 20)Xab, (6.102a)2 5 2 5
− √3 1D X 2 27
2 5 [2] ab
+ √ D[2]Yab = C2 5 g
− κ + 20 Yab. (6.102b)
From section 6.3, we know that D[2]Yab = 0 since Yab is divergence-free with vanishing
7-part. Using this, the above equations can be combined into
√ ( ) √7 2 5( 27)
D 2 22Xab − 3D2Yab = 2 5 Cg − κ + 20 Xab − 3 Cg − κ + 20 Yab. (6.103)
Recall from section 6.4, that (6.38)
2 = − √1D2 Xab CgXab D2Xab, (6.104)5
95
6. Eigenvalue spectra of the squashed seven-sphere
for transverse traceless symmetric rank-2 tensors Xab. A short calculation shows
that this holds for transverse 2-forms as well. Thus, acting with D2 on (6.103) and
antisymmetrising the free indices,
√
2 5( 9 )
CgY
2
ab = − 3 Cg − κ + 20 D[2]Xab. (6.105)
Eliminating D[2]Xab with (6.10(2b), we find tha20 9 )(
t )
C 2gYab = 9 Cg − κ + 20 Cg −
2 + 27κ 20 Yab (6.106)
whence either Yab = 0 or √
κ2 = Cg +
9 ± √3 + 910 Cg 20 . (6.107)2 5
In the case Yab = 0, (6.102) gives us the situation in section 6.4 but with the
extra information D[2]Xab = 0. As in section 6.5, this implies that we only get two
eigenvalues, given in (6.108d) below. The possi√ble eigenvalues are, therefore,
2 = + 14 ± √1 49κ Cg 5 √C +2 5 g 20 , (6.108a)
31
κ2 = Cg + ± √
5 49
10 C2 5√ g + 20 , (6.108b)
κ2 = + 9Cg ± √
3 9
10 √ Cg + 20 , (6.108c)2 5
2 = + 2 ± √1 + 1κ Cg 5 Cg 20 . (6.108d)2 5
Here, the top two lines apply to modes with nonzero 7-part, the third one to modes
with vanishing 7-part but nonzero 14-part and the last line to modes with only a
27-part.
6.8 Spectrum summary
In table 6.2, we give a summary of the eigenvalues found in sections 6.1 to 6.7. Recall
from section 3.2.4 that the Laplacian ∆ is the Hodge–de Rham operator ∆p when
acting on p-forms, the Lichnerowicz operator ∆L when acting on traceless symmetric
rank-2 tensors and related to (iD/ )2 through
(iD/ 21/2) = ∆ +
189
80 , (iD
2 27/ 3/2) = ∆− 80 , (6.109)
in the dimensionless system in which m2 = 9/20, when acting on spinors and vector-
spinors, respectively. For 3-forms, the operator directly related to the AdS4 masses is
Q, related to ∆3 by Q2 = ∆3 on transverse 3-forms. The possible eigenvalues of Q are
thus ±κ3, where κ 23 are the eigenvalues of ∆3. Note that the eigenvalues of ∆3 are
96
6.8. Spectrum summary
perfect squares, which we have used to simplify the expressions for the eigenvalues of
Q. This applies to (iD/ )2 as well. We have switched back to the dimensionful system
by inserting appropriate powers of 20m2/9. Due to the limitations of the method we
have used, all eigenvalues of the operators should be present in the table but some
of the listed possibilities might not be eigenvalues.107
Table 6.2: Summary of eigenvalues of differential operators on the squashed
S7 in conventions in which Rab = 6m2δab. Note that we have not proven that
the listed eigenvalues exist but rather that all eigenvalues are in the list. In
chapter 7, we find that some roots do not fit into supermultiplets and, hence,
are false. These are indicated by parentheses on ± or ∓ although there are
possibly some exceptions, see (7.7). Note that the sign in front of the square
root in the eigenvalues of the linear operators determine whether the eigenvalue
belongs to the positive or negative part of the spectrum.
Operator Possible eigenvalues
∆ m
2
0 9 20C( g )
∆ m
2 √
1 9 20Cg + 14± 2 20Cg + 49( ) ( )
∆ m
2 2 2 √
2 9 20
m
Cg 9 20
m
Cg + 72 9 20Cg + 44± 4 20Cg + 49
m2
( ) 2( √ )
∆L 9 20Cg + 36
m
9 20Cg + 32± 4 20C + 1
| |( √ ) ( √ )
g
m
Q (∓) 3 1± 20
|m|
Cg + 1 (±) 3 1± 20Cg + 49
± |m|
( √ )
( ) 3 3± 20Cg + 81
| |(m 1 √ ) ( √ )iD/ 1/2 (∓) 3 2 ± 20 + 49 ± |m| 3Cg ( ) 3 2 ± 20Cg + 81
| |(1 √ ) | |(3 √ )iD m m/ 3/2 (±) 3 (2
± 20Cg + 1 (∓) 3 2 ± 20Cg + 9
± |m| 1
√ ) ( √ )
( ) 3 2 ± 20Cg + 49
|m| 5
(±) 3 2 ± 20Cg + 49
107When comparing the spinor eigenvalues to, for instance, [19], note that what is referred to as
the Dirac operator in [19] is −iD/ in our conventions.
97
6. Eigenvalue spectra of the squashed seven-sphere
98
7. Mass spectrum and
supermultiplets
Having found the eigenvalue spectra of the various operators on the squashed seven-
sphere, see table 6.2, we return to eleven-dimensional supergravity. Since the squashed
seven-sphere is an Einstein space with positive curvature, the background with AdS4
as the spacetime and the squashed seven-sphere as the internal manifold is a solution
to the field equations in the Freund–Rubin ansatz. There are actually two solutions
related by reversing the direction of the flux, that is, by skew-whiffing, m 7→ −m.
As remarked in section 3.1, at most one of these can have unbroken supersymmetry.
Here, we begin by demonstrating this explicitly for the two squashed seven-sphere
vacua and then turn to the mass spectrum and supermultiplets of the N = 1 vacuum.
Recall from (6.2) that there is a sign choice when relating the dimensionless and
dimensionful unit systems. This comes from the fact that one can let m→7 −m in
the Freund–Rubin ansatz to obtain another solution to the field equations. However,
the geometry and eigenvalue spectra of the squashed seven-sphere is independent of
the flux-direction in the compactification. We, therefore, use the relation
| | = √3m (7.1)
2 5
√
and insert appropriate powers of 2 5|m|/3 to translate from the dimensionless to the
dimensionful system. The solution with m > 0 will be referred to as the left-squashed
vacuum and the one with m < 0 as the right-squashed vacuum.
7.1 Unbroken supersymmetry
Recall that the number of unbroken supersymmetries is given by the number of
linearly independent Killing spinors η, satisfying (3.13b)
D̃ maη := Daη + i 2 Γaη = 0. (7.2)
To investigate whether there are any unbroken supersymmetries we start by consid-
ering the holonomy of D̃m and the integrability condition (3.17)
W cdabη := Wab Γcdη = 0. (7.3)
99
7. Mass spectrum and supermultiplets
By (3.16), the Weyl tensor is given by W cdab = R cdab − 2m2δcdab. With the Riemann
tensor of the squashed seven-sphere from (5.22), we find
= 8
( )
W m20i 9 2(Γ0i + ijkΓ
̂k̂ ,) (7.4a)
W0ı̂ = −
8 2
9m 2Γ0ı̂ + ijkΓ
jk̂ , (7.4b)
= 16
( )
W m2ij 9 (Γij + Γı̂̂ ,16 )
(7.4c)
W 2 0kı̂̂ = 9 m (2Γı̂̂ + Γij + ijkΓ , ) (7.4d)
Wi̂ = −
8 2
9m 2Γi̂ + Γjı̂ +  Γ
0k̂
ijk − δijΓ k̂k , (7.4e)
where we have used the index split a = (̂ı, 0, i) and that λ2 = 1/5 for the Einstein-
squashed sphere. Here, we see that Wab are linear combinations of the generators
of G from (C.26) to (C.29).1082 Thus, the holonomy of D̃m is G2 [19] and there is
exactly one linearly independent solution to (7.3), namely the G2-invariant η.
To check whether η is a Killing spinor, we have to consider (7.2) and not only the
integrability condition. Recall from section 6.6 that Ďaη = 0 since h ⊂ g2, where
Ď = D |m| bca a − 3 aabcΣ , (7.5)
by (6.5). Using that the Γ-matrices can be represented by octonion multiplication,
Γa = −iLoa , as described in appendix C.1, and that the G2 invariant then is identified
with the real unit o0̂ = 1 ∈ O, we find
0 = Ďaη = Dao0̂ +
|m| abc
12 a ob(oco0̂) = D
a + |m|o a a0̂ 2 o = D η + i
|m|Γa2 η. (7.6)
The right-hand side is D̃aη for m > 0 and differs from D̃aη by imΓaη 6= 0 for m < 0.
Hence, D̃aη = 0 only for m > 0. Note that S7 is simply connected, whence there are
no global obstructions from the nonrestricted holonomy group. Thus, we conclude
that the left-squashed vacuum has one unbroken supersymmetry, N = 1, while the
right-squashed vacuum has none, N = 0, [19].
7.2 The left-squashed N = 1 vacuum
We proceed by analysing the left-squashed vacuum with one unbroken supersymmetry.
The possible particle masses, presented in table 7.1, are calculated from the possible
eigenvalues in table 6.2 and the mass operators in table 3.1. Then, when calculating
the dimensionless energy E0, we restrict to G-representations with suffi√ciently large
quadratic Casimir Cg to be able to simplify expressions like | − 10 + 20Cg + 49|,
in which the absolute value can be dropped if the root is larger than 10. The
strongest restriction needed for such simplifications is Cg ≥ 243/20. Additionally,
108Specifically, W0i ∝ T 0k 0k̂0i, W0ı̂ ∝ T0ı̂, Wij ∝ Tij , Wı̂̂ ∝ Tij + ijkT /2 and Wi̂ ∝ Ti̂ + ijkT .
100
7.2. The left-squashed N = 1 vacuum
there is a choice of sign in E0 for s = 0, 1/2 for small values of Cg. To be able to,
unambiguously, choose the plus sign we further restrict our attention to Cg > 99/4.
We refer to this as the asymptotic part of the spectrum. The resulting possibilities
for E0 are presented in table 7.2.
Table 7.1: Possible particle masses (M2 for bosons, M for fermions) in the
left-squashed vacuum based on table 3.1 and the eigenvalues in table 6.2. To get
a mass (squared) one value should be picked from each pair of braces. When
there are multiple braces in an expression, the same position must be chosen
in all of them. Each column in a pair of braces corresponds to one eigenvalue-
expression from table 6.2, while the rows correspond to different signs in the
expressions. The ordering is the same as in table 6.2. When applicable, the first
(second) subscript corresponds to the top (bottom) sign.
spt Possible masses
2
2+ m9 20
{
Cg } √ { }
3 √m 11 9 √∓√20 + 49 81C 

21,2 3  10 12
g 49 81 
m2  { } { }104 16 √ 1−1 9 20Cg + 140 − 20 20Cg + 49
m2  { } { }140 20 √ 1−2 9 20Cg + 104 + 16 20Cg + 49
2  { } { } 1+ m 20 + 0 72 44 + 0 0 +4 √9 Cg 44 −4 20C + 49  g√
1 m  { } √ { }− 13 15 √±√20 + 49 81 2 3 14 12 Cg 49 81 1,4 { } 
1 m  5 3 5 7 ∓√
√√√ { }
2 3 4 6 4 2 20Cg +
1 9 49 49 
2,3 ( 1) 9 49 49

0+ m
2 √
1 9 (20Cg + 396− 36√20Cg + 812 )
0+ m3 9 20Cg + 3{96 + 3}6 22  {
0Cg + 8}1 
0+ m 20 − 0 4 + 0 +4
√
C 2 9  g { 4 }−4
20Cg + 1
{ }√√ { }2
0− m  56 140 216 16 20 24 √√ 1 49 81 1,2 9 20Cg + 92 104 108 ∓ 20 16 12 20Cg + 1 49 81 
101
7. Mass spectrum and supermultiplets
Table 7.2: Possible asymptotic (Cg > 99/4) values of E0 in the left-squashed
vacuum based on table 3.2 with notation and masses from table 7.1.
spt Po(ssible√values of E)0
2+ 16 9 + 20Cg + 81
3 1 { } √

4 6 +√√
√ { }
2 6 5 3 20
49 81 
Cg + 
1  49 81
{ } √√√
√ { }
3 1 14 12 + 20 + 49 81 2 Cg2 6 13 15 49 81 
1− 1 
{ } 
+1 √
1 6 −1 + 20C + 49

 g{ } 
1− 1  19 √2 6 17 + 20Cg + 49
{ } √√√ { }1+ 1  9 9 11 +√20 + 9 81 49 6 7 Cg 49  { }
1 1  4 6 √√
√ { }
− +√20 + 49 81 2 6 5 3 Cg 49 81 1 { } √√√√ { }

1 1 22 24 + 20 + 49 81 24 6  23 21
Cg 49 81 
{ } √√√ { }1 1  4 6 4 22 6 5 3 5 7 +√20 1 9 49 49Cg + 1 9 49 49 2 { }
1 1  √
√ { }
14 12 14 16 √√
2 6 13 15 13 11 + 20Cg +
1 9 49 49 
3 ( √ ) 1 9 49 49

0+ 11,3 6 (9∓ 18) + 20C{ } g
+ 81 
0+ 1  9 11 +√√√
√ { }
2 6 7 20
9 1 
Cg + 1 

{ } √√ { }
0− 1 +1 −1 −3 +√√1 6 −1 +1 +3 20Cg + 1 49 81  √ 1 49 81

0− 1 { } √ { }

17 19 21 √√ 1 49 81 
2 6 19 17 15 + 20Cg + 1 49 81 
102
7.2. The left-squashed N = 1 vacuum
Now that we have the possible values of E0 (for large Cg), we investigate how
they fit into the N = 1 supermultiplets from table 3.3. Since the supersymme-
try generators are G-singlets, all fields of a supermultiplet must carry the same
irreducible G-representation [10]. That the irreducible pieces of the induced G-
representations fit exactly into supermultiplets has been verified in [10]. The irre-
ducible G-representations are labelled by Dynkin labels (p, q; r). In [10], they also
found that, for sufficiently large and fixed p, q, there is one s = 2, six s = 3/2, six
s = 1− and eight s = 1+ massive higher spin supermultiplets and 14 Wess–Zumino
supermultiplets.
For sufficiently large Dynkin labels, the multiplicities of the irreducible G-representa-
tions appearing in the harmonics are independent of the exact values of p and q [10].
The G-representations can be arranged as in table 7.3. If an eigenvalue expression
from table 6.2 is assigned to a box in one of these diagrams, there is a family of
eigenmodes of the corresponding operator with eigenvalues given by this expression.
Each such family has a constant difference between r and p and contains exactly one
irreducible G-representation (p, q; r) of eigenmodes for each sufficiently large pair
(p, q). If one eigenvalue expression is assigned to n different boxes, we say that it
has multiplicity n. Note that this does not imply that there are multiple irreducible
G-representations with identical eigenvalue since two boxes with the same eigenvalue
expression can have different r − p.
Table 7.3: Multiplicities, found in [10], of irreducible G-representations (p, q; r)
in the G-representation induced by the various Spin(7)-representations (in bold).
For large values of the Dynkin labels, the multiplicities are independent of p
and q and are given, for the indicated value of r and Spin(7)-representation,
by the number of boxes in a row. To give the complete content of the induced
representations, not only for large p and q, each box can be replaced by a
diagram that specifies which values of p and q that do not appear. Such
diagrams can be found in [10]. By the below supermultiplet analysis, eigenvalue
expressions can be assigned to the boxes marked by a cross without significant
ambiguity.
r = p+ 4 7 8 21 27
r = p+ 2 1 × × × × × ×
r = p × × × × × × × × × ×
r = p− 2 × × × × × ×
r = p− 4
r = p+ 4 35 48
r = p+ 2 × × × × × ×
r = p × × × × × × × ×
r = p− 2 × × × × × ×
r = p− 4
103
7. Mass spectrum and supermultiplets
The massive higher spin supermultiplet families are presented in table 7.4 and
the Wess–Zumino supermultiplet families in table 7.5. Each family of fields in a
family of supermultiplets, for instance the 2+ fields in the s = 2 massive higher
spin supermultiplet family, has a specific eigenvalue expression. Analogous to the
multiplicities of the eigenvalue expressions, there is a multiplicity associated with
each supermultiplet family specifying the number of supermultiplets in that family
for fixed (p, q) but arbitrary r. By the above, the multiplicity is independent of the
exact values of p, q as long as they are sufficiently large.
To deduce some of these multiplicities, we use that there is only one eigenvalue
expression for ∆0 and it has multiplicity one. Furthermore, the multiplicities of
the eigenvalue expressions of iD/ 1/2 are known from [105], where the spinor eigen-
modes were constructed explicitly from the scalar eigenmodes. Still, we cannot
deduce the multiplicity of the 1+ supermultiplet family and two of the Wess–Zumino
supermultiplet families.
The multiplicities found in this way are consistent with table 7.3. Assuming that
we have not missed anything in our calculations (see below), this implies that the
question mark in table 7.4 should be 8 and that the two question marks in table 7.5
should add to 12. Except for the possibility that one of the latter two is 0, it follows
that the only false roots in the asymptotic part of table 6.2 are those marked with
parentheses.
Table 7.4: Spins, parities and towers of fields in asymptotic massive higher spin
supermultiplet families that can be formed from the values of E0 in table 7.2.
A bracket [r, c] indicates which row, r, and column, c, are used in the braces in
table 7.2. The multiplicity of the supermultiplet family is denoted by n. The
case n = 3 applies only when p, q are sufficiently large, see [10], [105].
n Massive higher spin supermultiplets
1 2+ 3 32 [1, 2] 2 [1, 2] 1
+[1, 2]
2 1
3 3 12 [1, 1] 1
+[2, 3] 1−1 [1, 1] 2 [1, 3]1 2
3 32 [1, 1] 1
− +
2 [1, 2] 1 [1, 3]
1
2 [1, 3]2 3
3 1−1 [2, 1]
1 1
2 [1, 4] 2 [1, 1] 0
−
1 [1, 2]
2 1
3 1−2 [1, 1]
1 1 −
2 [1, 1] [1, 4] 02 [1, 2]4 23
? 1+[1 1] 1 [1 2] 1, 2 , 2 [1, 2] 0
+
2 [1, 1]
3 2
104
7.2. The left-squashed N = 1 vacuum
Table 7.5: Spins, parities and towers of fields in asymptotic Wess–Zumino
supermultiplet families, with notation as in table 7.4, that can be formed from
the values of E0 in table 7.2.
n Wess–Zumino supermultiplets
1 12 [1, 2] 0
−
1 [1, 3] 0+1
1
1 12 [1, 2] 0
+ −
3 02 [1, 3]
4
? 1 + −2 [1, 1] 02 [2, 2] 01 [1, 1]2
? 12 [1, 1] 0
−
2 [1, 1] 0+2 [1, 2]
3
The first thing to notice in tables 7.4 and 7.5 is which values of E0 are being used
and which eigenvalues these correspond to. We see that, in the towers corresponding
to linear operators on the squashed S7, that is, the spinorial fields and pseudoscalars,
only the top rows in the braces in table 7.2 fit into supermultiplets. The bottom
rows in the braces correspond to the signs in parentheses in table 6.2, which thus
are false roots. We also note that all possibilities in the towers corresponding to
quadratic operators on the squashed S7 fit into a supermultiplet. This implies that
every eigenvalue in the asymptotic part of the spectrum of the Laplacian ∆ that we
found in chapter 6 fits into a supermultiplet. Furthermore, every value that fits into
a supermultiplet fits only into one supermultiplet.
There is a reason to believe that we have not found all operator eigenvalues. This
would imply that there are errors in the calculations in chapter 6 such that we have
missed some eigenvalues in multiple calculations. Since all eigenvalues of ∆ that we
did find fit into supermultiplets, the errors would likely have to be systematic.
The reason to believe that we might have missed something is the following. In the
case of a spinor on the squashed S7, that is, the representation 8 of Spin(7), [105]
found that each of the four columns in the corresponding diagram in table 7.3 can
be assigned a single eigenvalue expression, see [19].109,110 If one assumes that this
continues to hold for all Spin(7)-representations and that no two columns belonging to
the same Spin(7)-representations have the same eigenvalue expressions, the question
mark in table 7.4 can only be 3 or 5, not 8. Similar remarks apply to the question
marks in table 7.5. Thus, if our results are complete, some eigenvalue expressions
must be assigned to multiple columns and there are unexplained degeneracies. In this
case, the multiplicities can, however, be consistently and essentially unambiguously
assigned to the remaining boxes in the table without violating the assumption that
each column should have a single eigenvalue expression.
109The eigenvalue expressions are the ones of iD/1/2 without parentheses in table 6.2.
110The heights of these columns were used to deduce some multiplicities in tables 7.4 and 7.5.
105
7. Mass spectrum and supermultiplets
Lastly, we turn to the low end of the spectrum. When Cg ≤ 99/4 there are possibilities
to choose the negative sign in the expressions for E0 in table 3.2 for s = 0 and
s = 1/2. Also, as explained above, some simplifications of the expressions in table 7.2
are only valid in the asymptotic part of the spectrum. However, since p, q and r
take nonnegative integer values, there are only a finite number of special cases. To
proceed with the analysis we have to assume that we have found all eigenvalues. In
the low part of the spectrum, it seems like not only the supermultiplets that are part
of one of the families in tables 7.4 and 7.5 are possible. In some of these, there are
even pseudoscalars of the type 0−1,2[2, c], that is, with a sign within parenthesis in
table 6.2.
The majority of the special cases occur for p = q = r = 0. Some of these are,
however, easily excluded. Firstly, the G-singlet, of course, gives a H-singlet when
restricted to H. Thus, p = q = r = 0 only occurs in the G-representation induced
by the H-singlet, by Frobenius reciprocity. The H-singlet only occurs in the G2-
representations 1 and 27 [10], whence there are two G-singlet 3-form modes. Both
of these are transverse [10]. One of them is Yabc = aabc. From the Killing spinor
equation (7.2), it immediately follows that ∇aabcd = mcabcd. Thus, the 3-form aabc
is transverse and has Q-eigenvalue 4m. This is a [r, c] = [1, 3] eigenvalue of Q and
hence not an exception to the parentheses in the eigenvalue table.
By orthogonality, the other G-singlet 3-form mode has only a 27-part. The analysis
in section 6.5 implies that the eigenvalue has [r, c] = [r, 1]. Since S7 has Betti number
b3 = 0, there are no 0-eigenvalues of Q, whence the possibilities are ±2m/3. Thus,
the AdS4 field is of type 0−[1, 1] or 0−1 2 [2, 1]. However, no supermultiplet can be
formed using 0−2 [2, 1], even for small p and q. Thus, the only remaining possibility
for the second G-singlet 3-form is QY = −2m/3Y . The corresponding field belongs
to 0−1 [1, 1] which is not an exception from the asymptotic case.
The remaining special cases that fit into supermultiplets are
0−1 [2, 2] : (p, q; r) = (1, 0; 1), (7.7a)
0−1 [2, 1] : (p, q; r) ∈ { (0, 0; 4), (0, 3; 0), (2, 1; 2), (0, 1; 0) } , (7.7b)
0−2 [2, 3] : (p, q; r) ∈ { (0, 0; 4), (0, 3; 0), (2, 1; 2) } . (7.7c)
We have neither confirmed nor excluded the existence of modes with the corresponding
eigenvalues.
In the low part of the spectrum, there is also the possibility of massless supermultiplets
and Dirac singletons. By analysing the possibilities and using the above remark
regarding the occurrence of G-singlets in the induced representations, we find that
there is one massless 2+ supermultiplet for (p, q; r) = (0, 0; 0), two massless 1−
supermultiplets with (p, q; r) ∈ { (2, 0; 0), (0, 0; 2) } and no Dirac singletons. This is
precisely what is expected since there is always exactly one massless spin 2 particle, the
graviton, and the massless 1− fields correspond to Killing vector fields that generate
isometries ofM7 [19]. These massless supermultiplets are multiplet-shortened special
cases of the 2+ and 1−1 supermultiplets in table 7.4.
106
8. Conclusions
We have studied M-theory, or rather its low-energy limit, eleven-dimensional super-
gravity, compactified on the squashed seven-sphere, motivated by the AdS instability
swampland conjecture. There are two vacua, the left-squashed N = 1 vacuum and
the right-squashed N = 0 vacuum, related by skew-whiffing. By the aforementioned
conjecture, the N = 0 vacuum should be unstable. One possible instability is related
to a tadpole and, in the dual conformal field theory, a global singlet marginal operator
(GSMO). To investigate whether such an instability can occur, considerable parts
of the mass spectrum of the theory are needed, which we thus aimed to derive. We
realised that all mass operators in the Freund–Rubin ansatz are related to a uni-
versal Laplacian, which, in particular, enabled significant simplifications by relating
Weyl tensor terms to group invariants. This is the main advancement of the thesis
compared to [21], [106].
We have found possible eigenvalue spectra of all operators of interest on the squashed
seven-sphere. By requiring consistency with supersymmetry, some false roots of
the first-order operators were excluded and the asymptotic part of the eigenvalue
spectra, including the multiplicities of the eigenvalue expressions, could be almost
completely determined. From the perspective of the spectrum of irreducible isometry
representations, derived in [10], our results indicate that there are degeneracies that
we have not been able to explain. This could be taken as evidence that our results
are incomplete. That would, however, require a series of systematic errors in the
calculations in such a way that all eigenvalues of the Laplacian ∆ that we did find
still fit into supermultiplets. We hope to address this in [22].
As explained in section 1.2.3, GSMO-related instabilities may occur when there is
a gauge singlet field, possibly composite, with E0 = 3. Thus, as in [18], one has to
look for fields that can be combined into a gauge singlet scalar composite such that
the energies of the elementary fields add up to 3. From the unitarity bounds on E0
in table 3.2, we see that s ≥ 1 is immediately excluded by requiring that the field
is a spacetime scalar. Two spin-1/2 fields can be combined with up to two scalar
fields to form a composite scalar that, a priori, could have E0 = 3. Also, there is the
possibility of using only spin-0 fields. With the bound E0 ≥ 1/2 there could be as
many as six scalar fields in the composite.
As discussed in section 3.2, the unitarity bounds for spins 0 and 1/2 correspond to
singletons and can only arise in 0+1 and 1/21. This implies, by the skew-whiffing
theorem [19], that there are no singletons in the left-squashed vacuum, consistent
with what we found in section 7.2, and only a single spin-1/2 singleton in the right-
squashed vacuum. Thus, there can actually be at most one scalar in a field dual to a
107
8. Conclusions
GSMO containing two spin-1/2 fields and at most five when there are only scalars.
For large v√alues of the quadratic Casimir Cg, the possible energies E0 grow approxi-
mately as Cg. The number of scalars with E0 < 3 and spinors with E0 < 3/2 is
therefore finite. Hence, there are only finitely many combinations that could possibly
produce E0 = 3. Without paying attention to which G-representations appear in
the various towers, there seems to be plenty of candidates. However, many of these
might be easily excluded by a more careful analysis. A problem that remains is that
there are masses that are small enough that both signs in the expressions for E0 are
viable. If any of these have a multiplicity of at least two, the corresponding E0 values
add to 3 if different signs are used. To settle this, more work is needed. Note also
that the presence of a GSMO does not imply that there is an instability. Similarly, if
it turns out that there are no GSMO-related instabilities, other types of instabilities
would have to be considered to strengthen or weaken the AdS instability swampland
conjecture.
The swampland program aims to distinguish low-energy effective theories that are
consistent when coupled to gravity from those that are not. As we have seen,
swampland criteria can have significant implications for low-energy physics and
cosmology, including the role of de Sitter space in string theory. Thus, the swampland
program can bring string theory closer to experiment. As long as there is no complete,
nonperturbative description of M-theory and the stringy swampland conjectures
remain unproven, the question of whether such experiments test string/M-theory
or only the conjectures remains open. Still, it is possible to investigate which
conjectures are physically implemented in the observable part of the universe. This
could hopefully stimulate further theoretical developments.
108
A. Conventions and representations
In this appendix, we present some conventions and notation used throughout the
thesis. We will always work in natural units in which c = ~ = 1. For the metric,
we use the mostly-plus signature. Although most equations are covariant and
valid in any basis, we use the basis (if nothing else is specified) in which ηαβ =
diag(−1,+1, . . . ,+1)αβ when a basis is needed.
When symmetrising and antisymmetrising tensors, we employ the weight-one defini-
tions and use parenthesis and bracket notation, respectively. For instance,
u(µvν) = 1(uµvν + uνvµ2 ), (A.1a)
u[µvν] = 1(uµvν − uν2 v
µ). (A.1b)
More generally,
( ) 1 ∑T µ1...µn = T µσ−1(1)...µσ−1(n)! , (A.2a)n σ∑∈Sn[µ ...µ ] = 1T 1 n sign σ T µσ−1(1)...µσ−1(n)! . (A.2b)n σ∈Sn
In this notation, we define the generalised Kronecker delta as
δµ1 ...µp
µ
ν1 ...ν
:= δµ1[ν . . . δ
p
ν ]. (A.3)p 1 p
In the superspace setting, we use (. . .] and [. . .) to denote graded symmetrisation and
graded antisymmetrisation of indices, respectively. The grading means that there is
an additional sign when fermionic indices pass through each other.
A.1 Representations and index notation
We use index notation and employ Einstein’s summation convention throughout the
text. When elements of a vector space111 are denoted with lower indices (for instance
vα), dual vectors (covectors) are denoted with upper indices (for instance uα). If the
vector space carries a (left-)representation of some group G, the dual vector space
carries the dual representation and a group element g acts like
g · vα = g βα v , g · uαβ = uβ(g−1) αβ , (A.4)
111Here we use “vector” in the general sense, not in the sense of an SO-vector, and, in the following,
α, β, . . . are not spinor indices but indices for an arbitrary vector space.
109
A. Conventions and representations
where we, by abuse of notation, use the same symbol for the group element and
its representation. Thus, the dual representation is given by gα = (g−1) α.112β β
Note that right-multiplication by the inverse, g−1, is a left-representation since
g−11 g
−1 −1
2 = (g2g1) . Also, we use the word representation to refer not only to the
actual representation but the representation space (module) as well.
Suppose that the vector space is complex. The complex conjugate vector, which is
an element of the complex conjugate vector space and whose coordinates are the
complex conjugates of the coordinates of the original vector, is then denoted by
(v )∗α = v̄ᾱ. The complex conjugate vector space and its dual carry representations
of G and a group element g acts like
g · v̄ᾱ = ḡ β̄ᾱ v̄ g · ūᾱ = ūβ̄(ḡ−1β̄ ) ᾱβ̄ . (A.5)
These transformation rules are summarised in table A.1.
Table A.1: Transformation of a vector v , a dual vector vαα , a complex
conjugated vector v̄ᾱ and a complex conjugated dual vector v̄ᾱ under a group
element g and a Lie algebra element T . Here, we use the convention g = exp(T )
without an i in the exponent. To switch to the convention with an i in the
exponent, let T 7→ iT (T̄ 7→ −iT̄ ).
Quantity Finite Infinitesimal
vα g · vα = g βα vβ δTvα = T βα vβ
vα g · vα = vβ(g−1) α α ββ δTv = v (−T αβ )
v̄ᾱ g · v̄ᾱ = ḡ β̄ᾱ v̄β̄ δT v̄ᾱ = T̄
β̄
ᾱ v̄β̄
v̄ᾱ g · v̄ᾱ = v̄β̄(ḡ−1) ᾱ δ ᾱ β̄ ᾱ
β̄ T
v̄ = v̄ (−T̄ )
β̄
A.2 The Lorentz group and special orthogonal
groups
The Lorentz group113 in d + 1 spacetime dimensions, denoted SO(d, 1), is defined
through the (d+1)-dimensional vector representation consisting of matrices with unit
determinant that leave ηab invariant. Therefore, we use ηab and its inverse to raise
and lower vector indices as usual. We define the generators Lab of the corresponding
Lie algebra representation by
(Lab)cd = δabcd . (A.6)
Note that we use the geometrical convention that a group element is Λ = expL,
without an i in the exponent. With this normalisation of the generators, the Lie
112If one raises and lowers indices using an invertible invariant Mαβ , one must define gαδ :=
Mαβg γ αβ Mγδ for g β to be the dual representation.
113We use “Lorentz group”, a bit carelessly, to refer to the identity component SO+(d, 1) and its
double cover Spin(d, 1) as well.
110
A.3. Quadratic Casimirs
bracket reads
[ [a b]Lab, Lcd] = −2δ[cL d]. (A.7)
The Lorentz group is, of course, the special case of Lorentzian signature of the more
general special orthogonal group SO(p, q) with arbitrary signature. We use the same
normalisation of the generators in the general case, that is, (A.6) and (A.7) are still
valid. Note, however, that the generators of the vector representation (Lab)cd differ
depending on the signature since indices are raised and lowered using the metric of
the corresponding signature.
A.3 Quadratic Casimirs
The Casimir operators of a finite-dimensional semisimple Lie algebra g are special
elements of the centre of the universal enveloping algebra, Z(U(g)). They are
elements of the form
C(n) a1...ang = t Ta1 . . . Tan (A.8)
that commute with all elements in g [88]. Due to the relation [Ta, Tb] = f cab Tc, one
can, without loss of generality, take ta1...an to be completely symmetric. It is easy
to see that C(n)g commutes with all of g if and only if ta1...an is an invariant tensor.
One can show that, for an algebra of rank r, there are precisely r algebraically
independent Casimir operators, which together with 1 generate the centre of U(g)
via multiplication and linear combinations [88]. Since C(n)g commutes with all of g,
it acts on an irreducible representation (n) (n)ρ by a constant, ρ(Cg ) = Cg (ρ) · 1. The
eigenvalues (n)Cg of the algebraically independent Casimirs can be used to uniquely
specify an irreducible representation [88], but for practical purposes we use Dynkin
labels for this.
In this thesis, we will only be concerned with quadratic Casimirs, that is, the above
t is a symmetric rank-2 tensor. A canonical choice is thus tab = κab, where κab is
the Cartan–Killing metric of g. For a semisimple g consisting of multiple simple
Lie algebras, there is, however, one quadratic Casimir corresponding to each simple
Lie algebra. Still, we call the Casimir corresponding to the canonical tab = κab the
quadratic Casimir of g. The most well-known example of a quadratic Casimir is
perhaps J2 of so(3), the square of the angular momentum.
We are interested in the quadratic Casimirs of four Lie algebras: g = sp(2)⊕ sp(1)C ,
h = sp(1)A ⊕ sp(1)B+C , so(7) and g2. See section 5.2 for an explanation of the
subscripts. The normalisations of the Casimirs of the relevant simple Lie algebras
are given in table A.2 and agree with [103]. As mentioned in section 5.2, we define
the G-Casimir by
Cg = 6κABTATB, (A.9)
which implies adg(Cg) = 6 · 1. As described above, g has two independent quadratic
Casimirs corresponding to sp(2) and sp(1). Thus, Cg is a linear combination of these.
111
A. Conventions and representations
Since adg ' adsp(2)⊕ adsp(1) we find
Cg(p, q; r) = 2Csp(2)(p, q) + 3Csp(1)(r), (A.10)
where (p, q; r) are the Dynkin labels of g, see [10].
Since we are considering h as a subalgebra of g, it will be convenient to use the
restriction κRS of the Cartan–Killing metric κAB of g and normalise the Casimir as
C RSh = 6κ TRTS. (A.11)
There is another independent quadratic Casimir of h, since h consists of two simple
Lie algebras. It is, however, only this one we will be using.114 With the index-split
R = (r, ṙ) for h = sp(1)A ⊕ sp(1)B+C we see from (5.28) that κrs = −3δrs and
κṙṡ = −5δ 115ṙṡ, whence Ch is proportional to Csp(1) /3 + Csp(1) + /5. To determineA A B
the constant of proportionality, we compute ρ7|h(Ch) = 12/5 · 1, where ρ7|h is
the restriction of the vector representation ρ7 of so(7) to h, using (4.1) and (5.37).
Comparing this with the above formula, using that 7→ (1, 1)⊕(0, 2) when restricting
to h,116 we find
6
Ch(p, q) = 2Csp(1) (p) + 5Csp(1) + (q). (A.12)A B C
Turning to Cso(7), we define117
C = −δb1 b2 Σa1a2so(7) a1a2 Σb1b2 . (A.13)
Computing ρ7(Cso(7)), where ρ7 is the 7 = (1, 0, 0)-representation of so(7), we find
that this agrees with the normalisation in table A.2.
Lastly, the generators of g2 are
T (g2) = (P ) b1b2a1a2 14 a1a2 Σb1b2 , (A.14)
where P14 is the g2-projector onto 14 in 7∧2 ' 7⊕ 14, since 14 = adg2 . Hence, Cg2
is proportional to (P b b14) 1 2Σa1a2a1a2 Σb1b2 . Using the 7 = (1, 0)-representation we find
that the normalisation that agrees with table A.2 is
C = −(P ) b1b2 a1a2g2 14 a1a2 Σ Σb1b2 . (A.15)
114For arbitrary squashing parameter, the relevant Casimir would be obtained by restricting gAB
from section 5.2, rather than κAB , to h.
115Recall that κAB is −3 on sp(1)A,B and −2 on sp(1)C which implies that it is −5 on sp(1)B+C .
116This decomposition is immediate from the structure of the generators presented in section 5.2.
It can also be found in [10].
117We use the analogous normalisation for so(5) ' sp(2) but not so(3) ' sp(1). For the latter, we
use the conventional S2 = s(s+ 1).
112
A.3. Quadratic Casimirs
Table A.2: Casimir eigenvalues of the four simple Lie algebras of interest in
terms of Dynkin labels. In the rightmost column, the values on the adjoint
representations are given for convenience.
Casimir C(ad)
1 3
Cso(7)(p, q, r) = 2p(p+ 2q + r + 5) + q(q + r + 4) + 8r(r + 6) 5
1
Cg2(p, q) = 3p(p+ 3q + 5) + q(q + 3) 4
1
Csp(2)(p, q) = 4p(p+ 2q + 4) +
1
2q(q + 3) 3
1
Csp(1)(p) = 4p(p+ 2) 2
113
A. Conventions and representations
114
B. Spinors
The vector representation of so(r, s) = Lie(SO(r, s)) is not the most elementary
so(r, s)-representation in the sense that there is another representation, the spinor
representation, which cannot be constructed from it but can be used to construct
it.118 At the group level, the spinor representations are only projective representations
of SO(r, s) but ordinary representations of Spin(r, s), the double cover of SO(r, s).
The Dirac spinor representation can be defined in terms of an irreducible complex
representation of the Clifford algebra C r̀,s [70], that is, Γ-matrices Γa satisfying
{Γa,Γb} = 2ηab1. (B.1)
Since this implies that
[Γab,Γcd] = −8 [a b]δ[cΓ d], (B.2)
where Γa1...an := Γ[a1 . . .Γan], we get a representation S of so(r, s) by
1
S(Lab) = ab4Γ . (B.3)
This is the Dirac spinor representation. Many properties of spinors depend on the
dimension, d = r + s, modulo eight119 and signature. Here, we focus on Spin(3, 1)
and Spin(10, 1) but begin with some general remarks. Spin(7), which is also of
importance for this thesis, is treated in appendix C. Furthermore, all considerations
in this appendix are local. To be able to define spinor fields globally on a manifold,
a spin structure is needed [70].
Arbitrary products of Γ-matrices can be computed by combinatorics. This is most
easily seen in a basis in which η = diag(−1, . . . ,−1, 1, . . . , 1) but is valid in any
basis, as long as the results are written in a basis-independent way, and follows
directly from (B.1). The simplification comes from the fact that, in this basis, Γa
squares to ±1 and Γa1...an = Γa1 . . .Γan as long as the indices are distinct. As an
example, consider ΓabΓcd. Here, either all indices are distinct (1 possibility); one of
a, b coincides with one of c, d (4 possibilities) or both of a, b coincide with one of c,
d each (2 possibilities). Thus,
ΓabΓ ab [a b]cd = Γ cd − 4δ[cΓ d] − 2δ
ab
cd1, (B.4)
where the signs come from anticommuting Γ-matrices with distinct indices. This
can be generalised to cases with contracted indices and more than two factors. For
118Note that with signature (r, s), we mean that the metric has r positive and s negative eigenvalues
in any basis.
119This is related to Bott periodicity [70].
115
B. Spinors
instance, ΓaΓa = d1 and
Γ Γa1...anb Γb = (−1)n(d− 2n)Γa1...an . (B.5)
In the latter, there are d− n values of b that do not coincide with any of the a’s and
n values of b that coincide with one of the a’s. These identities can be used to see
that
d tr Γa1...an = tr(ΓbΓ Γa1...an) = tr(Γ a1...an b n a1...anb bΓ Γ ) = (−1) (d− 2n) tr Γ , (B.6)
which implies that tr Γa1...an = 0 if n 6= 0 and, for odd d, n =6 d.
B.1 Spinors in arbitrary even dimension
Let us, for a moment, consider the case of even dimension, d = r + s = 2k. The
dimension of the Dirac spinor representation is 2k [70]. A basis of End(VS, VS), that
is, 2k × 2k matrices after a choice of basis for the Dirac representation space VS,
is provided by 1,Γa1 , . . . ,Γa1...ad when restricted to antisymmetrically independent
index combinations. To see this, consider
∑d
X = x Γa1...aia1...a , (B.7)i
i=0
where all xa1...a are completely antisymmetric. From the above remarks on how toi
compute products and traces of Γ-matrices, we see that xa1...a ∝ tr(XΓi a1...a ). Thus,i
X = 0 implies that all the x’s are 0, whence we have established linear independence.
By counting, we now see that there are 2d = 2k · 2k linearly independent components
of X, which is precisely the dimension of End(VS, VS).
Chirality and Weyl spinors
Since there is only one antisymmetrically independent index combination with d
indices, we may define γ by120
γa1...ad = ik+sΓa1...ad . (B.8)
It is easy to see that γ2 = 1 and {γ,Γa} = 0. Note that (Γa) BA , where A,B are
Dirac spinor indices, is a so(r, s)-invariant, as seen from
L c B
1 c B 1 c B cd B
ab · (Γ )A = 4(ΓabΓ )A − 4(Γ Γab)A + δab(Γd)A = 0. (B.9)
Thus, γ is also an invariant and the Dirac spinor representation is reducible by Shur’s
lemma since γ is not proportional to 1. We may form projection operators
= 1± γP± 2 , (B.10)
120In d = 4, this is commonly denoted by γ5.
116
B.1. Spinors in arbitrary even dimension
projecting onto the invariant subspaces with γ-eigenvalue ±1, respectively. These
smaller dimensional representations are known as the left and right-handed Weyl
spinor representations. Note that, if γΨ = ±Ψ, then γΓaΨ = ∓ΓaΨ, that is,
Γa interchanges left and right-handed spinor. Thus, writing a Dirac spinor as
ΨA = (ψα, χα̃)A, where ψ and χ are (Weyl spinors of t)he two kinds, this implies that
a B
(Γa) B = 0 (Γ )αβ̃A (Γa)α̃β 0 , (B.11)
A
where we have introduced chiral Γ-matrices. In this basis, we see explicitly that the
Dirac spinor representation of so(r, s) is reducible since Γab is block-diagonal. For
even d, one can prove that there is only one irreducible representation of the Clifford
algebra and that the Weyl spinor representations of so(r, s) are irreducible [70].
Invariant tensors and Majorana spinors
There are a few other invariants apart from γ. To see this, note that we can construct
a new set of matrices satisfying (B.1) by taking the negative, complex conjugate,
transpose or any combination thereof of all matrices Γa. Consider first the negative.
Since there is only one inequivalent irreducible Clifford algebra representation, there
exists a matrix M BA such that −Γa = MΓaM−1, relating the two representations.
Note that this equation is linear in M , which is seen by writing it as MΓa = −ΓaM .
The space of solutions is one-dimensional by Shur’s lemma. From the above, we see
that M = γ provides a canonical choice of M .
Similarly, there are two one-dimensional spaces of solutions (B B±)Ā to
B a a ∗±Γ = ±(Γ ) B±. (B.12)
These two spaces are related by B− = B+γ. Taking the complex conjugate of (B.12),
we find that (B∗ )−1± are also solutions whence (B∗ )−1± = z±B± for some z± ∈ C.
Complex conjugating this relation, we find z± ∈ R. Since the left and right-handed
spinor representations are inequivalent,121 B± are either block diagonal or block
antidiagonal in the Weyl basis, γ = diag(1,−1). With B− = B+γ, we find that
z− = ±z+ in the two cases, respectively. By rescaling B±, z± is rescaled by a positive
real number whence we may scale B± such that z± ∈ {−1,+1 } while maintaining
B− = B+γ.
Due to the index structure of B, we may ask whether there are any solutions to
ψ = B−1ψ∗. To this end, define the antilinear (conjugate-linear) map R : VS → VS
by ψ 7→ B−1ψ∗. Squaring this gives
R2(ψ) = R(B−1ψ∗) = B−1(B∗)−1ψ = zψ, (B.13)
where z = ±1 after the above rescaling of B. Consider first z = +1 so that R2 is the
identity map. We claim that there is a basis in which B coincides numerically with
121Note that γ ∝  a1b1 akbka1b1...a b Γ . . .Γ is a k’th-order Casimir with different eigenvalue on thek k
two Weyl representations.
117
B. Spinors
the unit matrix, which motivates the notation B B BĀ = δ . To see this, pick any ψ andĀ
let χ = R(ψ). Suppose that ψ and χ are linearly dependent so that χ = cψ for some
constant c. By antilinearity and R2 = 1, ψ = R(χ) = c̄χ = |c|2ψ whence c = eiθ.
Now, with ψ′ = eiϕψ, R(ψ′) = ei(θ−2ϕ)ψ′. Thus, for an appropriate choice of ϕ, we
get R(ψ′) = ψ′. Taking ψ′ as our first basis vector, B becomes block diagonal with a
1 in the upper left corner. If, on the other hand, ψ and χ are linearly independent,
we find
R(ψ + χ) = χ+ ψ, R(iψ − iχ) = −iχ+ iψ. (B.14)
In this case, we get two basis vectors, ψ + χ and iψ − iχ, which are mapped to
themselves by R. Proceeding by induction, we conclude that there is a basis such
that B is diagonal with 1s on the diagonal. Since BΓa = ±(Γa)∗B, the Γ-matrices
are real (+) or imaginary (−) in this basis. In both cases, the generators of so(p, q)
are real. Furthermore, ψ = R(ψ) reduces to the condition that the components of ψ
are real. This is known as a Majorana basis and the reality condition, ψ = R(ψ),
is the Majorana condition. If B is block diagonal in the Weyl basis, we can define
Majorana–Weyl spinors by requiring both γψ = ±ψ and R(ψ) = ψ. If B is block
antidiagonal, the Majorana condition gives a relation between the left and right-
handed components and there are no Majorana–Weyl spinors. In the above, B is
only well-defined up to a global complex phase. If B is redefined by B 7→ eiθB, the
real subspace defined by the Majorana condition is rotated by θ/2.
In the case z = −1, there is clearly no solution to ψ = R(ψ) whence it is not possible
to define Majorana spinors. However, given two Dirac spinors ψ and χ, one may
impose the relation χ = R(ψ), known as the symplectic Majorana condition.122 In
this case, R can be used to turn VS into a quaternionic space by defining jψ = R(ψ).123
This has applications in extended supersymmetry, see for instance [108].
There are also invariants (A )ĀB± and (C AB±) satisfying
A Γa = ±(Γa)†± A±, C a±Γ = ±(Γa)TC±. (B.15)
Note that these equations are linear in A and C whence there are one-dimensional
spaces of solutions. Again, the different signs can be related by A− = A+γ and
C− = C+γ. There is also a relation A ∝ (B−1)TC, as seen from the index structure
or a straightforward calculation. By defining the Dirac conjugate ψ̄ = ψ†A and the
Majorana conjugate ψ̃ = ψTC, we find that the relation A = (B−1)TC is necessary
for the Majorana condition ψ = B−1ψ∗ to take its usual form ψ̄ = ψ̃.
It is easy to see that A†± also satisfies (B.15), whence A†± ∝ A±. When A is rescaled,
the phase of the proportionality constant changes. Thus, we can choose A Hermitian,
which makes it well-defined up to a real constant factor. In the case we are dealing
with Majorana spinors, this is not always convenient. In that case, instead note
that BTC∗B = zCC± by an analogous argument. With the above normalisation
122When considering Grassmann-odd spinors, there is another generalisation of the Majorana
condition, the graded Majorana condition, which can be imposed on a single Dirac spinor [107].
123Since R is antilinear, ij = −ji.
118
B.2. Spinors in arbitrary odd dimension
(B∗)−1 = ±B (with the plus sign in the Majorana case), it follows that |zC | = 1. By
rescaling C, we set zC = 1. In the Majorana basis, B = 1, this means that C is real.
Insisting that A = (B−1)TC, we find A† = (B−1)TCT. By transposing the equation
for C in (B.15), it is easy to see that CT = ±C by using that the solution space is
one-dimensional. Thus, A is (anti-)Hermitian when C is (anti-)symmetric. Again, A
and C are only well-defined up to a real constant after imposing these conditions.
B.2 Spinors in arbitrary odd dimension
Consider now d = 2k + 1. We may construct a representation of the Clifford
algebra by using the Γ-matrices from d = 2k and taking a multiple of γ as the
last Γ-matrix. By adding ±γ to a set of Γ-matrices with signature (r, s) we end
up with signature (r + 1, s). To instead get (r, s + 1), ±iγ should be used as the
last Γ-matrix. Since there is only one inequivalent representation of C r̀,s but a sign
choice for the last Γ-matrix when we go up in dimension, there seems to be two
inequivalent representations of C r̀+1,s and C r̀,s+1. To see that the sign choice really
gives inequivalent representations of the d = 2k + 1 Clifford algebras, note that
Γd = cγ =⇒ Γa1...ad = c(−i)k+sa1...ad1. (B.16)
Thus, c, which can take two values once the signature is fixed, distinguishes the
two representations and they are indeed inequivalent. These representations are
irreducible [70]. The inequivalent Clifford representations are related by Γa →7 −Γa.
Note that this implies that the so-representations generated by Γab/4 are the same.
Since there is no longer an invariant γ, the situation is reversed compared to
even dimensions; in odd dimensions, there are two inequivalent irreducible Clifford
algebra representations but only one irreducible spinor representation, the Dirac
representation.
Due to (B.16), Γa1...an and Γan+1...ad are not linearly independent but related by a con-
traction with a1...ad . Thus, a basis for End(VS, VS) is provided by 1,Γa1 , . . . ,Γa1...ak
where, in contrast to the even-dimensional case, there are at most k indices. Lin-
ear independence is proved analogously to the case of even dimension. Counting
the antisymmetrically independent index combinations, we find that the span of
1,Γa1 , . . . ,Γa1...ak is 22k-dimensional, which agrees with the dimension of End(VS, VS).
In odd dimension, there are only half as many invariants as in even dimension. More
specifically, only one of the signs in each of A±, B± and C± is viable. Since these
invariants intertwine representations of the Clifford algebra, it is easy to see which of
them exists by using that the inequivalent representations are distinguished by the
sign of Γa1...ad . In appendix B.4, we demonstrate this explicitly for C± in d = 11.
119
B. Spinors
B.3 Spinors in four dimensions
In four dimensions, a Dirac spinor, ΨA, has four components and consists of a
left-handed and a right-handed irreducible Weyl spinor, ψα and χ̄α̇.124
A chiral basis
In the chiral (Weyl) basis, ΨA = (ψα, χ̄α̇) and (note the factor of i) A  B0 σa(γa) B = i αβ̇A aα̇β 0 , (B.17)σ̄
A
where, in the basis we choose and with index structure σa , the Pauli matrices are
( ) ( ) ( ) αβ̇ ( )
σ0 = −1 0 1 = 0 1, σ , σ2 0 −i 3 1 00 −1 1 0 = i 0 , σ = 0 −1 . (B.18)
In four spacetime dimensions, there is a useful exceptional isomorphism Spin(3, 1) '
SL(2,C). This means that the Weyl spinor representation of Spin(3, 1) is the defining
representation of SL(2,C), whence the antisymmetric tensor αβ is invariant. This
isomorphism is indicated by the fact that there is a one-to-one correspondence
between real-valued vectors va and Hermitian matrices V := vaσa. Given V , we can
construct another Hermitian matrix V ′ by
V ′ = ΛV Λ† (B.19)
In this transformation, a global phase of Λ is irrelevant whence we can demand
det Λ ∈ R 2 2≥0. Since detV = −v , the transformations that preserve v , or equivalently
ηab, are precisely those with det Λ = 1, that is, Λ ∈ SL(2,C). However, Λ = −1 is
not effective on V , whence SL(2,C) is a double cover of SO(3, 1).125
The above also explains the index structure σa , which is needed for (B.19) to make
αβ̇
sense in index-notation, V ′ = Λ γ δ̇
αβ̇ α
Λ̄ Vγδ̇. This being a Lorentz transformation, thatβ̇
is, V ′ = v′aσa where v′ = Λ ba a vb, further implies that σa is an invariant tensor underαβ̇
Spin(3, 1).
As already explained, ( ) ( )
αβ = 0 1
αβ
= 0 −1 −1 0 , αβ 1 0 , (B.20)
αβ
124Here, we use a dot instead of a bar on complex conjugated indices, as is common in Van der
Waerden notation.
125Note, however, that Λ = −1 is effective in ψ′ = Λψ, in perfect agreement with Spin(3, 1) being
the double cover of SO(3, 1).
120
B.3. Spinors in four dimensions
are invariant tensors due to det Λ = 1 in the spinor representation. Therefore, we can
use them to raise and lower spinor indices.126 We do this by left-multiplication, that is,
it is always the rightmost index of  which is contracted with the quantity whose index
is being raised or lowered (ψα = αβψβ). Note that αγβδ = βαγδ =6 αβ, whence
we cannot raise or lower indices on the -tensors themselves. This peculiarity is not
a problem since two contracted -tensors can always be written with a Kronecker
delta. Furthermore, these conventions imply that
ψ χαα = ψααβχ ββ = −ψ χβ. (B.21)
Because of this, we need a convention for how to place the indices when switching
between index notation and index-free notation. We use the convention that undotted
indices are contracted up-down, while dotted indices are contracted down-up, that is,
ψχ = ψαχ αα = χ ψα = χψ, ψ̄χ̄ = ψ̄ α̇α̇χ̄ = χ̄ ψ̄α̇α̇ = χ̄ψ̄, (B.22)
for anticommuting (Grassmann-odd) spinors. Due to how complex conjugation is
defined on Grassmann numbers, see appendix F, this implies that (ψχ)∗ = ψ̄χ̄.
The complex conjugated Pauli matrices σ̄aα̇β are obtained from σa by complex
αβ̇
conjugation and raising the indices. Numerically, in our basis and with the above
index structure, they are σ̄a = (−1,−σi)a. In index notation, the statement that σa
are Hermitian reads σa = σ̄a .
αβ̇ β̇α
One can show that
(a
σ σ̄b)β̇γ = −ηabδγ , σ̄(a|α̇β |b)σ ab α̇
αβ̇ α βγ̇
= −η δβ̇ , (B.23)
where the latter is obtained by complex conjugation of the former. This is equivalent
to the Dirac algebra {γa, γb} = 2ηab, with γa from (B.17). Using (B.23), we find
σa σ̄bβ̇α = −2ηab since the left-hand side is symmetric in a b due to σa being Hermitian.
αβ̇
We can also derive a type of Fierz identity by writing V = v σaαβ̇ a . Contracting withαβ̇
σ̄bβ̇α, we find −2vb = V bβ̇ααβ̇σ̄ and hence
σa σ̄δ̇γαβ̇ a = −2δ
γ
αδ
δ̇
β̇. (B.24)
Now define ( ab) β = [aσ σ b]γ̇βα αγ̇σ̄ . Using (B.23), it is straightforward to show that
[σab [a b], σcd] = 8δ σ , whence −σab[c d] /4 are the Lorentz generators in the left-handed
spinor representation. Due to the subtleties when raising and lowering spinor
indices, ( ab)α̇ = − [a|α̇γ |b] and [ ab ] = 8 [a b]σ̄ σ̄ σ σ̄ , σ̄cd δ[c σ̄ d]. Hence, σ̄ab/4 are the Lorentzβ̇ γβ̇
generators in the right-handed spinor representation, which is consistent with table A.1
since (σab) abαβ and (σ̄ )α̇β̇ are symmetric in αβ (α̇ β̇).127 This is precisely what is
needed for γab/4 to be the Lorentz generators in the Dirac spinor representation.
126Dotted indices are raised and lowered using the complex conjugates ̄α̇β̇ and ̄α̇β̇ , which, in our
basis, coincide numerically with αβ and αβ .
127Note that, due to how we raise and lower indices, −(Lab)α , rather than (Labβ )αβ , are the
generators of the dual of the left-handed spinor representation.
121
B. Spinors
We may define the invariant
5 = − iγ  abcd4! abcdγ , (B.25)
with the properties (γ5)2 = 1 and {γ5, γa } = 0.
128
In the Weyl basis,
β
(γ5) B = δα 0  BA 0 − α̇ . (B.26)δ
β̇
A
Hence, γ5 may be used to form projection operators P± = (1± γ5)/2 onto the two
chiralities.
A real basis
In eleven-dimensional supergravity, we will use Majorana spinors. Thus, for the
compactification to four dimensions, it is convenient with a Majorana basis, in which
the γ-matrices are real. Such a basis can easily be constructed for instance by letting
γ0 = iσ2 ⊗ σ1, γ1 = σ1 ⊗ σ1, γ2 = σ3 ⊗ σ1, γ3 = 1⊗ σ3, (B.27)
where the Pauli matrices are numerically the same as in (B.18). These satisfy the
Dirac algebra since the Pauli matrices anticommute and square to 1. From (B.25),
it is clear that the chirality projectors P± = (1± γ5)/2 are not real in any basis in
which γa are real. Hence, there are no Majorana–Weyl spinors in four dimensions
with Lorentzian signature.
B.4 Spinors in eleven dimensions
A spinor in eleven dimensions has 32 components. In a basis in which the eleven-
dimensional ηÂB̂ splits block-diagonally to a four-dimensional ηab and a seven-
dimensional δAB, we may construct eleven-dimensional Γ-matrices Γ̂Â as
Γ̂Â = (γa ⊗ 1, −γ5 ⊗ ΓA)Â, (B.28)
where γa are the four-dimensional γ-matrices and ΓA are the seven-dimensional
Γ-matrices.129 If we use the Majorana basis from appendix B.3 for γa and the basis
for ΓA given in appendix C.1, we get a basis for Γ̂Â in which they are real, that is, a
Majorana basis.130
Note that
Γ̂Â1...Â11 = −Â1...Â111. (B.29)
This specifies which of the two inequivalent representations of the Clifford algebra
Γ̂Â generate.
128In the context of Spin(3, 1), the superscript 5 is not an index.
129That γ5 enters in Γ̂A corresponds to the fact that the Clifford algebra of the eleven-dimensional
space is the Z2-graded tensor product of the four and seven-dimensional Clifford algebras [70].
130Note that γa are real while Γa and γ5 are imaginary in these bases.
122
B.4. Spinors in eleven dimensions
Since there is precisely one irreducible Dirac spinor representation in eleven dimen-
sions, there must be precisely one irreducible invertible tensor (up to a constant
factor) Cαβ, where α and β are spinor indices, by Shur’s lemma. Suppose that
there exists C± such that C±Γ̂Â = ±(Γ̂Â)TC±. Since ±(Γ̂Â)T satisfy the Clifford
algebra, C± are intertwiners between different representations of the Clifford algebra.
However, with Γ̂T Â1...Ân := (Γ̂[Â1)T . . . (Γ̂Ân])T,
Γ̂T Â1...Â11 = (Γ̂Â11...Â1)T = +Â1...Â111, (B.30)
whence Γ̂T Â generate the other, inequivalent, irreducible representation of the Clifford
algebra. Thus, C+ cannot exist.131 By a similar argument, C− must exist since
there are only two inequivalent irreducible representations of the Clifford algebra
and Γ̂Â and −Γ̂T Â generate equivalent representations since the representations are
distinguished by the sign difference between (B.29) and (B.30). In the following, we
write C instead of C− since C+ does not exist.
Since Cαβ is the only nonvanishing invariant with that index structure, it must be
either symmetric or antisymmetric. Also, if CT = ±C,132
(CΓ̂Â1...Ân)T = ±(ΓÂn)T . . . (ΓÂ1)TC = ±(−1)n(n+1)/2CΓ̂Â1...Ân . (B.31)
Using that (CΓ̂Â(i))5i=0, where Â(i) = (Â1, . . . Âi) is a multi-index, is a basis for all
linear maps from the space of spinors to itself (that is, 32× 32 matrices once a basis
has been chosen), we find that Cαβ = −Cβα, see table B.1.
Table B.1: Number of symmetric (S) and antisymmetric (A) matrices in the
Γ-basis, found by (B.31), depending on the sign in CT = ±C. Since there are
528 (496) (anti)symmetric 32× 32 matrices, we conclude that CT = −C.
CT = +C CT = −C
S A S A
C 1 1
CΓ̂Â(1) 11 11
CΓ̂Â(2) 55 55
CΓ̂Â(3) 165 165
CΓ̂Â(4) 330 330
CΓ̂Â(5) 462 462
The above algebraic properties of C is all that we will need in calculations. However,
they only define C up to a nonzero constant factor. In the above Majorana basis,
131Assuming that C exists, we find Γ̂T Â1...Â11 = C Γ̂[A1C−1 . . . C Γ̂An]C−1 = −Â1...Â+ + + 11+ + 1 which
contradicts (B.30).
132The formula analogous to (B.31) for C has a factor (−1)n(n−1)/2+ instead of (−1)n(n+1)/2.
123
B. Spinors
a particular choice of C coincides numerically with Γ̂0̂. This is, however, merely a
coincidence since C and Γ̂0̂ transform differently under a change of basis.
We use Cαβ and its inverse to raise and lower spinor indices. Indices on spinors
are raised and lowered by left-multiplication (ψα = Cαβψβ). We define raising and
lowering of the left (right) spinor index on a linear map M βα by multiplication from
the left (right), that is Mαβ = CαγM βγ and M γαβ = Mα Cγβ. This ensures that
contracted indices can be raised and lowered without picking up a sign as long they
stand next to each other, for instance, M βα ψβ = Mαβψβ andM βα αβαβ M̃ = M M̃βα .
124
C. Octonions
Here, we introduce the octonions and, in subsequent sections, relate them to Spin(7)
and the exceptional Lie group G2. The interested reader can find more details in
[109]. The octonions O are a real vector space spanned by one real unit, 1, and seven
imaginary units, oa, with multiplication defined by
oaob = −δab + a cab oc (C.1)
and 1 acting as a multiplicative identity both from the left and right. Here, the
structure constants aabc are totally antisymmetric with independent nonzero compo-
nents133
aabc = 1, for abc = 123, 257, 536, 374, 761, 642, 415, (C.2)
and the multiplication is extended to all of O as to be distributive over addition.
Using the index split a = (̂ı, 0, i), this may be written as
aı̂̂k̂ = ijk, aijk̂ = −ijk, a0i̂ = −δij. (C.3)
As presented, the construction might seem arbitrary but the octonions fit into the
sequence R, C, H, O, S, . . . where H are the quaternions, S the sedenions and every
entry is obtained from the previous one through the Cayley–Dickson construction
[109]. In each step in this sequence, some structure is lost. For instance, the
complex numbers cannot be ordered in a way compatible with multiplication and
the quaternions do not commute.134 In the step to the octonions, associativity is
lost. This means that the associator
[x, y, z] = (xy)z − x(yz) (C.4)
is, in general, nonzero for x, y, z ∈ O. However, the associator is completely antisym-
metric, whence
x(xy) = (xx)y, (xy)y = x(yy), (C.5)
for arbitrary x, y ∈ O. Thus, the octonions are said to be alternative. This property
is lost in the next step; the sedenions are nonalternative.135
Octonion conjugation is defined as
1∗ = 1, o∗a = −oa. (C.6)
133We use δab and its inverse to raise and lower indices.
134Another, purely algebraic, structure that is lost is that not every complex number is real in the
sense that x∗ = x is not generally true for x ∈ C.
135The sedenions do, however, satisfy the weaker property of power-associativity, x(xx) = (xx)x.
125
C. Octonions
This lets us define a scalar product as
〈x, y〉 = Re(x∗y), (C.7)
which coincides with the standard scalar product on R8 in the basis we have taken
and induces a norm
‖x‖2 = 〈x, x〉 = x∗x = xx∗, (C.8)
which is then the standard norm on R8. This norm satisfies
‖xy‖ = ‖x‖‖y‖, (C.9)
whence the octonions are said to be a normed division algebra [109]. This implies
that there are no zero-divisors; if x and y are nonzero, xy is also nonzero. That there
are no zero-divisors can also be seen by noting that, due to alternativity,
x∗(xy) = (x∗x)y = ‖x‖2y, (C.10)
whence multiplication by a nonzero x is inverted by multiplying by
∗
x−1 = x 2 . (C.11)‖x‖
Similarly, the proof of (C.9) is
‖xy‖2 = (xy)(y∗x∗) = x(yy∗)x∗ = ‖x‖2‖y‖2. (C.12)
Here, we have used (xy)∗ = y∗x∗ and the fact that x, y, x∗ and y∗ all belong to the
associative subalgebra generated by Im x and Im y [109].
An automorphism of the octonion algebra is, per definition, an R-linear invertible
map g : O→ O preserving the octonion multiplication, that is,
∀x, y ∈ O g(xy) = g(x)g(y). (C.13)
The automorphisms naturally form a group, AutO, with composition as group
multiplication. This is one way to define the exceptional Lie group G2, and the
definition we choose in this thesis. It is worth pointing out that g(1) = 1 for any
g ∈ G2, which is immediate from the definition. Also, automorphisms preserve the
scalar product, δab and aabc. The Lie algebra g2 = Lie(G2) is the derivation algebra
der(O) of G2 = Aut(O), that is, the linear transformations D : O→ O satisfying
∀x, y ∈ O D(xy) = xD(y) +D(x)y, (C.14)
and is the compact real form of the exceptional Lie algebra with the same name.
126
C.1. Spin(7), octonions and G2
C.1 Spin(7), octonions and G2
Denote the R-linear map from O to itself defined by left-multiplication by x ∈ O by
Lx. Two such maps can be composed but since the associator is nonvanishing
LxLy(z) := Lx(Ly(z)) = x(yz) =6 (xy)z = Lxy(z), (C.15)
in general. However, due to alternativity
{Lx, Ly}z = x(yz) + y(xz) = (xy)z − [x, y, z] + (yx)z − [y, x, z] =
= (xy + yx)z = L{x,y}z. (C.16)
Thus, since {oa, ob} = −2δab,
{Loa , Lo } = −2δb ab. (C.17)
This is almost identical to the anticommutator of two gamma matrices. To fix the
sign, consider the complexified octonions C⊗O and define
Γa := −iLoa . (C.18)
Since these satisfy the correct anticommutation relations, C⊗O can be identified with
the Dirac spinor representation of Spin(7). To derive the matrix representation of
Γa consider ΓaoA where A = (0̂, a) and o0̂ = 1 ∈ O. From the definition, Γa1 = −ioa,
whence
(Γ 0̂a) 0̂ = 0, (Γa)
b
0̂ = −iδ
b
a, (C.19)
and Γaob = −ioaob = iδab − ia cab oc, whence
(Γa)0̂b = iδab, (Γ )ca b = −ia cab . (C.20)
Thus, (Γa)AB is antisymmetric in its spinor indices and the independent nonvanishing
components are136
(Γa)b0̂ = −iδab, (Γa)bc = +iaabc. (C.21)
Here, we have used the invariant δAB to lower indices. Note that
δAB(Γa)BC = −(Γa)BAδBC , δĀB(Γ )Ba C = (Γ̄ )B̄a ĀδB̄C , (C.22)
where bars denote complex conjugation, compare to (B.15). Thus, CAB = δAB is
symmetric. Γa and Γab are antisymmetric while Γabc is symmetric and the ones with
more indices are related to these using a1...a7 .
(C.22) also implies that O is identified with Majorana spinors. Note, however, that
if ψ is Majorana, then Γaψ is not Majorana due to the different signs in (C.22).
Since there is only one spinor representation of Spin(7) and [Γa1...a7 ,Γb] = 0, it follows
that Γa1...a7 ∝ a1...a71 due to Shur[’s lemma. Using (C.2) we find]
(Γ1...7) 70̂0̂ = (−i) Re 1(o1(o2(o3(o4(o5(o6(o71))))))) = i, (C.23)
136With Γa := ±iLo one gets (Γa)b0̂ = ±iδab and (Γa)bc = ∓iaabc where the latter sign changes toa
± if one uses right-multiplication (Γa := ±iRo where Rxy = yx) instead of left-multiplication.a
127
C. Octonions
whence
Γa1...a7 = ia1...a71. (C.24)
Consider now the subgroup H of Spin(7) leaving o0̂ = 1 ∈ O invariant. Since the
gamma matrices are invariant under Spin(7), it follows that H is a subgroup of
G2. To find the generators of the corresponding Lie algebra we want to find linear
combinations kabΓab of Γab such that kabΓabη = 0 where η = o0̂. Note that kabΓabη
is an arbitrary homogeneous quadratic polynomial in the imaginary units where in
no term ocod is c = d, due to the antisymmetrisation. The polynomial must vanish
using the octonion multiplication but should be nontrivial in formal variables. Using
this, it is straightforward to construct the generators by inspecting (C.3). If we start
from o0oi we can cancel the result using
2o0oi +  jki o̂ok̂ = (−2 + 2)oı̂ = 0, (C.25)
whence
T0i = 2Γ jk0i + i Γ̂k̂ (C.26)
are three generators.137 Starting from o o = − k[i j] ij ok̂ we can similarly cancel the
result using o[̂ıo̂], resulting in generators
Tij = Γij + Γı̂̂. (C.27)
Since o0oi, oioj and oı̂o̂ are the only ways to produce ok̂, all vanishing linear combi-
nations of them can be expressed as linear combinations of T0i and Tij . Analogously,
we find
T jk0ı̂ = 2Γ0ı̂ + i Γjk̂, (C.28)
which is the only way to produce cancelling oi terms. Lastly, o0 may be produced
from oio̂, oı̂o k̂j and oko . However, the two former also produce an ok term. We can
demand that this vanishes since the only other way to cancel it is by using o0oı̂ which
can always be substituted by oio̂ and oı̂oj terms by adding an appropriate multiple
of T0ı̂. Cancellation of the remaining o0 terms determines the prefactor of o ok̂k . We
find
Ti̂ = 3Γ − 3Γ − 2δ Γ k̂i̂ ı̂j ij k . (C.29)
T0i, T0ı̂, Tij and Ti̂ clearly span Lie(H). Since T0i and T0ı̂ are the only ones containing
Γ0i and Γ0ı̂, respectively, these are linearly independent. Tij is antisymmetric and
independent of the previous ones while Ti̂ is symmetric, traceless and independent
of the other generators. In total, we get 3 + 3 + 3 + 5 = 14 generators for the
subgroup H. Recalling that H is a subgroup not only of Spin(7) but also of G2,
the corresponding Lie algebra is a subalgebra of g2. Thus, since dim g2 = 14, the
Lie algebras are actually the same. This holds at the group level as well: G2 is the
subgroup of Spin(7) leaving 1 ∈ O invariant.
137Note that i, j, k, . . . and ı̂, ̂, k̂, . . . transform under the same SU(2) whence, for instance, ı̂ can
be contracted with either i or ı̂.
128
C.2. Structure constant identities
C.2 Structure constant identities
In this section we present some useful relations for the octonion structure constants.
Recall (C.21) and (C.24), which we repeat here for convenience
(Γa)b0̂ = −iδab, (Γa)bc = +iaabc, (C.30a)
Γa1...a7 = ia1...a71. (C.30b)
Let η = o0̂ = 1 ∈ O be the G2-invariant spinor. We then have
aabc = iη̄Γabcη, (C.31)
since
Γ 3abcη = (−i) oa(oboc) = −iδbcoa − iaabc + iabcdaadeoe (C.32)
and η̄ picks out the o0̂ term.
Now define the dual of the structure constants
1 1
cabcd := (?a) efg efgabcd = 3!efgabcda = 6abcdefga . (C.33)
Clearly, cabcd is completely antisymmetric and the independent components are
cabcd = 1, for abc = 4567, 1274, 2354, 3164, 1265, 1375, 2376, (C.34)
or, with the index split a = (̂ı, 0, i),
c0ijk =  , c = − , c klijk 0ı̂̂k ijk ı̂̂ = −2δklij . (C.35)
From (C.30b) it follows that
Γ = − i  Γefgabcd 6 abcdefg , (C.36)
whence
cabcd = −η̄Γabcdη. (C.37)
The outer product ηη̄ can be expanded in terms of gamma matrices. Lowering
the index on η̄, this product is ηAηB which is symmetric in AB since we consider
commuting spinors.138 Hence, only terms containing 1 and Γabc can enter in the
expansion. Thus we write
ηη̄ = x1 + xabcΓabc. (C.38)
Contracting the spinor indices with 1 and Γabc, respectively, gives
η̄η = 8x =⇒ = 1x 8 η̄η, (C.39a)
η̄Γabcη = xdef tr(ΓabcΓ ) = −48xabc =⇒ xabc = − 1 η̄Γabcdef 48 η. (C.39b)
138In the compactification of D = 11 supergravity, η is a Grassmann-even spinor on the internal
7-dimensional manifold while spinors on the 4-dimensional spacetime are Grassmann-odd.
129
C. Octonions
Using the normalisation η̄η = 1 we get
ηη̄ = 1 − 18 48Γ
abc(η̄Γabcη). (C.40)
This implies, using ΓaΓ abcdΓ = −(4− 3)Γbcd,
Γ 7aηη̄Γa = 8 +
1
48Γ
abc(η̄Γabcη). (C.41)
Adding these yields the Fierz identity
Γaηη̄Γa = 1− ηη̄. (C.42)
Since, Γa and Γab are antisymmetric, η̄Γaη = 0 = η̄Γabη. Using the Fierz identity, we
find
η̄Γabcηη̄Γcdeη = η̄Γab(1− ηη̄)Γdeη = −2δdeab + η̄Γ deab η. (C.43)
Thus, using (C.31) and (C.37),
a acde = 2δdeabc ab + c deab , (C.44a)
a bcdabca = 6δda, (C.44b)
a abcabca = 42, (C.44c)
where the latter two follows from contracting the former. Through analogous
calculations, we find
[ef g]
cabcdc
defg = a efg efgabca − 9c[ab δc] − 6δabc , (C.45a)
cabcdc
cdef = 8δefab + 2c
ef
ab , (C.45b)
c cbcdeabcd = −24δea, (C.45c)
c abcdabcdc = 168, (C.45d)
and
def [e f ]cabcda = 6a[ab δc] , (C.46a)
c cde eabcda = 4aab , (C.46b)
c abcdabcd = 0. (C.46c)
Using the above, it is easy to show that
= 1δ a b1b2 b3b4a1a2 4! a1 aa2 cb1b2b3b4 . (C.47)
Lastly, there is a useful identity
[b1 b2b3] = 1 b1b2b3 − 2 [b1b2 b ]a[a a a a a c δ
3 , (C.48)
1 2 a3] 3 a1a2a3 [a1a2 a3]
which can be proven straightforwardly by checking it for all index combinations.
130
D. Differential forms
Differential forms are essentially antisymmetric tensors. Here, we give a brief
introduction to some relevant concepts, state our conventions and introduce an index-
free formalism. Consider a connected oriented pseudo-Riemannian d-dimensional
manifold M without boundary and let Ωp(M) denote the space of (sufficiently
smooth) p-forms onM. On a coordinate chart, we may write α ∈ Ωp(M) as
α = 1!dx
m1 ∧ . . . ∧ dxmpαm1...mp(x), (D.1)p
where αm1...mp is completely antisymmetric.139 Since the manifold is oriented, there
is a canonical volume form √
|g|
vol = m1! εm1...m dx ∧ . . . ∧ dx
md , (D.2)
d ¯ d
where ε is the covariant Levi-Civita symbol, the covariant tensor density of
weight¯
m1...md
+1 with ε1...d = +1, and g is the determinant of the metric of tensor-density-
weight −2. Equ¯ivalently√, this may be written as
|g| dxm1 ∧ . . . ∧ dxmd = ε̄m1...mdvol, (D.3)
where ε̄m1...md is the contravariant Levi-Civita symbol, that√is, the contravariant
tensor density of weight −1 with ε̄1...d = +1. Note that, since |g| is a pseudo-tensor
density of weight −1, the volume form is a pseudo-tensor density of weight 0. Also, we
use different symbols ε and ε̄ to distinguish between the covariant and contravariant
permutation symbols¯since
g . . . g ε̄n1...ndm1n1 m = ε̄dnd m1...m = g εd ¯m1...m
. (D.4)
d
Using the metric, we may define a pointwise inner product of p-forms α and β as140
(α, β) = 1!αm1...mpβ
m1n1 mpnp
n
p 1
...npg . . . g . (D.5)
The normalisation here is chosen such that, in the flat Euclidean case, gmn = δmn,
the pointwise norm ‖α‖2 = (α, α) is 1 for α = dx1 ∧ . . . ∧ dxp where p ≤ d. To see
this, note that the tensor representation of dx1∧ . . .∧dxp has a ±1 on every position
which is a permutation of 1, . . . , p, whence the contraction with the metric in (D.5)
gives p! which is then cancelled by the prefactor 1/p!.
139Note that we use different conventions for superdifferential forms in superspace, see section 2.2.3.
140For this to be positive definite, we need to restrict to Euclidean signature.
131
D. Differential forms
D.1 The Hodge dual
Using the volume form and the pointwise inner product, we can define the Hodge
star operator ? : Ωp(M)→ Ωd−p(M) by
∀α, β ∈ Ωp(M) : α ∧ (?β) = (α, β)vol. (D.6)
Here, ?β is referred to as the Hodge dual of β. The Hodge star operator is linear
and well defined [110]. Note that ?1 = vol since, by the definition,
?1 = 1 ∧ ?1 = (1, 1) vol = vol, (D.7)
where 1 is interp√reted as the constant function onM with value 1. Explicitly,
|g|
?α = ε gm1n1!( )! m1...m . . . g
mpnpα mp+1n1...npdx ∧ . . . ∧ dxmd . (D.8)p d− p ¯ d
This is seen from
α ∧ ?β = 1 α dxm1! m√1...mp ∧ . . . dx
mp∧
p
|g|
∧ ε gn1q1 . . . gnpqpβ dxnp+1 ∧ . . . ∧ dxnd
p!( =d− )!¯n1...nd q1...qp p
= 1 α βn1...npε ε̄m1...mpnp+1...nd!2( − )! m1...mp n1...n vol =p d p ¯ d
= 1 m1...mp!αm1...mpβ vol = (α, β)vol, (D.9)p
where, in the second to last step, we have used
ε ε̄m1...mpnp+1...ndn1...n = p!(d− p)! δm1 ...mp¯ d n1 ...n
. (D.10)
p
Since ?α is a (d− p)-form, its componen√ts are read off as
|g|
(?α) = ε αm1...mpmp+1...m ! m1...m . (D.11)d p ¯ d
From this, it follo√ws that
2 |g|? α = !( )!εm1...m (?α)
m1...md−pdxmd−p+1 ∧ . . . ∧ dxmd =
p d− p ¯ d
= |g| ε εn1...npm1...md−p md−p+1 md!2( − )!¯m1...m
αn ...n
p d p d¯ 1 p
dx ∧ . . . ∧ dx =
= (−1)p(d−p) sign g ε ε̄n1...npm1...md−p
p!2(d− mp)!¯ d−p+1...mdm1...m
α
d−p n1...np·
· dxmd−p+1 ∧ . . . ∧ dxmd =
= (−1)p(d−p) sign 1g !α
n1 np
n1...npdx ∧ . . . ∧ dx = (−1)p(d−p) sign g α,p
∴ ?2 = (−1)p(d−p) sign g. (D.12)
Thus, the Hodge star operator gives a natural isomorphism Ωp(M) ' Ωd−p(M).
Note that sign g only depends on the signature of the metric and is +1 (−1) for
Euclidean (Lorentzian) signature.
132
D.2. The exterior derivative and de Rham cohomology
D.2 The exterior derivative and de Rham
cohomology
Differential forms can be differentiated in a coordinate-independent manner without
the use of a covariant derivative. The exterior derivative, d, is the unique linear
operator d: Ωp(M)→ Ωp+1(M) satisfying
f ∈ Ω0(U), df = dxm∂mf, (D.13a)
d2 = 0, (D.13b)
α ∈ Ωp(M), d(α ∧ β) = dα ∧ β + (−1)pα ∧ dβ, (D.13c)
where U ⊆M is any coordinate patch [110]. In local coordinates, for a p-form α,
dα = 1 ∂ α dxn m1! n m1...mp ∧ dx ∧ . . . ∧ dx
mp , (D.14a)
p
(dα)nm1...mp = (p+ 1)∂[nαm1...mp]. (D.14b)
Note that we may replace the partial derivative, ∂n, by the Levi-Civita connection,
that is, the unique metric-compatible torsion-free affine connection, ∇n. This follows
from the Christoffel symbols being symmetric in their lower indices.
A p-form α is said to be closed if dα = 0 and exact if α = dβ for some (p − 1)-
form β. Note that all exact p-forms are closed since d2 = 0. Hence, the image of
d: Ωp−1(M)→ Ωp(M) is a linear subspace of the kernel of d: Ωp(M)→ Ωp+1(M)
and we may define ( )
ker(d: Ωp(M)→ Ωp+1(M)Hp(M) := ) . (D.15)
im d: Ωp−1(M)→ Ωp(M)
Hp(M) is a (quotient) vector space known as the p’th de Rham cohomology group
ofM [110]. Each element of Hp(M) is an equivalence class, known as a cohomology
class, [α] = {α + dβ : β ∈ Ωp−1(M) } where dα = 0, that is, it is a closed p-form
modulo exact p-forms. Based on intuition from Rn, one may think that all closed
p-forms, for p > 0, are exact, which would render Hp(M) trivial. Indeed, this is
true for sufficiently nice open subsets of Rn (star-shaped) by Poincaré’s lemma [110].
However, it is not true in general, the most obvious counterexample being dθ on a
circle, where θ is the usual angular coordinate.141
Due to properties of the exterior derivative and since we do not need a metric on
M to define the cohomology groups, b = dimHpp (M) are topological invariants
of the manifold [110]. For compact manifolds, bp < ∞ is known as the p’th Betti
number. Since there are no −1-forms and the only closed 0-forms are locally constant,
dimH0(M) is the number of connected components ofM (1 for connectedM).
141Note that θ is not a global coordinate on S1. Clearly, dθ is exact on the coordinate chart.
However, when writing dθ, we refer to the unique smooth global extension of this local form.
133
D. Differential forms
Another useful result, which we wil∫l use belo∫w, is the (generalised) Stokes’ theorem
dα = α, (D.16)
M ∂M
where ∂M is the boundary ofM and α is a (d− 1)-form.142 In particular, ifM has
no boundary, the integral of an exact form vanishes. This explains why dθ on the
circle cannot be exact.
D.3 The codifferential
Define the codifferential on p-forms by [78]
δ := (−1)p ?−1d ? = sign g (−1)d(p+1)+1 ?d ?. (D.17)
Note that, while d raises the form-degree by one unit, δ lowers it by one unit. Clearly,
δ2 = 0 by the analogous property of d. Using the (indefinite) pointwise inner product
on Ωp(M), we may define143 ∫
〈α, β〉 = vol (α, β). (D.18)
The codifferential is the formal adjoint of d since, by Stokes’ theorem and the
definition of ?, ∫ ∫ ∫
0 = d(α ∧ ?β) = dα ∧ ?β − (−1)p α ∧ d(?β) =
= 〈dα, β〉 − 〈α, δβ〉. (D.19)
where α is a (p− 1)-form and β a p-form. If δα = 0, α is said to be coclosed and, if
α = δβ, it is said to be coexact.
To find an index-expression for δα, we first define the pseudo-tensor
m
1
1...md = √ ε̄m1...md . (D.20)
|g|
It easy to see that m1...md is covariantly constant with respect to any metric-
compatible connection since  m1...mdm1...m is a constant. From (D.11) and (D.14),d
we see that √
|g|
(?dα) [m1 ...mp+1]mp+2...m =d ( + 1)!ε¯m1...m
(p+ 1)∇ α =
p d
= sign g!  ∇
[m1α...mp+1]m1...m . (D.21)p d
By using this to compute ?d ?α, keeping track of all signs, one finds
(δα) nm1...mp−1 = −∇ αnm1...mp−1 . (D.22)
142Note that any top form α, that is, a d-form, can be written as α(x) = f(x)vol.
143When we do not write out the integration domain, it should be understood that the integral is
over all ofM.
134
D.4. The Hodge–de Rham operator and harmonic forms
D.4 The Hodge–de Rham operator and harmonic
forms
Here, we restrict to the case of compact manifolds with Euclidean signature, so that
(D.18) is the positive definite inner product of the Hilbert space of L2-integrable
p-forms [78]. Using the differential, d, and the codifferential, δ, we can define the
Hodge–de Rham operator, or Hodge Laplacian,
∆p := δd + dδ, (D.23)
which is a second-order differential operator from Ωp(M) to itself. Note that
∆p is self-adjoint. Also, ∆p is nonnegative in the sense that 〈α,∆pα〉 ≥ 0 since
〈α, δdα〉 = 〈dα, dα〉 ≥ 0 and, similarly, 〈α, dδα〉 = 〈δα, δα〉 ≥ 0.
If a p-form α satisfies ∆pα = 0, we say that it is harmonic.144 Clearly, a closed,
coclosed p-form α is harmonic. The converse is also true since, if ∆pα = 0,
0 = 〈α,∆pα〉 = 〈α, δdα〉+ 〈α, dδα〉 = 〈dα, dα〉+ 〈δα, δα〉, (D.24)
which implies dα = 0 = δα, since the L2 inner product is positive definite [78].
There is an orthogonal decomposition Ωp(M) = ker dp⊕ im δp+1, where the subscripts
denote the form-degrees of the differential forms d and δ are acting on. This is seen
by noting that α ∈ ker d, that is, dpα = 0, is equivalent to 0 = 〈β, dpα〉 = 〈δp+1β, α〉
for all β, which per definition means that α ∈ (im δ ⊥p+1) . Hence, ker dp = (im δp+1)⊥,
the orthogonal complement of im δp+1. By a completely analogous argument, ker δp =
(im dp−1)⊥. Since all exact forms are closed, im dp−1 is a linear subspace of ker dp.
Thus, we can make the decomposition ker dp = im d pp−1 ⊕ H , where Hp is the
orthogonal complement of im dp−1 in ker dp, that is, Hp = ker dp ∩ (im d )⊥.145p−1
Putting this together, we have found the orthogonal decomposition
Ωp(M) = Hp ⊕ im dp−1 ⊕ im δp+1, Hp = ker dp ∩ ker δp, (D.25)
known as the Hodge decomposition [78]. Since Hp contains all closed, coclosed
p-forms, it is the space of harmonic p-forms. Also, ker d pp = im dp−1 ⊕ H implies
that every cohomology class [α] contains precisely one harmonic form and
p(M) = ker dpH pim d ' H . (D.26)p−1
Hence, the Betti number bp is the dimension of the space of harmonic forms or,
equivalently, the dimension of the 0-eigenspace of ∆p. Since connected manifolds
have b0 = 1, the only harmonic functions onM are constants. This depends crucially
onM being compact so that the integrals converge; the space of harmonic functions
on Rn is infinite-dimensional.
144This should not be confused with the harmonics of section 4.2.
145Similarly, ker δ pp = im δp+1 ⊕H .
135
D. Differential forms
136
E. Bundles, gauge theory and
gravity
In this appendix, we give a brief introduction to the concept of bundles and Einstein–
Cartan gravity, also known as Cartan’s formulation of general relativity. To set the
stage for Einstein–Cartan gravity, we present a brief review of some aspects of gauge
theory after the introduction to fibre bundles. Although we give some mathematical
details, we do not attempt at a complete or mathematically rigorous presentation
but rather to give some intuition for the concepts.
E.1 Fibre bundles
A fibre bundle over a manifold is a space that locally looks like the product of
the manifold and a fibre but may have a different structure globally. Formally, it
consists of a total space E, a base space M, a typical fibre F and a projection
map π : E → M such that for each x ∈ M there is an open neighbourhood Ux
of x and a diffeomorphism146 ϕ : π−1(Ux) → Ux × F satisfying π1 ◦ ϕ = π, where
π1 : Ux × F → Ux, is the natural projection onto the first factor, (y, f) 7→ y, [98],
[110]. This means that Ux × F and the preimage π−1(Ux), that is, the subset of
E that maps onto Ux under the projection, are indistinguishable spaces. Such a
diffeomorphism ψ is called a local trivialisation and is the bundle analogue of a
coordinate chart of a manifold. We think of the total space E as glued-together
fibres with one fibre Fx ' F for each x ∈M.
A simple example of a fibre bundle is a cylinder, S1 × [0, 1]. Here, the (typical) fibre
is [0, 1] and the base space S1. Since this bundle is globally, and not only locally, a
product, it is said to be a trivial bundle. An example of a nontrivial bundle with the
same base space, S1, and the same fibre, [0, 1], is provided by the Möbius loop. This
only looks like a product S1 × [0, 1] locally.
There are two types of bundles we are especially interested in, namely, vector bundles
and principal bundles. A vector bundle E is a fibre bundle whose fibres, V = π−1x (x),
and typical fibre, V = F , are vector spaces. Further, it is required that there is
a trivialising cover, that is, an open cover ofM consisting of local trivialisations,
such that the maps v 7→ ϕ−1(x, v) are linear maps between the vector spaces V and
π−1(x) [110]. An example of a vector bundle is the tangent bundle TM, consisting of
all tangent spaces of the manifold. Given two local trivialisations ϕi,j on a pair Ui,j
of intersecting open sets inM, we may consider ϕij = ϕ ◦ ϕ−1i j which is a map from
146In the setting of topological spaces this is instead a homeomorphism.
137
E. Bundles, gauge theory and gravity
U × F to U × F . By the properties of ϕ, we see that ϕij(x, v) = (x, tij(x)v). Here,
tij is known as a transition function [98]. For the tangent bundle, the transition
functions are GL(d,R)-valued, where d = dimM, and correspond to local changes
of bases on the tangent spaces.
A vector field is a special case of what is known as a section of a bundle. Technically,
it is a map X : M→ TM such that π ◦X = idM. Whereas an arbitrary function
from M to TM can assign a tangent vector in TyM to a point x ∈ M, the last
requirement ensures that X assigns a tangent vector in TxM to x. Differential forms
are sections of exterior powers of the cotangent bundle.
The other type of bundles we are interested in is principal bundles. For these, the fibre
is a Lie group F = G, known as the structure group, which acts transitively, freely
and smoothly from the right on the total space P [98]. The group action is required
to be compatible with the bundle structure in the sense that the fibres are preserved,
that is, π(p ·g) = π(p), and that, for the local trivialisations ϕi : π−1(Ui)→ Ui×G, if
ϕi(p) = (x, g1) then ϕi(p·g2) = (x, g1g2) or, equivalently, ϕ−1i (x, g1g2) = ϕ−1i (x, g1)·g2.
This means that the fibres G = π−1x (x) are G-torsors, that is, they are diffeomorphic
to G but lack a preferred choice of identity element. The transition functions of a
principal G-bundle are G-valued.
A local trivialisation of a principal bundle ϕ : P → U×G determines an embedding φ
of U in P by φ(x) = ϕ−1(x, e), where e is the identity element of G.147 This goes the
other way too, given an embedding φ : U → P there is a (unique) local trivialisation
ϕ defined by ϕ−1(x, g) = φ(x) · g. Hence, a principal bundle is trivial if, and only if,
it admits a global smooth section.
An example of a principal GL(d,R)-bundle is the frame bundle, consisting of all
bases of all tangent spaces ofM [78]. If we have a metric g onM, we may consider
the bundle of orthonormal frames. The transition functions are then restricted to
O(d,R). The orthonormal frame bundle is a subbundle of the frame bundle and
we say that we have a reduction of the structure group from GL(d,R) to O(d,R).
Such a reduction is always possible since all manifolds admit a Riemannian metric.
In general, there may, however, be obstructions to structure group reductions. For
instance, consider the further reduction from O(d,R) to SO(d,R) of the structure
group of TM. This is only possible if there is a bundle of frames such that for every
x ∈M, all frames in the fibre over x have the same orientation, which is equivalent
to the manifold being orientable. The Möbius loop, now considered as the base
manifold, is not orientable. Another example is provided by considering a reduction
of the tangent group to { e }, the trivial group. If such a reduction exists, the tangent
bundle is trivial and we say that the manifold is parallelisable.
Similar to the above, one can construct a frame bundle associated with any vector
147This is true for any fibre bundle and one may choose any element of the fibre. For principal
bundles, there is a canonical choice provided by e. What follows does, however, not hold for
arbitrary fibre bundles.
138
E.1. Fibre bundles
bundle, not only the tangent bundle. It is also possible to go in the opposite
direction and construct an associated vector bundle from a principal bundle. To this
end, suppose that we have a principal G-bundle with total space P . To construct
the associated vector bundle, we additionally need a vector space V and a left-
representation ρ : G→ GL(V ). Define the right-action (p, v) ·g = (p ·g, ρ(g−1)v) of G
on P × V . This gives an equivalence relation (p, v) ∼ (p, v) · g and we denote the set
of equivalence classes [p, v] by E = P ×ρ V . E can be given a differentiable structure
[98] and is a vector bundle overM with fibre V [78]. The projection πE : E →M is
given by πE([p, v]) = π(p). This is well-defined since, choosing another representative,
πE([p · g, ρ(g−1)v]) = π(p · g) = π(p). To give a vector space structure to the fibres
of E, note that any two points in the fibre over x can uniquely be written as [p, v1]
and [p, v2], for any p ∈ π−1(x). This follows immediately from the requirements on
the group action of G on P . Now define [p, v1] + c[p, v2] = [p, v1 + cv2]. That this
addition and scalar multiplication are well-defined is easily seen by taking other
representatives of [p, v1,2] and using that ρ is a linear representation.
A local trivialisation ϕ : π−1(U) → U × G of P induces a local trivialisation of E,
ϕE : π−1E (U)→ U × V , by ( )
ϕE([p, v]) = π1 ◦ ϕ(p), ρ(π2 ◦ ϕ(p))v , (E.1)
where π1,2 are the natural projections onto the first and second factors of U × G,
respectively. By the properties of ϕ and ρ, this is well-defined. Also, by restricting
to a single fibre, which amounts to fixing p ∈ π−1(x) by the above remark, we get a
linear map π2 ◦ ϕE : π−1E (x) → V (the inverse is v 7→ ϕ−1E (x, v)), which shows that
the fibres are isomorphic to V . Lastly, the transition functions of E are ρ(tij), where
tij are the transition functions of P , and take values in ρ(G) ⊆ GL(V ).
Sections of an associated bundle E = P ×ρ V are in one-to-one correspondence with
G-equivariant functions P → V . A function f : P → V is said to be G-equivariant
if f(p · g) = ρ(g−1)f(p). To see the correspondence, suppose that we have such a
function f . To construct a section of E, simply let x 7→ [p, f(p)] for any p ∈ Px.
This is well-defined since any other point in the same fibre Px is of the form p · g
and [p · g, f(p · g)] = [p, ρ(g)f(p · g)] = [p, f(p)] by the equivariance of f . Since
πE([p, f(p)]) = π(p) = x, this is indeed a section. To go in the other direction,
suppose that we have a section s of E. Note that s ◦ π(p), which is an element of
Eπ(p), has a unique representative of the form (p, v) for some v ∈ V . Thus, we can
define f(p) = v. Now, [p, v] = [p · g, ρ(g−1)v] whence f(p · g) = ρ(g−1)v, that is, f is
equivariant. These two constructions are clearly inverse.
As explained above, a local trivialisation ϕ : π−1(U)→ U ×G of P induces a local
trivialisation ϕ : π−1E E (U)→ U × V . For x ∈ U we then have ϕE ◦ s(x) = (x, v(x)),
whence we may, locally, think of the section s of E as a V -valued function v(x) on U .
Using that s(x) = [p, f(p)] for any p ∈ Px and (E.1), we find v(x) = ρ(π2 ◦ϕ(p))f(p).
By choosing p = ϕ−1(x, e) =: φ(x), where φ is the canonical embedding of U in P as
above, we get
v(x) = f(φ(x)). (E.2)
139
E. Bundles, gauge theory and gravity
E.2 Gauge theory
In a gauge theory with structure group G, the fundamental geometrical object is
a principal G-bundle over M.148 Fields that are charged under the gauge group
correspond to sections of associated vector bundles. Thus, to be able to, for instance,
write down a kinetic Lagrangian, we need a way of differentiating such sections.
The tangent space TpP of the principal bundle P at a point p contains a subspace
that is tangent to the fibre of P through p, called the vertical subspace and denoted
Vp. A principal connection on P is defined by assigning horizontal subspaces Hp to
each p such that Vp ⊕Hp = TpP , Hp depends smoothly on p and is equivariant in
the sense that Hp·g agrees with the pushforward of Hp along p 7→ p · g [98]. There is
a natural way of identifying Vp with g = Lie(G) and one can, by using this, define a
g-valued 1-form on P as the projection onto Vp in the decomposition TpP = Vp ⊕Hp
[98]. Given a local trivialisation over U ⊆M, this 1-form can be pulled back to a
local g-valued 1-form A on U . In what follows, we use this local description since it
is more well-suited for the calculations we are concerned with in the main text, even
though this somewhat obscures the geometrical nature of the subject.
Using the local connection form A, we define a covariant exterior differential,
D = d + A, (E.3)
acting on the tensor product of the bundle of differential p-forms and a bundle
associated with the principal bundle. Locally, a section of such a bundle can
be written as V i = dxm1 ∧ . . . ∧ dxmpV im1...mp /p!, where i is an index of some
representation of G.149 The covariant exterior differential acting on V i reads
DV i = dV i + Aij ∧ V j. (E.4)
Note that the special case of a 0-form is a section of an associated bundle. A change
of local trivialisation of the principal bundle induces a change of trivialisation of
the associated vector bundles corresponding to V i 7→ V ′i = gi V jj , where g ∈ G
depends on the spacetime point.150 Demanding that the covariant exterior derivative
is covariant, that is, DV 7→ D′V ′ = gDV , we get, dropping indices,
D′V ′ = d(gV ) + A′ ∧ gV = g dV + dg ∧ V + A′ ∧ gV = g(dV + A ∧ V ) (E.5)
whence
A′ = gAg−1 + g dg−1, (E.6)
where we have used that 0 = d(gg−1) = (dg)g−1 + g dg−1. This transformation law
can also be derived from the properties of the global connection form on the principal
bundle [98].
148In physics, G is often referred to as the gauge group (for instance the U(1)× SU(2)× SU(3) of
the Standard Model). We reserve the term gauge group for the gauged structure group, that is,
the group of gauge transformations.
149We always put the form-indices closest to the symbol on components of p-forms with additional
indices.
150A change of local trivialisation is a passive gauge transformation.
140
E.3. Einstein–Cartan gravity
Given a principal connection, we define its field strength, or curvature 2-form, as
F = dA+ A ∧ A. (E.7)
Define the operator F by FV = F ∧ V . Then,
D2V = D(dV +A∧V ) = (dA∧V −A∧dV )+(A∧dV +A∧A∧V ) = F ∧V, (E.8)
that is, D2 = F , which is known as the Bianchi identity of the first type. Since D 7→
D′ = gDg−1, this implies that F 7→ F ′ = gFg−1 under a change of local trivialisation.
Thus, F is a tensorial 2-form transforming under the adjoint representation. This
can also be seen by a straightforward calculation,
F ′ = d(gAg−1 + g dg−1) + (gAg−1 + g dg−1) ∧ (gAg−1 + g dg−1) = gFg−1. (E.9)
Clearly, [D,F ] = 0, which for the 2-form F is equivalent to DF = 0. This is the
Bianchi identity of the second type and can also be shown as
DF = D(dA+ A ∧ A) = (dA ∧ A− A ∧ dA) + [A ∧ F ] = 0, (E.10)
where [A ∧ F ] = A ∧ F − F ∧ A appears since F transforms under the adjoint
representation.
E.3 Einstein–Cartan gravity
In Cartan’s formulation of general relativity, one uses the language of principal
bundles and gauge theory to formulate Einstein’s theory of gravity. On a spacetime
of dimension d, the structure group is Spin(d − 1, 1). The principal connection is
referred to as the spin connection and is denoted by ω. The curvature 2-form is
denoted by R. Given a metric of signature (d− 1, 1), the structure group GL(d) of
TM may always be reduced to SO(d− 1, 1), providedM is orientable. Essentially,
this amounts to restricting the frame bundle to the subbundle of positively oriented
orthonormal frames. Spinor fields are, however, sections of a vector bundle associated
not to SO(d− 1, 1) but Spin(d− 1, 1). Thus, we need a lift of the structure group
SO(d− 1, 1) to the double cover Spin(d− 1, 1), that is, a spin structure, to be able
to define spinors. Apart from the obstructions to orientability [70], [77] and the
existence of a Lorentzian metric [77], there can be further obstructions to such
a lift. Specifically, there is such a lift to Spin(d − 1, 1) if and only if the second
Stiefel–Whitney class ofM vanishes [70].
Assuming that we have a spin structure, there is a bundle of spin frames and we
can take a local section ea = e ma ∂m. There is a dual frame of the cotangent bundle
ea = dxme a m bm such that ea em = δb and e ae na m a = δnm. These are known as vielbeins
or, in the case d = 4, vierbeins.151 The Lorentz indices a, b, c, . . . are referred to as
flat and coordinate indices m,n, p, . . . as curved. The vielbeins are used to convert
between the two types of indices. However, the Lorentz covariant derivate D acts
151“Viel” is German for many while “vier” means four.
141
E. Bundles, gauge theory and gravity
only on flat indices. This means that we distinguish between, for instance, TM and
the associated vector bundle of Spin(d− 1, 1) carrying the vector representation and
view e ma as an isomorphism between them. In Einstein–Cartan gravity, we therefore
only use curved indices as form-indices.
Since we have reduced GL(d) to SO(d− 1, 1) by restricting to orthonormal frames,
the vielbeins are orthonormal and
gmn = e ae bm n ηab. (E.11)
The word orthonormal is perhaps only appropriate for ηab = diag(−1,+1, . . . ,+1)ab.
However, any metric tensor with flat indices and the correct signature can be used.
The torsion 2-form of the spin connection is defined by
T a := Dea, =⇒ T a a a bmn = 2∂[men] + 2ω[m| be|n] , (E.12)
which implies that T nabc = 2e[b ∂a]enc − 2ω[ab]c. Since the spin connection is antisym-
metric in its last two indices, which follows from it being a principal so(d − 1, 1)
connection, we get
ω̊abc = e n n[b (∂a]enc − e[c ∂a]enb −) e n [c
∂b]ena,
ωabc = ω̊abc + κabc,  1 (E.13)κabc = −2 Tabc − Tacb − Tbca ,
where ω̊ is the unique torsion-free spin connection and κabc is referred to as the
contorsion tensor. Note that Tabc = −2κ[ab]c. As we will see, this formalism reduces
to the ordinary formalism of general relativity if the torsion is constrained to 0 so
that ω = ω̊.
Next, we define the affine connection152 ∇ by
∇mV n = e na DmV a. (E.14)
We can express ∇mV n as
∇ n nmV = ea (∂ V a + ω am m bV b) = e n∂ (V pe aa m p ) + e n a b pa ωm bep V =
= ∂ V n + (e nω am a m be bp + e na ∂ amep )V p =
= ∂mV n + Γ n V nm p , (E.15)
where Γ is the gl(d)-valued connection form of the affine connection.153 Thus, the
Christoffel symbols of the affine connection, that is, components of the connection
form, are154
Γ nm p = e nω aa m be b + e np a ∂ amep . (E.16)
152We refer to a linear connection on the tangent bundle as an affine connection. The term affine
connection could perhaps more appropriately be used for what [98] refers to as a generalised affine
connection, namely a principal connection on the bundle of affine frames with an aff(d)-valued
connection form.
153We use Γ rather than Γ to distinguish the Christoffel symbols from Γ-matrices with three indices.
154The placement of indices on Γ is not conventional. We follow the previously stated convention
that form-indices are placed first.
142
E.3. Einstein–Cartan gravity
We write ∇ also for the total covariant derivative, ∇m = ∂m + Γm + ωm, acting on
both flat and curved indices. Since the covariant derivate obeys Leibniz’s rule and
the definition of the affine connection reads, in terms of the total covariant derivative,
∇mV n = e na ∇ V am , this definition implies that
∇me na = 0, ∇ e am n = 0. (E.17)
This is sometimes referred to as the vielbein postulate. Since gmn = η a babem en ,
it immediately follows that ∇mgnp = 0, that is, the affine connection is metric
compatible. Note that the Lorentz covariant derivate D can always be replaced by
the total covariant derivate ∇ but the opposite direction is only possible if all indices
that ∇ acts on are Spin(d− 1, 1)-indices, that is, flat. In particular, T a = ∇ea and,
by the vielbein postulate,
Tm := e ma T a = ∇dxm = Γm ∧ dxnn . (E.18)
Hence, the torsion vanishes precisely when Γ p[m n] = 0. This is just the statement that
the unique metric-compatible torsion-free affine connection, that is, the Levi-Civita
connection, is symmetric in its lower indices.
The curvature 2-form R b = dω b + ω c ∧ ω ba a a c has components given by
R b b c bmna = 2∂[mωn]a + 2ω[m|a ω|n]c . (E.19)
This expression is structurally identical to the expression for the Riemann tensor in
terms of the Christoffel symbols of the Levi-Civita connection. As remarked in [111],
the relation (E.16) between the affine connection and the spin connection has the
structure of the gauge transformation in (E.6). The calculation that the curvature
2-form transforms tensorially goes through even though ena is not a local Lorentz
transformation. Thus, the Riemann tensor is obtained by converting the flat indices
on the curvature 2-form of the torsion-free spin connection to curved ones using the
vielbeins, as expected. Hence, the framework of general relativity is obtained by
demanding that the torsion vanishes.
Lastly, consider the action
1 ∫
S = 2 √ d
Dx eR, (E.20)
κ
where e is the determinant of e am , e = |g|, and R = R aa = R abab is the curvature
scalar. This is known as the Palatini action. Clearly, it reduces to the Einstein–
Hilbert action under the constraint of vanishing torsion since, then, all quantities
can be expressed in terms of the metric. However, we may view the vielbein and
spin connection as a priori independent, thereby obtaining a first-order formulation
of gravity. To see that this is classically equivalent to the Einstein–Hilbert action,
we need to show that the equation of motion for the spin connection forces it to
be torsion-free. To this end, let ω̊ be the torsion-free spin connection and write
ω = ω̊ + κ, where κ is the contorsion. We can make a change of variables and view
the contorsion and the vielbein as the fundamental quantities. Thus, we wish to
show that κ = 0 on-shell. By the definition of the curvature,
R = d(ω̊ + κ) + (ω̊ + κ) ∧ (ω̊ + κ) = R̊ + D̊κ+ κ ∧ κ = R̊ +Dκ− κ ∧ κ, (E.21)
143
E. Bundles, gauge theory and gravity
where R̊ is the curvat∫ure 2-form of ω̊.
155 Thus, the action can be written as
1 ( )
S = dDx ee me n R̊ ab + 2D̊ κ ab2 a b mn [m n] + 2κ
ac
[m κ
b
n]c . (E.22)κ
Since the contorsion is antisymmetric in its last two indices, the middle term can be
written as 2∇̊ κ mnm n , which is a total derivative not contributing to the equation of
motion. Thus, the equation of motion for κ is given by the variation of the third
term in (E.22). A straightforward calculation shows that κmab = 0 on-shell.
When matter is added to the Palatini action, there can be extra terms in the equation
of motion for the contorsion. As long as there are no new terms containing derivatives
of κ, we can still solve for the contorsion in terms of the other fields through its
equation of motion. This means that there are no independent propagating degrees of
freedom in the contorsion and that we can eliminate it from the theory by substitution
in the Lagrangian. The interaction terms arising in this way would not be present
if we instead constrain the torsion to 0. However, keeping κ dynamical, we could
just add the negative of these terms to the original Lagrangian to eliminate these
interactions ad hoc. This is essentially the method of Lagrange multipliers.
155Since κ is a 1-form in the adjoint representation, Dκ = dκ+ ω ∧ad κ = dκ+ ω ∧ κ+ κ ∧ ω.
144
F. Grassmann numbers
Grassmann numbers are graded-commutative objects crucial to, for instance, the
path integral formulation of theories with fermions and superspace formulations of
supersymmetric theories and supergravity. The Grassmann numbers form a graded
algebra over the real or complex numbers. Here, we introduce some conventions re-
lated to Grassmann numbers, in particular concerning differentiation and integration
with respect to Grassmann variables.
Let θα, α = 1, . . . , n, be Grassmann variables with a multiplication satisfying
θαθβ = −θβθα, (F.1)
generating the Grassmann algebra. The grading of the algebra simply counts
the numbers of Grassmann variables appearing multiplicatively in an expression.
Elements that are even in the grading, for instance 1 and θαθβ, are called Grassmann-
even while elements that are odd in the grading, for instance θα and θαθβθγ, are
Grassmann-odd. For instance in D = 11 supergravity, we use real Grassmann
variables but here we focus on the complex case since the real case is easily inferred.
We define complex conjugation such that
(θαθβ)∗ = θ̄β̇ θ̄α̇. (F.2)
We define differentiation with respect to θα and θ̄α̇ by
∂α :=
∂ ∂ ∂
, θβ := δβ β̇
∂θα ∂θα α
, θ̄ := 0,
∂θα (F.3)
∂̄α̇ :=
∂ ∂ ∂
, θ̄β̇ := δβ̇, θβα̇ := 0,
∂θ̄α̇ ∂θ̄α̇ ∂θ̄α̇
linearity and the graded Leibniz rule. Thus,
∂ (θβ1 . . . θβn) = ∂ (θβ1 . . . θβn) = nδ[β1θβ2α α . . . θβn]. (F.4)∂θα
If we insist on (F.2) being valid for Grassmann operators and functions as well, this
is consistent with
(∂α)∗ = −∂̄α̇ (F.5)
since( then ) ( )
∂ (θβ
∗ ∗
1 βn
α . . . θ ) = nδ[β1θβ2 [β̇α . . . θβn] = nδ 1 θ̄β̇n . . . θ̄β̇2]α̇
= −(−1)n∂̄ (θ̄β̇n . . . θ̄β̇1) = (−1)n+1 [β̇nδ n θ̄β̇n−1 . . . θ̄β̇1]α̇ α̇ , (F.6)
145
F. Grassmann numbers
and an n-cycle has parity (−1)n+1.
In d = 4, we can raise and lower the indices using the antisymmetric αβ. Contrary
to how we raise and lower indices on other quantities, the indices on ∂α and ∂̄α̇ are
raised and lowered from the right. This ensures that
∂αθβ = δαβ ∂̄α̇θ̄β̇ = δα̇β̇ . (F.7)
We also want to be able to integrate over Grassmann variables. For this, we use the
Berezin integral ∫ ∫
dθ θβα := δβα, dθ̄ θ̄β̇α̇ := δ
β̇
α̇. (F.8)
Note that dθα should be interpreted as the integration measure for θα, even though
the index position is different. Also, the dimension of the measure is opposite to
that of the Grassmann variable to make the integral dimensionless. The integration
measure dθα should not be confused with the superdifferential form dθα; it should
always be clear from the context which of the two is being referred to. If we make
a change of variables θα 7→ Sαβθβ we see that the measure has to transform like
dθ 7→ dθ S−1βα β α. This also motivates
dθα = dθ βαβ , (F.9)
similar to how the index on ∂α was raised.
By requiring graded linea∫rity we get
dθ β1α θ . . . θβn = nδ[β1θβ2 . . . θβn]α . (F.10)
Note that this means that the integr∫ation operator and differential operator can beidentified
dθα = ∂α. (F.11)
These considerations go through completely analogously for dotted indices. In
particular, this means that (∫ )∗ ∫
dθα = − dθ̄α̇. (F.12)
In d = 4 dimensions we∫also define ∫
d2θ θ2 := 1, d2θ̄ θ̄2 := 1. (F.13)
This is consistent with ∫ 1 ∫ ∫d2θ = αβ4 dθα dθβ (F.14)
since ∫ ∫ ∫
αβ d [γ θ dθ  θγ δα β γδ θ = 2αβ dθ δ θδ]γδ α β = 2αβ
δγ
γδδαβ = 4. (F.15)
146
F. Grassmann numbers
From the above, it is also clear t(ha∫t )∗ ∫
d2θ = d2θ̄. (F.16)
Hence, we can i∫dentify ∫
d2θ = 1∂α4 ∂α =
1 2 d2 14∂ , θ̄ = 4 ∂̄
α̇
α̇∂̄ =
1
∂̄24 . (F.17)
147
F. Grassmann numbers
148
G. Solving the supergravity
Bianchi identities
In this appendix, we solve the Bianchi identities
D[AT DBC) + T E D D[AB T|E|C) = R[ABC) , D[AHBCDE) + 2T F[AB H|F |CDE) = 0,
(G.1)
see (2.88), of eleven-dimensional supergravity subject to the constraints that the
only nonzero components of HABCD and T CAB are
Habcd, Habγδ (= 2i(Γab)γδ, ) (G.2a)
T γ c c γ [b cde] γ bcde γab , Tαβ = 2i(Γ )αβ, Taβ = Hbcde k1δa (Γ )β + k2(Γa )β , (G.2b)
as in (2.85) and (2.86). We begin by writing the Bianchi identities for all combinations
of bosonic and fermionic indices. Using the constraints, these read
(ABC,D) : D D E D D[ATBC) + T[AB T|E|C) = R[ABC) ,
(αβγ, d) : 0 + 0 = 0, (G.3a)
(αβγ, δ) : 0 + 2iΓe T δ(αβ |e|γ) = R δ(αβγ) , (G.3b)
(aβγ, d) : 0 + 4iT ε d da(β Γ|ε|γ) = Rβγa , (G.3c)
(aβγ, δ) : 2D(βT δ e δ δγ)a + 2iΓβγTea = 2Ra(βγ) , (G.3d)
(abγ, d) : 0 + 2iT εab Γdεγ = 2R dγ[ab] , (G.3e)
(abγ, δ) : 2D T δ δ ε δ δ[a b]γ +DγTab + 2Tγ[a T|ε|b] = Rabγ , (G.3f)
(abc, d) : 0 + 0 = R d[abc] , (G.3g)
(abc, δ) : D T δ + T εT δ[a bc] [ab |ε|c] = 0, (G.3h)
and
(ABCDE) : D F[AHBCDE] + 2T[AB H|F |CDE] = 0,
(αβγδε) : 0 + 0 = 0, (G.4a)
(aβγδε) : 0 − 8Γf(βγΓ|fa|δε) = 0, (G.4b)
(abγδε) : 0 + 0 = 0, (G.4c)
(abcδε) : 0 + 4i(Γf ζδεHfabc − 6T(ε|[a Γbc]|δ)ζ) = 0, (G.4d)
(abcdε) : DεHabcd + 12iT ζ[ab Γcd]ζε = 0, (G.4e)
(abcde) : D[aHbcde] + 0 = 0, (G.4f)
where we have dropped the parentheses around Γ to save space. The result of the
following analysis is presented in appendix G.1.
149
G. Solving the supergravity Bianchi identities
(G.4b)
We start with (G.4b), which does not even contain any dynamical field. Contracting
with all symmetric matrices ΓcIγδ, where cI is a multi-index,
Γc γδΓb Γ = 1Γb Γc 1 1I ) Iγδ b cIδγ b cIγδ(βγ |ba|δε 6 βγ Γbaδε + 6ΓβδΓ Γbaγε + 6ΓβεΓ Γbaδγ
+ 1Γb c 1Iγδ b cIδγ 1 b cIδγ[6 εγΓ Γbaδβ + 6ΓεδΓ Γbaγβ + 6ΓγδΓ Γbaβ1 ]ε
=
= 4(ΓbΓcI6 Γba)(βε) + tr (Γ
cIΓ b b cIba)Γβε + tr (Γ Γ )Γbaβε , (G.5)
and computing the terms
tr (ΓbΓc1)Γ c1ba βε = 32Γ a βε, (G.6a)
tr (ΓbΓc1c2)Γba βε = 0, (G.6b)
tr (ΓbΓc1...c5)Γba βε = 0, (G.6c)
tr (Γc1Γ bba)Γβε = 0, (G.6d)
tr (Γc1c2Γ b [c1 c2]ba)Γβε = 64δa Γβε , (G.6e)
tr (Γc1...c5Γba)Γbβε = 0, (G.6f)
(ΓbΓc1Γ c1ba)(βε) = −8Γ a βε, (G.6g)
(ΓbΓc1c2Γba)(βε) = −16δ[c1 c2]a Γβε , (G.6h)
(ΓbΓc1...c5Γba)(βε) = −(5− 5)Γc1...c5a βε = 0, (G.6i)
we see that (G.4b) is indeed an identity.
(G.4d)
Next, we use (G.4d) to solve for k1 and k2 in (G.2b). Expanding T ζaε , using (G.2b),
and contracting with all symmetric ΓdIεδ
0 = ΓdIεδΓaδεHab1b2b3 + 6H Γd(Iδε(k δc1 Γ...c4 ζc1...c4 1 [b + k Γ c1...c4 ζ)Γ = (G.7)1| ε 2 [b1| ε |b2b3]ζδ )
= tr (ΓdIΓa) [cH 1 |dI | c2c3c4] dI c1...c4ab1b2b3 + 6Hc1...c4 k1δ[b tr (Γ Γ Γ1 b2b3]) + k2 tr (Γ Γ[b Γ ) .1 b2b3]
Calculating the terms
tr (Γd1Γa)H d1ab1b2b3 = 32H b (G.8a)1b2b3 ,
tr (Γd1d2Γa)Hab1b2b3 = 0, (G.8b)
tr (Γd1...d5Γa)Hab1b2b3 = 0, (G.8c)
[c
δ 1[b tr (Γ
|d1|Γc2c3c4]Γb2b3]) = −32 · 6
[c
δ 1 |d1|c2
c3c4] d1[c1 c2c3c4]
[b η δb b ] = 192η δb b b , (G.8d)1 1 2 3 1 2 3
[c
δ 1 tr (Γ|d1d2|Γc2c3c4][b Γb2b3]) = 0, (G.8e)1
η tr (Γ|d1...d5|Γ d1d2d3d4d5[b1|[c1 c2c3c4]|Γb2b3]) = −32 · 5!η[b1|[c1δc c c ]|b b ], (G.8f)2 3 4 2 3
tr (Γd1Γ c1...c4[b Γb2b3]) = 0,1 (G.8g)
150
G. Solving the supergravity Bianchi identities
tr (Γd1d2Γ c1...c4[b Γ ) = 0,1 b2b3] (G.8h)
tr (Γd1...d5Γ[b1|c1...c4|Γb2b3]) = −8η[b1|[c1 tr (Γ
d1...d5Γc2c3c4]|b2b3]) =
= 32 · 5! · 8η δd1d2d3d4d5[b1|[c1 c2c3c4]|b b ]. (G.8i)2 3
Thus, (G.7) becomes
0 = 32Hd1b1b2b3 (1 + 36
1
k1) =⇒ k1 = −36 , (G.9a)
0 = 0, (G.9b)
0 = −192 · 5! [d1d2 ...d5]δ[b b Hb ] (k1 − 8k2) =⇒ =
k1 1
k
1 2 3 2 8 = −288 . (G.9c)
This solves (G.4d) completely.
(G.4e)
From (G.4e), we immediately find
DεH ζabcd = −12iT[ab Γcd]ζε. (G.10)
Here, one could act with another covariant derivative and use the Bianchi identity
of the first type. However, DH = 0 implies D2H = 0 whence no information not
already contained in (G.3) and (G.4) can be extracted in this way. Similar remarks
apply to (G.4f) whence we now turn to (G.3).
(G.3c)
This equation gives R dβγa in terms of Habcd as
i
R b1 b2b3b4 ε b1...b4 εβγad = −72Hb1...b4(8δa Γ (β| + Γa (β| )Γd ε|γ) =
= − i H (24δb1b2Γb3b472 b1...b4 a d + Γ
b1...b4
ad )βγ. (G.11)
Note that the right-hand side is antisymmetric in a d, which means that (G.3c) puts
no constraint on H and that we have solved it completely. Had we not put in H in
the theory by hand, (G.3c) would have constrained some irreducible components of
T βaα .
(G.3b)
Since R is Lie algebra-valued in its two last indices, R δ 1 ad δβγα = 4RβγadΓ α . Hence,
using (G.11), (G.3b) becomes
2H Γb (k δa1Γ...a4 δ + k Γ a1...a4 δa1...a4 (αβ| 1 b |γ) 2 b |γ) ) =
= H a1a2 a3a4 a1...a4 bc δa1...a4(3k1δbc Γ (αβ| + k2Γbc (αβ|)Γ |γ) . (G.12)
Contracting with all symmetric ΓdIβγ, using (G.9c) and
ΓdIβγΓb Γa 1J K δ(αβ γ) = 3(2Γ
bJΓdIΓaK + tr (ΓbJΓdI )ΓaK ) δα , (G.13)
151
G. Solving the supergravity Bianchi identities
we get (
0 = Ha1...a4 32Γa1Γd Γ...a4 + 16 tr (Γa1Γ )Γ...a4I d +I
+ 4ΓbΓ Γ a1...a4d b + 2 tr (ΓbΓd )Γ a1...a4b +I I
− 48Γa1a2Γ Γa3a4 − 24 tr (Γa1a2Γ a3a4d d )Γ ) +I I
− 2Γ a1...a4Γ Γbc − tr (Γ a1...a4bc d bc Γd )Γbc . (G.14)I I
There are eight terms to compute for each number of d-indices. The first one is
Γ[a1Γ Γ...a4] = −Γ a1...a4 [a [ad1 d1 + δ
1 ...a4] 1 ...a4]
d1 Γ − 3δd1 Γ =
= −Γ a1...a4 − 2 [aδ 1Γ...a4]d1 d1 , (G.15a)
Γ[a1Γ Γ...a4] = Γ a1...a4 + 2 [a1Γ ...a4] + 6 [a1Γa2 ...a4]d1d2 d1d2 δ[d δ1 d2] [d1 d +2]
+ 6 [a1aδ 2Γ...a4] − 6 [aδ 1a2d1d2 d1d2 Γ
...a4] =
= Γ a1...a4d1d2 − 4
[a1Γ ...a4]δ[d d ] , (G.15b)1 2
Γ[a1Γ Γ...a4] = −Γ a1...a4 + 5 [a1Γ ...a4] − 15 [a1 aδ 2 ...a4]d1...d5 d1...d5 [d1 ...d δ5] [d Γ ...d ] +1 5
− 60 [a aδ 1 2Γ ...a4] [a1a2 a3 a4][d1d2 ...d − 60δ5] [d Γ1d2 ...d5] +
− 180 [a aδ 1 2a3Γ a4] + 60 [a1a2a3Γaδ 4] + 120δa1a2a3a3[d d d ...d ] [d d d ...d ] [d d d d Γ1 2 3 5 1 2 3 5 1 2 3 4 d5] =
= −Γ a1...a4 [a1 ...a4] [a1a2a3 a4]d1...d5 − 10δ[d Γ1 ...d − 120δ5] [d d d Γ1 2 3 ...d +5]
+ 120δa1a2a3a4[d1d2d Γ .3d4 d5]
(G.15c)
The second one is
tr (Γ[a1Γ ...a4] [a1 ...a4]d1)Γ = 32δd1 Γ , (G.16a)
tr (Γ[a1Γd1d2)Γ...a4] = 0, (G.16b)
tr (Γ[a1Γ ...a4]d1...d5)Γ = 0. (G.16c)
The third one is
ΓbΓ Γ a1...a4 = −(6− 1)Γ a1...a4 − 4 · 7 [aδ 1 ...a4]d1 b d1 d1 Γ =
= −5Γ a1...a4 − 28 [aδ 1Γ...a4]d1 d1 , (G.17a)
ΓbΓ Γ a1...a4d1d2 b = (5− 2)Γ a1...a4
[a1 ...a4] [a1a2 ...a4]
d1d2 − 8(6− 1)δ[d Γd ] − 12 · 7δ1 2 d1d2 Γ =
= 3Γ a1...a4 − 40 [a1Γ ...a4]d1d2 δ[d d ] − 84
[a1aδ 2 ...a4]
1 2 d1d2
Γ ,
(G.17b)
ΓbΓ Γ a1...a4 = −(2− 5)Γ a1...a4 − 20(3− 4) [a1d1...d5 b d1...d5 δ[d Γ
...a4]
1 ...d ] +5
+ 120(4− 3) [a1a2Γ ...aδ 4] [a1a3a3 a4][d1d2 ...d5] + 240(5− 2)δ[d Γ +1d2d3 ...d5]
− 120(6− 1)δa1a2a3a4[d1d2d3d Γd5] =4
= 3Γ a1...a4 + 20 [a1Γ ...a4] + 120 [a1a2Γ ...a4]d1...d5 δ[d1 ...d ] δ5 [d1d2 ...d +5]
+ 720 [a1aδ 3a3 a4] a1a2a3a4[d d d Γ − 600δ1 2 3 ...d5] [d1d d d Γ2 3 4 d5].
(G.17c)
152
G. Solving the supergravity Bianchi identities
The fourth one is
tr (ΓbΓ )Γ a1...a4 = 32Γ a1...a4d1 b d1 , (G.18a)
tr (ΓbΓ a1...a4d1d2)Γb = 0, (G.18b)
tr (ΓbΓ a1...a4d1...d5)Γb = 0. (G.18c)
The fifth term is
Γ[a1a2Γ Γa3a4] = Γ a1...a4 − 2 [aδ 1Γ...a4] + 2 [aδ 1Γ...a4]d1 d1 d1 d1 =
= Γ a1...a4d1 , (G.19a)
Γ[a1a2Γ Γa3a4] = Γ a1...a4 − 4 [aδ 1Γa2 ...a4] [a a a a ]d1d2 d1d2 [d d ] − 4δ
1Γ 2 3 4[d +1 2 1 d2]
− 2 [a1a2Γ...a4] − 2 [a a [a aδ δ 1 2Γ...a4] + 8δ 1 2Γ...a4]d1d2 d1d2 d1d2 =
= Γ a1...a4 + 4 [a aδ 1 2Γ...a4]d1d2 d1d2 (G.19b)
Γ[a1a2Γ Γa3a4] = Γ a1...a4 − 5 [aδ 1Γa2 ...a4] [a1 a2a3 a4]d1...d5 d1...d5 [d1 ...d ] + 5δ5 [d Γ1 ...d +5]
− 20 [a1a2Γ ...a4] [aδ 1a2 ...a4][d d ...d ] − 20δ[d d Γ ...d ] − 80
[a1aδ 2[d d Γ
a3 a4]
1 2 5 1 2 5 1 2 ...d5] +
− 120 [a1a2a3Γ a4] + 120 [a1a2a3Γa4]δ δ + 120δa1a2a3a4[d1d2d3 ...d5] [d1d2d3 ...d5] [d1d2d3d Γ4 d5] =
= Γ a1...a4 + 40 [aδ 1a2Γ ...a4] + 120δa1a2a3a4d1...d5 [d1d2 ...d5] [d1d2d3d Γ4 d5]
(G.19c)
The sixth one is
tr (Γ[a1a2Γ )Γa3a4]d1 = 0, (G.20a)
tr (Γ[a1a2Γ )Γa3a4]d1d2 = −64
[a
δ 1
a2 ...a4]
d1d2 Γ , (G.20b)
tr (Γ[a1a2Γ a3a4]d1...d5)Γ = 0. (G.20c)
The seventh term is
Γ a1...a4Γ Γbc = −(6 · 5− 2 · 6)Γa1...a4 + 4(7 · 6) [aδ 1 ...a4]bc d1 d1 d1 Γ =
= −18Γ a1...a4 + 168 [aδ 1Γ...a4]d1 d1 , (G.21a)
Γ a1...a4Γ Γbc = −(5 · 4− 4 · 5 + 2)Γa1...a4 + 8(6 · 5− 2 · 6) [aδ 1Γ...a4]bc d1d1 d1d2 [d1 d2]+
+ 12(7 · 6) [a aδ 1 2Γ...a4]d1d2 =
= −2Γ a1...a4 − 144 [a1Γ ...a4] + 504 [a1a2 ...a4]d1d2 δ[d1 d2] δd1d2 Γ ,
(G.21b)
Γ a1...a4 bc a1...a4bc Γd1...d5Γ = −(2 · 1− 10 · 2 + 5 · 5)Γd1...d5 +
+ 20(3 · 2− 8 · 3 + 4 · 3) [a1Γ...aδ 4][d1 ...d ]+5
+ 120(4 · 3− 6 · 4 + 3 · 2) [a a ...a ]δ 1 2 4[d d Γ ...d ]+1 2 5
− 240(5 · 4− 4 · 5 + 2 · 1) [a1aδ 2a3 a4][d1d d Γ ...d ]+2 3 5
− 120(6 · 5− 2 · 6 + 1 · 0)δa1a2a3a4[d d d d Γ1 2 3 4 d5] =
= −2Γ a1...a5 [a1 ...a4]d1...d5 − 120δ[d Γ...d ] − 720
[a1a2 ...a4]δ
1 5 [d Γ1d2 ...d ] +5
− 480 [a1a2aδ 3[d d d Γ
a4] − 2160δa1a2a3a4
1 2 3 ...d5] [d1d2d3d Γ4 d5].
(G.21c)
153
G. Solving the supergravity Bianchi identities
Finally, the eighth term is
tr (Γ a1...a4 bcbc Γd1)Γ = 0, (G.22a)
tr (Γ a1...a4bc Γ bcd1d2)Γ = 0, (G.22b)
tr (Γ a1...a4bc Γ bcd1...d5)Γ = −32 a1...a4 bcbc d1...d5Γ =
= −64Γ a1...a4d1...d5 , (G.22c)
where we have used
n! Γa1...a11−n = −Γb1...bn a1...a 11−nbn...b1 , (G.23)
which follows immediately from (B.29). Inserting (G.15) to (G.22) in (G.14) the
parenthesis vanishes in all three cases. Hence, (G.3b) follows from what we already
knew and does not constrain Habcd.
(G.3e)
Since Rγabd is antisymmetric in its last two indices, Rγabd = Rγ[ab]d−Rγ[ad]b−Rγ[bd]a.
Thus, (G.3e) can equivalently be written as
R ε ε εγabd = iTab Γdεγ − iTad Γbεγ − iTbd Γaεγ. (G.24)
Since the right-hand side is antisymmetric in b d, this solves (G.3e) completely.
(G.3d)
This equation can be expressed only in terms of T γab since, from (G.10) and (G.24)
δ = −1Raβγ (4R
cd δ
βacdΓ γ =
= i
)
T ε4 cd Γaεβ − T
ε
ac Γ εdεβ + Tad Γcεβ Γcd δγ , (G.25a)
D T δ = −D H (k δc1Γc2c3c4 δ + k Γ c1c2c3c4 δβ γa β c1...c4 1 a γ 2 a γ ) =
= 12iT ζΓ (k δc1Γc2c3c4 δ + k Γ c1c2c3c4 δ[c ). (G.25b)1c2 c3c4]ζβ 1 a γ 2 a γ
Inserting (G.25) in (G.3d), contracting with all symmetric Γ βγd and suppressing theI
spinor indices, we get
48 [cT[c1c2Γc3c4]Γd (k 1 ...c4] c1...c4 aI 1δb Γ + k2Γb ) + 4 tr (Γ Γd )TI ab =
= TcaΓ Γ ca cab d Γ − 2TI abΓcΓd Γ . (G.26)I
Splitting the first term as
48 [cδ 1b T[c1c2Γc3c4]Γ Γ...c4]d = 24Tbc2Γc3c4Γ c2c3c4d Γ + 24T Γ Γ Γc2c3c4 , (G.27)I I c3c4 bc2 dI
and using (G.9a) and (G.9c), (G.26) becomes
0 = −4T Γ c2c3c4 c2c3c4 c1...c4bc2 c3c4Γd Γ − 4Tc3c4Γbc2Γd Γ − TI I c1c2Γc3c4Γd ΓI b +
+ 24 tr (ΓaΓd )Tab − 6Tc1c2ΓbΓd Γc1c2 + 12Tc2bΓc1Γ Γc1c2d . (G.28)I I I
154
G. Solving the supergravity Bianchi identities
We have six terms to compute for each number of d-indices. First, we decompose
T γab into its irreducible components
T γab = T̃
γ
ab + 2T̃ δ
γ δ γ
[a Γb]δ + T̃ Γabδ , (G.29)
where
T̃ γΓbab γα = 0, T̃ β aa Γβα = 0. (G.30)
When computing the six terms above, we will need to contract one or both bosonic
indices on T with Γ-matrices with various numbers of indices. To not have to redo
the calculation, we compute the general contractions here. First,
TabΓb = (T̃ab + T̃aΓb − T̃ bbΓa + T̃Γab)Γ = 9T̃a + 10T̃Γa. (G.31)
Now,
T Γbc1...cnab = Tab(ΓbΓc1...cn − nηb[c1Γ...cn]) =
= (9T̃ + 10T̃Γ )Γc1...cn − nT [c1Γ...cn]a a a =
= 9T̃ Γc1...cn + 10T̃Γ c1...cn + 10nT̃ δ[c1Γ...cn]a a a +
− nT̃ [c1Γ...cn]a − nT̃ Γ[c1Γ...cn]a + nT̃ [c1ΓaΓ...cn] − nT̃Γ [c1Γ...cn]a =
= −nT̃ [c1Γ...cn]a +
+ (9− n)T̃ Γc1...cn + nT̃ [c1Γ ...cn] + n(n− 1)T̃ [c1δc2Γ...cn]a a a +
+ (10− n)T̃Γ c1...cna + n(11− n)T̃ δ[c1Γ...cn]a . (G.32)
When contracting(both indices, we get )
T Γba ba [a b] a bab c1...c = Tn ab Γ Γc1...cn − 2nδ[c Γ1 ...c ] − n(n− 1)δn [c1c2 Γ...cn] = (G.33)
= n(n− 1)T̃ 2[c1c2Γ...cn] − 2n(10− n)T̃[c1Γ...cn] + (110− 21n+ n )T̃Γc1...cn ,
where we have used
T Γbaab = T b aabΓ Γ = (9T̃a + 10T̃Γa)Γa = 110T̃, (G.34a)
[a b]
Tabδ[c Γ ...c ] = −(n− 1)T̃1 n [c1c2Γ...cn] + (11− 2n)T̃[c1Γ...cn] + (11− n)T̃Γc1...cn ,
(G.34b)
Tabδ
a b
[c1c2 Γ...cn] = T̃[c1c2Γ...cn] + 2T̃[c1Γ...cn] + T̃Γc1...cn . (G.34c)
Since we might get constraints on some of the irreducible components of T , we
calculate all six terms with a single d-index first
T Γ Γ Γc2c3c4 = T (−90δc2 + 54Γc2bc2 c3c4 d1 bc2 d1 d1) =
= −90(T̃bd1 + T̃bΓd1 − T̃d1Γb + T̃Γbd1)+
+ 54(−T̃bd1 + 8T̃bΓd1 + T̃d1Γb + 9T̃Γbd1 + 10T̃ ηbd1) =
= −144T̃bd1 + 342T̃bΓd1 + 144T̃d1Γb + 396T̃Γbd1 + 540T̃ ηbd1 ,
(G.35a)
155
G. Solving the supergravity Bianchi identities
T Γ Γ Γc2c3c4 = T c3c4c3c4 bc2 d1 c3c4(−6Γbd1 − 8η Γ
c3c4 − 14 [c cδ 3Γ 4] + 16 [c3 c4]bd1 b d1 δd1 Γ b + 18δ
c3c4
b d1) =
= +6(2T̃bd1 − 16T̃bΓd1 + 16T̃d1Γb + 72T̃Γbd1)+
+ 8(110T̃ ηbd1)+
− 14(−T̃bd1 + 8T̃bΓd1 + T̃d1Γb + 9T̃Γbd1 + 10T̃ ηbd1)+
+ 16(T̃bd1 + T̃bΓd1 + 8T̃d1Γb − 9T̃Γbd1 + 10T̃ ηbd1)+
+ 18(T̃bd1 + T̃bΓd1 − T̃d1Γb + T̃Γbd1) =
= 60T̃bd1 − 174T̃bΓd1 + 192T̃d1Γb + 180T̃Γbd1 + 900T̃ ηbd1 ,
(G.35b)
T c1...c4c1c2Γc3c4Γd1Γb = T (28Γc1c2c1c2 bd1 − 112
[c1Γc2]δd1 b − 56ηbd1Γ
c1c2) =
= −28(2T̃bd1 − 16T̃bΓd1 + 16T̃d1Γb + 72T̃Γbd1)+
− 112(T̃bd1 + T̃bΓd1 + 8T̃d1Γb − 9T̃Γbd1 + 10T̃ ηbd1)+
+ 56(110T̃ ηbd1) =
= −168T̃bd1 + 336T̃bΓd1 − 1344T̃d1Γb − 1008T̃Γbd1 + 5040T̃ ηbd1 ,
(G.35c)
tr (ΓaΓd1)Tab = −32Tbd1 =
= −32T̃bd1 − 32T̃bΓd1 + 32T̃d1Γb − 32T̃Γbd1 , (G.35d)
Γ Γ Γc1c2 = (Γ c1c2 + Γc1c2 − 4 [c1 c2]T c1c2c1c2 b d1 Tc1c2 bd1 ηbd1 δ[b Γd ] − 2δb d1) =1
= −1(2T̃bd1 − 16T̃bΓd1 + 16T̃d1Γb + 72T̃Γbd1)+
− 1(110T̃ ηbd1)+
+ 4(−T̃bd1 + 7T̃[bΓd1])+
− 2(T̃bd1 + T̃bΓd1 − T̃d1Γb + T̃Γbd1) =
= −8T̃bd1 + 28T̃bΓd1 − 28T̃d1Γb − 38T̃Γbd1 − 110T̃ ηbd1 , (G.35e)
T Γ Γ Γc1c2 = T (−8Γ c2 c2c2b c1 d1 c2b d1 − 10δd1) =
= −8(−T̃bd1 + 8T̃bΓd1 + T̃d1Γb + 9T̃Γbd1 + 10T̃ ηbd1)+
− 10(T̃bd1 + T̃bΓd1 − T̃d1Γb + T̃Γbd1) =
= 18T̃bd1 − 54T̃bΓd1 − 18T̃d1Γb − 62T̃Γbd1 − 80T̃ ηbd1 . (G.35f)
Collecting the terms, (G.28) becomes
0 = 0T̃bd1 − 2592T̃bΓd1 + 720T̃d1Γb − 2580T̃Γbd1 − 11100T̃ ηbd1 . (G.36)
Contracting b d1 immediately gives T̃ = 0 whence 2592T̃bΓd1 = 720T̃d1Γb. Contracting
the latter with Γb we find T̃a = 0 since 2 · 2592 6= 11 · 720.
Having found that T γab = T̃
γ
ab , we move on to the case with two d-indices. When
contracting T with Γ-matrices, we now only get the first terms in (G.32) and (G.33).
156
G. Solving the supergravity Bianchi identities
Hence,
Tbc2Γc3c4Γ Γc2c3c4d1d2 = Tbc2(−26Γ c2d1d2 + 108δ
c2
[d Γd2]) =1
= −26(−2T̃b[d1Γd2]) + 108(T̃b[d1)Γd2] =
= 160T̃(b[d1Γd2], (G.37a)
T Γ Γ Γc2c3c4c3c4 bc2 d1d2 = Tc3c4 4Γ c3c4d1d2b +
+ 12 c3c4 [c3 c4]ηb[d1Γd ] + 10δb Γd1d2 + 24
[c c ]
δ 3 4
2 [d Γ1 d) +2]b
+ 32 [cη δ 2Γc4] − 28δc3c4 c3c4b[d1 d2] b [d Γ1 d2] − 16δd1d2Γb =
= 4(−6T̃[d1d2Γb]) + 12 · 0 + 10(−2T̃b[d1Γd2])+
+ 24(−T̃d1d2Γb + T̃[d1|b|Γd2]) + 32 · 0− 28T̃b[d1Γd2]+
− 16T̃d1d2Γb =
= −48T̃d1d2Γb − 88T̃b[d1Γd2],
( (G.37b)
T Γ Γ Γ c1...c4 = T −8Γ c1c2 c1c2c1c2 c3c4 d1d2 b c1c2 d1d2b + 56ηb[d1Γd ] −)112 [cδ 1[d Γ c2]+2 1 d2]b
+ 224 [cη δ 1b[d1 d ]Γ
c2] + 112δc1c2d1d2Γb =2
= −8(−6T̃[d1d2Γb]) + 56 · 0− 112(−T̃d1d2Γb + T̃[d1|b|Γd2])+
+ 224 · 0 + 112(T̃d1d2Γb) =
= 240T̃d1d2Γb + 144T̃b[d1Γd2],
(G.37c)
tr (ΓaΓd1d2)Tab = 0, ( (G.37d)
T Γ Γ Γc1c2 = T Γ c1c2c1c2 b d1d2 c1c2 bd1d2 + 2η
c1c2 [c1 c2] [c1 c2]
b[d1Γd2] + 2δb Γd1d2 )− 4δ[d Γ1 d ] b+2
+ 4 [cη 1 c2] c1c2 c1c2b[d1δd2]Γ − 4δb [d Γd2] − 2δ1 d1d2Γb =
= 1(−6T̃[d1d2Γb]) + 2 · 0 + 2(−2T̃b[d1Γd2])+
− 4(−T̃d1d2Γb + T̃[d1|b|Γd2]) + 4 · 0− 4(T̃b[d1Γd2])− 2(T̃d1d2Γb) =
= −8T̃d1d2Γb − 16T̃b[d1Γd2],
(G.37e)
T Γ Γ Γc1c2 = T (6Γc2 − 16δc2c2b c1 d1d2 c2b d1d2 [d Γ ) =1 d2]
= 6(2T̃b[d1Γd2])− 16(−T̃b[d1Γd2]) =
= 28T̃b[d1Γd2]. (G.37f)
The terms sum to 0, so we get no new information.
Lastly, we do the calculation with five d-indices
T Γ Γ Γc2c3c4 = T (10Γ c2 + 30δc2bc2 c3c4 d1...d5 bc2 d1...d5 [d Γ1 ...d5]) =
= 10(5T̃b[d1Γ...d5]) + 30(T̃b[d1Γ...d5]) =
= 80T̃b[d1Γ...d5], (G.38a)
157
G. Solving the supergravity Bianchi identities
(
T Γ Γ Γc2c3c4 = T −2Γ c3c4 + 0η Γ c3c4 [c3 c4]c3c4 bc2 d1...d5 c3c4 d1...d5b b[d1 ...d ] − 2δ5 b Γd1...d5 +
+ 0 [cδ 3 Γ c4] + 10δc3c4 [c3 c4][d1 ...d5]b b [d Γ1 ...d5] + 80η)b[d1δd2 Γ...d ] +5
− 40δc3c4 c3c4[d1d Γ2 ...d5]b + 240ηb[d1δd2d3Γ...d5] =
= −2(−30T̃[d1d2Γ...d5]b)− 2(5T̃b[d1Γ...d5]) + 10(T̃b[d1Γ...d5])
+ 80(3ηb[d1T̃d2d3Γ...d5])− 40(T̃[d1d2Γ...d5]b)
− 240(ηb[d1T̃d2d3Γ...d5]) =
= −20T̃b[d1Γ...d5] + 0T̃[d1d2Γ...d5]b + 480ηb[d1T̃d2d3Γ...d5],
( (G.38b)
T Γ Γ Γ c1...c4 = T 4Γ c1c2 + 40η Γ c1c2 [c1 c2]c1c2 c3c4 d1...d5 b c1c2 d1...d5b b[d1 ...d ] − 80δ5 [d Γ1 ...d ]b +5
− 160 [c c ]η δ 1 2 c1c2b[d1 d2 Γ...d5] )− 80δ[d d Γ1 2 ...d5]b+
+ 480η δc1c2b[d1 d2d3Γ...d5] =
= 4(−30T̃[d1d2Γ...d5b]) + 40(−12ηb[d1T̃d2d3Γ...d5])
− 80(5T̃[d1d2Γ...d5b])− 160(3ηb[d1T̃d2d3Γ...d5])
− 80(T̃[d1d2Γ...d5]b) + 480(ηb[d1T̃d2d3Γ...d5]) =
= 120T̃b[d1Γ...d5] − 480T̃[d1d2Γ...d5]b − 480ηb[d1T̃d2d3Γ...d5],
(G.38c)
tr (ΓaΓd1...d5)Tab = 0, ( (G.38d)
T Γ Γ c1c2 c1c2 [c1 c2]c1c2 b d1...d5Γ = Tc1c2 Γbd1...d5 − 2δb Γd1...d5 + 10
[c
δ 1
c2]
[d Γ1 ...d5]b +
+ 5η Γ c1c2 − 10δc1c2 c1c2b[d1 ...d5] b [d Γ + 20δ Γ +1 ...d5] [d)1d2 ...d5]b
− 40 [cη δ 1 c2] c1c2b[d1 d2 Γ...d ] − 60ηb[d1δ5 d2d3Γ...d5] =
= (−30T̃[bd1Γ...d5])− 2(5T̃b[d1Γ...d5]) + 10(5T̃[d1d2Γ...d5b])
+ 5(−60ηb[d1T̃d2d3Γ...d5])− 10(T̃b[d1Γ...d5]) + 20(T̃[d1d2Γ...d5]b)
− 40(3ηb[d1T̃d2d3Γ...d5])− 60(ηb[d1T̃d2d3Γ...d5]) =
= −40T̃b[d1Γ...d5] + 80T̃[d1d2Γ...d5]b − 240ηb[d1T̃d2d3Γ...d5],
(G.38e)
T c1c2c2bΓc1Γd1...d5Γ = Tc2b(0Γ c2d1...d5 − 10δ
c2
[d Γ1 ...d5]) =
= 10T̃b[d1Γ...d5]. (G.38f)
Again, the terms sum to 0. To conclude, we have solved (G.3d) completely and
found that T γab consists of the single irreducible part T̃
γ
ab . Equivalently, this can be
expressed as
T γ abc δab Γ γ = 0, (G.39)
as can be seen from (G.32) and (G.33). This is the equation of motion for T γab .
158
G. Solving the supergravity Bianchi identities
(G.3f)
The equation reads
D T δ = R δ + 2T εT δ − 2D T δγ ab abγ γ[a b]ε [a b]γ , (G.40)
where the right-hand side can be expressed in terms of Rabcd and Habcd by using
(G.2b) and that Rabcd is Lie algebra-valued.
Contracting (G.40) with δγδ , we find
DγT γab + 2T εT
γ
γ[a |ε|b] = 0. (G.41)
Contracting (G.40) with Γb γdδ , using T bcabΓ = −T ca , (G.41) and Rab := R d 156dab ,
− 16Rad = 2T ε γ ε δ b γγ[a T|ε|d] + 2Tγ[a T|ε|b] Γ dδ + 2D[aT
δ b γ
b]γ Γ dδ . (G.42)
Using the previous results, only Γ-algebra remain to get the equation of motion for
R. First,
D δ b γ[aTb]γ Γ dδ = 0, (G.43)
since T δbγ only contains Γ(3) and Γ(5), that is, Γ-matrices with three and five indices.
Next, [ ]
T ε γγ[a T|ε|d] = H H
c1...c4
b1...b4 [k
2
1 tr (δb1Γ...b4η Γ ) + k2 tr (Γ b1...b4[a d]c1 ...c4 2 [a Γd]c1...c4)] =
= H Hc1...c4 −32 · 6k2δb1η δb2b b 4 · 5!3 4b1...b4 1 [a d]c1 c2 c3c4 + 32
2 b1b2b3b4
5 k2ηc1[aδd] c2c3c4 =
= 192 c2c3c4 − 3072H H H H c2c3c4362 c2c3c3[a d] 2882 c2c3c3[a d] =
= 0. (G.44)
The second term in the right-hand side of (G.42) is[ ]
2T εT δΓb γ b bγ[a |ε|b] dδ = tr (TaTbΓ d − TbTaΓ d) = tr T (T Γb − Γba b d dTb) , (G.45)
where we have suppressed spinor indices and treated T γaβ as a matrix (T
γ
a)β . Since
T only contains Γ(3) and Γ(5), we need only keep Γ(3,5,6,8)a -terms in the parenthesis.
Dropping other terms that do not contribute to the trace, indicated by ' below, we
compute ( )
T Γb = H [k δa1Γ...a4Γb + k Γ a1...a4 bb d a1...a4 1 b d 2 b Γ d ' ]
' Ha1...a4((3k − 28k )δa1Γ...a41 2 d + (−k1 )+ 6k )Γ a1...a42 d , (G.46a)
Γb T = H [k δa1Γb Γ...a4d b a1...a4 1 b d + k2Γb Γ a1...a4d b ' ]
' Ha1...a4 (−3k1 − 28k2)δa1d Γ...a4 + (−k a1...a41 − 6k2)Γd , (G.46b)
156We contract the outermost indices of the curvature tensor to define the Ricci tensor due to the
right-action convention. Thus, R gets its usual sign: for instance, R < 0 for AdS.
159
G. Solving the supergravity Bianchi identities
whence[ ] [ ]
tr T (T Γb − Γb T[ ) = tr T H (6k δc1Γ...c4 + 12k Γ c1...c4a b d d b a c1...c4 1 d 2 d ) = ]
= H Hc1...c4 6k2δe1η tr (Γe2e3e4Γ ) + 12k2 tr (Γ e1...e4e1...e4 1 a dc1 c2c3c4 2 a Γdc1...c4) =
= −3! · 32 · 6k2H c2c3c41 ac2c3c4Hd +
+ 4! · 32 · 12k2 22ηabH − 4! · 32 · 4 · 12k2H c2c3c42 dc2c3c4Ha =
= −4H c2c3c43 ac2c3c4Hd +
1
9ηadH
2, (G.47)
where H2 := H Ha1...a4a1...a4 and we used, in the second to last step,
tr (Γ e1...e4a Γ
[e
dc1...c4) = η tr (Γ
e1...e4Γ 1 ...e4]ad c1...c4)− 4δd tr (Γa Γc1...c4). (G.48)
Inserting the above in (G.42), we find
− 16 1 4Rad = 9η H
2
ad − 3H
c2c3c4
ac2c3c4Hd . (G.49)
To write this with the Einstein tensor in the left-hand side, we contract this and find
−16R = −H2/9. Thus,
Rab −
1
2ηabR =
1 cde 1
12HacdeHb − 96ηabH
2. (G.50)
We now turn to the equation of motion for H. The strategy is similar to the R-
equation but we contract (G.40) with other combinations of Γ-matrices. First contract
with Γ γcδ and then antisymmetrise a b c. Since the curvature tensor is Lie algebra-
valued in its last pair of indices, R δΓ γ[ab|γ| c]δ = 0. Similarly, 2D δ
γ
[aTb|γ| Γc]δ = 0 since
T δ (3,5)bγ only contains Γ . Lastly,
2T εT δΓ γ = 2k2H H tr (Γ d1...d4Γ e1...e4γ[a |ε|b c]δ 2 d1...d4 e1...e4 [a d Γc]) =
= −64k2 d1...d4 e1...e42Hd1...d4He1...e4a b c =
= − 1  d1...d4e1...e41296 abc Hd1...d4He1...e4 , (G.51)
whence the contracted and antisymmetrised (G.40) becomes
DγT δ[ab Γ γ =
1
 d1...d4e1...e4c]δ 1296 abc Hd1...d4He1...e4 . (G.52)
Now we contract (G.40) with Γb γcdδ and then antisymmetrise a c d. The term
containing R is again zero since Rab is Lie algebra-valued. Using (G.52), the term in
the left-hand side of (G.40) gives
D δ b γ δ γ 1 e1...e4f1...f4γT[a|b| Γ cd]δ = −2DγT[ac Γd]δ = −648acd He1...e4Hf1...f4 . (G.53)
The last term in the right-hand side of (G.40) splits into
D T δ b γb [a|γ| Γ cd]δ = k1D
bH e2e3e4[a|e2e3e4 tr (Γ Γb|cd]) = −32 · 6k1DbHabcd =
= −16Db3 Hbacd, (G.54)
160
G. Solving the supergravity Bianchi identities
and
−D T δΓb γ[a| bγ |cd]δ = −k1D H
e2e3e4
[a| be2e3e4 tr (Γ Γb|cd]) = 0. (G.55)
The second term in the right-hand side of (G.40) splits into T εT δΓb γγ[a| bε |cd]δ and
T εT δΓb γγb ε[a cd]δ . Since the components of T only contain Γ(3) and Γ(5) the only po-
tentially nonvanishing contributions come from products Γ(3)Γ(5)Γ(3) and Γ(5)Γ(3)Γ(3).
However,
H e1 ...e4 f1...f4 be1...e4Hf1...f4 tr (δ[aΓ Γ|b| Γ cd]) = 0, (G.56)
since this is really only 9 Γ-matrices due to the contracted b’s. On the other hand,
H H tr (Γ e1...e4δf1Γ...f4Γb ) = 32 e1...e4f1...f4e1...e4 f1...f4 [a |b| cd] acd He1...e4Hf1...f4 , (G.57)
whence
T εT δΓb γ = − 1  e1...e4f1...f4γ[a| bε |cd]δ 324 acd He1...e4Hf1...f4 . (G.58)
T εγb T
δ b γ
ε[a Γ cd]δ is similar. As above, only the term where the b-index is on the δ is
nonzero. Hence
T εT δΓb γ = k k H H tr (δe1Γ...e4Γ f1...f4 bγb ε[a cd]δ 1 2 e1...e4 f1...f4 b [a Γ cd]) = (G.59)
= − 1  e1...e4f1...f4324 acd He1...e4Hf1...f4 . (G.60)
Inserting the above terms in (G.40), we get
Dd 1H = −  e1...e4f1...f4dabc 1152 abc He1...e4Hf1...f4 . (G.61)
The Bianchi identity of the second type
Although the Bianchi identity of the second type, DR AB = 0, is not independent of
the above, it can be used to extract DεRabcd. Alternatively, this could be obtained
from (G.40) by applying another covariant derivative and using the Bianchi identity
of the first type. From (2.88b),
D ζεRabcd + 2D[bR|ε|a]cd + Tab Rζεcd + 2T
ζ
ε[a R|ζ|b]cd = 0. (G.62)
Using (G.2b), (G.11) and (G.24), we see that DεRabcd can be expressed in terms of
Habcd, T γab and their Da-derivatives.
161
G. Solving the supergravity Bianchi identities
G.1 Solution to the supergravity Bianchi
identities
To conclude, we have found ((G.2b) and (G.9))
1 ( ] )γ [b cde γT bcde γaβ = −288Hbcde 8δa Γ β + Γa β , (G.63)
the Bianchi identities for the field strengths T γ dab , Rabc and Habcd ((G.3g), (G.3h)
and (G.4f))
D T δ[a bc] + T ε δ[ab T|ε|c] = 0, (G.64a)
R d[abc] = 0, (G.64b)
D[aHbcde] = 0, (G.64c)
the equations of motion ((G.39), (G.50) and (G.61))
T γ abc δab Γ γ = 0, (G.65a)
1 1
R cde
1 2
ab − 2ηabR = 12HacdeHb − 96ηabH , (G.65b)
Dd 1H = −  e1...e4f1...f4dabc 1152 abc He1...e4Hf1...f4 , (G.65c)
as well as equations relating the Dα-derivatives of the field strengths to their values
and Da-derivatives ((G.10), (G.40) and (G.62)). This implies that only the θ = 0
components of the field strengths are independent component fields.
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