Scalar Potential of the Squashed Seven-sphere Stability, the Swampland and Kähler Geometry

Abstract

Finding a consistent theory of quantum gravity has been a long-standing problem in physics and all attempts made share a common feature: quantum gravitational effects become important only at very high energies, much higher than the energy scales at which current particle physics are done. Thus, connecting a particle phenomenological description to a theory of quantum gravity has proven to be incredibly difficult. A recently proposed solution to this is known as the Swampland program which aims to construct a set of conjectures dictating how theories consistent with quantum gravity must behave. Theories not fulfilling these conjectures are said to lie in the ”swampland”, while consistent theories are said to lie in the ”landscape”. In this thesis the compactification of 11-dimensional supergravity on a squashed sevensphere is studied. This scenario seems to contradict the Non-AdS SUSY conjecture, at least at a perturbative level, and a better of understanding of it is therefore essential for the swampland program. A thorough description of the geometry of a squashed seven-sphere is provided, which then is extended to spacetime dependent parameters of the sphere described by scalar fields. The stability in terms of these scalars is analysed, which leads to the contradiction mentioned above. Finally, an attempt to generalise the setup using complex geometry is done, treating the scalar fields as coordinates on a Kähler manifold. It is found that a simple and natural coset structure for the Kähler manifold, SL(2,R)/U(1), might not be the correct ansatz. Hence, further studies of the full 11-dimensional theory would be beneficial in order to determine the true coset structure.