Algebraic Structures in M-theory

Abstract

In this thesis we reformulate the bosonic sector of eleven dimensional supergravity as a simultaneous nonlinear realisation based on the conformal group and an enlarged affine group called G_{11}. The vielbein and the gauge fields of the theory appear as space-time dependent parameters of the coset representatives. Inside the corresponding algebra g_{11} we find the Borel subalgebra of e_7, whereas performing the same procedure for the Borel subalgebra of e_8 we have to add some extra generators. The presence of these new generators lead to a new formulation of gravity, which includes both a vielbein and its dual. We make the conjecture that the final symmetry of M-theory should be described by a coset space, with the global and the local symmetry given by the Lorentzian Kac-Moody algebra e_{11} and its subalgebra invariant under the Cartan involution, respectively. The pure gravity itself in $D$ dimensions is argued to have a coset symmetry based on the very extended algebra A^{+++}_{D-3}. The tensor generators of a very extended algebra are divided into representations of its maximal finite dimensional subalgebra. The space-time translations are thought to be introduced as weights in a basic representation of the Kac-Moody algebra. Some features of the root system of a general Lorentzian Kac-Moody algebra are discussed, in particular those related to even self-dual lattices.