E theory

: its algebra, gauge fixing, and 7D equations of motion

Abstract

It has been proposed that all strings and branes are contained in the non-linear realisation of the Kac-Moody algebra E₁₁ with the semi-direct product of its fundamental representation l₁, denoted E₁₁ ⊗_s l₁. In the process of building the non-linear realisation, the dynamics of strings and branes are naturally invariant under symmetries which are determined by the relevant decomposition of the E₁₁ algebra. At low levels, these dynamics are exactly those of D dimensional maximal supergravity, constructed from the subgroup GL(D) ⊗E_11-D, with the fields and coordinates of the theory appearing as parameters of the generators of the algebra. 
In this thesis, we first construct relevant decompositions of the E₁₁ ⊗_s l₁ algebra. We begin by extending previous calculations of the algebra of the 11D decomposition up to levels 5 and 6 in the adjoint representation. We then calculate the algebra of the 7D decomposition in the adjoint representation, vector representation, and Cartan involution invariant subalgebra up to level 5. The final algebra that we calculate is the 10 dimensional IIB Cartan involution invariant subalgebra up to level 4. 
We then build a tangent space metric in the 11D, 5D, and 4D decompositions of the E₁₁ algebra, and additionally for the A₁⁺⁺⁺  algebra, which leads to a description of 4 dimensional gravity. These tangent space metrics are then used to build a set of gaugefixing conditions which allow gauge symmetries to be fixed in an E₁₁ covariant way. 
The final part of the thesis constructs the non-linear realisation in the 7 dimensional decomposition of E₁₁ at low levels, using the algebra derived in an earlier chapter. This is then used to find a set of dynamical equations, which, when truncated, agree exactly with the equations of motion for the graviton, the scalar, the one-form, and the two-form in 7D maximal supergravity. Finally, we derive the duality relations of the scalar and the graviton with their corresponding dual fields, from the duality relations of the 1-form and 2-form found in the previous section.

Date of Award	1 Jun 2019
Original language	English
Awarding Institution	King's College London
Supervisor	Peter West (Supervisor), Sameer Murthy (Supervisor) & Nikolay Gromov (Supervisor)