Title:

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

It has been proposed that all strings and branes are contained in the nonlinear realisation of the KacMoody algebra E_{11} with the semidirect product of its fundamental representation l_{1}, denoted E_{11} ⊗_{s }l_{1}. In the process of building the nonlinear realisation, the dynamics of strings and branes are naturally invariant under symmetries which are determined by the relevant decomposition of the E_{11} algebra. At low levels, these dynamics are exactly those of D dimensional maximal supergravity, constructed from the subgroup GL(D) ⊗E_{11D}, 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 E_{11} ⊗_{s }l_{1} 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 E_{11} algebra, and additionally for the A_{1}^{+++} 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 E_{11} covariant way. The final part of the thesis constructs the nonlinear realisation in the 7 dimensional decomposition of E_{11} 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 oneform, and the twoform 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 1form and 2form found in the previous section.
