Abstract
We discuss the historical evidence for the conjecture that the nonlinear realisation of the KacMoody algebra E8+++, which will be referred to as E11, describes the extension of elevendimensional supergravity known as Mtheory. The algebraic background is presented and some of the consequences of the conjecture are explored. In particular, we present the construction of halfBPS branes using the E11 solution generating element with lowlevel roots before discussing the role of general roots in the solution generating method. The correspondence between roots within E11 and brane solutions is used to reproduce the rules for brane congurations which lead to bound and marginal states. Using these rules, we present the embeddings of simplylaced algebras within E11 with their supergravity solution interpretation.The use of nonlinear sigmamodels with symmetric spaces to describe the hidden symmetries of gravity, as well as extended gravitational theories, is reviewed. Examples include: the original work of Ehlers, the more general construction of axisymmetric stationary solutions and theories which are consistent truncations of elevendimensional supergravity. It is shown that these symmetries generate nonlinear transformations of solutions, of which many have wellunderstood physical interpretations. Applications of the target space symmetries are described and used to generate and transform between solutions.
Motivated by the use of null geodesics on symmetric spaces to describe solutions of theories with hidden symmetries, we construct onedimensional sigma models. These models are built with cosets of normal real forms of the nite, simplylaced algebras and general involution invariant subalgebras. The SL(n;R)=SO(p; q) models, with n 4, are reproduced before we present our work with general An(n), Dn(n) and En(n). Solutions are presented for algebras of low rank and used, iteratively, to construct solutions of arbitrary rank. These models are embedded into A+++ n(n) algebras to generate dualgravity solutions and E11(11) to generate supergravity solutions with the congurations which are classied in the preliminary material.
A special set of maximal codimension A2(2) solutions embedded within A+++ n(n) are considered which form a telescopic series of nite algebras within A+ 2(2). These are shown to interpolate between known gravity solutions and exotic objects. We discuss and interpret the gravitational theory extensions which these objects are solutions of. We show that bound states of branes with various algebraic congurations are shown to possess symmetries which are easily described within the sigma model. Several examples are provided and we discuss the role of these symmetries in solution generation.
Date of Award  2014 

Original language  English 
Awarding Institution 

Supervisor  Paul Cook (Supervisor) & Peter West (Supervisor) 