Sigma-models in Kac-Moody algebras and M-theory

Student thesis: Doctoral ThesisDoctor of Philosophy


We discuss the historical evidence for the conjecture that the non-linear realisation of the Kac-Moody algebra E8+++, which will be referred to as E11, describes the extension of eleven-dimensional supergravity known as M-theory. The algebraic background is presented and some of the consequences of the conjecture are explored. In particular, we present the construction of half-BPS branes using the E11 solution generating element with low-level 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 simply-laced algebras within E11 with their supergravity solution interpretation.

The use of non-linear sigma-models 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 eleven-dimensional supergravity. It is shown that these symmetries generate non-linear transformations of solutions, of which many have well-understood 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 one-dimensional sigma models. These models are built with cosets of normal real forms of the nite, simply-laced 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 dual-gravity solutions and E11(11) to generate supergravity solutions with the congurations which are classied in the preliminary material.

A special set of maximal co-dimension 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 Award2014
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorPaul Cook (Supervisor) & Peter West (Supervisor)

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