Applications of Kac-Moody Algebras to Gravity and String Theory

It has previously been proposed that the the theory of strings and branes possesses a large symmetry group generated by the Kac-Moody algebra $E_{11}$. It has also previously been proposed that the the theory of gravitation in four dimensions possesses a large symmetry group generated by the Kac-Moody algebra $A_1^{+++}$. These symmetry groups predict the existence of an infinite collection of fields in each theory, including a field that can be interpreted as a dual graviton. In this thesis we will discuss developments related to these approaches to string theory and gravity. We begin by reviewing the method of nonlinear realizations, then introducing the Kac-Moody algebra $A_1^{+++}$, along with its first fundamental representation representation which introduces space-time, to low levels. We will then consider a nonlinear realization involving their semi-direct product, and use the associated Cartan forms to construct dynamics for the low level fields of this nonlinear realization. We will then propose duality relations between fields of the theory at different levels, and propose a nonlinear equation of motion for the dual graviton arising from $A_1^{+++}$. We then consider the analogous results for $E_{11}$, and describe a proposed nonlinear dual graviton equation in the context of string theory. We next discuss irreducible representations of $E_{11}$. We will construct isotropy groups associated to massive and massless particles, branes such as the M2 and M5 branes, and the IIA string. Finally we present an approach for introducing spinorial representations of the Kac-Moody algebra $E_{11}$.