ORIGINAL

Abstract

In this thesis, we investigate the mathematical correspondence between three modularly invariant systems -- a chaotic free particle on the hyperbolic Fundamental Domain, 2+1 dimensional quantum gravity on the torus universe model, and 1+1 conformal field theories living on a torus manifold. We aim to use formalisms developed for the free particle system, and the torus CFT to develop our understanding of the classical and quantum dynamics of the Anti de-Sitter torus universe evolving under Einstein gravity. To this end, we show that the classical dynamics of the quantum gravity system are identical to the chaotic free particle, with a different parametrization of the time variable. The classical torus universe has a maximal volume, and its ergodicity implies that the shape changes chaotically in time. Using the Maass forms introduced to solve the quantized free particle system, we formulate a set of separable solutions to the quantized gravity system. As Maass forms do not have an explicit construction, we invoke the AdS/CFT correspondence to explore other solutions constructed from CFT partition functions. We introduce a \( T \bar T \) deformation into our CFT system, and show that the deformed partition functions are solutions to the AdS Hamiltonian. Furthermore, we use the deformed CFT formalism to develop a method for time-evolving any modular form under the AdS Hamiltonian, resulting in non-separable, time-dependant eigenstates. These waveforms give us a better understanding of the quantum dynamics of the torus universe, and represent a new entry in the AdS/CFT dictionary between gravity in Anti de-Sitter space-time and ( \(T \bar T \) deformed) conformal field theory.