Applications of Random Graphs to 2D Quantum Gravity

TL;DR

This thesis explores two-dimensional quantum gravity using random graph models, demonstrating a UV dimensional reduction in simplified models and challenging the Seiberg-Shih conjecture through higher genus amplitude calculations.

Contribution

It introduces a continuum limit model showing UV dimensional reduction and provides evidence against the Seiberg-Shih conjecture using random graph formulations.

Findings

Effective spacetime dimension reduces in UV in the simplified model.

Higher genus amplitudes with a single boundary violate the Seiberg-Shih conjecture.

Coupling to matter allows multiple boundary states in dynamical triangulation.

Abstract

The central topic of this thesis is two dimensional Quantum Gravity and its properties. The term Quantum Gravity itself is ambiguous as there are many proposals for its correct formulation and none of them have been verified experimentally. In this thesis we consider a number of closely related approaches to two dimensional quantum gravity that share the property that they may be formulated in terms of random graphs. In one such approach known as Causal Dynamical Triangulations, numerical computations suggest an interesting phenomenon in which the effective spacetime dimension is reduced in the UV. In this thesis we first address whether such a dynamical reduction in the number of dimensions may be understood in a simplified model. We introduce a continuum limit where this simplified model exhibits a reduction in the effective dimension of spacetime in the UV, in addition to having rich cross-over behaviour. In the second part of this thesis we consider an approach closely related to causal dynamical triangulation; namely dynamical triangulation. Although this theory is less well-behaved than causal dynamical triangulation, it is known how to couple it to matter, therefore allowing for potentially multiple boundary states to appear in the theory. We address the conjecture of Seiberg and Shih which states that all these boundary states are degenerate and may be constructed from a single principal boundary state. By use of the random graph formulation of the theory we compute the higher genus amplitudes with a single boundary and find that they violate the Seiberg-Shih conjecture. Finally we discuss whether this result prevents the replacement of boundary states by local operators as proposed by Seiberg.