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UBC Theses and Dissertations

Emergence of Riemannian manifolds from graphs and aspects of Chern-Simons theory Chen, Si

Abstract

This thesis is divided into three chapters. Chapter 1 is a study of statistical models of graphs, in order to explore possible realizations of emergent manifolds. Graphs with given numbers of vertices and edges are considered, governed by a Hamiltonian that favors graphs with near-constant valency and local rotational symmetry. The model is simulated numerically in the canonical ensemble. It is found that the model exhibits a first-order phase transition, and that the low energy states are almost triangulations of two dimensional manifolds. The resulting manifold shows topological "handles" and surface intersections in a higher embedding space as well as non-trivial fractal dimension. The model exhibits a phase transition temperature of zero in the bulk limit. We explore the effects of adding long-range interactions to the model, which restore a finite transition temperature in the bulk limit. In Chapter 2, aspects of Chern-Simons theory are studied. The relations between Chern-Simons theory, a model known as BF theory named after the fields that appear in the actions, and 3D gravity, are explored and generalized to the case of non-orientable spacetime manifolds. U(1) Chern-Simons theory is quantized canonically on orientable manifolds, and U(1) BF theory is similarly quantized on non-orientable manifolds. By requiring the quantum states to form a representation of the deformed holonomy group and the deformed large gauge transformation group, we find that the mapping class group of the spacetime manifold can be consistently represented, provided the prefactor k of the Chern-Simon action satisfies quantization conditions which in general are non-trivial. We also find a k 1/k duality for the representations. Motivated by open questions about interpreting the finite size results from Chapter 1, models of finite size scaling for systems with a first-order phase transition are discussed in Chapter 3. Three physics models -- the Potts model, the Go model for protein folding, and the graph model in Chapter 1 -- are simulated. Several finite size scaling models, including three functional forms to fit the energy distributions, and a capillarity model, are compared with simulations of the corresponding physics models.

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