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UBC Theses and Dissertations

(2+1)-dimensional gravity over a two-holed torus, T²#T² Newbury, Peter R.


Research into the relationships between General Relativity, topology, and gauge theory has, for the most part, produced abstract mathematical results. This thesis is an attempt to bring these powerful theories down to the level of explicit geometric examples. Much progress has recently been made in relating Chern-Simons gauge field theory to (2+1)-dimensional gravity over topologically non-trivial surfaces. Starting from the dreibe informalism, we reduce the Einstein action, a functional of geometric quantities, down to a functional only of the holonomies over flat compact surfaces, subject to topological constraints. We consider the specific examples of a torus T2, and then the two-holedtorus, T2#T2. Previous studies of the torus are based on the fact that the torus, and onlythe torus, can support a continuous, non-vanishing tangent vector field. The results we produce here, however, are applicable to all higher genus surfaces. We produce geometric models for both test surfaces and explicitly write down the holonomies, transformations in the Poincare group, ISO(2,1). The action over each surface is very nearly canonical, and we speculate on the phase space of dynamical variables. The classical result suggests the quantum mechanical version of the theory exists on curved space time.

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