Noncommutative geometry and matrix models

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Date
2017
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University of Alabama Libraries
Abstract

Noncommutative geometry is a proposed description of spacetime at energies near or beyond the Planck scale. A particularly intriguing representation of a noncommutative algebra is the matrix representation. Matrix models have been shown to include geometry, gravity and nonperturbative aspects of string theory. We wish to use matrix models to study noncommutative aspects of cosmological models. We begin by studying a matrix model analog of the BTZ black hole, which is a solution of 2+1 general relativity. We propose a Chern-Simons type theory constructed from finite dimensional matrices. After introducing the notion of a rotationally invariant boundary, we count the degeneracy of physical degrees of freedom associated with the boundary. This number coincides with the number of degrees of freedom needed to reproduce the Bekenstein-Hawking entropy relation. Next we study rotationally invariant solutions to d dimensional matrix models. We find d-1 dimensional solutions which have desirable cosmological features, in particular, we find matrix models that resolve cosmological singularities. In the last section, we restrict our attention to a Lorentzian analog of the complex projective plane, which is a four-dimensional solution of an eight-dimensional matrix action.

Description
Electronic Thesis or Dissertation
Keywords
Physics
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