Date of Award

Spring 5-16-2011

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Tony Pantev

Abstract

We study moduli spaces of boundary conditions in 2D topological field theories. To a compactly generated linear infinity-category X, we associate a moduli functor M_X parametrizing compact objects in X. The Barr-Beck-Lurie monadicity theorem allows us to establish the descent properties of M_X, and show that M_X is a derived stack. The Artin-Lurie representability criterion makes manifest the relation between finiteness conditions on X, and the geometricity of M_X. If X is fully dualizable (smooth and proper), then M_X is geometric, recovering a result of Toën-Vaquie from a new perspective. Properness of X does not imply geometricity in general: perfect complexes with support is a counterexample. However, if X is proper and perfect (symmetric monoidal, with ``compact = dualizable''), then M_X is geometric.

The final chapter studies the moduli of Noncommutative Calabi-Yau Spaces (oriented 2D-topological field theories). The Cobordism Hypothesis and Deligne's Conjecture are used to outline an approach to proving the unobstructedness of this space, and constructing a Frobenius structure on it.

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