Moduli Problems in Derived Noncommutative Geometry
نویسندگان
چکیده
Moduli Problems in Derived Noncommutative Geometry Pranav Pandit Tony Pantev, Advisor We study moduli spaces of boundary conditions in 2D topological field theories. To a compactly generated linear∞-category X , we associate a moduli functor MX parametrizing compact objects in X . Using the Barr-Beck-Lurie monadicity theorem, we show that MX is a flat hypersheaf, and in particular an object in the ∞-topos of derived stacks. We find that the Artin-Lurie representability criterion makes manifest the relation between finiteness conditions on X , and the geometricity of MX . If X is fully dualizable (smooth and proper), then MX is geometric, recovering a result of Toën-Vacqiue 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 MX is geometric. The final chapter studies the moduli of oriented 2D topological field theories (Noncommutative Calabi-Yau Spaces). The Cobordism Hypothesis, Deligne’s conjecture, and formal En-geometry are used to outline an approach to proving the unobstructedness of this space and constructing a Frobenius structure on it.
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