A discretization of a derived moduli space arising from gauge theory

One could say that an ulterior motive for this talk is to understand, through an example, how to construct derived moduli spaces out of L-infinity algebras (which are homotopical generalizations of Lie algebras). L-infinity algebras produce derived moduli spaces or stacks, and we will see how one can change (up to homotopy) the underlying L-infinity algebra and compare their corresponding derived spaces/stacks. In this talk, we will use a discretization of matrix valued differential forms to produce an ind-finite (i.e., inductive limit of finite dimensional pieces) model of a derived moduli space whose classical locus is the moduli space of complex vector bundles with flat connections on a (closed, oriented) topological 3-manifold M with a triangulation K_M.

It is known that a natural derived enhancement of the moduli space of flat bundles of a closed, oriented 3-manifold should carry a -1 shifted symplectic structure. We will see how such a -1 shifted symplectic structure can be constructed on our ind-finite model out of L-infinity data. In fact, this -1 shifted symplectic structure can be generated out of the Chern–Simons action functional and some homotopy data associated with it. One can think of this lift of the -1 shifted symplectic structure as an example of -1 shifted prequantization. The existence of this associated data implies that a certain homotopy version of the Chern–Simons functional induces a d-critical structure on the classical moduli space of vector bundles with flat connection on M.

This talk is based on ongoing joint work with Kai Behrend.

Time: 4:10-5:10pm

Location: MATH 126

Seminar Website: https://yifeng-huang-math.github.io/seminar_ubc_ag_23w.html