L-infinity homotopies and a discretization of Chern-Simons theory
We will describe a "discretization of Chern-Simons theory", using Whitney forms and homotopies between L-infinity algebras. Derived moduli problems can be (locally) described by L-infinity algebras, and it is often of interest to understand a moduli problem in terms of a "governing L-infinity algebra". This description in terms of a governing L-infinity algebra may depend on a choice of a homotopy model for such a L-infinity algebra. In this particular example, we utilize the well-known Dupont homotopy operator to define a discretization of the infinite-dimensional DGLA controlling the moduli problem relevant to Chern-Simons theory. In doing so, we can describe an ( ind-) finite-dimensional model for a derived enhancement of the moduli space of simple flat connections on an oriented closed 3-manifold M equipped with a triangulation K. This derived moduli space has a -1 shifted symplectic structure. In our discretized model, we can describe a guise of prequantization data for this -1 shifted symplectic structure, and orientation (in the spin sense) for our moduli space. This data determines local presentations of our moduli space as critical loci, and a way to glue a perverse sheaf of vanishing cycles of these critical loci