Globalization of perturbative Chern-Simons theory on the moduli space of flat connections in the BV formalism

Abstract

We study the perturbative path integral of Chern-Simons theory (the effective BV action on zero-modes) in Lorenz gauge, expanded around a (possibly non-acyclic) flat connection, as a family over the smooth irreducible stratum ${\mathcal{M}}^{\prime }\subset \mathcal{M}$ of the moduli space of flat connections. We prove that it is horizontal with respect to the Grothendieck connection up to a BV-exact term. From it, we construct a volume form on ${\mathcal{M}}^{\prime }$ - the "global partition function" - whose cohomology class is independent of the metric, and so is a 3-manifold invariant. As an element of the construction, we construct an extension of the perturbative partition function to a nonhomogeneous form on the space of triples $(A,A^{\prime },g)$ consisting of (1) a "kinetic" flat connection $A$ around which Chern-Simons action is expanded, (2) a "gauge-fixing" flat connection $A^{\prime }$, (3) a metric $g$. This extension is horizontal with respect to an appropriate Gauss-Manin superconnection (which involves the BV operator as a degree zero component).

115pages, 15 figures. v2: 1-extended smoothness condition has been removed, explicit formula for the Grothendieck connection added v3: changed according to referee's suggestions