Globalization of Perturbative Chern-Simons Theory on the Moduli Space of Flat Connections in the BV Formalism

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 $M^{\text{'}}\subset M$ 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 $M^{\text{'}}$ -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 $(A,A^{\text{'}},g)$ consisting of (1) a "kinetic" flat connection A around which Chern-Simons action is expanded, (2) a "gauge-fixing" flat connection $A^{\text{'}}$ , (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).