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

Abstract

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 $${\mathcal {M}}'\subset {\mathcal {M}}$$ M ′ ⊂ 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 $${\mathcal {M}}'$$ M ′ —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 $$(A,A',g)$$ ( A , A ′ , g ) consisting of (1) a “kinetic” flat connection A around which Chern–Simons action is expanded, (2) a “gauge-fixing” flat connection $$A'$$ A ′ , (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).