Quantum Chern–Simons Theories on Cylinders: BV-BFV Partition Functions

Abstract

We compute partition functions of Chern–Simons type theories for cylindrical spacetimes I times Sigma , with I an interval and dim Sigma = 4l+2, in the BV-BFV formalism (a refinement of the Batalin–Vilkovisky formalism adapted to manifolds with boundary and cutting–gluing). The case dim Sigma = 0 is considered as a toy example. We show that one can identify—for certain choices of residual fields—the “physical part” (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton–Jacobi action computed in the companion paper (Cattaneo et al., Constrained systems, generalized Hamilton–Jacobi actions, and quantization, arXiv:2012.13270), without any quantum corrections. This Hamilton–Jacobi action is the action functional of a conformal field theory on Sigma . For dim Sigma = 2, this implies a version of the CS-WZW correspondence. For dim Sigma = 6, using a particular polarization on one end of the cylinder, the Chern–Simons partition function is related to Kodaira–Spencer gravity (a.k.a. BCOV theory); this provides a BV-BFV quantum perspective on the semiclassical result by Gerasimov and Shatashvili.