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

We compute partition functions of Chern-Simons type theories for cylindrical spacetimes $$ , with I an interval and $$ , in the BV-BFV formalism (a refinement of the Batalin-Vilkovisky formalism adapted to manifolds with boundary and cutting-gluing). The case $$ is considered as a toy example. We show that one can identify-for certain choices of residual fields-the "physical part" (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton-Jacobi action computed in the companion paper (Cattaneo et al., Constrained systems, generalized Hamilton-Jacobi actions, and quantization, arXiv:2012.13270), without any quantum corrections. This Hamilton-Jacobi action is the action functional of a conformal field theory on $$ . For $$ , this implies a version of the CS-WZW correspondence. For $$ , 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.