Quantum Chern-Simons theories on cylinders: BV-BFV partition functions

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 [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.