  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.
