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

TL;DR

This paper computes Chern-Simons partition functions on cylindrical spacetimes using the BV-BFV formalism, revealing connections to Hamilton-Jacobi actions, conformal field theories, and Kodaira-Spencer gravity across different dimensions.

Contribution

It extends the BV-BFV formalism to compute partition functions on cylinders, linking them to known theories like WZW and BCOV, and clarifies the classical-quantum correspondence in these settings.

Findings

Identification of the physical part of the BV-BFV effective action with Hamilton-Jacobi action.

Establishment of a CS-WZW correspondence for 2-dimensional cases.

Relation of Chern-Simons partition functions to Kodaira-Spencer gravity in 6 dimensions.

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 [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.