Classical BV theories on manifolds with boundary

TL;DR

This paper extends the BV formalism to gauge theories on manifolds with boundary, connecting bulk and boundary constructions and exploring higher codimension cases, with applications to various classical field theories.

Contribution

It develops a comprehensive BV-BFV framework for manifolds with boundary and corners, facilitating consistent quantization of gauge theories in these settings.

Findings

Unified BV-BFV construction for boundary and bulk theories

Extension to manifolds with corners and higher codimension strata

Application to classical gauge theories like Chern-Simons and BF theories

Abstract

In this paper we extend the classical BV framework to gauge theories on spacetime manifolds with boundary. In particular, we connect the BV construction in the bulk with the BFV construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners. We present several examples including electrodynamics, Yang-Mills theory and topological field theories coming from the AKSZ construction, in particular, the Chern-Simons theory, the $BF$ theory, and the Poisson sigma model. This paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing.