General Boundary Formulation for $n$-Dimensional Classical Abelian Theory with Corners

TL;DR

This paper develops a general boundary formulation for $n$-dimensional abelian gauge theories with corners, extending the axiomatic approach inspired by TQFTs to classical field theories, paving the way for future geometric quantization.

Contribution

It introduces a reduction procedure for abelian gauge theories within the GBF framework, including manifolds with corners, and constructs abelian Yang-Mills theories in this setting.

Findings

Framework accommodates manifolds with smooth boundaries and corners.

Provides a classical analogue of extended TQFTs.

Lays groundwork for geometric quantization of abelian theories.

Abstract

We propose a general reduction procedure for classical field theories provided with abelian gauge symmetries in a Lagrangian setting. These ideas come from an axiomatic presentation of the general boundary formulation (GBF) of field theories, mostly inspired by topological quantum field theories (TQFT). We construct abelian Yang-Mills theories using this framework. We treat the case for space-time manifolds with smooth boundary components as well as the case of manifolds with corners. This treatment is the GBF analogue of extended TQFTs. The aim for developing this classical formalism is to accomplish, in a future work, geometric quantization at least for the abelian case.