Background fields in the presymplectic BV-AKSZ approach

TL;DR

This paper extends the presymplectic BV-AKSZ framework to include background fields, providing a geometric approach to gauge theories with backgrounds, including applications to higher spin fields and symmetries.

Contribution

It develops a novel geometric formulation of gauge theories with background fields within the presymplectic BV-AKSZ approach, generalizing existing models to more complex backgrounds.

Findings

Describes gauge theories with background fields as presymplectic gauge PDEs over background spaces.

Introduces homogeneous gauge PDEs for higher spin fields on symmetric spaces.

Connects higher-form symmetries and gauging to the presymplectic BV-AKSZ framework.

Abstract

The Batalin-Vilkovisky formulation of a general local gauge theory can be encoded in the structure of a so-called presymplectic gauge PDE -- an almost-$Q$ bundle over the spacetime exterior algebra, equipped with a compatible presymplectic structure. In the case of a trivial bundle and an invertible presymplectic structure, this reduces to the well-known AKSZ sigma model construction. We develop an extension of the presympletic BV-AKSZ approach to describe local gauge theories with background fields. It turns out that such theories correspond to presymplectic gauge PDEs whose base spaces are again gauge PDEs describing background fields. As such, the geometric structure is that of a bundle over a bundle over a given spacetime. Gauge PDEs over backgrounds arise naturally when studying linearisation, coupling (gauge) fields to background geometry, gauging global symmetries, etc. Less obvious examples involve parametrised systems, Fedosov equations, and the so-called homogeneous (presymplectic) gauge PDEs. The latter are the gauge-invariant generalisations of the familiar homogeneous PDEs and they provide a very concise description of gauge fields on homogeneous spaces such as higher spin gauge fields on Minkowski, (A)dS, and conformal spaces. Finally, we briefly discuss how the higher-form symmetries and their gauging fit into the framework using the simplest example of the Maxwell field.