Background fields in the presymplectic BV-AKSZ approach
Ivan Dneprov, Maxim Grigoriev
TL;DR
This work extends the presymplectic BV-AKSZ framework to theories with background fields by formulating presymplectic gPDEs over background, i.e. bundles whose bases are themselves gPDEs encoding background geometry. It develops a consistent set of conditions ensuring the BV master equation holds when background fields obey their own gPDEs, and shows background symmetries are captured by the total Q-structure. The paper provides a range of concrete constructions and examples: from gravity (Palatini-Cartan-Weyl) and Fedosov geometry to Fronsdal higher-spin theory, Dirac and Maxwell systems with background couplings, and scalar conformal geometry, including the gauging of internal and spacetime symmetries and the role of homogeneous presymplectic gPDEs. By connecting linearization, gauging, and homogeneous structures within this framework, it establishes a unified, local BV description of gauge theories in nontrivial background geometries with explicit action functionals and symmetry realizations. The approach facilitates systematic treatment of background couplings and higher-form symmetries in a geometrically natural, AKSZ-like setting, with potential applications to background-field methods and symmetry-gauging in advanced gauge theories.
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.