Weak Gauge PDEs

TL;DR

This paper introduces weak gauge PDEs, a finite-dimensional approach to describing local gauge theories that relaxes certain mathematical conditions, providing clearer physical interpretation and potential for classifying complex gauge systems.

Contribution

It proposes a novel notion of weak gauge PDEs that simplifies the description of gauge theories and connects them to BV formalism, with uniqueness and minimality properties.

Findings

Weak gauge PDEs can be derived from weak presymplectic formulations.

Any weak gauge PDE corresponds to a standard BV formulation.

Examples include non-Lagrangian self-dual Yang-Mills theory.

Abstract

Gauge PDEs generalise the AKSZ construction when dealing with generic local gauge theories. Despite being very flexible and invariant, these geometrical objects are usually infinite-dimensional and are difficult to define explicitly, just like standard infinitely-prolonged PDEs. We propose a notion of a weak gauge PDE in which the nilpotency of the BRST differential is relaxed in a controllable way. In this approach a nontopological local gauge theory can be described in terms of a finite-dimensional geometrical object. Moreover, among the equivalent weak gauge PDEs describing a given system, a minimal one can usually be found and is unique in a certain sense. In the case of a Lagrangian system, the respective weak gauge PDE naturally arises from its weak presymplectic formulation. We prove that any weak gauge PDE determines the standard jet-bundle Batalin-Vilkovisky formulation of the underlying gauge theory, giving an unambiguous physical interpretation of these objects. The formalism is illustrated by a few examples, including the non-Lagrangian self-dual Yang-Mills theory and a finite jet-bundle. We also discuss possible applications of the approach to the characterisation of those infinite-dimensional gauge PDEs that correspond to local theories.