Consistent deformations in the presymplectic BV-AKSZ approach

TL;DR

This paper introduces a finite-dimensional presymplectic BV-AKSZ framework for analyzing consistent gauge theory interactions, simplifying the process by avoiding quotient spaces and extending the frame-like approach.

Contribution

It develops a novel finite-dimensional presymplectic approach to study gauge interactions, extending the BV-AKSZ formalism and describing an unusual homological deformation theory.

Findings

Rederived Chern-Simons and YM theories from linearized versions.

Provided a simplified method for analyzing consistent interactions.

Extended the BV-AKSZ approach with a new deformation theory.

Abstract

We develop a framework for studying consistent interactions of local gauge theories, which is based on the presymplectic BV-AKSZ formulation. The advantage of the proposed approach is that it operates in terms of finite-dimensional spaces and avoids working with quotient spaces such as local functionals or functionals modulo on-shell trivial ones. The structure that is being deformed is that of a presymplectic gauge PDE, which consists of a graded presymplectic structure and a compatible odd vector field. These are known to encode the Batalin--Vilkovisky (BV) formulation of a local gauge theory in terms of the finite dimensional supergeometrical object. Although in its present version the method is limited to interactions that do not deform the presymplectic structure and relies on some natural assumptions, it gives a remarkably simple way to analyse consistent interactions. The approach can be considered as the BV-AKSZ extension of the frame-like approach to consistent interactions. We also describe the underlying homological deformation theory, which turns out to be slightly unusual compared to the standard deformations of differential graded Lie algebras. As an illustration, the Chern-Simons and YM theories are rederived starting from their linearized versions.