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Consistent deformations in the presymplectic BV-AKSZ approach

Jordi Frias, Maxim Grigoriev

TL;DR

This paper develops a deformation theory for local gauge theories within the presymplectic BV-AKSZ framework, emphasizing a finite-dimensional, on-shell description that avoids quotient spaces of local functionals. It separates deformations into an infinitesimal Hamiltonian change and a subsequent deformation of the Q-structure while keeping the presymplectic form fixed, and it identifies potential obstructions via $Q_0$-cohomology. The authors provide a concrete recursive procedure to construct consistent deformations order by order, and illustrate the approach by rederiving Chern-Simons and Yang-Mills theories from their linearized counterparts. The work offers a streamlined, frame-like route to consistent interactions with potential applications to higher spin and gravity-like theories, with caveats about when presymplectic deformations are required. Overall, the framework connects BV-BRST, AKSZ, and covariant phase space ideas into a practical tool for analyzing local gauge interactions.

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.

Consistent deformations in the presymplectic BV-AKSZ approach

TL;DR

This paper develops a deformation theory for local gauge theories within the presymplectic BV-AKSZ framework, emphasizing a finite-dimensional, on-shell description that avoids quotient spaces of local functionals. It separates deformations into an infinitesimal Hamiltonian change and a subsequent deformation of the Q-structure while keeping the presymplectic form fixed, and it identifies potential obstructions via -cohomology. The authors provide a concrete recursive procedure to construct consistent deformations order by order, and illustrate the approach by rederiving Chern-Simons and Yang-Mills theories from their linearized counterparts. The work offers a streamlined, frame-like route to consistent interactions with potential applications to higher spin and gravity-like theories, with caveats about when presymplectic deformations are required. Overall, the framework connects BV-BRST, AKSZ, and covariant phase space ideas into a practical tool for analyzing local gauge interactions.

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.
Paper Structure (15 sections, 4 theorems, 56 equations)

This paper contains 15 sections, 4 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Deformation theory
  4. BV formulation of gauge theories
  5. BV for local gauge theories and consistent deformations
  6. Presymplectic BV-AKSZ formulation
  7. Consistent deformations of presymplectic gPDEs
  8. General setup
  9. Recursive construction
  10. Lowest orders in more details
  11. Trivial deformations
  12. Examples
  13. Chern Simons theory
  14. Yang-Mills theory
  15. Conclusions

Key Result

Proposition 3.1

Given a linear pgPDE $(E,Q_0,\omega,\mathcal{L}_0)$, consider its deformation $(E,Q_{(N)},\omega,\mathcal{L}_{(N)})$ such that $Q_i, \mathcal{L}_i$ fulfill for all $1\,{\leqslant}\, i\,{\leqslant}\, N$. Then the defining relations of pgPDE are satisfied up to order $N+3$:

Theorems & Definitions (10)

  • Definition 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof