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Weak Gauge PDEs

Maxim Grigoriev, Dmitry Rudinsky

TL;DR

The paper tackles the challenge that gauge PDEs are typically infinite-dimensional and hard to handle explicitly. It introduces weak gauge PDEs (wgPDEs) by allowing $Q^2$ to lie in an involutive distribution $\\mathcal{K}$, yielding finite-dimensional models that retain the essential gauge data. A main result is that every wgPDE determines a jet-bundle Batalin-Vilkovisky (BV) formulation on a quotient, providing an AXSZ-like BV interpretation for non-Lagrangian local theories; the authors illustrate this with the non-Lagrangian self-dual Yang-Mills theory and a finite jet example, and show how weak presymplectic formulations arise naturally in this framework. The work offers a versatile, invariant framework to characterize and convert infinite-dimensional gauge PDEs into tractable finite descriptions, with potential applications to classifying local theories and linking BV-BRST formalisms to AKSZ-type constructions in new contexts.

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.

Weak Gauge PDEs

TL;DR

The paper tackles the challenge that gauge PDEs are typically infinite-dimensional and hard to handle explicitly. It introduces weak gauge PDEs (wgPDEs) by allowing to lie in an involutive distribution , yielding finite-dimensional models that retain the essential gauge data. A main result is that every wgPDE determines a jet-bundle Batalin-Vilkovisky (BV) formulation on a quotient, providing an AXSZ-like BV interpretation for non-Lagrangian local theories; the authors illustrate this with the non-Lagrangian self-dual Yang-Mills theory and a finite jet example, and show how weak presymplectic formulations arise naturally in this framework. The work offers a versatile, invariant framework to characterize and convert infinite-dimensional gauge PDEs into tractable finite descriptions, with potential applications to classifying local theories and linking BV-BRST formalisms to AKSZ-type constructions in new contexts.

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.
Paper Structure (18 sections, 5 theorems, 45 equations)

This paper contains 18 sections, 5 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\boldsymbol{Q}$-manifolds and gauge systems
  4. Jet-bundle BV description
  5. Gauge PDEs
  6. Weak gauge PDEs
  7. Weak $\boldsymbol{Q}$-manifolds
  8. Vector fields along sections (maps)
  9. Vertical jets
  10. Weak gauge PDEs
  11. Local BV system from wgPDE
  12. Inequivalent solutions
  13. Examples
  14. Scalar field
  15. Self-Dual Yang--Mills
  16. ...and 3 more sections

Key Result

Proposition 3.3

Let $(M,Q,\mathcal{K})$ be a weak $Q$-manifold and let $\mathcal{A}\subset C^{\infty}(M)$ be a subalgebra of functions annihilated by $\mathcal{K}$. Then $Q$ preserves $\mathcal{A}$ and $Q^2(f)=0$ for any $f\in\mathcal{A}$.

Theorems & Definitions (26)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Example 3.4
  • Example 3.5
  • ...and 16 more