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Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles

Maxim Grigoriev

Abstract

A gauge PDE is a geometrical object underlying what physicists call a local gauge field theory defined at the level of equations of motion (i.e. without specifying Lagrangian) in terms of Batalin-Vilkovisky (BV) formalism. This notion extends the BV formulation in terms of jet-bundles on the one hand and the geometrical approach to PDEs on the other hand. In this work we concentrate on gauge PDEs equipped with a compatible presymplectic structure and show that under some regularity conditions this data defines a jet-bundle BV formulation. More precisely, the BV jet-bundle arises as the symplectic quotient of the super jet-bundle of the initial gauge PDE. In this sense, presymplectic gauge PDEs give an invariant geometrical approach to Lagrangian gauge systems, which is not limited to jet-bundles. Furthermore, the presymplectic gauge PDE structure naturally descends to space-time submanifolds (in particular, boundaries, if any) and, in this respect, is quite similar to AKSZ sigma models which are long known to have this feature. We also introduce a notion of a weak presymplectic gauge PDE, where the nilpotency of the differential is replaced by a presymplectic analog of the BV master equation, and show that it still defines a local BV system. This allows one to encode BV systems in terms of finite-dimensional graded geometry, much like the AKSZ construction does in the case of topological models.

Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles

Abstract

A gauge PDE is a geometrical object underlying what physicists call a local gauge field theory defined at the level of equations of motion (i.e. without specifying Lagrangian) in terms of Batalin-Vilkovisky (BV) formalism. This notion extends the BV formulation in terms of jet-bundles on the one hand and the geometrical approach to PDEs on the other hand. In this work we concentrate on gauge PDEs equipped with a compatible presymplectic structure and show that under some regularity conditions this data defines a jet-bundle BV formulation. More precisely, the BV jet-bundle arises as the symplectic quotient of the super jet-bundle of the initial gauge PDE. In this sense, presymplectic gauge PDEs give an invariant geometrical approach to Lagrangian gauge systems, which is not limited to jet-bundles. Furthermore, the presymplectic gauge PDE structure naturally descends to space-time submanifolds (in particular, boundaries, if any) and, in this respect, is quite similar to AKSZ sigma models which are long known to have this feature. We also introduce a notion of a weak presymplectic gauge PDE, where the nilpotency of the differential is replaced by a presymplectic analog of the BV master equation, and show that it still defines a local BV system. This allows one to encode BV systems in terms of finite-dimensional graded geometry, much like the AKSZ construction does in the case of topological models.
Paper Structure (22 sections, 3 theorems, 51 equations)

This paper contains 22 sections, 3 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $Q$-manifolds
  4. $Q$-manifolds as gauge systems
  5. Local gauge theories as gauge PDEs
  6. Gauge PDE
  7. Examples
  8. PDE
  9. AKSZ sigma model
  10. BV-formulation at the level of equations of motion
  11. Presymplectic gauge PDEs
  12. Lagrangian BV system
  13. Descent-completed BV systems
  14. Presymplectic BV systems
  15. Presymplectic gauge PDEs
  16. ...and 7 more sections

Key Result

Proposition 4.1

Let $J^{\infty}(\mathcal{E})$ be equipped with a symplectic structure ${\overset{n}{\omega}}$ determined (as explained above) by a nondegenerate $\omega^\mathcal{E}$ and a vertical evolutionary vector field $s$, $\mathrm{gh}(s)=1$ (not necessarily nilpotent) such that equations descent2 and master a

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Proposition 4.1
  • proof
  • Definition 4.2
  • Proposition 4.3
  • Proposition 5.1
  • proof