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A Poincare lemma for sigma models of AKSZ type

Glenn Barnich, Maxim Grigoriev

TL;DR

The paper addresses the challenge of understanding local BRST cohomology for AKSZ-type sigma models. By exploiting the jet-space (variational bicomplex) framework, it proves that in coordinate neighborhoods the local BRST cohomology of such models is isomorphic to the $Q$-cohomology of the target space, and it extends this correspondence to the cohomology of functional multivectors. This reduces field-theoretic invariants, deformations, and symmetries to purely target-space data, providing a powerful tool for the inverse problem of the calculus of variations in general gauge systems. The results have concrete implications for classifying Lagrange and weak Poisson structures in AKSZ-type theories, with explicit examples illustrating the equivalence between spacetime and target-space cohomologies.

Abstract

For a sigma model of AKSZ-type, we show that the local BRST cohomology is isomorphic to the cohomology of the target space differential when restricted to coordinate neighborhoods both in the base and in the target. An analogous result is shown to hold for the cohomology in the space of functional multivectors. Applications of these latter cohomology classes in the context of the inverse problem of the calculus of variation for general gauge systems are also discussed.

A Poincare lemma for sigma models of AKSZ type

TL;DR

The paper addresses the challenge of understanding local BRST cohomology for AKSZ-type sigma models. By exploiting the jet-space (variational bicomplex) framework, it proves that in coordinate neighborhoods the local BRST cohomology of such models is isomorphic to the -cohomology of the target space, and it extends this correspondence to the cohomology of functional multivectors. This reduces field-theoretic invariants, deformations, and symmetries to purely target-space data, providing a powerful tool for the inverse problem of the calculus of variations in general gauge systems. The results have concrete implications for classifying Lagrange and weak Poisson structures in AKSZ-type theories, with explicit examples illustrating the equivalence between spacetime and target-space cohomologies.

Abstract

For a sigma model of AKSZ-type, we show that the local BRST cohomology is isomorphic to the cohomology of the target space differential when restricted to coordinate neighborhoods both in the base and in the target. An analogous result is shown to hold for the cohomology in the space of functional multivectors. Applications of these latter cohomology classes in the context of the inverse problem of the calculus of variation for general gauge systems are also discussed.

Paper Structure

This paper contains 17 sections, 2 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. AKSZ construction
  3. BRST differential
  4. Bracket and BV master action
  5. Examples
  6. Generalities on jet-spaces and local BRST cohomology
  7. Horizontal complex
  8. BRST differential
  9. Local BRST cohomology
  10. BRST cohomology for functional multivectors
  11. Local BRST cohomology for AKSZ-type sigma models
  12. Cohomology of space-time part
  13. Cohomology of complete differential
  14. Cohomology for functional multivectors
  15. Functional multivectors, symmetries and generalized Poisson structures
  16. ...and 2 more sections

Key Result

Proposition 4.1

The local BRST cohomology $H(s,\widehat{\mathcal{F}})$ is isomorphic to the cohomology $H(Q)$ in target space functions for coordinate neighborhoods of the base and the target space, where with $[\Theta^g_{\alpha_g}(\Psi^A)]$ denoting representatives of $H^g(Q)$.

Theorems & Definitions (2)

  • Proposition 4.1
  • Proposition 4.1