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Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket

Glenn Barnich, Marc Henneaux

TL;DR

This work analyzes how the BV antibracket on local BRST cohomology relates to the Dickey bracket on conserved currents and the Poisson bracket on conserved charges. Using BRST cohomology modulo $d$ and the extended BFV Hamiltonian formalism, it proves precise isomorphisms among these algebras: in non-gauge theories, the ghost-number $-1$ sector of the BV cohomology, the Dickey algebra of currents, and the Poisson algebra of charges are canonically equivalent; in gauge theories, currents without charge modify the structure via an ideal $I$, but the Dickey algebra modulo $I$ remains isomorphic to the charges’ Poisson algebra. The general ghost-number case is treated through a decomposition in the extended formalism, showing that the antibracket map is governed by the spatial Poisson bracket on a distinguished sector, with an abelian ideal encoding gauge-dependent redundancies. These results connect the BV/BRST framework to canonical Hamiltonian structures, providing a unified understanding of conserved quantities in both non-gauge and gauge theories and offering practical tools for computing their algebras.

Abstract

One may introduce at least three different Lie algebras in any Lagrangian field theory : (i) the Lie algebra of local BRST cohomology classes equipped with the odd Batalin-Vilkovisky antibracket, which has attracted considerable interest recently~; (ii) the Lie algebra of local conserved currents equipped with the Dickey bracket~; and (iii) the Lie algebra of conserved, integrated charges equipped with the Poisson bracket. We show in this paper that the subalgebra of (i) in ghost number $-1$ and the other two algebras are isomorphic for a field theory without gauge invariance. We also prove that, in the presence of a gauge freedom, (ii) is still isomorphic to the subalgebra of (i) in ghost number $-1$, while (iii) is isomorphic to the quotient of (ii) by the ideal of currents without charge. In ghost number different from $-1$, a more detailed analysis of the local BRST cohomology classes in the Hamiltonian formalism allows one to prove an isomorphism theorem between the antibracket and the extended Poisson bracket of Batalin, Fradkin and Vilkovisky.

Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket

TL;DR

This work analyzes how the BV antibracket on local BRST cohomology relates to the Dickey bracket on conserved currents and the Poisson bracket on conserved charges. Using BRST cohomology modulo and the extended BFV Hamiltonian formalism, it proves precise isomorphisms among these algebras: in non-gauge theories, the ghost-number sector of the BV cohomology, the Dickey algebra of currents, and the Poisson algebra of charges are canonically equivalent; in gauge theories, currents without charge modify the structure via an ideal , but the Dickey algebra modulo remains isomorphic to the charges’ Poisson algebra. The general ghost-number case is treated through a decomposition in the extended formalism, showing that the antibracket map is governed by the spatial Poisson bracket on a distinguished sector, with an abelian ideal encoding gauge-dependent redundancies. These results connect the BV/BRST framework to canonical Hamiltonian structures, providing a unified understanding of conserved quantities in both non-gauge and gauge theories and offering practical tools for computing their algebras.

Abstract

One may introduce at least three different Lie algebras in any Lagrangian field theory : (i) the Lie algebra of local BRST cohomology classes equipped with the odd Batalin-Vilkovisky antibracket, which has attracted considerable interest recently~; (ii) the Lie algebra of local conserved currents equipped with the Dickey bracket~; and (iii) the Lie algebra of conserved, integrated charges equipped with the Poisson bracket. We show in this paper that the subalgebra of (i) in ghost number and the other two algebras are isomorphic for a field theory without gauge invariance. We also prove that, in the presence of a gauge freedom, (ii) is still isomorphic to the subalgebra of (i) in ghost number , while (iii) is isomorphic to the quotient of (ii) by the ideal of currents without charge. In ghost number different from , a more detailed analysis of the local BRST cohomology classes in the Hamiltonian formalism allows one to prove an isomorphism theorem between the antibracket and the extended Poisson bracket of Batalin, Fradkin and Vilkovisky.

Paper Structure

This paper contains 7 sections, 8 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. The antibracket map induced in local BRST cohomology
  3. The Dickey bracket
  4. Isomorphisms in the case of non-degenerate Lagrangian field theory
  5. Gauge theories. Ghost number $-1$
  6. Gauge theories. General analysis
  7. Conclusion

Key Result

Theorem 4.1

The odd Lie algebra ${\cal S}=(H^{-1,n}(s|d),\{\cdot,\cdot\})$ is isomorphic to the algebra of inequivalent variational symmetries equipped with the bracket induced by the commutator for vector fields.

Theorems & Definitions (8)

  • Theorem 4.1
  • Theorem 4.2
  • Corollary 4.1
  • Theorem 4.3
  • Theorem 5.1
  • Theorem 6.1
  • Theorem 6.2
  • Theorem 6.3