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BRST Noether Theorem and Corner Charge Bracket

Laurent Baulieu, Tom Wetzstein, Siye Wu

Abstract

We provide a proof of the BRST Noether 1.5th theorem, conjectured in [JHEP 10 (2024) 055], for a broad class of rank-1 BV theories including supergravity and 2-form gauge theories. The theorem asserts that the BRST Noether current of any BRST invariant gauge fixed Lagrangian decomposes on-shell into a sum of a BRST-exact term and a corner term that defines Noether charges. This extends the holographic consequences of Noether's second theorem to gauge fixed theories and, in particular, offers a universal gauge independent Lagrangian derivation of the invariance of the S-matrix under asymptotic symmetries. Furthermore, we show that these corner Noether charges are inherently non-integrable. To address this non-integrability, we introduce a novel charge bracket that accounts for potential symplectic flux and anomalies, providing an honest canonical representation of the asymptotic symmetry algebra. We also highlight a general origin of a BRST cocycle associated with asymptotic symmetries.

BRST Noether Theorem and Corner Charge Bracket

Abstract

We provide a proof of the BRST Noether 1.5th theorem, conjectured in [JHEP 10 (2024) 055], for a broad class of rank-1 BV theories including supergravity and 2-form gauge theories. The theorem asserts that the BRST Noether current of any BRST invariant gauge fixed Lagrangian decomposes on-shell into a sum of a BRST-exact term and a corner term that defines Noether charges. This extends the holographic consequences of Noether's second theorem to gauge fixed theories and, in particular, offers a universal gauge independent Lagrangian derivation of the invariance of the S-matrix under asymptotic symmetries. Furthermore, we show that these corner Noether charges are inherently non-integrable. To address this non-integrability, we introduce a novel charge bracket that accounts for potential symplectic flux and anomalies, providing an honest canonical representation of the asymptotic symmetry algebra. We also highlight a general origin of a BRST cocycle associated with asymptotic symmetries.

Paper Structure

This paper contains 15 sections, 1 theorem, 161 equations, 1 table.

Table of Contents

  1. Introduction
  2. BRST Noether current
  3. The basic set-up
  4. The 1.5th theorem
  5. Holographic Ward identities
  6. Symplectic structure
  7. Preliminaries
  8. BRST Noetherian flux
  9. Gauge fixed fundamental canonical relation
  10. Charge bracket
  11. BRST cocycle for asymptotic symmetries
  12. Application: Abelian 2-form coupled to Chern--Simons
  13. Towards a generalization for rank-2 BV systems
  14. Constraints on the BRST transformations
  15. Review of the BRST covariant phase space

Key Result

Theorem 2.1

Let $s$ be the off-shell nilpotent BRST operator BRST_trans, defining a class of rank-1 BV theories. Consider the corresponding gauge fixed Lagrangian which is invariant under $s$. Then, on-shell of the gauge fixed equations of motion derived from this Lagrangian, the ghost number one conserved Noether current $J^{\mu}_{\rm BRST}$ associated with the BRST invariance of L_GF_thm decomposes as whe

Theorems & Definitions (2)

  • Theorem 2.1: BRST Noether 1.5
  • proof