BRST Noether Theorem and Corner Charge Bracket

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.

💡 Research Summary

This paper establishes a rigorous proof of what the authors call the “BRST Noether 1.5th theorem,” a statement that extends Noether’s second theorem to the realm of gauge‑fixed, BRST‑invariant Lagrangians for a wide class of rank‑1 Batalin‑Vilkovisky (BV) theories. Rank‑1 BV theories are those whose gauge algebra closes off‑shell, encompassing Yang–Mills, general relativity (both first‑ and second‑order formulations), abelian 2‑form gauge fields, N = 1 supergravity, and many supersymmetric extensions.

The authors begin by introducing the full BRST multiplet (\Phi^{I}=(\phi^{i},c^{A},\bar c^{A},b^{A})) and write down a very general nilpotent BRST transformation (eq. 2.3) that includes ghost‑number‑one fields (c^{A}) and, where needed, ghost‑of‑ghost fields (c^{P}) of ghost number two. By expanding the nilpotency condition (s^{2}=0) they derive a set of algebraic constraints on the structure functions (Appendix A), which are essentially the L∞ relations governing the gauge algebra.

With this machinery in place, the central result is the decomposition of the BRST Noether current associated with the gauge‑fixed Lagrangian (\tilde L = L + s\Psi): \