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Local BRST cohomology in the antifield formalism: I. General theorems

G. Barnich, F. Brandt, M. Henneaux

TL;DR

The paper develops general theorems for local BRST cohomology in the antifield formalism, linking H^*(s|d) to the Koszul–Tate sector via δ and to conserved currents through H^n_1(δ|d), thereby providing a cohomological reformulation of Noether’s theorem. It proves vanishing results for higher antighost numbers (H^p_k(δ|d) for k>p) in theories with finite Cauchy order and explicitly computes H^n_2(δ|d) for electromagnetism, Yang–Mills, and gravity, finding nontrivial classes only in electromagnetism (and in special abelian sectors). The analysis shows nonminimal sectors and auxiliary fields do not alter the local BRST cohomology, and it establishes isomorphisms between cohomologies of theories related by auxiliary content. These general results set the stage for concrete computations of H^k(s|d) in Yang–Mills theory with sources in a companion paper. Overall, the work bridges local BRST cohomology with characteristic cohomology and provides practical descent tools for studying gauge theories.

Abstract

We establish general theorems on the cohomology $H^*(s|d)$ of the BRST differential modulo the spacetime exterior derivative, acting in the algebra of local $p$-forms depending on the fields and the antifields (=sources for the BRST variations). It is shown that $H^{-k}(s|d)$ is isomorphic to $H_k(δ|d)$ in negative ghost degree $-k\ (k>0)$, where $δ$ is the Koszul-Tate differential associated with the stationary surface. The cohomological group $H_1(δ|d)$ in form degree $n$ is proved to be isomorphic to the space of constants of the motion, thereby providing a cohomological reformulation of Noether theorem. More generally, the group $H_k(δ|d)$ in form degree $n$ is isomorphic to the space of $n-k$ forms that are closed when the equations of motion hold. The groups $H_k(δ|d)$ $(k>2)$ are shown to vanish for standard irreducible gauge theories. The group $H_2(δ|d)$ is then calculated explicitly for electromagnetism, Yang-Mills models and Einstein gravity. The invariance of the groups $H^{k}(s|d)$ under the introduction of non minimal variables and of auxiliary

Local BRST cohomology in the antifield formalism: I. General theorems

TL;DR

The paper develops general theorems for local BRST cohomology in the antifield formalism, linking H^*(s|d) to the Koszul–Tate sector via δ and to conserved currents through H^n_1(δ|d), thereby providing a cohomological reformulation of Noether’s theorem. It proves vanishing results for higher antighost numbers (H^p_k(δ|d) for k>p) in theories with finite Cauchy order and explicitly computes H^n_2(δ|d) for electromagnetism, Yang–Mills, and gravity, finding nontrivial classes only in electromagnetism (and in special abelian sectors). The analysis shows nonminimal sectors and auxiliary fields do not alter the local BRST cohomology, and it establishes isomorphisms between cohomologies of theories related by auxiliary content. These general results set the stage for concrete computations of H^k(s|d) in Yang–Mills theory with sources in a companion paper. Overall, the work bridges local BRST cohomology with characteristic cohomology and provides practical descent tools for studying gauge theories.

Abstract

We establish general theorems on the cohomology of the BRST differential modulo the spacetime exterior derivative, acting in the algebra of local -forms depending on the fields and the antifields (=sources for the BRST variations). It is shown that is isomorphic to in negative ghost degree , where is the Koszul-Tate differential associated with the stationary surface. The cohomological group in form degree is proved to be isomorphic to the space of constants of the motion, thereby providing a cohomological reformulation of Noether theorem. More generally, the group in form degree is isomorphic to the space of forms that are closed when the equations of motion hold. The groups are shown to vanish for standard irreducible gauge theories. The group is then calculated explicitly for electromagnetism, Yang-Mills models and Einstein gravity. The invariance of the groups under the introduction of non minimal variables and of auxiliary

Paper Structure

This paper contains 17 sections, 20 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Cohomological groups ${\bf H^{k}(s)}$ and ${\bf H^k(s|d)}$
  3. Regularity conditions. Examples
  4. Local $p$-forms and antifields
  5. BRST differential
  6. Homological Perturbation Theory and $H^*(s|d)$
  7. Constants of the motion and ${\bf H^n_1(\delta|d)}$.
  8. Results on ${\bf H^p_k(\delta|d)}$ (p, k arbitrary)
  9. Linear gauge theories
  10. Normal theories
  11. Results on $H^n_2(\delta|d)$
  12. Calculation of $H^n_2(\delta|d)$ for electromagnetism
  13. Calculation of $H^n_2(\delta|d)$ for Yang-Mills models and gravity
  14. Non minimal sector - Shift symmetry
  15. Auxiliary fields
  16. ...and 2 more sections

Key Result

Theorem 5.1

: There exists a derivation $s$ of total ghost number equal to $1$ such that (i) $s=\delta +\gamma+s_1+$"higher orders", antigh ("higher orders")$\geq 2$ ; (ii) $s^2 =0$ ($s$ is a differential).

Theorems & Definitions (20)

  • Theorem 5.1
  • Theorem 5.2: on the cohomology of s
  • Theorem 5.3: on the cohomology of d
  • Theorem 5.4: on the cohomology of ${\bf \delta}$ modulo $d$
  • Theorem 6.1: on the cohomology of $s$ modulo $d$
  • Theorem 7.1
  • Theorem 8.1: descent equations for $\delta$ and $d$
  • Theorem 8.2
  • Theorem 8.3
  • Theorem 9.1
  • ...and 10 more