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Local BRST cohomology in gauge theories

Glenn Barnich, Friedemann Brandt, Marc Henneaux

TL;DR

This paper provides a comprehensive algebraic analysis of local BRST cohomology for gauge theories of the Yang-Mills type. By decomposing the BRST differential into Koszul-Tate δ and longitudinal γ, the authors derive deep results for H(s|d), H(δ|d), and H(γ|d), linking anomalies, counterterms, and deformations to characteristic and equivariant cohomologies. The work systematically employs the descent equations and the small algebra to map the space of possible deformations and anomalies, showing in particular that for semisimple gauge groups the BRST inverse problem is highly constrained and anomalies are fixed by a small set of characteristic classes. The results extend to effective field theories and provide a robust framework for understanding the renormalization and consistency of gauge-invariant operators, as well as the structure of global symmetries and conserved currents. The treatment includes special cases such as free abelian gauge fields and three-dimensional Chern-Simons theory, illustrating the versatility and scope of the cohomological approach.

Abstract

The general solution of the anomaly consistency condition (Wess-Zumino equation) has been found recently for Yang-Mills gauge theory. The general form of the counterterms arising in the renormalization of gauge invariant operators (Kluberg-Stern and Zuber conjecture) and in gauge theories of the Yang-Mills type with non power counting renormalizable couplings has also been worked out in any number of spacetime dimensions. This Physics Report is devoted to reviewing in a self-contained manner these results and their proofs. This involves computing cohomology groups of the differential introduced by Becchi, Rouet, Stora and Tyutin, with the sources of the BRST variations of the fields ("antifields") included in the problem. Applications of this computation to other physical questions (classical deformations of the action, conservation laws) are also considered. The general algebraic techniques developed in the Report can be applied to other gauge theories, for which relevant references are given.

Local BRST cohomology in gauge theories

TL;DR

This paper provides a comprehensive algebraic analysis of local BRST cohomology for gauge theories of the Yang-Mills type. By decomposing the BRST differential into Koszul-Tate δ and longitudinal γ, the authors derive deep results for H(s|d), H(δ|d), and H(γ|d), linking anomalies, counterterms, and deformations to characteristic and equivariant cohomologies. The work systematically employs the descent equations and the small algebra to map the space of possible deformations and anomalies, showing in particular that for semisimple gauge groups the BRST inverse problem is highly constrained and anomalies are fixed by a small set of characteristic classes. The results extend to effective field theories and provide a robust framework for understanding the renormalization and consistency of gauge-invariant operators, as well as the structure of global symmetries and conserved currents. The treatment includes special cases such as free abelian gauge fields and three-dimensional Chern-Simons theory, illustrating the versatility and scope of the cohomological approach.

Abstract

The general solution of the anomaly consistency condition (Wess-Zumino equation) has been found recently for Yang-Mills gauge theory. The general form of the counterterms arising in the renormalization of gauge invariant operators (Kluberg-Stern and Zuber conjecture) and in gauge theories of the Yang-Mills type with non power counting renormalizable couplings has also been worked out in any number of spacetime dimensions. This Physics Report is devoted to reviewing in a self-contained manner these results and their proofs. This involves computing cohomology groups of the differential introduced by Becchi, Rouet, Stora and Tyutin, with the sources of the BRST variations of the fields ("antifields") included in the problem. Applications of this computation to other physical questions (classical deformations of the action, conservation laws) are also considered. The general algebraic techniques developed in the Report can be applied to other gauge theories, for which relevant references are given.

Paper Structure

This paper contains 146 sections, 37 theorems, 348 equations.

Table of Contents

  1. Conventions and notation
  2. Spacetime.
  3. Spacetime.
  4. Gauge theories of the Yang-Mills type.
  5. General.
  6. Introduction
  7. Purpose of report
  8. Gauge theories of the Yang-Mills type
  9. Relevance of BRST cohomology
  10. Cohomology and antifields
  11. Further comments
  12. Appendix \ref{['Intro']}.A: Gauge-fixing and antighosts
  13. Appendix \ref{['Intro']}.B: Contractible pairs
  14. Outline of report
  15. Locality - Algebraic Poincaré lemma: $H(d)$
  16. ...and 131 more sections

Key Result

Theorem 4.1

(i) A local function is a total derivative iff it has vanishing Euler-Lagrange derivatives with respect to all fields, (ii) Two local functionals ${\cal F},{\cal G}$ agree for all sections $s$, ${\cal F}(f,s)={\cal G}(g,s)$ iff their integrands differ by a total derivative, $f=g+\partial_\mu j^\mu$, for some local functions $j^\mu$, whose boundary integral vanishes, $\oint_{\partial M}j=0$.

Theorems & Definitions (37)

  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 5.1
  • Theorem 5.1
  • Lemma 6.1
  • Theorem 6.1
  • Theorem 6.2
  • Theorem 6.3
  • Corollary 6.1
  • ...and 27 more