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Consistent couplings between fields with a gauge freedom and deformations of the master equation

G. Barnich, M. Henneaux

TL;DR

The paper analyzes consistent couplings for gauge fields through BRST/BV methods by recasting the problem as deformations of the master equation $(S,S)=0$. It proves the antibracket map is trivial on the space of all functionals but can yield obstructions when locality is imposed, formalized via $H^0(s|d)$. In 3D Chern–Simons theory, locality constrains consistent local interactions to nonabelian CS extensions, and these deformations exhibit rigidity, with no further independent local deformations. Overall, the BV master equation framework provides precise cohomological criteria for when and how gauge-covariant couplings can exist locally, as well as when they are rigid.

Abstract

The antibracket in BRST theory is known to define a map $\rm{H^p \times H^q \longrightarrow H^{p+q+1}}$ associating with two equivalence classes of BRST invariant observables of respective ghost number p and q an equivalence class of BRST invariant observables of ghost number p+q+1. It is shown that this map is trivial in the space of all functionals, i.e., that its image contains only the zeroth class. However it is generically non trivial in the space of local functionals. Implications of this result for the problem of consistent interactions among fields with a gauge freedom are then drawn. It is shown that the obstructions to constructing such interactions lie precisely in the image of the antibracket map and are accordingly inexistent if one does not insist on locality. However consistent local interactions are severely constrained. The example of the Chern-Simons theory is considered. It is proved that the only consistent, local, Lorentz covariant interactions for the abelian models are exhausted by the non-abelian Chern-Simons extensions.

Consistent couplings between fields with a gauge freedom and deformations of the master equation

TL;DR

The paper analyzes consistent couplings for gauge fields through BRST/BV methods by recasting the problem as deformations of the master equation . It proves the antibracket map is trivial on the space of all functionals but can yield obstructions when locality is imposed, formalized via . In 3D Chern–Simons theory, locality constrains consistent local interactions to nonabelian CS extensions, and these deformations exhibit rigidity, with no further independent local deformations. Overall, the BV master equation framework provides precise cohomological criteria for when and how gauge-covariant couplings can exist locally, as well as when they are rigid.

Abstract

The antibracket in BRST theory is known to define a map associating with two equivalence classes of BRST invariant observables of respective ghost number p and q an equivalence class of BRST invariant observables of ghost number p+q+1. It is shown that this map is trivial in the space of all functionals, i.e., that its image contains only the zeroth class. However it is generically non trivial in the space of local functionals. Implications of this result for the problem of consistent interactions among fields with a gauge freedom are then drawn. It is shown that the obstructions to constructing such interactions lie precisely in the image of the antibracket map and are accordingly inexistent if one does not insist on locality. However consistent local interactions are severely constrained. The example of the Chern-Simons theory is considered. It is proved that the only consistent, local, Lorentz covariant interactions for the abelian models are exhausted by the non-abelian Chern-Simons extensions.

Paper Structure

This paper contains 8 sections, 2 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. The master equation and the antibracket map
  3. Higher order maps
  4. Constructing consistent couplings as a deformation problem
  5. Spacetime locality of the deformation - The example of free abelian Chern-Simons models
  6. Rigidity of non-abelian Chern-Simons theory
  7. Conclusion
  8. Acknowledgements

Key Result

Theorem 1

the BRST cohomologies $H^*(s)$ and $H^*(s^\prime)$ associated with two formulations of a theory differing in the auxiliary field content are isomorphic. Furthermore, the isomorphism i : $H^*(s)\longrightarrow H^*(s^\prime)$ commutes with the antibracket map.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2