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

TL;DR

This paper investigates the properties of the antibracket in BRST theory, showing its triviality in the space of all functionals but non-triviality in local functionals, and explores implications for constructing consistent gauge field interactions.

Contribution

It demonstrates the triviality of the antibracket map in the space of all functionals and its non-triviality in local functionals, impacting the understanding of gauge interactions.

Findings

Antibracket map is trivial in all functionals but non-trivial locally.

Obstructions to gauge interactions are in the image of the antibracket map.

Only non-abelian Chern-Simons interactions are consistent and local for abelian models.

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.