BRST quantization and equivariant cohomology: localization with asymptotic boundaries

TL;DR

This paper develops a BRST quantization framework for gauge theories with asymptotic boundaries, enabling localization calculations in supergravity and related theories by incorporating equivariant cohomology.

Contribution

It introduces a background field formalism with a nilpotent BRST charge that accounts for asymptotic boundary conditions and residual symmetries, extending localization techniques to supergravity.

Findings

Constructed a background field formalism for soft gauge algebras.

Defined a nilpotent BRST charge acting on background and quantum fields.

Demonstrated applicability to supergravity on asymptotic backgrounds.

Abstract

We develop BRST quantization of gauge theories with a soft gauge algebra on spaces with asymptotic boundaries. The asymptotic boundary conditions are imposed on background fields, while quantum fluctuations about these fields are described in terms of quantum fields that vanish at the boundary. This leads us to construct a suitable background field formalism that is generally applicable to soft gauge algebras, and therefore to supergravity. We define a nilpotent BRST charge that acts on both the background and the quantum fields, as well as on the background and quantum ghosts. When the background is restricted to be invariant under a residual isometry group, the background ghosts must be restricted accordingly and play the role of the parameters of the background isometries. Requiring in addition that the background ghosts will be BRST invariant as well then converts the BRST algebra into an equivariant one. The background fields and ghosts are then invariant under the equivariant transformations while the quantum fields and ghosts transform under both the equivariant and the background transformations. We demonstrate how this formalism is suitable for carrying out localization calculations in a large class of theories, including supergravity defined on asymptotic backgrounds that admit supersymmetry.