Equivariant Symplectic Geometry of Gauge Fixing in Yang-Mills Theory

TL;DR

This paper interprets gauge fixing in Yang-Mills theory through equivariant symplectic geometry, linking BRST cohomology to equivariant cohomology and applying localization to compute path integrals.

Contribution

It introduces a novel geometric framework connecting gauge fixing, BRST cohomology, and equivariant localization in Yang-Mills theory.

Findings

Reformulation of Faddeev-Popov gauge fixing as equivariant localization

Identification of ghost operator as a (pre)symplectic form

Application of the method to compute the partition function of supersymmetric quantum mechanics

Abstract

The Faddeev-Popov gauge fixing in Yang-Mills theory is interpreted as equivariant localization. It is shown that the Faddeev-Popov procedure amounts to a construction of a symplectic manifold with a Hamiltonian group action. The BRST cohomology is shown to be equivalent to the equivariant cohomology based on this symplectic manifold with Hamiltonian group action. The ghost operator is interpreted as a (pre)symplectic form and the gauge condition as the moment map corresponding to the Hamiltonian group action. This results in the identification of the gauge fixing action as a closed equivariant form, the sum of an equivariant symplectic form and a certain closed equivariant 4-form which ensures convergence. An almost complex structure compatible with the symplectic form is constructed. The equivariant localization principle is used to localize the path integrals onto the gauge slice. The Gribov problem is also discussed in the context of equivariant localization principle. As a simple illustration of the methods developed in the paper, the partition function of N=2 supersymmetric quantum mechanics is calculated by equivariant localization