BRST Quantization of Gauge Theory in Noncommutative Geometry: Matrix Derivative Approach

TL;DR

This paper develops a BRST quantization framework for gauge theories within noncommutative geometry using a matrix derivative approach, deriving invariant actions and analyzing symmetry structures.

Contribution

It introduces a novel BRST quantization method in noncommutative geometry via the superconnection formalism and applies it to specific gauge groups.

Findings

Quantum action matches 't Hooft gauge with spontaneous symmetry breaking.

Only the even part of the supergroup acts as a gauge symmetry.

Explicit treatment of SU(2/1) and SU(2/2) cases.

Abstract

The BRST quantization of a gauge theory in noncommutative geometry is carried out in the ``matrix derivative" approach. BRST/anti-BRST transformation rules are obtained by applying the horizontality condition, in the superconnection formalism. A BRST/anti-BRST invariant quantum action is then constructed, using an adaptation of the method devised by Baulieu and Thierry-Mieg for the Yang-Mills case. The resulting quantum action turns out to be the same as that of a gauge theory in the 't Hooft gauge with spontaneously broken symmetry. Our result shows that only the even part of the supergroup acts as a gauge symmetry, while the odd part effectively provides a global symmetry. We treat the general formalism first, then work out the $SU(2/1)$ and $SU(2/2)$ cases explicitly.