BRST Quantization of 2d-Yang Mills Theory with Anomalies

TL;DR

This paper explores the BRST quantization of anomalous 2d-Yang Mills theory, revealing how anomalies influence the algebraic structure and cohomology, especially when coupled with chiral fermions.

Contribution

It demonstrates the impact of anomalies on the BRST cohomology and establishes the equivalence with finite-dimensional gauge group cohomology for certain fermion representations.

Findings

Anomalies cause the constraints to form a Kac-Moody algebra with negative central charge.

BRST cohomology aligns with finite gauge group cohomology for large fermion representations.

Pure YM or small-representation fermion couplings exhibit infinitely many inequivalent cohomology classes.

Abstract

In this paper we discuss the BRST-quantization of anomalous 2d-Yang Mills (YM) theory. Since we use an oscillator basis for the YM-Fock-space the anomaly appears already for a pure YM-system and the constraints form a Kac-Moody algebra with negative central charge. We also discuss the coupling of chiral fermions and find that the BRST-cohomology for systems with chiral fermions in a sufficiently large representation of the gauge group is completely equivalent to the cohomology of the finite dimensional gauge group. For pure YM theory or YM theory coupled to chiral fermions in small representations there exists an infinite number of inequivalent cohomology classes. This is discussed in some detail for the example of $SU(2)$.