(3+1)-Dimensional Schwinger Terms and Non-commutative Geometry

TL;DR

This paper rigorously derives the Schwinger term in (3+1)-dimensional chiral QCD using non-commutative geometry, connecting Lie algebra cocycles to gauge theory anomalies and generalizing calculus to non-commutative spaces.

Contribution

It explicitly relates the Mickelsson-Faddeev-Shatashvili cocycle to operator algebra cocycles, bridging gauge theory anomalies with non-commutative geometry.

Findings

Explicit calculation of the cocycle's cohomology class.

Connection between gauge theory anomalies and non-commutative geometry.

Development of a new calculus for non-commutative forms.

Abstract

We discuss 2-cocycles of the Lie algebra $\Map(M^3;\g)$ of smooth, compactly supported maps on 3-dimensional manifolds $M^3$ with values in a compact, semi-simple Lie algebra $\g$. We show by explicit calculation that the Mickelsson-Faddeev-Shatashvili cocycle $\f{\ii}{24\pi^2}\int\trac{A\ccr{\dd X}{\dd Y}}$ is cohomologous to the one obtained from the cocycle given by Mickelsson and Rajeev for an abstract Lie algebra $\gz$ of Hilbert space operators modeled on a Schatten class in which $\Map(M^3;\g)$ can be naturally embedded. This completes a rigorous field theory derivation of the former cocycle as Schwinger term in the anomalous Gauss' law commutators in chiral QCD(3+1) in an operator framework. The calculation also makes explicit a direct relation of Connes' non-commutative geometry to (3+1)-dimensional gauge theory and motivates a novel calculus generalizing integration of $\g$-valued forms on 3-dimensional manifolds to the non-commutative case.