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

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.