Edwin Langmann, Jouko Mickelsson
(3+1)-Dimensional Schwinger Terms and Non-commutative Geometry
The paper is published: Phys. Letters B 338 (1994) 241-248

MSC:

17B55 Homological methods in Lie algebras

17B65 Infinite-dimensional Lie algebras, See also {22E65}

17B81 Applications to physics

46L87 Noncommutative differential geometry, See also {58B30,

58B30 Noncommutative differential geometry and topology, See also {46L30, 46L87, 46L89}

81R10 Representations of infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody and other current algebras, See also {17B65, 17B67, 22E65, 22E67, 22E70}

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.