Wodzicki residue and anomalies of current algebras

TL;DR

This paper links anomalies in current algebras to the Wodzicki residue of pseudodifferential operators, providing a geometric perspective on quantum anomalies in gauge theories.

Contribution

It introduces a novel approach to compute current algebra anomalies using the Wodzicki residue and relates it to a geometric renormalization framework.

Findings

Anomalies expressed via Wodzicki residue of pseudodifferential operators

Connection between anomalies and Radul 2-cocycle in Lie algebra of PSDO's

Geometric interpretation of current algebra renormalization

Abstract

The commutator anomalies (Schwinger terms) of current algebras in $3+1$ dimensions are computed in terms of the Wodzicki residue of pseudodifferential operators; the result can be written as a (twisted) Radul 2-cocycle for the Lie algebra of PSDO's. The construction of the (second quantized) current algebra is closely related to a geometric renormalization of the interaction Hamiltonian $H_I=j_{\mu} A^{\mu}$ in gauge theory.