On the Topological Interpretation of Gravitational Anomalies

TL;DR

This paper provides a topological interpretation of gravitational anomalies using noncommutative geometry, specifically through $K$-theory and $K$-homology of $C^*$-algebras, and applies the Connes-Moscovici index theorem.

Contribution

It introduces a novel topological framework for understanding gravitational anomalies via noncommutative algebraic tools and index theory.

Findings

Topological interpretation of gravitational anomalies

Application of Connes-Moscovici index theorem to anomalies

Connection between $K$-theory, $K$-homology, and anomalies

Abstract

We consider the mixed gravitational-Yang-Mills anomaly as the coupling between the $K$-theory and $K$-homology of a $C^*$-algebra crossed product. The index theorem of Connes-Moscovici allows to compute the Chern character of the $K$-cycle by local formulae involving connections and curvatures. It gives a topological interpretation to the anomaly, in the sense of noncommutative algebras.