On equivariant Chern-Weil forms and determinant lines

TL;DR

This paper develops a differential-geometric approach to G-equivariant characteristic classes, linking equivariant curvature and Chern-Weil forms to standard forms, and applies it to determinant lines of Dirac operators, illuminating quantum anomalies.

Contribution

It introduces a differential-geometric model of the classifying space BG for Lie groups, translating equivariant forms into nonequivariant ones, and applies this to analyze determinant lines and anomalies.

Findings

Derived the moment map of the determinant line for G-equivariant Dirac operators.

Connected equivariant curvature forms to standard Chern-Weil forms via a geometric model.

Provided insights into anomaly formulas in quantum field theory.

Abstract

A strong from of invariance under a group G is manifested in a family over the classifying space BG. We advocate a differential-geometric avatar of BG when G is a Lie group. Applied to G-equivariant connections on smooth principal or vector bundles, the equivariance-->families principle converts the G-equivariant extensions of curvature and Chern-Weil forms to the standard nonequivariant versions. An application of this technique yields the moment map of the determinant line of a G-equivariant Dirac operator, which in turn sheds light on some anomaly formulas in quantum field theory.