Surveys in Differential Geometry

Volume 22 (2017)

On equivariant Chern–Weil forms and determinant lines

Pages: 125 – 132



Daniel S. Freed (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)


A strong form of invariance under a group $G$ is manifested in a family over the classifying space $BG$. We advocate a differentialgeometric avatar of $BG$ when $G$ is a Lie group. Applied to $G$-equivariant connections on smooth principal or vector bundles, the equivariance $\to$ 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.

The work of D.S.F. is supported by the National Science Foundation under grant DMS-1207817.

Published 13 September 2018