Abstract: In a succession of papers, physicists and mathematicians have achieved an explicit parameterization of Bianchi-IX gravitational instantons in terms of theta functions with characteristics. By exploiting the latter, in this talk, I will shed light on a rationality phenomena in the spectral action of SU(2)-invariant Bianchi-IX metrics. This will be done by showing that for the instantons, each term in the expansion of their spectral action gives rise to a modular form of weight 2 that can be written explicitly in terms of well-known modular forms, namely the Eisenstein series and the modular discriminant. An elegant proof of the rationality result will also be presented, which is based on expressing Seeley-de Witt coefficients as noncommutative residues of Laplacians. This talk is based on joint works with Wentao Fan and Matilde Marcolli.

Abstract: For a smooth manifold M with a Lie group G action, we can define equivariant cohomology based on differential forms on M. That is Weil model. This construction is analogous to Borel construction on the level of differential forms. The Cartan model is then derived from the Weil model. The Cartan model provides an explicit way to compute equivariant cohomology. We will introduce both models and prove that they are equivalent.

This is the third and final part of this talk.