An extended variational formula for the Bismut-Cheeger eta form and its applications

TL;DR

This paper extends the variational formula for the Bismut-Cheeger eta form to include twisted spin$^c$ Dirac operators and proves its additivity, leading to new proofs of key results in differential K-theory.

Contribution

It generalizes the variational formula for the eta form to twisted operators and establishes its additivity, enhancing the understanding of differential K-theory and index theory.

Findings

Extended variational formula for eta form without kernel bundle assumption

Proved $Z_2$-graded additivity of the eta form

Provided alternative proofs of the index homomorphism and Riemann-Roch-Grothendieck theorem

Abstract

The purpose of this paper is to extend our previous work on the variational formula for the Bismut-Cheeger eta form without the kernel bundle assumption by allowing the spin$^c$ Dirac operators to be twisted by isomorphic vector bundles, and to establish the $\mathbb{Z}_2$-graded additivity of the Bismut-Cheeger eta form. Using these results, we give alternative proofs of the fact that the analytic index in differential $K$-theory is a well defined group homomorphism, and the Riemann-Roch-Grothendieck theorem in $\mathbb{R}/\mathbb{Z}$ $K$-theory.