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

Man-Ho Ho

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.

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

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 Dirac operators to be twisted by isomorphic vector bundles, and to establish the -graded additivity of the Bismut-Cheeger eta form. Using these results, we give alternative proofs of the fact that the analytic index in differential -theory is a well defined group homomorphism, and the Riemann-Roch-Grothendieck theorem in -theory.
Paper Structure (11 sections, 13 theorems, 191 equations)

This paper contains 11 sections, 13 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. Some primary and secondary characteristic forms
  5. An extended variational formula for the Bismut--Cheeger eta form
  6. Local index theory for twisted spin$^c$ Dirac operators: the Miščenko--Fomenko--Freed--Lott approach
  7. A proof of the extended variational formula for the Bismut--Cheeger eta form
  8. Applications of the extended variational formula for the Bismut--Cheeger eta form
  9. $\mathbb{Z}_2$-graded additivity of the Bismut--Cheeger eta form
  10. The analytic index in differential and $\mathbb{R}/\mathbb{Z}$ $K$-theory
  11. The RRG theorem in $\mathbb{R}/\mathbb{Z}$ $K$-theory for the twisted spin$^c$ Dirac operators

Key Result

Proposition 1.1

($=$ Proposition prop 3.2) Let $\pi:X\to B$ be a submersion with closed, oriented and spin$^c$ fibers of even dimension, equipped with a Riemannian and differential spin$^c$ structure $(T^HX, g^{T^VX}, g^\lambda, \nabla^\lambda)$. Let $(E, g^E)$ be a Hermitian bundle and $(F, g^F, \nabla^F)$ a Hermi

Theorems & Definitions (26)

  • Proposition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.2
  • Theorem 1.3
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • ...and 16 more