Eta forms and the odd pseudodifferential families index

TL;DR

This paper develops a geometric framework for eta forms associated with families of pseudodifferential operators, linking them to index theory, K-theory, and eta invariants, with applications to Dirac operators and classifying spaces.

Contribution

It introduces the total eta form for parameter-dependent pseudodifferential families, providing explicit representatives of the odd Chern character of their index and connecting to eta invariants and K-theory gerbes.

Findings

The eta form's differential equals the odd Chern character of the index.

The 1-form part relates to the eta invariant determinant.

The 2-form part acts as a B-field on the K-theory gerbe.

Abstract

Let $A(t)$ be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration $\phi$ with base $Y.$ The standard example is $A+it$ where $A$ is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and $t\in\bbR$ is the `suspending' parameter. Let $\pi_{\cA}:\cA(\phi)\longrightarrow Y$ be the infinite-dimensional bundle with fibre at $y\in Y$ consisting of the Schwartz-smoothing perturbations, $q,$ making $A_y(t)+q(t)$ invertible for all $t\in\bbR.$ The total eta form, $\eta_{\cA},$ as described here, is an even form on $\cA(\phi)$ which has basic differential which is an explicit representative of the odd Chern character of the index of the family: % d\eta_{\cA}=\pi_{\cA}^*\gamma_A, \Ch(\ind(A))=[\gamma_{A}]\in H^{\odd}(Y). \tag{*}\label{efatoi.5} % The 1-form part of this identity may be interpreted in terms of the $\tau$ invariant (exponentiated eta invariant) as the determinant of the family. The 2-form part of the eta form may be interpreted as a B-field on the K-theory gerbe for the family $A$ with \eqref{efatoi.5} giving the `curving' as the 3-form part of the Chern character of the index. We also give `universal' versions of these constructions over a classifying space for odd K-theory. For Dirac-type operators, we relate $\eta_{\cA}$ with the Bismut-Cheeger eta form.