Surveys in Differential Geometry

Volume 15 (2010)

Eta forms and the odd pseudodifferential families index

Pages: 279 – 322

DOI: http://dx.doi.org/10.4310/SDG.2010.v15.n1.a9

Authors

Richard Melrose (Department of Mathematics, Massachusetts Institute of Technology.)

Frédéric Rochon (Department of Mathematics, Australian National University.)

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:

\begin{equation} d\eta_{\cA}=\pi_{\cA}^*\gamma _A,\ \Ch(\ind(A))=[\gamma_{A}]\in H^{\odd}(Y). \tag{*}\label{efatoi.5}\end{equation}

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.

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