Eta forms and the odd pseudodifferential families index

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 ϕ 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∈\bbR is the `suspending' parameter. Let π\cA:\cA(ϕ)⟶Y be the infinite-dimensional bundle with fibre at y∈Y consisting of the Schwartz-smoothing perturbations, q, making Ay(t)+q(t) invertible for all t∈\bbR. The total eta form, η\cA, as described here, is an even form on \cA(ϕ) which has basic differential which is an explicit representative of the odd Chern character of the index of the family:

dη\cA=π∗\cAγA, \Ch(\ind(A))=[γA]∈H\odd(Y).(*)

The 1 -form part of this identity may be interpreted in terms of the τ 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 (*) 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 η\cA with the Bismut-Cheeger eta form.