Eta Forms and the Odd Pseudodifferential Families Index

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 ∈ R is the ‘suspending’ parameter. Let πA : A(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 ∈ R. The total eta form, ηA, as described here, is an even form on A(φ) which has basic differential which is an explicit representative of the odd Chern character of the index of the family: (*) dηA = π ∗ AγA, Ch(ind(A)) = [γA] ∈ H (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 ηA with the Bismut-Cheeger eta form. Introduction Eta forms, starting with the eta invariant itself, appear as the boundary terms in the index formula for Dirac operators [2], [5], [4], [21], [22]. One aim of the present paper is to show that, with the freedom gained by working in the more general context of families of pseudodifferential operators, these forms appear as universal transgression, or connection, forms for the cohomology class of the index. That these forms arise in the treatment of boundary problems corresponds to the fact that boundary conditions amount to the explicit inversion of a suspended (or model) problem on the boundary. The odd index of the boundary family is trivial and the eta form is an explicit trivialization of it in cohomology. To keep the discussion within bounds we work here primarily in the ‘odd’ setting of a family of self-adjoint elliptic pseudodifferential operators, taken to be of order 1, on the fibres of a fibration of compact manifolds