Eta Forms and the Chern Character

The semi-topological nature of the eta-invariant of a self-adjoint elliptic differential operator derives from a relative identification with a Chern character. This remarkable semi-locality property of the eta-invariant can be seen in spectral flow formulae and many other applications [APS2, APS3, BC1, DZ1, L1]. In this paper we prove two geometric index theorems for a family of first-order elliptic operators over a manifold with boundary by computing eta form representatives for the Chern character classes of the index bundle. The eta forms occur as relative and regularized traces on infinite-dimensional vector bundles realized as the limiting values of superconnection character forms. The formulas are non-local and general, they do not require spin structures, compatibility with Clifford actions, or dimensional restrictions.