Index Theory for Short-Ranged Fields in Higher Dimensions

We set out to inverstigate the L2-index theory of Dirac operators on even dimensional open spin manifolds with warped ends, coupled to vector potentials with compactly supported field strength. These operators are not Fredholm, in general. The standard example is furnished by the Euclidean space. The problem turns out to be equivalent to a boundary value problem with nonlocal boundary conditions. The main result expresses the index in terms of the usual Atiyah-Singer local contribution and an eta invariant of a twisted Dirac operator on the "boundary at infinity." The Chern character in the local contribution can be transgressed to a local boundary term associated to the tangential component of the vector potential.