Index theorems on open infinite manifolds

Abstract

A new index theorem for Dirac operators defined on open infinite manifolds is derived. The derivation is based on the use of trace identifies and assumes the validity of the standard chiral anomaly equation. The theorem generalizes the Callias index theorem for arbitrary background fields and is analogous to the Atiyah-Patodi-Singer index theorem for Dirac operators on open compact manifolds. The essential differences between compact-and infinite-manifold index theorems is clarified by demonstrating that on an open compact manifold the standard form of the chiral anomaly is altered. A family index theorem is then derived and used to devise a technique for the computation of the η-invariant of arbitrary hermitian Dirac operators. This provides a complete solution to the mathematical problem of fermion number fractionization. Finally, elements of cohomology are developed by showing how the adiabatic approximation, when supplemented by a spectral flow analysis, is related to the index theorem.