Topological invariants for Dirac operators on open spaces

Topological invariants associated with a class of Dirac operators on odd-dimensional open Euclidean spaces are discussed. An invariant is introduced which, analogously to the one of Callias, depends on a free parameter. For examples considered, the above invariant with the parameter set equal to zero turns out to be harmonic conjugate to the Atiyah-Patodi-Singer eta invariant as a function of two physical parameters. Quantum field-theoretical implications of these invariants are studied