Spectral asymmetry on an open space

The spectral asymmetry is evaluated for a family of Dirac operators interacting with a topological background field and defined on an open infinite space. For these operators the spectral asymmetry is given by an integral over a local quantity that relates only to the homotopy properties of the background field. $\zeta$-function regularization is employed and a possible simple pole in the limit where the regulator is removed is shown to vanish. The spectral asymmetry can be computed in a closed form in specific models. This is exemplified in various cases involving solitons, vortices, magnetic monopoles, and instantons as background fields.