Higher spectral flow for Dirac operators with local boundary conditions

Abstract

We consider a one parameter family [Formula: see text] of families of fiberwise twisted Dirac type operators on a fibration with the typical fiber an even dimensional compact manifold with boundary, which verifies [Formula: see text] with [Formula: see text] being a smooth map from the fibration to a unitary group [Formula: see text]. For each [Formula: see text], we impose on [Formula: see text] a certain fixed local elliptic boundary condition [Formula: see text] and get a self-adjoint extension [Formula: see text]. Under the assumption that [Formula: see text] has vanishing [Formula: see text]-index bundle, we establish a formula for the higher spectral flow of [Formula: see text], [Formula: see text]. Our result generalizes a recent result of [A. Gorokhovsky and M. Lesch, On the spectral flow for Dirac operators with local boundary conditions, Int. Math. Res. Not. IMRN (2015) 8036–8051.] to the families case.