Higher spectral flow for Dirac operators with local boundary conditions

We consider a one parameter family {D-u}, u is an element of [0, 1] of families of fiberwise twisted Dirac type operators on a fibration with the typical fiber an even dimensional compact manifold with boundary, which verifies D-1 = gD(0)g(-1) with g being a smooth map from the fibration to a unitary group U(N). For each u is an element of [0, 1], we impose on D-u a certain fixed local elliptic boundary condition F and get a self-adjoint extension D-u,D- F. Under the assumption that D-0,D- F has vanishing K-1-index bundle, we establish a formula for the higher spectral flow of {D-u,D- F}, u is an element of [0, 1]. 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.