Higher Spectral Flow for Dirac Operators with Local Boundary Conditions

We consider a gauge invariant one parameter family of families of fiberwise twisted Dirac type operators on a fiberation with the typical fiber an even dimensional compact manifold with boundary, i.e., a family ${D_u}, u\in [0,1]$with$D_1=gD_0g^{-1}$forasuitableunitaryautomorphism$g$ of the twisted bundle. Suppose all the operators $D_u$ are imposed with a certain \emph{local elliptic} boundary condition $F$ and $D_{u,F}$ is the self-adjoint extension of $D_u$. We establish a formula for the higher spectral flow of $\{D_{u,F}\}$, $u\in [0,1]$. Our result generalizes a recent result of Gorokhovsky and Lesch to the families case.