Determinant bundles, manifolds with boundary and surgery: II. Spectral sections and surgery rules for anomalies

Let φ : M → B be a closed fibration of Riemannian manifolds and let O = (Oz), z ∈ B, be a family of generalized Dirac operators. Let H ⊂ M be an embedded hypersurface fibering over B; χ : H → B. Let OH = (OH,z) be the Dirac family induced on χ : H → B. Each fiber φ-1(z) = Mz in φ : M → B is the union along χ-1(z) = Hz of two manifolds with boundary M0z, M1z. In this paper, generalizing our previous work [16], we prove general surgery rules for the local and global anomalies of the Bismut-Freed connection on the determinant bundle associated to O. Our results depend heavily on the b-calculus [12], on the surgery calculus [11] and on the APS family index theory developed in [13], in particular on the notion of spectral section for the family OH.