DETERMINANT BUNDLES, MANIFOLDS WITH BOUNDARY AND SURGERY

We define determinant bundles associated to the following data: (i) a family of generalized Dirac operators on even dimensional manifolds wi th boundary, (ii) the choice of a spectral section for the family of D irac operators induced on the boundary. Under the assumption that the operators of the boundary family have null spaces of constant dimensio n we define, through the notion of b-zeta function, a Quillen metric. We also introduce the analogue of the Bismut-Freed connection. We prov e that the curvature of a natural perturbation of the Bismut-Freed con nection equals the 2-form piece in the right-hand side of the family i ndex formula, thus extending to manifolds with boundary results of Qui llen, Bismut and Freed. Given a closed fibration, we investigate the b ehaviour of the Quillen metric and of the Bismut-Freed connection unde r the operation of surgery along a fibering hypersurface. We prove, in particular, additivity formulae for the curvature and for the logarit hm of the holonomy.