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Communications in Mathematical Physics

, Volume 178, Issue 3, pp 597–626 | Cite as

Determinant bundles, manifolds with boundary and surgery

  • Paolo Piazza
Article

Abstract

We define determinant bundles associated to the following data: (i) a family of generalized Dirac operators on even dimensional manifolds with boundary, (ii) the choice of a spectral section for the family of Dirac operators induced on the boundary. Under the assumption that the operators of the boundary family have null spaces of constant dimension we define, through the notion ofb-zeta function, a Quillen metric. We also introduce the analogue of the Bismut-Freed connection. We prove that the curvature of a natural perturbation of the Bismut-Freed connection equals the 2-form piece in the right-hand side of the family index formula, thus extending to manifolds with boundary results of Quillen, Bismut and Freed. Given a closed fibration, we investigate the behaviour of the Quillen metric and of the Bismut-Freed connection under the operation of surgery along a fibering hypersurface. We prove, in particular, additivity formulae for the curvature and for the logarithm of the holonomy.

Keywords

Neural Network Manifold Complex System Nonlinear Dynamics Quantum Computing 
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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Paolo Piazza
    • 1
  1. 1.Istituto Matematico “G. Castelnuovo”Università degli Studi di Roma “La Sapienza”RomaItaly

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