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The spectral $\zeta$-function regularized geometry of the determinant line bundle for a family of first-order elliptic operators over a closed manifold encodes a subtle relation between the local family's index theorem and fundamental non-local spectral invariants. A great deal of interest has been directed towards a generalization of this theory to families of elliptic boundary value problems. We give here precise formulas for the relative zeta metric and curvature in terms of Fredholm determinants and traces of operators over the boundary. This has consequences for anomalies over manifolds with boundary.

Copyright 2001 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 07 (2001), pp. 8-16
• Publisher Identifier: S 1079-6762(01)00089-0
• 2000 Mathematics Subject Classification. Primary 58G20, 58G26, 11S45; Secondary 81T50
• Received by the editors December 15, 1999
• Posted on April 2, 2001
• Communicated by Michael Taylor
• Comments (When Available)

Simon Scott

Department of Mathematics, King's College, London WC2R 2LS, U.K.

E-mail address: sscott@mth.kcl.ac.uk