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Relative zeta determinants and the geometry of the determinant line bundle
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## Relative zeta determinants and the geometry of the determinant line bundle

### Simon Scott

**Abstract.**
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`

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