Relative zeta determinants and the geometry of the determinant line bundle

Scott, Simon

doi:10.1090/S1079-6762-01-00089-0

Relative zeta determinants and the geometry of the determinant line bundle

Author: Simon Scott
Journal: Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 8-16
MSC (2000): Primary 58G20, 58G26, 11S45; Secondary 81T50
DOI: https://doi.org/10.1090/S1079-6762-01-00089-0
Published electronically: April 2, 2001
MathSciNet review: 1826991
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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.

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Sc2000 Scott, S. G., In preparation.

ScWo99 Scott, S.G., and Wojciechowski, K.P., ‘The $\zeta$-determinant and Quillen’s determinant for a Dirac operator on a manifold with boundary’, Geom. Funct. Anal. 10 (2000), 1202–1236.