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.
BiFr86 Bismut, J. M., and Freed, D., ‘The analysis of elliptic families: (I) Metrics and connections on determinant bundles’, Commun. Math. Phys. 106 (1986), 159–176.
Fo87 Forman, R., ‘Functional determinants and geometry’, Invent. Math. 88 (1987), 447–493.
Gr99 Grubb, G., ‘Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems’, Ark. Mat. 37 (1999), 45–86.
GrSe96 Grubb, G., and Seeley, R., ‘Zeta and eta functions for Atiyah-Patodi-Singer operators’, J. Geom. Anal. 6 (1996), 31–77.
LeTo98 Lesch, M., and Tolksdorf, J., ‘On the determinant of one-dimensional elliptic boundary value problems’, Comm. Math. Phys. 193 (1998), 643–660.
PrSe86 Pressley, A. and Segal, G. B., Loop Groups, Oxford, Clarendon Press, 1986.
Qu85 Quillen, D. G., ‘Determinants of Cauchy-Riemann operators over a Riemann surface’, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 37–41; English transl., Functional Anal. Appl. 19 (1985), 31–34.
Sc99 Scott, S. G., ‘Splitting the curvature of the determinant line bundle’, Proc. Amer. Math. Soc. 128 (2000), 2763–2775.
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.
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Additional Information
Simon Scott
Affiliation:
Department of Mathematics, King’s College, London WC2R 2LS, U.K.
Email:
sscott@mth.kcl.ac.uk
Received by editor(s):
December 15, 1999
Received by editor(s) in revised form:
September 15, 2000
Published electronically:
April 2, 2001
Communicated by:
Michael Taylor
Article copyright:
© Copyright 2001
American Mathematical Society