Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762



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
Published electronically: April 2, 2001
MathSciNet review: 1826991
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. 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. MR 88h:58110a
  • 2. Forman, R., `Functional determinants and geometry', Invent. Math. 88 (1987), 447-493. MR 89b:58212
  • 3. Grubb, G., `Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems', Ark. Mat. 37 (1999), 45-86. MR 2000c:35265
  • 4. Grubb, G., and Seeley, R., `Zeta and eta functions for Atiyah-Patodi-Singer operators', J. Geom. Anal. 6 (1996), 31-77. MR 97g:58174
  • 5. Lesch, M., and Tolksdorf, J., `On the determinant of one-dimensional elliptic boundary value problems', Comm. Math. Phys. 193 (1998), 643-660. MR 2000b:58060
  • 6. Pressley, A. and Segal, G. B., Loop Groups, Oxford, Clarendon Press, 1986. MR 88i:22049
  • 7. 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. MR 86g:32035
  • 8. Scott, S. G., `Splitting the curvature of the determinant line bundle', Proc. Amer. Math. Soc. 128 (2000), 2763-2775. MR 2000m:58055
  • 9. Scott, S. G., In preparation.
  • 10. 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. CMP 2001:05

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (2000): 58G20, 58G26, 11S45, 81T50

Retrieve articles in all journals with MSC (2000): 58G20, 58G26, 11S45, 81T50

Additional Information

Simon Scott
Affiliation: Department of Mathematics, King’s College, London WC2R 2LS, U.K.

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

American Mathematical Society