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Splitting the curvature
of the determinant line bundle
Author:
Simon Scott
Journal:
Proc. Amer. Math. Soc. 128 (2000), 2763-2775
MSC (1991):
Primary 58G20, 58G26; Secondary 81T50
Posted:
December 7, 1999
MathSciNet review:
1662210
Full-text PDF Free Access
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References |
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Additional Information
Abstract: It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold  has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families of Dirac operators over the two halves.
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Scott, S.G.: 1997, in preparation.
- 13.
Scott, S.G.: 1999, `Determinants of higher-order elliptic boundary value problems and the Quillen metric', preprint.
- 14.
Scott, S.G., and Torres, F.: 1998, `Elliptic families in dimension one: geometry of the determinant line bundle', preprint.
- 15.
Scott, S.G., and Wojciechowski, K.P.: 1998, `The -Determinant and Quillen's determinant on the Grassmannian of elliptic self-adjoint boundary conditions', C. R. Acad. Sci. Paris, t. 328, Serie I, 139-144.
- 16.
Wojciechowski, K.P.: 1997, `The -determinant and the additivity of the
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- 1.
- Bismut, J.M.: 1986, `The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs', Invent. Math. 83, 91-151. MR 87g:58117
- 2.
- Bismut, J.M., Freed,D.: 1986 `The analysis of elliptic families:(I) Metrics and connections on determinant bundles', Commun. Math. Phys. 106, 159-176. MR 88h:58110a
- 3.
- Booß-Bavnbek, B., and Wojciechowski, K.P.: 1993, Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston. MR 94h:58168
- 4.
- Booß-Bavnbek, B., Scott, S. G., and Wojciechowski, K. P.: 1998, `The -determinant and -determinant on the Grassmannian in dimension one', Letters in Math. Phys., to appear.
- 5.
- Grubb, G.: 1999, `Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems', Ark. Mat. 37, 45-86.
- 6.
- Melrose, R.B., Piazza, P.: 1997, `Families of Dirac operators, boundaries and the
 -calculus', J. Diff. Geom. 46, 99-167. MR 99a:58144
- 7.
- Piazza, P.: 1996, `Determinant bundles, manifolds with boundary and surgery', I Comm. Math. Phys. 178, 597-626; 1998, II, Comm. Math. Phys. 193, 105-124. MR 98a:58169
- 8.
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- 9.
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- 11.
- Scott, S.G.: 1995, `Determinants of Dirac boundary value problems over odd-dimensional manifolds', Commun. Math. Phys. 173, 43-76. MR 96g:58205
- 12.
- Scott, S.G.: 1997, in preparation.
- 13.
- Scott, S.G.: 1999, `Determinants of higher-order elliptic boundary value problems and the Quillen metric', preprint.
- 14.
- Scott, S.G., and Torres, F.: 1998, `Elliptic families in dimension one: geometry of the determinant line bundle', preprint.
- 15.
- Scott, S.G., and Wojciechowski, K.P.: 1998, `The -Determinant and Quillen's determinant on the Grassmannian of elliptic self-adjoint boundary conditions', C. R. Acad. Sci. Paris, t. 328, Serie I, 139-144.
- 16.
- Wojciechowski, K.P.: 1997, `The -determinant and the additivity of the -invariant on the smooth, self-adjoint Grassmannian', Comm. Math. Phys. 201, 423-444.
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Additional Information
Simon Scott
Affiliation:
Department of Mathematics, King’s College, Strand, London WC2R 2LS, United Kingdom
Email:
sscott@mth.kcl.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05311-3
PII:
S 0002-9939(99)05311-3
Keywords:
Determinant line bundle,
elliptic family,
Grassmann section,
regularized determinant,
splitting principle
Received by editor(s):
September 30, 1998
Posted:
December 7, 1999
Dedicated:
Dedicado a la memoria de Hugo Rojas 1973-1997
Communicated by:
Peter Li
Article copyright:
© Copyright 2000 American Mathematical Society
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