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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
Published electronically: December 7, 1999
MathSciNet review: 1662210
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold $M=X^{0}\cup X^{1}$ 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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  • 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 $\zeta $-determinant and ${\mathcal C}$-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 $b$-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. Pressley, A. and Segal, G.B.: 1986, `Loop Groups', O.U.P. MR 88i:22049
  • 9. Quillen, D.G.: 1985, `Determinants of Cauchy-Riemann operators over a Riemann surface', Funk. Anal. i ego Prilozhenya 19, 37-41. MR 86g:32035
  • 10. Segal, G.B.: 1990, `The definition of conformal field theory', preprint.
  • 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 $\zeta$-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 $\zeta$-determinant and the additivity of the $\eta$-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

Keywords: Determinant line bundle, elliptic family, Grassmann section, regularized determinant, splitting principle
Received by editor(s): September 30, 1998
Published electronically: 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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