Communications in Mathematical Physics

, Volume 106, Issue 1, pp 159–176 | Cite as

The analysis of elliptic families. I. Metrics and connections on determinant bundles

  • Jean-Michel Bismut
  • Daniel S. Freed
Article

Abstract

In this paper, we construct the Quillen metric on the determinant bundle associated with a family of elliptic first order differential operators. We also introduce a unitary connection on λ and calculate its curvature. Our results will be applied to the case of Dirac operators in a forthcoming paper.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Differential Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [ABP]
    Atiyah, M. F., Bott, R., Patodi, V. K.: On the heat equation and the Index Theorem. Invent. Math.19, 279–330 (1973)Google Scholar
  2. [APS1]
    Atiyah, M. F., Patodi, V. K., Singer, I. M.: Spectral asymmetry and Riemannian Geometry, III. Math. Proc. Camb. Phil. Soc.79, 71–99 (1976)Google Scholar
  3. [AS1]
    Atiyah, M. F., Singer, I. M.: The Index of elliptic operators. IV. Ann. Math.93, 119–138 (1971)Google Scholar
  4. [B1]
    Bismut, J. M.: The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs. Invent. Math.83, 91–151 (1986)Google Scholar
  5. [BF1]
    Bismut, J. M., Freed D. S.: Fibré déterminant et invariant êta. C.R. Acad. Sci. Sér. 1.301, 707–710 (1985)Google Scholar
  6. [BF2]
    Bismut, J. M., Freed, D. S.: The analysis of elliptic families. Part II. (To appear in Commun. Math. Phys.)Google Scholar
  7. [F]
    Freed, D. S.: (To appear)Google Scholar
  8. [Gr]
    Greiner, P.: An asymptotic expansion for the heat equation. Arch. Ration. Mech. Anal.41, 163–218 (1971)Google Scholar
  9. [Q1]
    Quillen, D.: Superconnections and the Chern character. Topology24, 89–95 (1985)Google Scholar
  10. [Q2]
    Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, 31–34 (1985)Google Scholar
  11. [Se]
    Seeley, R. T.: Complex powers of an elliptic operator. Proc. Symp. Pure Math. AMS10, 288–307 (1967)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Jean-Michel Bismut
    • 1
  • Daniel S. Freed
    • 2
  1. 1.Départment de MathématiqueUniversité Paris-SudOrsayFrance
  2. 2.Department of MathematicsM.I.T.CambridgeUSA

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