Skip to main content
SpringerLink
Log in

A New Short Proof of the Local Index Formula and Some of Its Applications

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

An Erratum to this article was published on 18 June 2004

Abstract

We give a new short proof of the index formula of Atiyah and Singer based on combining Getzler’s rescaling with Greiner’s approach of the heat kernel asymptotics. As an application we can easily compute the CM cyclic cocycle of even and odd Dirac spectral triples, and then recover the Atiyah-Singer index formula (even case) and the Atiyah-Patodi-Singer spectral flow formula (odd case).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atiyah, M., Bott, R., Patodi, V.: On the heat equation and the index theorem. Invent. Math. 19, 279–330 (1973)

    MATH  Google Scholar 

  2. Atiyah, M., Singer, I.: The index of elliptic operators. I. Ann. of Math. (2) 87, 484–530 (1968)

    Google Scholar 

  3. Atiyah, M., Singer, I.: The index of elliptic operators. III. Ann. of Math. (2) 87, 546–604 (1968)

    Google Scholar 

  4. Atiyah, M., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Camb. Philos. Soc. 77, 43–69 (1975)

    MATH  Google Scholar 

  5. Atiyah, M., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry. III. Math. Proc. Camb. Philos. Soc. 79, 71–99 (1976)

    MATH  Google Scholar 

  6. Beals, R., Greiner, P., Stanton, N.: The heat equation on a CR manifold. J. Differ. Geom. 20, 343–387 (1984)

    MathSciNet  MATH  Google Scholar 

  7. Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Berlin-Heidelberg-New York: Springer-Verlag, 1992

  8. Bismut, J.-M.: The Atiyah-Singer theorems: A probabilistic approach. I. The index theorem. J. Funct. Anal. 57, 56–99 (1984)

    MATH  Google Scholar 

  9. Bismut, J.-M., Freed, D.: The analysis of elliptic families. Commun. Math. Phys 107, 103–163 (1986)

    MathSciNet  MATH  Google Scholar 

  10. Chern, S., Hu, X.: Equivariant Chern character for the invariant Dirac operator. Michigan Math. J. 44, 451–473 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Connes, A.: Noncommutative geometry. London-New York: Academic Press, 1994

  12. Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. GAFA 5, 174–243 (1995)

    MATH  Google Scholar 

  13. Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger operators with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Berlin-Heidelberg-New York: Springer-Verlag, 1987

  14. Getzler, E.: Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem. Commun. Math. Phys. 92, 163–178 (1983)

    MathSciNet  MATH  Google Scholar 

  15. Getzler, E.: A short proof of the local Atiyah-Singer index theorem. Topology 25, 111–117 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Getzler, E.: The odd Chern character in cyclic homology and spectral flow. Topology 32, 489–507 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gilkey, P.: Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Berkeley, CA: Publish or Perish, 1984

  18. Glimm, J., Jaffe, A.: Quantum physics. A functional integral point of view. New York: Springer-Verlag, 1987

  19. Greiner, P.: An asymptotic expansion for the heat equation. Arch. Rational Mech. Anal. 41, 163–218 (1971)

    MATH  Google Scholar 

  20. Guillemin, V.: A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55, 131–160 (1985)

    MathSciNet  MATH  Google Scholar 

  21. Hadamard, J.: Lectures on Cauchy’s problem in linear partial differential equations. New York: Dover Publications, 1953

  22. Higson, N.: On the Connes-Mocovici residue cocycle. Preprint

  23. Lescure, J.-M.: Triplets spectraux pour les variétés à singularité conique isolée. Bull. Soc. Math. France 129, 593–623 (2001)

    MathSciNet  MATH  Google Scholar 

  24. Lawson, B., Michelson, M.-L.: Spin Geometry. Princeton, NJ: Princeton Univ. Press, 1993

  25. Melrose, R.: The Atiyah-Patodi-Singer theorem. Bostan: A.K. Peters, 1993

  26. Moscovici, H.: Eigenvalue inequalities and Poincaré duality in noncommutative geometry. Commun. Math. Phys. 184, 619–628 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Piriou, A.: Une classe d’opérateurs pseudo-différentiels du type de Volterra. Ann. Inst. Fourier 20, 77–94 (1970)

    Google Scholar 

  28. Roe, J.: Elliptic operators, topology and asymptotic methods. Pitman Research Notes in Mathematics Series 395, New York: Longman, 1998

  29. Taylor, M.E.: Partial differential equations. II. Applied Mathematical Sciences, 116, Berlin-Heidelberg-New York: Springer-Verlag, 1996

  30. Wodzicki, M.: Local invariants of spectral asymmetry. Invent. Math. 75, 143–177 (1984)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA

    Raphaël Ponge

Authors
  1. Raphaël Ponge
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raphaël Ponge.

Additional information

Communicated by A. Connes

An erratum to this article can be found at http://dx.doi.org/10.1007/s00220-004-1109-4

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ponge, R. A New Short Proof of the Local Index Formula and Some of Its Applications. Commun. Math. Phys. 241, 215–234 (2003). https://doi.org/10.1007/s00220-003-0915-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-003-0915-4

Keywords

Search

Navigation