Inventiones mathematicae

, Volume 83, Issue 1, pp 91–151 | Cite as

The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs

  • Jean-Michel Bismut
Article

Summary

The purpose of this paper is to give two heat equation proofs of the Index Theorem of Atiyah-Singer for a family of Dirac operators.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alvarez-Gaume, L.: Supersymmetry and the Atiyah-Singer Index Theorem. Commun. Math. Physics90, 161–173 (1983)Google Scholar
  2. 2.
    Atiyah, M.F.:K-Theory. New York: Benjamin 1967Google Scholar
  3. 3.
    Atiyah, M.F.: Algebraic Topology and operators in Hilbert Space. In: Lect. in Mod. Anal. Appl. I. Lect. Notes Math. Berlin, Heidelberg, New York: Springer103, 101–120 (1969)Google Scholar
  4. 4.
    Atiyah, M.F.: Circular Symmetry and Stationary phase approximation. In: Proceedings of the Conference in Honor of L. Schwartz. Astérisque (1985, to appear)Google Scholar
  5. 5.
    Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes. I. Ann. Math.86, 374–407 (1967); II.88, 451–491 (1968)Google Scholar
  6. 6.
    Atiyah, M.F., Bott, R., Patodi, V.K.: On the heat equation and the Index Theorem. Invent. math.19, 279–330 (1973)Google Scholar
  7. 7.
    Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology3 (Supp. 1) 3–38 (1964)Google Scholar
  8. 8.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators. I. Ann. Math.87, 484–530 (1968); III. Ann. Math.87, 546–604 (1968)Google Scholar
  9. 9.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators. IV. Ann. Math.93, 119–138 (1971)Google Scholar
  10. 10.
    Azencott, R.: Une approche probabiliste du Théorème de l'indice d'Atiyah-Singer. D'après J. M. Bismut. Séminaire Bourbaki. Exposé633 (1984–1985) (à paraître)Google Scholar
  11. 11.
    Azencott, R., Baldi, P., Bellaiche, A., Bellaiche, C., Bougerol, P., Chaleyat, Maurel, M., Elie, L., Granara, J.: Géodésiques et diffusions en temps petit. Astérisque84-85, 1–281 (1981)Google Scholar
  12. 12.
    Berline, N., Vergne, M.: A computation of the equivariant index of the Dirac operator. Bull. Soc. Math. Fr. (to appear)Google Scholar
  13. 13.
    Bismut, J.M.: Mécanique aléatoire. Lect. Notes Math.866. Berlin, Heidelberg, New York: Springer 1981Google Scholar
  14. 14.
    Bismut, J.M.: An introduction to the stochastic calculus of variations. In: Kohlmann, M., Christopeit, N. (eds.) Stochastic Differential Systems. Lect. Notes Control Inf. Sci.43, 33–72 (1982)Google Scholar
  15. 15.
    Bismut, J.M.: Martingales, the Malliavin calculus and Hypoellipticity under general Hörmander's conditions. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 469–505 (1981)Google Scholar
  16. 16.
    Bismut, J.M.: Large deviations and the Malliavin calculus. Prog. Math. no 45, Basel-Boston-Stuttgart: Birkhäuser (1984)Google Scholar
  17. 17.
    Bismut, J.M.: Transformations différentiables du mouvement Brownien. In: Proceedings of the conference in honor of L. Schwartz. Astérisque (1985, to appear)Google Scholar
  18. 18.
    Bismut, J.M.: The Atiyah-Singer Theorems: A probabilistic approach. I. J. Funct. Anal.57, 56–99 (1984); II. J. Funct. Anal.57, 329–348 (1984)Google Scholar
  19. 19.
    Bismut, J.M.: Index Theorem and equivariant cohomology on the loop space. Commun. Math. Phys.98, 213–237 (1985)Google Scholar
  20. 20a.
    Bismut, J.M.: The infinitesimal Lefschetz formulas: A heat equation proof. J. Funct. Anal.62, 435–457 (1985)Google Scholar
  21. 20b.
    Bismut, J.M.: Le Théorème de l'indice des familles: Une démonstration par l'équation de la chaleur. CRAS SérieI, 300, no 20, 691–693 (1985)Google Scholar
  22. 21.
    Bismut, J.M.: Localization formulas, superconnections, and the Index Theorem for families. Commun. Math. Phys. (1985, to appear)Google Scholar
  23. 22.
    Bismut, J.M., Michel, D.: Diffusions conditionnelles. I. J. Funct. Anal.44, 174–211 (1981); II. Générateur conditionnel. Application au filtrage. J. Funct. Anal.45, 272–292 (1982)Google Scholar
  24. 23.
    Boutet de Montvel, L.: Systèmes presque elliptiques: Une autre démonstration de la formule de l'indice. Conference in honor of L. Schwartz. Astérisque (1985, to appear)Google Scholar
  25. 24.
    Boutet de Montvel, L., Malgrange, B. (to appear)Google Scholar
  26. 25.
    Friedan, D., Windey, H.: Supersymmetric derivation of the Atiyah-Singer Index and the Chiral Anomaly. Nuclear Physics B.235, 395–416 (1984)Google Scholar
  27. 26.
    Getzler, E.: Pseudodifferential operators on super manifolds and the Atiyah-Singer Index Theorem. Commun. Math. Phys.92, 163–178 (1983)Google Scholar
  28. 27.
    Getzler, E.: A short proof of the Atiyah-Singer Index Theorem. Topology (to appear)Google Scholar
  29. 28.
    Gilkey, P.: Curvature and the eigenvalues of the Laplacian. Adv. Math.10, 344–382 (1973)Google Scholar
  30. 29.
    Gilkey, P.: Lefschetz fixed point formulas and the heat equation. In: Byrnes, C. (ed.) Partial Differential Equations and Geometry. Proceeding, Park City Conf. 1977. Lect. Notes Pure Anal. Math.48, 91–147 (1979)Google Scholar
  31. 30.
    Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. Paris, Série I257, 7–9 (1963)Google Scholar
  32. 31.
    Malliavin, P.: Stochastic Calculus of variations and hypoelliptic operators. “Proceedings of the Conference on Stochastic Differential equations of Kyoto” 1976, 195–263. New York: Wiley 1978Google Scholar
  33. 32.
    Patodi, V.K.: Curvature and the eigenforms of the Laplace operator. J. Differ. Geom.5, 233–249 (1971)Google Scholar
  34. 33.
    Quillen, D.: Superconnections and the Chern Character. Topology24, 89–95 (1985)Google Scholar
  35. 34.
    Shih, W.: Fiber Cobordism and the index of a family of elliptic differential operators. Bull. Am. Math. Soc.72, 984–991 (1966)Google Scholar
  36. 35.
    Spivak, M.: Differential geometry. Vol. 5. Boston: Publish or Perish 1975Google Scholar
  37. 36.
    Treves, F.: Introduction to pseudodifferential operators and Fourier integral operators. Vol. 1. New York: Plenum Press 1980Google Scholar
  38. 37.
    Wess, J., Bagger, J.: Supersymmetry and supergravity. Princeton Series in Physics. Princeton: Princeton University Press 1983Google Scholar
  39. 38.
    Witten, E.: Fermion quantum number in Kaluza Klein Theory. (to appear)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Jean-Michel Bismut
    • 1
  1. 1.Département de MathématiqueUniversité Paris-SudOrsayFrance

Personalised recommendations