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Localization formulas, superconnections, and the index theorem for families

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Abstract

In this paper, we give a new proof of the localization formulas of Berline and Vergne [9] and Duistermaat and Heckman [18]. When interpreted in the framework of Atiyah [2], the probabilistic heat equation proof of the Index Theorem given in our paper [12] appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions. The results of Quillen [25] on superconnections are briefly presented. The heat equation proofs [15] of the Index Theorem for families are described. It is shown that in this framework, the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space.

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References

  1. Alvarez-Gaumé, L.: Supersymmetry and the Atiyah-Singer Index Theorem. Commun. Math. Phys.90, 161–173 (1983)

    Google Scholar 

  2. Atiyah, M.F.: Circular symmetry and stationary phase approximation. In: Proceedings of the conference in honor of L. Schwartz. Paris: Astérisque 1985 (to appear)

    Google Scholar 

  3. 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 

  4. Atiyah, M.F., Bott, R.: The moment map and equivariant cohomology. Topology23, 1–28 (1984)

    Google Scholar 

  5. Atiyah, M.F., Bott, R., Patodi, V.K.: On the heat equation and the Index Theorem. Invent. Math.19, 279–330 (1973)

    Google Scholar 

  6. Atiyah, M.F., Singer, I.M.: The index of elliptic operators. I. Ann. Math.87, 484–530 (1968); III.87, 546–604 (1968)

    Google Scholar 

  7. Atiyah, M.F., Singer, I.M.: The index of elliptic operators. IV. Ann. Math.93, 119–138 (1971)

    Google Scholar 

  8. Berline, N., Vergne, M.: A computation of the equivariant index of the Dirac operator (to appear)

  9. Berline, N., Vergne, M.: Zéros d'un champ de vecteurs et classes caractéristiques équivariantes. Duke Math. J.50, 539–549 (1983)

    Google Scholar 

  10. Bismut, J.-M.: Large deviations and the Malliavin calculus. Progress in Math. No. 45. Basel: Birkhäuser 1984

    Google Scholar 

  11. Bismut, J.-M.: Transformations différentiables du mouvement Brownien. In: Proceedings of the conference in honor of L. Schwartz. Paris: Astérisque 1985 (to appear)

    Google Scholar 

  12. Bismut, J.-M.: The Atiyah-Singer theorems: a probabilistic approach. I. J. Funct. Anal.57, 56–99 (1984); II.57, 329–348 (1984)

    Google Scholar 

  13. Bismut, J.-M.: Index theorem and equivariant cohomology on the loop space. Commun. Math. Phys.98, 213–237 (1985)

    Google Scholar 

  14. Bismut, J.-M.: The infinitesimal Lefschetz formulas: a heat equation proof. J. Funct. Anal.62, 435–457 (1985)

    Google Scholar 

  15. Bismut, J.-M.: The index theorem for families of Dirac operators: two heat equation proofs. Invent. Math. (to appear)

  16. Bismut, J.-M., Michel, D.: Diffusions conditionnelles. I. J. Funct. Anal.44, 174–211 (1981); II. Générateur conditionnel. Application au filtrage.45, 272–292 (1982)

    Google Scholar 

  17. Bott, R., Tu, L.H.: Differential forms in algebraic topology. Graduate texts in Math., Vol. 82. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  18. Duistermaat, J.J., Heckman, G.: On the variation of the cohomology of the reduced phase space. Invent. Math.69, 259–268 (1982); Addendum72, 153–158 (1983)

    Google Scholar 

  19. Friedan, D., Windey, H.: Supersymmetric derivation of the Atiyah-Singer index and the Chiral anomaly. Nucl. Phys. B235, 395–416 (1984)

    Google Scholar 

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

    Google Scholar 

  21. Getzler, E.: A short proof of the Atiyah-Singer Index Theorem. Topology (to appear)

  22. Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. Paris Ser. I257, 7–9 (1963)

    Google Scholar 

  23. Mathai, V., Quillen, D.: Superconnections, Thom classes and equivariant differential forms (to appear)

  24. McKean, H., Singer, I.M.: Curvature and the eigenvalues of the Laplacian. J. Differ. Geom.1, 43–69 (1967)

    Google Scholar 

  25. Quillen, D.: Superconnections and the Chern character. Topology24, 89–95 (1985)

    Google Scholar 

  26. Witten, E.: Supersymmetry and Morse theory. J. Diff. Geom.17, 661–692 (1982)

    Google Scholar 

  27. Berline, N., Vergne, M.: The equivariant Index and Kirillov's character formula. Trans. Am. Math. Soc. (to appear)

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  1. Départment de Mathématiques, Université Paris-Sud, F-91405, Orsay, France

    J. -M. Bismut

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  1. J. -M. Bismut
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Communicated by A. Jaffe

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Bismut, J.M. Localization formulas, superconnections, and the index theorem for families. Commun.Math. Phys. 103, 127–166 (1986). https://doi.org/10.1007/BF01464285

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  • DOI: https://doi.org/10.1007/BF01464285

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