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The equivariant family index theorem in odd dimensions

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Abstract

In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed’s odd dimensional index formula. Then we extend this theorem to the noncommutative geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.

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References

  1. Bismut, J. M.: The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs. Invent. Math., 83, 91–151 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators, Springer-Verlag, Berlin, 1992

    Book  MATH  Google Scholar 

  3. Berline, N., Vergne, M.: A computation of the equivariant index of the Dirac operators. Bull. Soc. Math. France., 113, 305–345 (1985)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Freed, D.: Two index theorems in odd dimensions. Commu. Anal. Geom., 6, 317–329 (1998)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Getzler, E., Szenes, A.: On the Chern character of theta-summable Fredholm modules. J. Funct. Aual., 84, 343–357 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Liu, K., Ma, X.: On family rigidity theorems. I. Duke Math. J., 102, 451–474 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lafferty, J. D., Yu, Y., Zhang, W.: A direct geometric proof of Lefschetz fixed point formulas. Trans. Amer. Math. Ser., 329, 571–583 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Liu, K., Wang, Y.: Rigidity theorems on odd dimensional manifolds. Pure Appl. Math. Q., 5, 1139–1159 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ponge, R.: A new short proof of the local index formula and some of its applications. Comm. Math. Phys., 241, 215–234 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ponge, R., Wang, H.: Noncommutative geometry, conformal geometry, and the local equivariant index theorem. arXiv:1210.2032

  13. Wang, Y.: Volterra calculus, local equivariant family index theorem and equivariant eta forms. arXiv: 1304.7354

  14. Wang, Y.: Chern-Connes character for the invariant Dirac operator in odd dimensions. Sci. China Ser. A, 48, 1124–1134 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wang, J., Wang, Y.: Geometric quantization of odd dimensional spinc manifolds. Bull. Korean Math. Soc., 49, 223–234 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Yu, Y.: Lectures on Atiyah-Singer index theorem. In: Nankai Institute of Mathematics, meomographic notes, Tianjin, 1986

    Google Scholar 

  17. Zhang, W.: Local Atiyah-Singer index theorem for families of Dirac operators. Differential Geometry and Topology, Lecture Notes in Math., 1369, Springer, Berlin, 1989, 351–366

    Google Scholar 

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Correspondence to Yong Wang.

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Supported by National Natural Science Foundation of China (Grant No. 11271062) and Program for New Century Excellent Talents in University (Grant No. 13-0721)

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Bao, K.H., Wang, J. & Wang, Y. The equivariant family index theorem in odd dimensions. Acta. Math. Sin.-English Ser. 31, 1149–1162 (2015). https://doi.org/10.1007/s10114-015-3637-6

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  • DOI: https://doi.org/10.1007/s10114-015-3637-6

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