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The Noncommutative Infinitesimal Equivariant Index Formula

Published online by Cambridge University Press:  03 July 2014

Yong Wang
Yong Wang*
Affiliation:
School of Mathematics and Statistics, Northeast Normal University, Changchun Jilin, 130024, China, [email protected]
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Abstract

In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regularity and give an explicit formula for them. We also establish an infinitesimal equivariant family index formula and introduce the infinitesimal equivariant eta forms as well as compare them with the equivariant eta forms.

Keywords

Infinitesimal equivariant Chern-Connes charactersinfinitesimal equivariant eta cochainsinfinitesimal equivariant family indexinfinitesimal equivariant eta forms58J2019K56
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 1 , August 2014 , pp. 73 - 102
Copyright
Copyright © ISOPP 2014 

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References

REFERENCES

Az.Azmi, F., The equivariant Dirac cyclic cocycle, Rocky Mountain J. Math. 30 (2000), 1171–1206.Google Scholar
BGS.Beals, R., Greiner, P. and Stanton, N., The heat equation on a CR manifold, J. Differential Geom. 20 (1984), 343–387.CrossRefGoogle Scholar
BGV.Berline, N., Getzler, E. and Vergne, M., Heat kernels and Dirac operators, Springer-Verlag, Berlin, 1992.Google Scholar
BV1.Berline, N. and Vergne, M., A computation of the equivariant index of the Dirac operators, Bull. Soc. Math. France 113 (1985), 305–345.Google Scholar
BV2.Berline, N. and Vergne, M., The equivariant index and Kirillov character formula, Amer. J. Math. 107 (1985), 1159–1190.Google Scholar
Bi.Bismut, J. M., The infinitesimal Lefschetz formulas: a heat equation proof, J. Func. Anal. 62 (1985), 435–457.Google Scholar
BlF.Block, J. and Fox, J., Asymptotic pseudodifferential operators and index theory, Contemp. Math. 105 (1990), 1–32.Google Scholar
CH.Chern, S. and Hu, X., Equivariant Chern character for the invariant Dirac operators, Michigan Math. J. 44 (1997), 451–473.CrossRefGoogle Scholar
Co.Connes, A., Entire cyclic cohomology of Banach algebras and characters of θ -summable Fredholm module, K-Theory 1 (1988), 519–548.Google Scholar
CM1.Connes, A. and Moscovici, H., Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), 345–388.Google Scholar
CM2.Connes, A. and Moscovici, H., Transgression and Chern character of finite dimensional K-cycles, Commun. Math. Phys. 155 (1993), 103–122.Google Scholar
Do1.Donnelly, H., Eta invariants for G-spaces, Indiana Univ. Math. J. 27 (1978), 889–918.Google Scholar
Fa.Fang, H., Equivariant spectral flow and a Lefschetz theorem on odd dimensional spin manifolds, Pacific J. Math. 220 (2005), 299–312.Google Scholar
Fe.Feng, H., A note on the noncommutative Chern character (in Chinese), Acta Math. Sinica 46 (2003), 57–64.Google Scholar
Go.Goette, S., Equivariant eta invariants and eta forms, J. reine angew Math. 526 (2000), 181–236.Google Scholar
Ge1.Getzler, E., The odd Chern character in cyclic homology and spectral flow, Topology 32 (1993), 489–507.Google Scholar
Ge2.Getzler, E., Cyclic homology and the Atiyah-Patodi-Singer index theorem, Contemp. Math. 148 (1993), 19–45.Google Scholar
GS.Getzler, E. and Szenes, A., On the Chern character of theta-summable Fredholm modules, J. Func. Anal. 84 (1989), 343–357.Google Scholar
Gr.Greiner, P., An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal. 41 (1971), 163–218.CrossRefGoogle Scholar
JLO.Jaffe, A., Lesniewski, A. and Osterwalder, K., Quantum K-theory: The Chern character, Comm. Math. Phys. 118 (1988), 1–14.Google Scholar
KL.Klimek, S. and Lesniewski, A., Chern character in equivariant entire cyclic cohomology, K-Theory 4 (1991), 219–226.Google Scholar
LYZ.Lafferty, J. D., Yu, Y. L. and Zhang, W. P., A direct geometric proof of Lefschetz fixed point formulas, Trans. AMS. 329 (1992), 571–583.CrossRefGoogle Scholar
LM.Liu, K.; Ma, X., On family rigidity theorems, I. Duke Math. J. 102(3) (2000), 451–474.Google Scholar
Po.Ponge, R., A new short proof of the local index formula and some of its applications, Comm. Math. Phys. 241 (2003), 215–234.Google Scholar
PW.Ponge, R. and Wang, H., Noncommutative geometry, conformal geometry, and the local equivariant index theorem, arXiv:1210.2032.Google Scholar
Wa1.Wang, Y., The equivariant noncommutative Atiyah-Patodi-Singer index theorem, K-Theory, 37 (2006), 213–233.Google Scholar
Wa2.Wang, Y., The Greiner's approach of heat kernel asymptotics, equivariant family JLO characters and equivariant eta forms, arXiv:1304.7354.Google Scholar
Wu.Wu, F., The Chern-Connes character for the Dirac operators on manifolds with boundary, K-Theory 7 (1993), 145–174.CrossRefGoogle Scholar
Zh.Zhang, W., A note on equivariant eta invariants, Proc. AMS 108 (1990), 1121–1129.Google Scholar