Skip to main content
×
×
Home

The Noncommutative Infinitesimal Equivariant Index Formula

  • Yong Wang (a1)
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.

Copyright
References
Hide All
Az.Azmi, F., The equivariant Dirac cyclic cocycle, Rocky Mountain J. Math. 30 (2000), 11711206.
BGS.Beals, R., Greiner, P. and Stanton, N., The heat equation on a CR manifold, J. Differential Geom. 20 (1984), 343387.
BGV.Berline, N., Getzler, E. and Vergne, M., Heat kernels and Dirac operators, Springer-Verlag, Berlin, 1992.
BV1.Berline, N. and Vergne, M., A computation of the equivariant index of the Dirac operators, Bull. Soc. Math. France 113 (1985), 305345.
BV2.Berline, N. and Vergne, M., The equivariant index and Kirillov character formula, Amer. J. Math. 107 (1985), 11591190.
Bi.Bismut, J. M., The infinitesimal Lefschetz formulas: a heat equation proof, J. Func. Anal. 62 (1985), 435457.
BlF.Block, J. and Fox, J., Asymptotic pseudodifferential operators and index theory, Contemp. Math. 105 (1990), 132.
CH.Chern, S. and Hu, X., Equivariant Chern character for the invariant Dirac operators, Michigan Math. J. 44 (1997), 451473.
Co.Connes, A., Entire cyclic cohomology of Banach algebras and characters of θ -summable Fredholm module, K-Theory 1 (1988), 519548.
CM1.Connes, A. and Moscovici, H., Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), 345388.
CM2.Connes, A. and Moscovici, H., Transgression and Chern character of finite dimensional K-cycles, Commun. Math. Phys. 155 (1993), 103122.
Do1.Donnelly, H., Eta invariants for G-spaces, Indiana Univ. Math. J. 27 (1978), 889918.
Fa.Fang, H., Equivariant spectral flow and a Lefschetz theorem on odd dimensional spin manifolds, Pacific J. Math. 220 (2005), 299312.
Fe.Feng, H., A note on the noncommutative Chern character (in Chinese), Acta Math. Sinica 46 (2003), 5764.
Go.Goette, S., Equivariant eta invariants and eta forms, J. reine angew Math. 526 (2000), 181236.
Ge1.Getzler, E., The odd Chern character in cyclic homology and spectral flow, Topology 32 (1993), 489507.
Ge2.Getzler, E., Cyclic homology and the Atiyah-Patodi-Singer index theorem, Contemp. Math. 148 (1993), 1945.
GS.Getzler, E. and Szenes, A., On the Chern character of theta-summable Fredholm modules, J. Func. Anal. 84 (1989), 343357.
Gr.Greiner, P., An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal. 41 (1971), 163218.
JLO.Jaffe, A., Lesniewski, A. and Osterwalder, K., Quantum K-theory: The Chern character, Comm. Math. Phys. 118 (1988), 114.
KL.Klimek, S. and Lesniewski, A., Chern character in equivariant entire cyclic cohomology, K-Theory 4 (1991), 219226.
LYZ.Lafferty, J. D., Yu, Y. L. and Zhang, W. P., A direct geometric proof of Lefschetz fixed point formulas, Trans. AMS. 329 (1992), 571583.
LM.Liu, K.; Ma, X., On family rigidity theorems, I. Duke Math. J. 102(3) (2000), 451474.
Po.Ponge, R., A new short proof of the local index formula and some of its applications, Comm. Math. Phys. 241 (2003), 215234.
PW.Ponge, R. and Wang, H., Noncommutative geometry, conformal geometry, and the local equivariant index theorem, arXiv:1210.2032.
Wa1.Wang, Y., The equivariant noncommutative Atiyah-Patodi-Singer index theorem, K-Theory, 37 (2006), 213233.
Wa2.Wang, Y., The Greiner's approach of heat kernel asymptotics, equivariant family JLO characters and equivariant eta forms, arXiv:1304.7354.
Wu.Wu, F., The Chern-Connes character for the Dirac operators on manifolds with boundary, K-Theory 7 (1993), 145174.
Zh.Zhang, W., A note on equivariant eta invariants, Proc. AMS 108 (1990), 11211129.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of K-Theory
  • ISSN: 1865-2433
  • EISSN: 1865-5394
  • URL: /core/journals/journal-of-k-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 8 *
Loading metrics...

Abstract views

Total abstract views: 95 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th July 2018. This data will be updated every 24 hours.