Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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GEOMETRIC QUANTIZATION OF ODD DIMENSIONAL SPINc MANIFOLDS

  • Wang, Jian (School of Mathematics and Statistics Northeast Normal University) ;
  • Wang, Yong (School of Mathematics and Statistics Northeast Normal University)
  • Received : 2009.10.10
  • Published : 2012.03.31
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Abstract

We prove a Guillemin-Sternberg geometric quantization formula for circle action on odd dimensional $spin^c$-manifolds. We prove two Kostant type formulas in this case. As a corollary, we get a cutting formula for the odd spinc quantization.

Keywords

References

  1. C. da S. Ana, Y. Karshon, and S. Tolman, Quantization of presymplectic manifolds and circle actions, Trans. Amer. Math. Soc. 352 (2000), no. 2, 525-552. https://doi.org/10.1090/S0002-9947-99-02260-6
  2. M. F. Atiyah and I. M. Singer, The index of elliptic operators III, Ann. of Math. (2) 87 (1968), 546-604. https://doi.org/10.2307/1970717
  3. H. Duistermaat, V. Guillemin, E. Meinrenken, and S. Wu, Symplectic reduction and Riemann-Roch for circle actions, Math. Res. Lett. 2 (1995), no. 3, 259-266. https://doi.org/10.4310/MRL.1995.v2.n3.a3
  4. H. Fang, Equivariant spectral flow and a Lefschetz theorem on odd-dimensional spin manifolds, Pacific J. Math. 220 (2005), no. 2, 299-312. https://doi.org/10.2140/pjm.2005.220.299
  5. D. Freed, Two index theorems in odd dimensions, Commu. Anal. Geom. 6 (1998), no. 2, 317-329. https://doi.org/10.4310/CAG.1998.v6.n2.a4
  6. S. Fuchs, Spin-c Quantization, Prequantization and Cutting, Thesis, University of Toronto, 2008.
  7. V. Guillemin, Reduced phase space and Riemann-Roch, Lie theory and geometry, 305-334, Progr. Math., 123, Birkhauser Boston, Boston, MA, 1994.
  8. V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group rep- resentations, Invent. Math. 67 (1982), no. 3, 515-538. https://doi.org/10.1007/BF01398934
  9. V. Guillemin, S. Sternberg, and J. Weitsman, Signature quantization, J. Differential Geom. 66 (2004), no. 1, 139-168. https://doi.org/10.4310/jdg/1090415031
  10. L. C. Jeffrey and F. C. Kirwan, Localization and quantization conjecture, Topology 36 (1997), no. 3, 647-693. https://doi.org/10.1016/S0040-9383(96)00015-8
  11. L. C. Jeffrey and F. C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995), no. 2, 291-327. https://doi.org/10.1016/0040-9383(94)00028-J
  12. J. D. Lafferty, Y. L. Yu, and W. P. Zhang, A direct geometric proof of Lefschetz fixed point formulas, Trans. Amer. Math. Soc. 329 (1992), no. 2 571-583. https://doi.org/10.2307/2153952
  13. H. Lawson and M. Michelsohn, Spin Geometry, Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989.
  14. E. Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247-258. https://doi.org/10.4310/MRL.1995.v2.n3.a2
  15. K. Liu and Y.Wang, Rigidity theorems on odd dimensional manifolds, Pure Appl. Math. Q. 5 (2009), no. 3, 1139-1159. https://doi.org/10.4310/PAMQ.2009.v5.n3.a7
  16. E. Meinrenken, On Riemann-Roch formulas for multiplicities, J. Amer. Math. Soc. 9 (1996) 373-389. https://doi.org/10.1090/S0894-0347-96-00197-X
  17. E. Meinrenken, Symplectic surgery and the $Spin^{c}$Spinc-Dirac operator, Adv. Math. 134 (1998), no. 2, 240-277. https://doi.org/10.1006/aima.1997.1701
  18. Y. Tian and W. Zhang, An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg, Invent. Math. 132 (1998), no. 2, 229-259. https://doi.org/10.1007/s002220050223
  19. M. Vergne, Multiplicity formula for geometric quantization, Part I., Duke Math. J. 82 (1996), no. 1, 143-179. https://doi.org/10.1215/S0012-7094-96-08206-X
  20. M. Vergne, Multiplicity formula for geometric quantization, Part II, Duke Math. J. 82 (1996), no. 1, 181-194. https://doi.org/10.1215/S0012-7094-96-08207-1
  21. E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303-368. https://doi.org/10.1016/0393-0440(92)90034-X
  22. W. Zhang, Lectures on Chern-weil Theory and Witten Deformations, Nankai Tracks in Mathematics Vol. 4, World Scientific, Singapore, 2001.

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