Communications in Mathematical Physics

, Volume 245, Issue 3, pp 449–489 | Cite as

Gerbes over Orbifolds and Twisted K-Theory

Article
First Online:
Received:
Accepted:

Abstract

In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3(X) via Chern-Weil theory. For an arbitrary gerbe , a twisting K orb(X) of the orbifold K-theory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], Atiyah-Segal [2] and Bowknegt et. al. [4] in the smooth case and by Adem-Ruan [1] for discrete torsion on an orbifold.

Keywords

Geometric Model Characteristic Class Smooth Case Discrete Torsion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adem, A., Ruan, Y.: Twisted orbifold K-theory. Commun. Math. Phys. 237(3), 533–556 (2003) MR 1 993 337Google Scholar
  2. 2.
    Atiyah, M.: K-Theory Past and Present. arXiv:math.KT/0012213Google Scholar
  3. 3.
    Behrend, K., Edidin, D., Fantechi, B., Fulton, W., Göttsche, L., Kresch, A.: Introduction to Stacks. In preparationGoogle Scholar
  4. 4.
    Bouwknegt, P., Carey, A.L., Mathai, V., Murray, M.K., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228(1), 17–45 (2002) MR 2003g:58049CrossRefMATHGoogle Scholar
  5. 5.
    Brylinski, J.-L.: Loop spaces characteristic classes and geometric quantization. Progress in Mathematics 107. Boston, MA: Birkhäuser Boston, Inc., 1993Google Scholar
  6. 6.
    Carey, A.L., Mickelsson, J.: The universal gerbe Dixmier-Douady class and gauge theory. Lett. Math. Phys. 59(1), 47–60 (2002) MR 2003e:58048CrossRefMATHGoogle Scholar
  7. 7.
    Chen, W.: A Homotopy Theory of Orbispaces. (2001) arXivmath.AT/0102020Google Scholar
  8. 8.
    Chen, W., Ruan, Y.: Orbifold Quantum Cohomology. (2000) arXiv:math.AG/0005198Google Scholar
  9. 9.
    Cohen, R.L., Jones, J.D.S., Segal, G.B.: Floer’s infinite-dimensional Morse theory and homotopytheory. In: The Floer Memorial Volume, Basel: Birkhäuser, 1995, pp. 297–325Google Scholar
  10. 10.
    Crainic, M., Moerdijk, I.: A homology theory forétale groupoids. J. Reine Angew. Math. 521, 25–46 (2000)MathSciNetMATHGoogle Scholar
  11. 11.
    Donovan, P., Karoubi, M.: Graded Brauer Groups and K-theory. IHES 38 (1970)Google Scholar
  12. 12.
    Freed, D.: The Verlinde algebra is twisted equivariant K-theory. arXiv:math.RT/0101038Google Scholar
  13. 13.
    Giraud, J.: Cohomologie non Abéliene. Berlin-Heidelberg-New York: Springer-Verlag, 1971Google Scholar
  14. 14.
    Grothendieck, A.: Dix exposés sur la cohomologie des schémas. Amsterdam: North Holland, 1968Google Scholar
  15. 15.
    Hitchin, N.: Lectures Notes on Special Lagrangian Submanifolds. (1999) arXiv:math.DG/9907034Google Scholar
  16. 16.
    Lupercio, E., Uribe, B.: Holonomy for grebes over orbifolds. arXiv:math.AT/0307114Google Scholar
  17. 17.
    Lupercio, E., Uribe, B.: Deligne Cohomology for Orbifolds Discrete Torsion and B-fields. In: Geometric and Topological methods for Quantum Field Theory, Singapore: World Scientific, 2002Google Scholar
  18. 18.
    Lupercio, E., Uribe, B.: Loop groupoids, gerbes, and twisted sectors on orbifolds. In: Orbifolds in mathematics and physics (Madison WI 2001), Contemp. Math. Vol. 310, Providence RI: Am. Math. Soc., 2002, 163–184 MR 1 950 946Google Scholar
  19. 19.
    May, P.: Simplicial Objects in Algebraic Topology. Van Nostrand Mathematical Studies 11, Princeton N.J.: D. Van Nostrand Co. Inc., 1967Google Scholar
  20. 20.
    Moerdijk, I.: Calssifying topos and foliations. Ann. Inst. Fourier 41, 189–209 (1991)MathSciNetMATHGoogle Scholar
  21. 21.
    Moerdijk, I.: Proof of a conjecture of A. Haefiger. Topology 37(4), 735–741 (1998)CrossRefMATHGoogle Scholar
  22. 22.
    Moerdijk, I.: Orbifolds as groupoids: an introduction. In: Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math., Vol. 310, Providence, RI: Am. Math. Soc. (2002), pp. 205–222 MR 1 950 948Google Scholar
  23. 23.
    Moerdijk, I., Pronk, D.A.: Orbifolds sheaves and groupoids. K-Theory 12(1), 3–21 (1997)CrossRefMATHGoogle Scholar
  24. 24.
    Moerdijk, I., Pronk, D.A.: Simplicial cohomology of orbifolds. Indag. Math. (N.S.) 10(2), 403–416 (1999)Google Scholar
  25. 25.
    Murray, M.K.: Bundle Gerbes. J. Lond. Math. Soc. 54(2), 403–416 (1996)MATHGoogle Scholar
  26. 26.
    Murray, M.K., Stevenson, D.: Bundle Gerbes. Stable isomorphism and local theory. J. Lond. Math. Soc. 62(2), 925–937 (2000)MATHGoogle Scholar
  27. 27.
    Pressley, A., Segal, G.: Loop groups. New York: The Clarendon Press Oxford University Press, Oxford Science Publications, 1986Google Scholar
  28. 28.
    Rosenberg, J.: Continuous trace C *-algebras from the bundle theoretic point of view. J. Aust. Math. Soc. A47, 368 (1989)Google Scholar
  29. 29.
    Ruan, Y.: Stringy geometry and topology of orbifolds. In: Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), Contemp. Math. Vol. 312, Providence RI: Am. Math. Soc. 2002, pp. 187–233 MR 1 941 583Google Scholar
  30. 30.
    Satake, I.: The Gauss -Bonet theorem for V-manifolds. J. Math. Soc. Japan 9, 464–492 (1957)MATHGoogle Scholar
  31. 31.
    Segal, G.: Classifying spaces and spectral sequences. Inst. Hautes Etudes Sci. Publ. Math. 34, 105–112 (1968)MATHGoogle Scholar
  32. 32.
    Segal, G.: Fredholm complexes. Quart. J. Math. Oxford Ser. 21(2), 385–402 (1970)MATHGoogle Scholar
  33. 33.
    Segal, G.: Categories and cohomology theories. Topology 13, 292–312 (1974)CrossRefGoogle Scholar
  34. 34.
    Thurston, W.: Three-dimensional geometry and topology. Vol. 1. Princeton Mathematical Series 35. Princeton NJ: Princeton University Press, 1997Google Scholar
  35. 35.
    Witten, E.: D-branes and K-theory. J. High Energy Phys. 12(19), pp. 41 (1998) (electronic). MR 2000e:81151Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Departamento de MatemáticasCINVESTAVApartado PostalMéxico, D.F. México
  2. 2.Department of MathematicsUniversity of Michigan Ann ArborUSA

Personalised recommendations