An introduction to gerbes on orbifolds

Ernesto Lupercio[1]; Bernardo Uribe[2]

  • [1] CINVESTAV Departamento de Matemáticas Apartado Postal 14-740 07000 México D. F. México
  • [2] University of MIchigan Department of Mathematics East Hall Ann Arbor, MI 48109 USA

Annales mathématiques Blaise Pascal (2004)

  • Volume: 11, Issue: 2, page 155-180
  • ISSN: 1259-1734

Abstract

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This paper is a gentle introduction to some recent results involving the theory of gerbes over orbifolds for topologists, geometers and physicists. We introduce gerbes on manifolds, orbifolds, the Dixmier-Douady class, Beilinson-Deligne orbifold cohomology, Cheeger-Simons orbifold cohomology and string connections.

How to cite

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Lupercio, Ernesto, and Uribe, Bernardo. "An introduction to gerbes on orbifolds." Annales mathématiques Blaise Pascal 11.2 (2004): 155-180. <http://eudml.org/doc/10503>.

@article{Lupercio2004,
abstract = {This paper is a gentle introduction to some recent results involving the theory of gerbes over orbifolds for topologists, geometers and physicists. We introduce gerbes on manifolds, orbifolds, the Dixmier-Douady class, Beilinson-Deligne orbifold cohomology, Cheeger-Simons orbifold cohomology and string connections.},
affiliation = {CINVESTAV Departamento de Matemáticas Apartado Postal 14-740 07000 México D. F. México; University of MIchigan Department of Mathematics East Hall Ann Arbor, MI 48109 USA},
author = {Lupercio, Ernesto, Uribe, Bernardo},
journal = {Annales mathématiques Blaise Pascal},
keywords = {gerbes; orbifolds},
language = {eng},
month = {7},
number = {2},
pages = {155-180},
publisher = {Annales mathématiques Blaise Pascal},
title = {An introduction to gerbes on orbifolds},
url = {http://eudml.org/doc/10503},
volume = {11},
year = {2004},
}

TY - JOUR
AU - Lupercio, Ernesto
AU - Uribe, Bernardo
TI - An introduction to gerbes on orbifolds
JO - Annales mathématiques Blaise Pascal
DA - 2004/7//
PB - Annales mathématiques Blaise Pascal
VL - 11
IS - 2
SP - 155
EP - 180
AB - This paper is a gentle introduction to some recent results involving the theory of gerbes over orbifolds for topologists, geometers and physicists. We introduce gerbes on manifolds, orbifolds, the Dixmier-Douady class, Beilinson-Deligne orbifold cohomology, Cheeger-Simons orbifold cohomology and string connections.
LA - eng
KW - gerbes; orbifolds
UR - http://eudml.org/doc/10503
ER -

References

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  1. A. Adem, Y. Ruan, Twisted Orbifold K -Theory 
  2. M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165-189 Zbl0317.14001MR399094
  3. M. Artin, A. Grothendieck, J.L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, (1972), Springer-Verlag, Berlin Zbl0234.00007MR354653
  4. P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray, D. Stevenson, Twisted K -theory and K -theory of bundle gerbes, Comm. Math. Phys. 228 (2002), 17-45 Zbl1036.19005MR1911247
  5. J-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, 107 (1993), Birkhäuser Boston Inc., Boston, MA Zbl0823.55002MR1197353
  6. L. Carey, S. Johnson, M. Murray, Holonomy on D-Branes 
  7. J. Cheeger, J. Simons, Differential characters and geometric invariants, Geometry and topology (College Park, Md., 1983/84) 1167 (1985), 50-80, Springer, Berlin Zbl0621.57010MR827262
  8. W. Chen, Y. Ruan, A New Cohomology Theory for Orbifold 
  9. M. Crainic, I. Moerdijk, A homology theory for étale groupoids, J. Reine Angew. Math. 521 (2000), 25-46 Zbl0954.22002MR1752294
  10. P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), 75-109 Zbl0181.48803MR262240
  11. L. Dixon, J. Harvey, C. Vafa, E. Witten, Strings on orbifolds. II, Nuclear Phys. B 274 (1986), 285-314 MR851703
  12. L. Dixon, J. Harvey, C. Vafa, E. Witten, Strings on orbifolds. II, Nuclear Phys. B 274 (1986), 285-314 MR851703
  13. P. Donovan, M. Karoubi, Graded Brauer groups and K -theory with local coefficients, Inst. Hautes Études Sci. Publ. Math. (1970), 5-25 Zbl0207.22003MR282363
  14. D. Freed, K -theory in quantum field theory, Current developments in mathematics, 2001 (2002), 41-87, Int. Press, Somerville, MA Zbl1036.19004
  15. D. Freed, M. Hopkins, On Ramond-Ramond fields and K -theory, J. High Energy Phys. (2000) Zbl0990.81624MR1769477
  16. D. Freed, E. Witten, Anomalies in string theory with D-branes, Asian J. Math. 3 (1999), 819-851 Zbl1028.81052MR1797580
  17. D. Freed, M. Hopkins, C. Teleman, Twisted equivariant K -theory with complex coefficients 
  18. K. Gomi, Y. Terashima, Higher-dimensional parallel transports, Math. Res. Lett. 8 (2001), 25-33 Zbl1008.53027MR1825257
  19. A. Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque (1984), 70-97 Zbl0562.57012MR755163
  20. N. Hitchin, Lectures on Special Lagrangian Submanifolds Zbl1079.14522MR1876068
  21. M.J. Hopkins, I.M. Singer, Quadratic functions in geometry, topology,and M-theory Zbl1116.58018
  22. T. Kawasaki, The signature theorem for V -manifolds, Topology 17 (1978), 75-83 Zbl0392.58009MR474432
  23. T. Kawasaki, The Riemann-Roch theorem for complex V -manifolds, Osaka J. Math. 16 (1979), 151-159 Zbl0405.32010MR527023
  24. T. Kawasaki, The index of elliptic operators over V -manifolds, Nagoya Math. J. 84 (1981), 135-157 Zbl0437.58020MR641150
  25. E. Lupercio, B. Uribe, Differential Characters for Orbifolds and string connections I Zbl1111.58031
  26. E. Lupercio, B. Uribe, Differential Characters for Orbifolds and string connections II Zbl1111.58031
  27. E. Lupercio, B. Uribe, Gerbes over Orbifolds and Twisted K-theory Zbl1068.53034MR2045679
  28. E. Lupercio, B. Uribe, Holonomy for gerbes over orbifolds Zbl1099.53040
  29. E. Lupercio, B. Uribe, Inertia orbifolds, configuration spaces and the ghost loop space Zbl1066.55006MR2068317
  30. E. Lupercio, B. Uribe, Deligne Cohomology for Orbifolds, discrete torsion and B-fields, Geometric and Topological methods for Quantum Field Theory (2002), World Scientific Zbl1055.81608MR2010004
  31. E. Lupercio, B. Uribe, Loop groupoids, gerbes, and twisted sectors on orbifolds, Orbifolds in mathematics and physics (Madison, WI, 2001) 310 (2002), 163-184, Amer. Math. Soc., Providence, RI Zbl1041.58008MR1950946
  32. I. Moerdijk, Classifying topos and foliations, Ann. Inst. Fourier 41 (1991), 189-209 Zbl0727.57029MR1112197
  33. I. Moerdijk, Proof of a conjecture of A. Haefliger, Topology 37 (1998), 735-741 Zbl0897.22003MR1607724
  34. I. Moerdijk, D. A. Pronk, Orbifolds, sheaves and groupoids, -Theory 12 (1997), 3-21 Zbl0883.22005MR1466622
  35. I. Moerdijk, D. A. Pronk, Simplicial cohomology of orbifolds, Indag. Math. (N.S.) 10 (1999), 269-293 Zbl1026.55007MR1816220
  36. Y. Ruan, Discrete torsion and twisted orbifold cohomology Zbl1089.57017
  37. Y. Ruan, Gerbes and twisted orbifold quantum cohomology Zbl1146.53069
  38. Y. Ruan, Stringy geometry and topology of orbifolds, Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000) 312 (2002), 187-233, Amer. Math. Soc., Providence, RI Zbl1060.14080MR1941583
  39. Y. Ruan, Stringy orbifolds, Orbifolds in mathematics and physics (Madison, WI, 2001) 310 (2002), 259-299, Amer. Math. Soc., Providence, RI Zbl1080.14500MR1950951
  40. I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359-363 Zbl0074.18103MR79769
  41. G. Segal, Classifying spaces and spectral sequences, Inst. Hautes Etudes Sci. Publ. Math. 34 (1968), 105-112 Zbl0199.26404MR232393
  42. E. Sharpe, Discrete torsion, quotient stacks, and string orbifolds, Orbifolds in mathematics and physics (Madison, WI, 2001) 310 (2002), 301-331, Amer. Math. Soc., Providence, RI Zbl1042.81075MR1950952
  43. W. Thurston, Three-dimensional geometry and topology. Vol. 1, 35 (1997), Princeton University Press, Princeton, NJ Zbl0873.57001MR1435975
  44. C. Vafa, E. Witten, On orbifolds with discrete torsion, J. Geom. Phys. 15 (1995), 189-214 Zbl0816.53053MR1316330
  45. A. Weil, Sur les théorèmes de de Rham, Comment. Math. Helv. 26 (1952), 119-145 Zbl0047.16702MR50280
  46. E. Witten, Overview of K -theory applied to strings, Strings 2000. Proceedings of the International Superstrings Conference (Ann Arbor, MI) 16 (2001), 693-706 Zbl1003.81020MR1827946

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