An introduction to gerbes on orbifolds
Annales Mathématiques Blaise Pascal, Tome 11 (2004) no. 2, pp. 155-180.

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.

@article{AMBP_2004__11_2_155_0,
     author = {Ernesto Lupercio and Bernardo Uribe},
     title = {An introduction to gerbes on orbifolds},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {155--180},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {11},
     number = {2},
     year = {2004},
     doi = {10.5802/ambp.190},
     mrnumber = {2109605},
     zbl = {1079.53040},
     language = {en},
     url = {ambp.centre-mersenne.org/item/AMBP_2004__11_2_155_0/}
}
Ernesto Lupercio; Bernardo Uribe. An introduction to gerbes on orbifolds. Annales Mathématiques Blaise Pascal, Tome 11 (2004) no. 2, pp. 155-180. doi : 10.5802/ambp.190. https://ambp.centre-mersenne.org/item/AMBP_2004__11_2_155_0/

[1] A. Adem; Y. Ruan Twisted Orbifold K-Theory (arXiv:math. AT/0107168)

[2] M. Artin Versal deformations and algebraic stacks, Invent. Math., Volume 27 (1974), pp. 165-189 | Article | MR 399094 | Zbl 0317.14001

[3] M. Artin; A. Grothendieck; J.L. Verdier Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Springer-Verlag, Berlin, 1972 (Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 269) | MR 354653

[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., Volume 228 (2002) no. 1, pp. 17-45 | Article | MR 1911247 | Zbl 1036.19005

[5] J-L. Brylinski Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics, Volume 107, Birkhäuser Boston Inc., Boston, MA, 1993 | MR 1197353 | Zbl 0823.55002

[6] L. Carey; S. Johnson; M. Murray Holonomy on D-Branes (arXiv:hep-th/0204199)

[7] J. Cheeger; J. Simons Differential characters and geometric invariants, Geometry and topology (College Park, Md., 1983/84) (Lecture Notes in Math.) Volume 1167, Springer, Berlin, 1985, pp. 50-80 | MR 827262 | Zbl 0621.57010

[8] W. Chen; Y. Ruan A New Cohomology Theory for Orbifold (arXiv:math.AG/000 4129)

[9] M. Crainic; I. Moerdijk A homology theory for étale groupoids, J. Reine Angew. Math., Volume 521 (2000), pp. 25-46 | Article | MR 1752294 | Zbl 0954.22002

[10] P. Deligne; D. Mumford The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969) no. 36, pp. 75-109 | Article | Numdam | MR 262240 | Zbl 0181.48803

[11] L. Dixon; J. Harvey; C. Vafa; E. Witten Strings on orbifolds. II, Nuclear Phys. B, Volume 274 (1986) no. 2, pp. 285-314 | Article | MR 851703

[12] L. Dixon; J. Harvey; C. Vafa; E. Witten Strings on orbifolds. II, Nuclear Phys. B, Volume 274 (1986) no. 2, pp. 285-314 | Article | MR 851703

[13] P. Donovan; M. Karoubi Graded Brauer groups and K-theory with local coefficients, Inst. Hautes Études Sci. Publ. Math. (1970) no. 38, pp. 5-25 | Article | Numdam | MR 282363 | Zbl 0207.22003

[14] D. Freed K-theory in quantum field theory, Current developments in mathematics, 2001, Int. Press, Somerville, MA, 2002, pp. 41-87 | Zbl 1036.19004

[15] D. Freed; M. Hopkins On Ramond-Ramond fields and K-theory, J. High Energy Phys. (2000) no. 5, Paper 44, 14 pages | MR 1769477 | Zbl 0990.81624

[16] D. Freed; E. Witten Anomalies in string theory with D-branes, Asian J. Math., Volume 3 (1999) no. 4, pp. 819-851 | MR 1797580 | Zbl 1028.81052

[17] D. Freed; M. Hopkins; C. Teleman Twisted equivariant K-theory with complex coefficients (arXiv:math.AT/0206257)

[18] K. Gomi; Y. Terashima Higher-dimensional parallel transports, Math. Res. Lett., Volume 8 (2001) no. 1-2, pp. 25-33 | MR 1825257 | Zbl 1008.53027

[19] A. Haefliger Groupoïdes d’holonomie et classifiants, Astérisque (1984) no. 116, pp. 70-97 (Transversal structure of foliations (Toulouse, 1982)) | MR 755163 | Zbl 0562.57012

[20] N. Hitchin Lectures on Special Lagrangian Submanifolds (arXiv:math.DG/9907034) | MR 1876068

[21] M.J. Hopkins; I.M. Singer Quadratic functions in geometry, topology,and M-theory (arXiv:math.AT/0211216)

[22] T. Kawasaki The signature theorem for V-manifolds, Topology, Volume 17 (1978) no. 1, pp. 75-83 | Article | MR 474432 | Zbl 0392.58009

[23] T. Kawasaki The Riemann-Roch theorem for complex V-manifolds, Osaka J. Math., Volume 16 (1979) no. 1, pp. 151-159 | MR 527023 | Zbl 0405.32010

[24] T. Kawasaki The index of elliptic operators over V-manifolds, Nagoya Math. J., Volume 84 (1981), pp. 135-157 | MR 641150 | Zbl 0437.58020

[25] E. Lupercio; B. Uribe Differential Characters for Orbifolds and string connections I (arXiv:math.DG/0311008)

[26] E. Lupercio; B. Uribe Differential Characters for Orbifolds and string connections II (In preparation)

[27] E. Lupercio; B. Uribe Gerbes over Orbifolds and Twisted K-theory (arXiv:math.AT/0105039, to appear in Comm. Math. Phys.) | MR 2045679 | Zbl 1068.53034

[28] E. Lupercio; B. Uribe Holonomy for gerbes over orbifolds (arXiv:math.AT/0307114)

[29] E. Lupercio; B. Uribe Inertia orbifolds, configuration spaces and the ghost loop space (arXiv:math.AT/0210222, to appear in Quart. Jour. of Math.) | MR 2068317 | Zbl 1066.55006

[30] E. Lupercio; B. Uribe Deligne Cohomology for Orbifolds, discrete torsion and B-fields, Geometric and Topological methods for Quantum Field Theory, World Scientific, 2002 | MR 2010004 | Zbl 1055.81608

[31] E. Lupercio; B. Uribe Loop groupoids, gerbes, and twisted sectors on orbifolds, Orbifolds in mathematics and physics (Madison, WI, 2001) (Contemp. Math.) Volume 310, Amer. Math. Soc., Providence, RI, 2002, pp. 163-184 | MR 1950946 | Zbl 1041.58008

[32] I. Moerdijk Classifying topos and foliations, Ann. Inst. Fourier, Volume 41 (1991), pp. 189-209 | Article | Numdam | MR 1112197 | Zbl 0727.57029

[33] I. Moerdijk Proof of a conjecture of A. Haefliger, Topology, Volume 37 (1998) no. 4, pp. 735-741 | Article | MR 1607724 | Zbl 0897.22003

[34] I. Moerdijk; D. A. Pronk Orbifolds, sheaves and groupoids, K-Theory, Volume 12 (1997) no. 1, pp. 3-21 | Article | MR 1466622 | Zbl 0883.22005

[35] I. Moerdijk; D. A. Pronk Simplicial cohomology of orbifolds, Indag. Math. (N.S.), Volume 10 (1999) no. 2, pp. 269-293 | Article | MR 1816220 | Zbl 1026.55007

[36] Y. Ruan Discrete torsion and twisted orbifold cohomology (arXiv:math.AG/0005299)

[37] Y. Ruan Gerbes and twisted orbifold quantum cohomology (Preprint)

[38] Y. Ruan Stringy geometry and topology of orbifolds, Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000) (Contemp. Math.) Volume 312, Amer. Math. Soc., Providence, RI, 2002, pp. 187-233 | MR 1941583 | Zbl 1060.14080

[39] Y. Ruan Stringy orbifolds, Orbifolds in mathematics and physics (Madison, WI, 2001) (Contemp. Math.) Volume 310, Amer. Math. Soc., Providence, RI, 2002, pp. 259-299 | MR 1950951 | Zbl 01942890

[40] I. Satake On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A., Volume 42 (1956), pp. 359-363 | Article | MR 79769 | Zbl 0074.18103

[41] G. Segal Classifying spaces and spectral sequences, Inst. Hautes Etudes Sci. Publ. Math., Volume 34 (1968), pp. 105-112 | Article | Numdam | MR 232393 | Zbl 0199.26404

[42] E. Sharpe Discrete torsion, quotient stacks, and string orbifolds, Orbifolds in mathematics and physics (Madison, WI, 2001) (Contemp. Math.) Volume 310, Amer. Math. Soc., Providence, RI, 2002, pp. 301-331 | MR 1950952 | Zbl 1042.81075

[43] W. Thurston Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, Volume 35, Princeton University Press, Princeton, NJ, 1997 (Edited by Silvio Levy) | MR 1435975 | Zbl 0873.57001

[44] C. Vafa; E. Witten On orbifolds with discrete torsion, J. Geom. Phys., Volume 15 (1995) no. 3, pp. 189-214 | Article | MR 1316330 | Zbl 0816.53053

[45] A. Weil Sur les théorèmes de de Rham, Comment. Math. Helv., Volume 26 (1952), pp. 119-145 | Article | MR 50280 | Zbl 0047.16702

[46] E. Witten Overview of K-theory applied to strings, Strings 2000. Proceedings of the International Superstrings Conference (Ann Arbor, MI), Volume 16 (2001) no. 5, pp. 693-706 | MR 1827946 | Zbl 1003.81020