An introduction to gerbes on orbifolds
Lupercio, Ernesto ; Uribe, Bernardo
Annales mathématiques Blaise Pascal, Tome 11 (2004), p. 155-180 / Harvested from Numdam

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 = {Lupercio, Ernesto and Uribe, Bernardo},
     title = {An introduction to gerbes on orbifolds},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {11},
     year = {2004},
     pages = {155-180},
     doi = {10.5802/ambp.190},
     zbl = {1079.53040},
     mrnumber = {2109605},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2004__11_2_155_0}
}
Lupercio, Ernesto; Uribe, Bernardo. An introduction to gerbes on orbifolds. Annales mathématiques Blaise Pascal, Tome 11 (2004) pp. 155-180. doi : 10.5802/ambp.190. http://gdmltest.u-ga.fr/item/AMBP_2004__11_2_155_0/

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

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

[3] Artin, M.; Grothendieck, A.; Verdier, J.L. 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] Bouwknegt, P.; Carey, A. L.; Mathai, V.; Murray, M. K.; Stevenson, D. Twisted K-theory and K-theory of bundle gerbes, Comm. Math. Phys., Tome 228 (2002) no. 1, pp. 17-45 | Article | MR 1911247 | Zbl 1036.19005

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

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

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

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

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

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

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

[13] Donovan, P.; Karoubi, M. 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] Freed, D. K-theory in quantum field theory, Current developments in mathematics, 2001, Int. Press, Somerville, MA (2002), pp. 41-87 | Zbl 1036.19004

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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