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.

DOI : 10.5802/ambp.190
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
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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/articles/10.5802/ambp.190/

[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 | DOI | MR | Zbl

[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

[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 | DOI | MR | Zbl

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

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

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

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

[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 | DOI | Numdam | MR | Zbl

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

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

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

[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 | Zbl

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

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

[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 | DOI | MR | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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