## An Introduction to Gerbes on Orbifolds. (2004)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Lupercio04anintroduction,

author = {Ernesto Lupercio and Bernardo Uribe},

title = {An Introduction to Gerbes on Orbifolds.},

year = {2004}

}

### OpenURL

### Abstract

### Citations

481 |
Théorie des topos et cohomologie étale des schémas, Lect
- Artin, Grotendieck, et al.
(Show Context)
Citation Context ...bifolds. The paper of Artin [2] is the place where a very explicit conection with groupoid atlases takes place for the first time. Implicitly these ideas are already present in Grothendieck’s toposes =-=[3]-=-. The groupoiod approach to orbifolds is finally carried out by Haefliger [18] and by Moerdijk and his collaborators [31, 34, 9, 33]. In this work they put forward the important concept of Morita equi... |

389 |
Three-Dimensional Geometry and Topology
- Thurston
- 1990
(Show Context)
Citation Context ... late 70’s Kawasaki generalizes index theory to the orbifold setting [21, 22, 23]. Another important work along these veins in the work of Thurston specially his concept of orbifold fundamental group =-=[42]-=- Somewhat independently the algebraic geometers developed the concept of stack in order to deal with moduli problems. As it happens orbifolds arise quite naturally from the very same moduli problems a... |

382 | The irreducibility of the space of curves of given genus
- Deligne, Mumford
(Show Context)
Citation Context ...dn’t take long to realize that the theory of stacks provided another way of understanding the category of orbifolds, and viceversa. For example, the Deligne-Mumford moduli stack Mg for genus g curves =-=[10]-=- is in fact an orbifold. This is one of the reasons for the importance of orbifolds, many moduli spaces are better understood as orbifolds. The paper of Artin [2] is the place where a very explicit co... |

142 | Lectures on special Lagrangian submanifolds
- Hitchin
(Show Context)
Citation Context ...of a line bundle with a connection. In spite of this, there is a geometric interpretation of the previous situation. This can be seen as a motivation for the introduction of the concept of gerbe (cf. =-=[19]-=-). (For more details on the physics see [15, 14].) Definition: Let M be a manifold. A gerbe with connection on M is given by the following data: i) A good Leray atlas U = {Ui}i of M. ii) Smooth maps g... |

134 |
A new cohomology theory for orbifold
- Chen, Ruan
(Show Context)
Citation Context ...lated to orbifolds in the physics community. The introduction to the mathematics side of the geometrization of many of these ideas and results is due to Chen and Ruan. Their highly influential papers =-=[8, 38]-=- introduced many concepts from the physics literature rigorously into symplectic and algebraic geometry. In this work orbifolds are completely general, not necessarily global quotients. In particular ... |

132 |
Differential characters and geometric invariants, from: “Geometry and topology
- Cheeger, Simons
- 1985
(Show Context)
Citation Context ...2-form ω on M we have obtained a pair (χ, ω) with χ: Z1(M) −→ R/Z and ∫ χ(∂c) = ω mod Z c whenever c is a smooth 2-chain (the pair (χ, ω) is called a differential character). Following Cheeger-Simons =-=[7]-=- we will denote by ˆ H 2 cs(M) the group of such differential characters of M. 18Gerbes on Orbifolds If we substitute the line bundle by a (q − 2)-gerbe with connection. The holonomy becomes now a ho... |

117 | Anomalies in string theory with D-branes
- Freed, Witten
- 1999
(Show Context)
Citation Context ...op groupoid (L[M/G])/G. The analogous result for a general orbifold is more subtle and we refer the reader to [25] for details. There we use this theorem to generalize the results of Freed and Witten =-=[16]-=- on anomaly cancellation in string theory to the orbifold case. To conclude let us mention that building on an idea of Hopkins and Singer [20] we have defined orbifold Chern-Simons cohomology. The mai... |

99 |
On a generalization of the notion of manifold
- Satake
- 1956
(Show Context)
Citation Context ...ultiplication by n. We refer the reader to the papers [4, 6] for gerbes from the point of view of bundle gerbes. 2 Orbifolds The notion of orbifold was first introduced by Satake in his seminal paper =-=[39]-=-. In this 1956 paper Satake defines for the very first time the concept 6Gerbes on Orbifolds of an orbifold by means of orbifold atlases whose charts Satake calls local uniformizing systems. The name... |

95 | On orbifolds with discrete torsion
- Vafa, Witten
- 1995
(Show Context)
Citation Context ... A, B]. An analogous definition can be made for n-gerbes yielding a (n + 2)-form ω. A discrete torsion on an orbifold G = [M/G] is a 2-cocycle θ: G × G → U(1) in the bar group cohomology complex of G =-=[43]-=- (cf. [41]). Proposition 3.5:[29] For a global orbifold [M/G] the map θ ↦→ (θ, 0, 0) injects the group of discrete torsions of an orbifold into the group of flat gerbes (=flat B-fields). In fact the i... |

90 | Twisted K-theory and Ktheory of bundle gerbes
- Bouwknegt, Carey, et al.
(Show Context)
Citation Context ... 3)] = H3 (M, Z) then α = β. In other words, the image of [M, BPU(n)] → H3 (M, Z) is exactly the subgroup of torsion elements that are killed by multiplication by n. We refer the reader to the papers =-=[4, 6]-=- for gerbes from the point of view of bundle gerbes. 2 Orbifolds The notion of orbifold was first introduced by Satake in his seminal paper [39]. In this 1956 paper Satake defines for the very first t... |

86 |
Classifying spaces and spectral sequences
- Segal
- 1968
(Show Context)
Citation Context ...efines the proper étale Leray groupoid G and by definition it is Morita equivalent to X. Given an orbifold a very important construction is that of its classifying space. The nerve of a groupoid (see =-=[40]-=-) is a semisimplicial set NG where the objects of G are the vertices, the morphisms the 1-simplexes, the triangular commutative diagram the 2-simplexes, and so on. We can define the boundary maps δi :... |

53 |
Graded Brauer groups and K-theory with local coecients Inst
- Donovan, Karoubi
- 1970
(Show Context)
Citation Context ...nnection is flat if and only if dd(g) is a torsion class in cohomology. This is the case if and only if the projective bundle E is finite dimensional. Proof: This is true because of a result of Serre =-=[13]-=- valid for any CWcomplex M. It states that if a class α ∈ H3 (M, Z) is a torsion element then there exists a principal bundle Z → M with structure group PU(n) so that when seen as an element β ∈ [M, B... |

52 | Quadratic functions in geometry, topology, and M-theory
- Hopkins, Singer
(Show Context)
Citation Context ...his theorem to generalize the results of Freed and Witten [16] on anomaly cancellation in string theory to the orbifold case. To conclude let us mention that building on an idea of Hopkins and Singer =-=[20]-=- we have defined orbifold Chern-Simons cohomology. The main difficulty here is to make sense of what an orbifold differential character should be [24]. We make a definition in such a way that we can p... |

51 | Twisted equivariant K-theory with complex coefficients
- Freed, Hopkins, et al.
- 2002
(Show Context)
Citation Context ...undle with connection over LG) is an inner local system on ∧G. 21In the case in which we have a Lie group acting with finite stabilizers these line bundles are the coefficients Freed-Hopkins-Teleman =-=[17]-=- used to twist the cohomology of the twisted sectors in order to get a Chern character isomorphism with the twisted K-theory of the orbifold. We have used gerbes in [26] to obtain twisted versions of ... |

48 |
String on Orbifolds
- Dixon, Harvey, et al.
- 1985
(Show Context)
Citation Context ...ators [31, 34, 9, 33]. In this work they put forward the important concept of Morita equivalence. The interest of orbifolds in physics can be traced back to the work of Dixon, Harvey, Vafa and Witten =-=[11, 12]-=- where motivated by superstring compactification they introduce a orbifold theory using a K3 with 27 singular points. It is there that the orbifold Euler characteristic is defined motivated by the phy... |

44 | Stringy geometry and topology of orbifolds
- Ruan
(Show Context)
Citation Context ...f the main results of [28] is the following theorem Theorem 4.6: The fixed suborbifold of LG under the natural S 1 -action (rotating the loops) is ∧G = (LG) S1 The following definition is due to Ruan =-=[37, 36, 35]-=-. He used this definition to obtain a twisted version of the Chen-Ruan cohomology [8] that has revived the interest in the theory of orbifolds in the last few years. Definition: An “inner local system... |

42 |
Versal deformations and algebraic stacks
- Artin
- 1974
(Show Context)
Citation Context ...d moduli stack Mg for genus g curves [10] is in fact an orbifold. This is one of the reasons for the importance of orbifolds, many moduli spaces are better understood as orbifolds. The paper of Artin =-=[2]-=- is the place where a very explicit conection with groupoid atlases takes place for the first time. Implicitly these ideas are already present in Grothendieck’s toposes [3]. The groupoiod approach to ... |

35 |
spaces, characteristic classes and geometric quantization, volume 107
- Loop
- 1993
(Show Context)
Citation Context ...type II orbifold superstring theories is the same as a gerbe with connection on the orbifold. The following theorem generalizes a result of Brylinski that he proved in the case of a smooth manifold M =-=[5]-=-. Theorem 3.4: We have the following classifications. • The group of isomorphism classes of line orbibundles with connection on G is isomorphic to H 2 (M, Z(2)). • The group of isomorphism classes of ... |

35 |
Sur les théorèmes de de Rham
- Weil
- 1952
(Show Context)
Citation Context ...eaves 0 −→ Z −→ R exp(2πi ) −→ U(1) −→ 1 3immediately implies an isomorphism H 1 (M, U(1)) ∼ = H 2 (M, Z) The class of [g] in H 2 (M, Z) is called the Chern class c1(L) of L. It is a theorem of Weil =-=[44]-=- that −[F] is the image of the Chern class c1(L) under the map H 2 (M, Z) → H 2 (M, R). The Chern class completely determines the isomorphism type of the line bundle L, but does not determine the isom... |

34 | A homology theory for étale groupoids
- Crainic, Moerdijk
(Show Context)
Citation Context ...irst time. Implicitly these ideas are already present in Grothendieck’s toposes [3]. The groupoiod approach to orbifolds is finally carried out by Haefliger [18] and by Moerdijk and his collaborators =-=[31, 34, 9, 33]-=-. In this work they put forward the important concept of Morita equivalence. The interest of orbifolds in physics can be traced back to the work of Dixon, Harvey, Vafa and Witten [11, 12] where motiva... |

28 |
The index of elliptic operators over V -manifolds
- Kawasaki
- 1981
(Show Context)
Citation Context ...hool carried out brilliantly the study of orbifolds. It deserves special mention the work of Tetsuro Kawasaki. In his papers of the late 70’s Kawasaki generalizes index theory to the orbifold setting =-=[21, 22, 23]-=-. Another important work along these veins in the work of Thurston specially his concept of orbifold fundamental group [42] Somewhat independently the algebraic geometers developed the concept of stac... |

25 | On Ramond-Ramond fields and K-theory
- Freed, Hopkins
(Show Context)
Citation Context ... of this, there is a geometric interpretation of the previous situation. This can be seen as a motivation for the introduction of the concept of gerbe (cf. [19]). (For more details on the physics see =-=[15, 14]-=-.) Definition: Let M be a manifold. A gerbe with connection on M is given by the following data: i) A good Leray atlas U = {Ui}i of M. ii) Smooth maps gijk: Uijk −→ U(1). iii) A collection (Aij) of 1-... |

24 |
The Riemann-Roch theorem for complex V -manifolds
- Kawasaki
- 1979
(Show Context)
Citation Context ...hool carried out brilliantly the study of orbifolds. It deserves special mention the work of Tetsuro Kawasaki. In his papers of the late 70’s Kawasaki generalizes index theory to the orbifold setting =-=[21, 22, 23]-=-. Another important work along these veins in the work of Thurston specially his concept of orbifold fundamental group [42] Somewhat independently the algebraic geometers developed the concept of stac... |

24 | Overview of K-theory applied to strings
- Witten
(Show Context)
Citation Context ...n character isomorphism with the twisted K-theory of the orbifold. We have used gerbes in [26] to obtain twisted versions of K-theory that act a recipients of the charges of D-branes in string theory =-=[45]-=- generalizing work of Adem and Ruan [1]. Returning to the subject of string connections we have the following result. Theorem 4.8: Take a global gerbe ξ with connection over [M/G] and let E be the lin... |

22 |
Groupöıdes d’holonomie et classifiants
- Haefliger
- 1984
(Show Context)
Citation Context ...with groupoid atlases takes place for the first time. Implicitly these ideas are already present in Grothendieck’s toposes [3]. The groupoiod approach to orbifolds is finally carried out by Haefliger =-=[18]-=- and by Moerdijk and his collaborators [31, 34, 9, 33]. In this work they put forward the important concept of Morita equivalence. The interest of orbifolds in physics can be traced back to the work o... |

20 |
Orbifolds as Groupoids: An
- Moerdijk
- 2002
(Show Context)
Citation Context ...irst time. Implicitly these ideas are already present in Grothendieck’s toposes [3]. The groupoiod approach to orbifolds is finally carried out by Haefliger [18] and by Moerdijk and his collaborators =-=[31, 34, 9, 33]-=-. In this work they put forward the important concept of Morita equivalence. The interest of orbifolds in physics can be traced back to the work of Dixon, Harvey, Vafa and Witten [11, 12] where motiva... |

19 | Discrete torsion and twisted orbifold cohomology
- Ruan
(Show Context)
Citation Context ...f the main results of [28] is the following theorem Theorem 4.6: The fixed suborbifold of LG under the natural S 1 -action (rotating the loops) is ∧G = (LG) S1 The following definition is due to Ruan =-=[37, 36, 35]-=-. He used this definition to obtain a twisted version of the Chen-Ruan cohomology [8] that has revived the interest in the theory of orbifolds in the last few years. Definition: An “inner local system... |

15 | K-theory in quantum field theory
- Freed
(Show Context)
Citation Context ... of this, there is a geometric interpretation of the previous situation. This can be seen as a motivation for the introduction of the concept of gerbe (cf. [19]). (For more details on the physics see =-=[15, 14]-=-.) Definition: Let M be a manifold. A gerbe with connection on M is given by the following data: i) A good Leray atlas U = {Ui}i of M. ii) Smooth maps gijk: Uijk −→ U(1). iii) A collection (Aij) of 1-... |

15 |
The signature theorem for V -manifolds
- Kawasaki
- 1978
(Show Context)
Citation Context ...hool carried out brilliantly the study of orbifolds. It deserves special mention the work of Tetsuro Kawasaki. In his papers of the late 70’s Kawasaki generalizes index theory to the orbifold setting =-=[21, 22, 23]-=-. Another important work along these veins in the work of Thurston specially his concept of orbifold fundamental group [42] Somewhat independently the algebraic geometers developed the concept of stac... |

11 | Simplicial cohomology of orbifolds
- Moerdijk, Pronk
- 1999
(Show Context)
Citation Context ...irst time. Implicitly these ideas are already present in Grothendieck’s toposes [3]. The groupoiod approach to orbifolds is finally carried out by Haefliger [18] and by Moerdijk and his collaborators =-=[31, 34, 9, 33]-=-. In this work they put forward the important concept of Morita equivalence. The interest of orbifolds in physics can be traced back to the work of Dixon, Harvey, Vafa and Witten [11, 12] where motiva... |

8 | Differential characters on orbifolds and string connections I, available at http://www.arxiv.org/abs/math.DG/0311008
- Lupercio, Uribe
(Show Context)
Citation Context ...r may imagine that these are two strings evolving and interacting in M if she prefers to do so. γ 1 Σ γ 3 γ 2 γ 4 t (4.1) 19Now consider in general an orbifold X. We will describe now the results of =-=[30, 27, 24, 25]-=- that refine the previous results to the case of orbifolds. First we have defined an infinite dimensional orbifold, the loop orbifold LX associated to X by giving an explicit groupoid representation o... |

8 |
groupoids, gerbes, and twisted sectors on orbifolds, Orbifolds in mathematics and physics
- Loop
- 2002
(Show Context)
Citation Context ...r may imagine that these are two strings evolving and interacting in M if she prefers to do so. γ 1 Σ γ 3 γ 2 γ 4 t (4.1) 19Now consider in general an orbifold X. We will describe now the results of =-=[30, 27, 24, 25]-=- that refine the previous results to the case of orbifolds. First we have defined an infinite dimensional orbifold, the loop orbifold LX associated to X by giving an explicit groupoid representation o... |

8 |
Calssifying topos and foliations
- Moerdijk
- 1991
(Show Context)
Citation Context |

8 | Discrete torsion, quotient stacks, and string orbifolds
- Sharpe
(Show Context)
Citation Context ... analogous definition can be made for n-gerbes yielding a (n + 2)-form ω. A discrete torsion on an orbifold G = [M/G] is a 2-cocycle θ: G × G → U(1) in the bar group cohomology complex of G [43] (cf. =-=[41]-=-). Proposition 3.5:[29] For a global orbifold [M/G] the map θ ↦→ (θ, 0, 0) injects the group of discrete torsions of an orbifold into the group of flat gerbes (=flat B-fields). In fact the induced map... |

7 |
Proof of a conjecture of A
- Moerdijk
- 1998
(Show Context)
Citation Context ... ≤ n − 1 ⎩ (x1, . . .,xn−1) if i = n NG determines G and its geometric realization BG is called the classifying space of the orbifold. This space is important to us because it is a result of Moerdijk =-=[32]-=- that H ∗ (X; Z) ∼ = H ∗ (BX; Z) where the left hand side is sheaf cohomology (to be defined) and the right hand side is simplicial cohomology of BX ≃ MG with coefficients in Z. Example: In the case i... |

6 | Inertia orbifolds, configuration spaces and the ghost loop space
- Lupercio, Uribe
(Show Context)
Citation Context ...ts (∧G)0: Elements v ∈ G1 such that s(v) = t(v). • Morphisms (∧G)1: For v, w ∈ (∧G)0 an arrow v α → w is an element α ∈ G1 such that v · α = α · w v �� ◦ �� α α −1 � ◦ w −1 One of the main results of =-=[28]-=- is the following theorem Theorem 4.6: The fixed suborbifold of LG under the natural S 1 -action (rotating the loops) is ∧G = (LG) S1 The following definition is due to Ruan [37, 36, 35]. He used this... |

4 | Deligne cohomology for orbifolds, discrete torsion and b-fields, Geometric and Topological methods for Quantum Field Theory, World Scientific
- Lupercio, Uribe
- 2002
(Show Context)
Citation Context ...f cohomology. We define now the so-called BeilinsonDeligne cohomology of G. For the purpose of exposition we will introduce this cohomology for the Leray groupoid of Example 2 and refer the reader to =-=[29, 27]-=- for the case of a general orbifold groupoid. 14�� �� �� � � �� �� �� �� �� �� Gerbes on Orbifolds A G-sheaf is a sheaf over G on which G acts continuously. Let A p G denote the G-sheaf of differenti... |

3 |
Gerbes over Orbifolds and Twisted K-theory, arXiv: math.AT/0110207
- Lupercio, Uribe
(Show Context)
Citation Context ... Segal [40]. If (M, U) is an atlas for M then the classifying space of the groupoid MU is BMU ≃ M. ⎫ ⎬ ⎭ 3 Gerbes over Orbifolds. In this section we discuss definitions and result first introduced in =-=[26]-=-. Example: Let us recast the definition of gerbe over a manifold (M, U), with Leray groupoid MU. Notice than in this case 13• (MU)0 = ∐ i Ui • (MU)1 = ∐ (i,j) Uij • (MU)2 = ∐ (i,j,k) Uijk and so on. ... |

2 |
Holonomy for gerbes over orbifolds, arXiv:math.AT/0307114
- Lupercio, Uribe
(Show Context)
Citation Context ...m ∗ A = − √ −1g −1 dg The G-invariant 3-form ω = dB ∈ Ω 3 (G0) is called the curvature of the gerbe with connection (g, A, B). Here by G-invariant we mean that s ∗ ω = t ∗ ω. The following theorem of =-=[26, 27]-=- describes the basic classification of gerbes over orbifold (without a connection). Theorem 3.1: The following holds. • Every gerbe on an orbifold has a representative of the form (G, g) where G is a ... |

1 |
Gerbes and twisted orbifold quantum cohomology
- Ruan
(Show Context)
Citation Context ...f the main results of [28] is the following theorem Theorem 4.6: The fixed suborbifold of LG under the natural S 1 -action (rotating the loops) is ∧G = (LG) S1 The following definition is due to Ruan =-=[37, 36, 35]-=-. He used this definition to obtain a twisted version of the Chen-Ruan cohomology [8] that has revived the interest in the theory of orbifolds in the last few years. Definition: An “inner local system... |