### Citations

787 |
Conformal field theory,
- Francesco, Mathieu, et al.
- 1997
(Show Context)
Citation Context ...with the integrable characters for the affine algebras SU(n) k of some integral weight k. Such a relation has been well-studied in the case of n = 2 and it falls into an A-D-E classification scheme =-=[7, 9, 11]-=-. Evidences for what might occur in the case of n = 3 were presented in [1] by computing the Clebsch-Gordan coefficients extensively 1for the subgroups of SU(3). Indications from the lattice integrab... |

545 | Chern-Simons gauge theory as a string theory,” hep-th/9207094 - Witten |

287 | Instantons on ALE spaces, quiver varieties and Kac-Moody algebras,
- Nakajima
- 1994
(Show Context)
Citation Context ...e Lie algebras may be arising so naturally in the study of two-dimensional quotient spaces. Based on the previous studies of Yang-Mills instantons on ALE spaces as in [45], Nakajima has introduced in =-=[46]-=- the notion of a quiver variety which is roughly a hyper-Kähler moduli space of representations of a quiver associated to a finite graph (We shall turn to quivers in the next section). There, he prese... |

164 |
The operator algebra of orbifold models
- Dijkgraaf, Vafa, et al.
- 1989
(Show Context)
Citation Context ...endieck group of coherent sheaves on π−1 (0), can be expressed as the Clebsch-Gordan coefficients. Furthermore, string theory analysis also predicts a similar relation between the two ring structures =-=[31]-=-. The geometric McKay Correspondence can thus be stated as CONJECTURE 3.2 (Geometric McKay Correspondence) Let Γ, M, and ˜ M be as in Conjecture 3.1. Then, there exist bijections Basis of H ∗ ( ˜ M, Z... |

160 |
Modular invariance and discrete torsion on orbifolds,” Nucl. Phys. B273
- Vafa
- 1986
(Show Context)
Citation Context ...its many crepant resolutions [48]. String theory so far has yielded two distinguished desingularizations: the traditional CFT resolution without discrete torsion and deformation with discrete torsion =-=[32]-=-. In this paper, we are concerned only with Kähler resolutions without discrete torsion. 7 See [40] for a discussion on this point. 10homology classes. Indeed in [42], Ito and Nakajima prove that for... |

144 |
Worldsheet approach to heterotic instantons and solitons,
- CaUan, Harvey, et al.
- 1991
(Show Context)
Citation Context ...ry fields of the WZW. This interpretation agrees with the analysis of Ooguri and Vafa, but we are not certain how to reproduce the result directly from Nakajima’s work. 194.3 T-duality and Branes In =-=[14, 15, 16, 17]-=-, the SU(2) k WZW theory arose in a different but equivalent context of brane dynamics. As shown in [36], the type IIA (IIB) string theory on an An−1 ALE space is T-dual to the type IIB (IIA) theory... |

113 |
Two-Dimensional Black Hole and
- Ooguri, Vafa
- 1996
(Show Context)
Citation Context ...backgrounds, such as in the case of gauge symmetry enhancement of the Type IIA string theory compactified on singular K3 where the B-field vanishes [34]. Later, we will discuss a tensored coset model =-=[36]-=- describing this singular non-linear sigma model and relate it to the algebraic McKay Correspondence. 6 In fact, a given Gorenstein singularity generally admits many crepant resolutions [48]. String t... |

103 |
Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry
- Batyrev, Dais
- 1996
(Show Context)
Citation Context ...umbers of cohomology classes, motivated by the works of string theorists and the fact that ˜ M has no odd-dimensional cohomology 7 , mathematicians have generalized the classical McKay Correspondence =-=[38, 40, 41]-=- to geometry. The geometric McKay Correspondence in 2-dimensions actually identifies the cohomology ring of ˜ M and the representation ring of Γ not only as vector spaces but as rings. Given a finite ... |

102 |
Construction geometrique de la correspondence de McKay,
- Gonzalez-Sprinberg, Verdier
- 1983
(Show Context)
Citation Context ...wo dimensions and hence beyond A-D-E. Indeed, algebraic geometers have done extensive research in generalizing McKay’s original correspondence to Gorenstein singularities of dimension greater than 2 (=-=[37]-=- to [44]); many standing conjectures exist in this respect. On the other hand, there is the conjecture mentioned above regarding the SU(n) k WZW and the subgroups of SU(n) in general. It is our hope... |

96 |
Vorlesungen uber das Ikosaeder und die Auflosung der Gleichungen vom fumften Grade,
- Klein
- 1884
(Show Context)
Citation Context ...ral Coefficients Correlation Functions Ring Table 1: A-D-E Correspondences in 2-dimensions. The same graphs and their affine extensions appear in different theories. the discrete subgroups Γ of SU(2) =-=[3]-=-. These were noted to be in 1-1 correspondence with the Platonic solids in IR 3 , and with some foresight, we write them as: • type A: the cyclic groups (the regular polygons); • type D: the binary di... |

93 |
Catastrophes and the Classification of Conformal Theories”,
- Warner
- 1989
(Show Context)
Citation Context ...ric McKay Correspondence and see how Calabi-Yau (CY) non-linear sigma models constructed on the Gorenstein singularities associated with the finite groups may be related to Kazama-Suzuki coset models =-=[20, 21, 22, 23, 27, 28]-=-, which in turn can be related to the WZW models. This link would provide a natural setting for the emergence of the A-D-E classification of the modular invariants. In due course, we will review and e... |

91 | Orbifold resolution by D-branes
- Douglas, Greene, et al.
- 1997
(Show Context)
Citation Context ...ir research on supersymmetric Yang-Mills theories resulting from the type IIB D-branes on orbifold singularities [1], as prompted by collective works in constructing (conformal) gauge theories (e.g., =-=[4]-=- [5] [6] and references therein), it was conjectured by Hanany and He that there may exist a McKay-type correspondence between the bifundamental matter content and the modular invariant partition func... |

75 |
lattice integrable models associated with graphs, Nucl.Phys
- Francesco, Zuber, et al.
- 1990
(Show Context)
Citation Context ... the case of n = 3 were presented in [1] by computing the Clebsch-Gordan coefficients extensively 1for the subgroups of SU(3). Indications from the lattice integrable model perspective were given in =-=[13]-=-. The natural question to pose is why there should be such correspondences. Indeed, why should there be such an intricate chain of connections among string theory on orbifolds, finite representation t... |

72 |
The McKay correspondence for finite subgroups of SL(3,C).
- Ito, Reid
- 1994
(Show Context)
Citation Context ...resolved orbifolds were also predicted by Vafa for CY manifolds realized as hypersurfaces in weighted projective spaces and by Zaslow for Kähler manifolds [29]. In dimension three, it has been proved =-=[38, 40]-=- that every Gorenstein singularity admits a crepant resolution 6 and that every crepant resolution satisfies the Conjecture 3.1 and the Vafa-Zaslow Hodge number formulae. For higher dimensions, there ... |

61 | Resolution of Orbifold Singularities in String Theory
- Aspinwall
(Show Context)
Citation Context ...etry of the target space. One 9example is that CFT is often 5 insensitive to Gorenstein singularities and quantum effects revolve the singularity so that the CFT is smooth. More precisely, Aspinwall =-=[33]-=- has shown that non-vanishing of the NS-NS B-field makes the CFT smooth. In fact, string theory predicts the Euler characteristic of the resolved orbifold [30]; the local form of the statement is CONJ... |

55 |
Enhanced Gauge Symmetries and K3
- Aspinwall
- 1995
(Show Context)
Citation Context ... are examples of singular CFT’s defined on singular backgrounds, such as in the case of gauge symmetry enhancement of the Type IIA string theory compactified on singular K3 where the B-field vanishes =-=[34]-=-. Later, we will discuss a tensored coset model [36] describing this singular non-linear sigma model and relate it to the algebraic McKay Correspondence. 6 In fact, a given Gorenstein singularity gene... |

55 |
String Theory and Black
- Witten
- 1991
(Show Context)
Citation Context ...uld be viewed as a particular non-conformal deformation of the superpotential appearing in Ooguri and Vafa’s model. 15 The SL(2, IR)/U(1) coset model describes the two-dimensional black hole geometry =-=[62]-=-, while the SU(2)/U(1) Kazama-Suzuki model is just the N = 2 minimal model. 164.1 Fusion Algebra, Cohomology and Representation Rings Comparing the Table 4 with the Table 3, we immediately see that t... |

54 |
The Coulomb branch of (4, 4) supersymmetric field theories in two-dimensions”,
- Diaconescu, Seiberg
- 1997
(Show Context)
Citation Context ... models also mysteriously falls into an A-D-E type [8]. There have been some recent attempts to explain this seeming accident from the supersymmetric field theory and brane configurations perspective =-=[15, 17]-=-. In this paper we push from the direction of the Geometric McKay Correspondence and see how Calabi-Yau (CY) non-linear sigma models constructed on the Gorenstein singularities associated with the fin... |

54 |
Exactly solvable string compactifications on manifolds
- Gepner
(Show Context)
Citation Context ...is as the Geometric McKay Correspondence. Now from physics, studies in conformal field theory (CFT) have prompted many beautiful connections among graph theory, fusion algebra, and modular invariants =-=[7, 9, 11, 20, 21]-=-. The classification of the modular invariant partition function associated with SU(2) WessZumino-Witten (WZW) models also mysteriously falls into an A-D-E type [8]. There have been some recent atte... |

52 |
String theory on group manifolds
- Gepner, Witten
- 1986
(Show Context)
Citation Context ...with the integrable characters for the affine algebras SU(n) k of some integral weight k. Such a relation has been well-studied in the case of n = 2 and it falls into an A-D-E classification scheme =-=[7, 9, 11]-=-. Evidences for what might occur in the case of n = 3 were presented in [1] by computing the Clebsch-Gordan coefficients extensively 1for the subgroups of SU(3). Indications from the lattice integrab... |

49 | NonAbelian finite gauge theories
- Hanany, He
- 1999
(Show Context)
Citation Context ...nsions and naturally incorporates diverse concepts in mathematics. In the course of their research on supersymmetric Yang-Mills theories resulting from the type IIB D-branes on orbifold singularities =-=[1]-=-, as prompted by collective works in constructing (conformal) gauge theories (e.g., [4] [5] [6] and references therein), it was conjectured by Hanany and He that there may exist a McKay-type correspon... |

43 | On an inner product in modular tensor categories,
- Kirillov
- 1996
(Show Context)
Citation Context ...irst exemplified by the so-called ribbon categories and the second, quiver categories. 5.1 Ribbon Categories as Modular Tensor Categories Prominent work in the first step has been done by A. Kirillov =-=[53]-=- and we shall adhere to his notations. We are interested in monoidal additive categories, in particular, we need the following: 17There is interesting work done to formalize to sub-factors and to inve... |

30 | On the landau-ginzburg description of n = 2 minimal models
- Witten
(Show Context)
Citation Context ...ric McKay Correspondence and see how Calabi-Yau (CY) non-linear sigma models constructed on the Gorenstein singularities associated with the finite groups may be related to Kazama-Suzuki coset models =-=[20, 21, 22, 23, 27, 28]-=-, which in turn can be related to the WZW models. This link would provide a natural setting for the emergence of the A-D-E classification of the modular invariants. In due course, we will review and e... |

29 | superconformal models as topological field theories. - Eguchi, Yang |

27 | Algebraic Geometry and Effective - Martinec - 1989 |

23 |
Integrable lattice models, graphs and modular invariant conformal field theories, Int. J.Mod.Phys. A 7
- Francesco
- 1992
(Show Context)
Citation Context ...toric resolution diagrams of C 3 /ZZn × ZZn singularities reproduce the graphs that classify the A-type modular invariants of the SU(3) k WZW models. For which k? It has been previously observed in =-=[63]-=- that there seems to be a correspondence, up to some truncation, between the subgroups ZZn × ZZn ⊂ SU(3) and the A-type SU(3) n−1 modular invariants, which do appear as subgraphs of the ˜C 3 toric d... |

20 | D-branes on nonabelian threefold quotient singularities - Greene, Lazaroiu, et al. - 1999 |

19 | D-branes on three-dimensional nonabelian orbifolds,” JHEP 9902, 008
- Muto
- 1999
(Show Context)
Citation Context ...singly, similar features have been noted in three dimensions. The ClebschGordan coefficients for the tensor product of irreducible representations for all discrete subgroups of SU(3) were computed in =-=[1, 50]-=-, and a possible correspondence was noted, and conjectured for n ≥ 3, between the resulting Dynkin-like diagrams and the graphic representations of the fusion rules and modular invariants of SU(3) k... |

18 | From Threebranes to Large N Gauge Theories - Klebanov |

17 |
Representations of partially ordered sets,
- Nazarova, Roiter
- 1972
(Show Context)
Citation Context ...m of the simple Lie algebras depended on integer solutions of 1/p + 1/q + 1/r ≥ 1. Since Gabriel’s theorem is so restrictive, extensions thereto have been done to relax certain assumptions (e.g., see =-=[56]-=-). This will hopefully give us give more graphs and in particular those appearing in finite group, WZW, orbifold theories or non-linear σmodels at higher dimensions. A vital step in the proof is that ... |

16 | Crepent Resolutions of Trihedral Singularities
- Ito
- 1994
(Show Context)
Citation Context ...etry side, the representation rings of these groups were related to the Groethendieck (cohomology) rings of the resolved manifolds constructed from the Gorenstein singularity of the respective groups =-=[38]-=-; we shall refer to this as the Geometric McKay Correspondence. Now from physics, studies in conformal field theory (CFT) have prompted many beautiful connections among graph theory, fusion algebra, a... |

15 |
D(n) quivers from branes
- Kapustin
- 1998
(Show Context)
Citation Context ...rch on supersymmetric Yang-Mills theories resulting from the type IIB D-branes on orbifold singularities [1], as prompted by collective works in constructing (conformal) gauge theories (e.g., [4] [5] =-=[6]-=- and references therein), it was conjectured by Hanany and He that there may exist a McKay-type correspondence between the bifundamental matter content and the modular invariant partition functions of... |

15 |
String vacua and orbifoldized L-G models, Mod. Phys. Lett. A4
- Vafa
- 1989
(Show Context)
Citation Context ...}| . (3.2) Furthermore, the Hodge numbers of resolved orbifolds were also predicted by Vafa for CY manifolds realized as hypersurfaces in weighted projective spaces and by Zaslow for Kähler manifolds =-=[29]-=-. In dimension three, it has been proved [38, 40] that every Gorenstein singularity admits a crepant resolution 6 and that every crepant resolution satisfies the Conjecture 3.1 and the Vafa-Zaslow Hod... |

13 | On the topology of desingularizations of Calabi-Yau orbifolds
- Joyce
(Show Context)
Citation Context ...set model [36] describing this singular non-linear sigma model and relate it to the algebraic McKay Correspondence. 6 In fact, a given Gorenstein singularity generally admits many crepant resolutions =-=[48]-=-. String theory so far has yielded two distinguished desingularizations: the traditional CFT resolution without discrete torsion and deformation with discrete torsion [32]. In this paper, we are conce... |

10 | The Cappelli-Itzyskon-Zuber A-D-E classification
- Gannon
(Show Context)
Citation Context ...with the integrable characters for the affine algebras SU(n) k of some integral weight k. Such a relation has been well-studied in the case of n = 2 and it falls into an A-D-E classification scheme =-=[7, 9, 11]-=-. Evidences for what might occur in the case of n = 3 were presented in [1] by computing the Clebsch-Gordan coefficients extensively 1for the subgroups of SU(3). Indications from the lattice integrab... |

10 |
Neveu-Schwarz Five-Branes at Orbifold Singularities and Holography
- Diaconescu, Gomis
(Show Context)
Citation Context ...ry fields of the WZW. This interpretation agrees with the analysis of Ooguri and Vafa, but we are not certain how to reproduce the result directly from Nakajima’s work. 194.3 T-duality and Branes In =-=[14, 15, 16, 17]-=-, the SU(2) k WZW theory arose in a different but equivalent context of brane dynamics. As shown in [36], the type IIA (IIB) string theory on an An−1 ALE space is T-dual to the type IIB (IIA) theory... |

9 |
Landau-Ginzburg Orbifolds. Nucl. Phys. B339
- Intrilligator, Vafa
- 1990
(Show Context)
Citation Context ...the tensor 14 The universality classes of the LG models are completely specified by their superpotentials W, and such a simple characterization leads to very powerful methods of detailed computations =-=[22, 25]-=-. Generalizations of these models have many important applications in string theory, and the OPE coefficients of topological LG theories with judiciously chosen non-conformal deformations yield the fu... |

7 |
A Correspondence dual to McKay’s. alg-geom/9612003
- Brylinski
(Show Context)
Citation Context ...sions and hence beyond A-D-E. Indeed, algebraic geometers have done extensive research in generalizing McKay’s original correspondence to Gorenstein singularities of dimension greater than 2 ([37] to =-=[44]-=-); many standing conjectures exist in this respect. On the other hand, there is the conjecture mentioned above regarding the SU(n) k WZW and the subgroups of SU(n) in general. It is our hope that Fi... |

6 |
Modular Invariants, Graphs and α-Induction for Nets and
- Böckenhauser, Evans
(Show Context)
Citation Context ...otations. We are interested in monoidal additive categories, in particular, we need the following: 17There is interesting work done to formalize to sub-factors and to investigate the graphs generated =-=[58]-=-. 20DEFINITION 5.1 A ribbon category is an additive category C with the following additional structures: • BRAIDING: A bifunctor ⊗ : C × C → C along with functorial associativity and commutativity is... |

5 | Orbifold Euler characteristic, in Mirror symmetry - Roan - 1997 |

4 |
A Unified Scheme for Modular Invariant Partition Functions of WZW Models
- Abolhassani, Ardalan
(Show Context)
Citation Context ...ants corresponding to SU(n) for n = 4, .., 7 respectively. Moreover, proposals for unified schemes for the modular invariants by considering orbifolds by abelian Γ in SU(2,3,..,6) have been made in =-=[12]-=-. Let us summarize what we have found. We see that the representation ring of finite groups with its associated (⊗, ⊕), the chiral ring of nonlinear σ-model with its (⊗, ⊕), the fusion ring of the WZW... |

3 |
Fiz 90
- B172
- 1986
(Show Context)
Citation Context ...is as the Geometric McKay Correspondence. Now from physics, studies in conformal field theory (CFT) have prompted many beautiful connections among graph theory, fusion algebra, and modular invariants =-=[7, 9, 11, 20, 21]-=-. The classification of the modular invariant partition function associated with SU(2) WessZumino-Witten (WZW) models also mysteriously falls into an A-D-E type [8]. There have been some recent atte... |

3 |
Introduction to Lie Algebras and Representation Theory
- Humphereys
- 1972
(Show Context)
Citation Context ...of nodes and edges of Γ, of the general type: ∑ 1 i pi ≥ c where c is some constant and {pi} is a set of integers characterizing the problem at hand. This inequality has a long history in mathematics =-=[57]-=-. In our context, we recall that the uniqueness of the five perfect solids in IR 3 (and hence the discrete subgroups of SU(2)) relies essentially on the equation 1/p + 1/q ≥ 1/2 having only 5 pairs of... |

3 | Topological Mirrors and Quantum Rings in Mirror Symmetry - Vafa - 1997 |

2 |
The Classification of SU(m)k Automorphism Invariants
- Gannon
(Show Context)
Citation Context ...lgebra construction has been extended to SU(3) and a similar classification of the modular invariants have in fact been done and are shown to correspond to the so-called generalized Dynkin Diagrams =-=[7, 10, 13]-=-. On the other hand, the Clebsch-Gordan coefficients of the McKay type for the discrete subgroups of SU(3) have been recently computed in the context of studying D3-branes on orbifold singularities [1... |

2 |
Unzerlegbare Darstellugen I. Manuscripta Math. 6
- Gabriel
- 1972
(Show Context)
Citation Context ...encoding procedure (at least a version thereof) have been done even before McKay’s result. In fact, in 1972, Gabriel has noticed that categorical studies of quivers lead to A-D-E-type classifications =-=[55]-=-. DEFINITION 5.3 We define the quiver category L(Γ, Λ), for a finite connected graph Γ with orientation Λ, vertices Γ0 and edges Γ1 as follows: The objects in this category are any collection (V, f) o... |