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34
Category theory for conformal boundary conditions
 FIELDS INST. COMMUN. AMER. MATH. SOC., PROVIDENCE, RI
, 2003
"... ... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the d ..."
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Cited by 50 (14 self)
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... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives rise to a NIMrep of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.
Chiral Structure of Modular Invariants for Subfactors
, 1999
"... In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of MM morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their “ambichiral ” intersection, and we show that the ambichiral braidin ..."
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Cited by 48 (21 self)
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In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of MM morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their “ambichiral ” intersection, and we show that the ambichiral braiding is nondegenerate if the original braiding of the NN morphisms is. Moreover, in this case the dimensions of the irreducible representations of the chiral fusion rule algebras are given by the chiral branching coefficients which describe the ambichiral contribution in the irreducible decomposition of αinduced sectors. We show that modular invariants come along naturally with several nonnegative integer valued matrix representations of the original NN Verlinde fusion rule algebra, and we completely determine their decomposition into its characters. Finally the theory is illustrated by various examples, including the treatment of all SU (2)k modular invariants.
On qanalog of McKay correspondence and ADE classification of sl (2) conformal field theories
 Adv. Math
"... Abstract. The goal of this paper is to classify “finite subgroups in Uq(sl2)” where q = e πi/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq(sl2); we show that this definition is a natural generalization of the notion of a subgroup i ..."
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Cited by 46 (4 self)
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Abstract. The goal of this paper is to classify “finite subgroups in Uq(sl2)” where q = e πi/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq(sl2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to ̂ sl2 at level k = l − 2. We show that “finite subgroups in Uq(sl2) ” are classified by Dynkin diagrams of types An, D2n, E6, E8 with Coxeter number equal to l, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in ( ̂ sl2)k conformal field theory.
Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors
 Commun. Math. Phys
, 2000
"... A braided subfactor determines a coupling matrix Z which commutes with the S and Tmatrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a blockdiagonal structure which can be reinterpreted as the diagonal coupling matrix with r ..."
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Cited by 35 (5 self)
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A braided subfactor determines a coupling matrix Z which commutes with the S and Tmatrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a blockdiagonal structure which can be reinterpreted as the diagonal coupling matrix with respect to a suitable extension. We show that there are always two intermediate subfactors which correspond to left and right maximal extensions and which determine “parent ” coupling matrices Z ± of type I. Moreover it is shown that if the intermediate subfactors coincide, so that Z + = Z − , then Z is related to Z + by an automorphism of the extended fusion rules. The intertwining relations of chiral branching coefficients between original and extended S and Tmatrices are also clarified. None of our results depends on nondegeneracy of the braiding, i.e. the S and Tmatrices need not be modular. Examples from SO(n) current algebra models illustrate that the parents can be different, Z + ̸ = Z − , and that Z need not be related to a type I invariant by such an automorphism. 1
Projections in string theory and boundary states for Gepner models
 Nucl. Phys. B
, 2000
"... In string theory various projections have to be imposed to ensure supersymmetry. We study the consequences of these projections in the presence of world sheet boundaries. Atype boundary conditions come in several classes; only boundary fields that do not change the class preserve supersymmetry. Our ..."
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Cited by 32 (6 self)
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In string theory various projections have to be imposed to ensure supersymmetry. We study the consequences of these projections in the presence of world sheet boundaries. Atype boundary conditions come in several classes; only boundary fields that do not change the class preserve supersymmetry. Our analysis takes in particular properly into account the resolution of fixed points under the projections. Thus e.g. the compositeness of some previously considered boundary states of Gepner models follows from chiral properties of the projections. Our arguments are model independent; in particular, integrality of all annulus coefficients is ensured by model independent arguments. 1 1
Algebraic orbifold conformal field theories
 Proceedings of National Academy of Sci. USA 97
, 2000
"... Abstract. We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categ ..."
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Cited by 22 (5 self)
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Abstract. We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also show that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time. Cosets and orbifolds are two methods of producing new two dimensional conformal field theories from given ones (cf. [MS]). In [X2, 3,4, 5], unitary coset conformal field theories are formulated in the algebraic quantum field theory framework and such a formulation is used to solve many questions beyond the reach of other approaches.
Canonical tensor product subfactors
"... Canonical tensor product subfactors (CTPS’s) describe, among other things, the embedding of chiral observables in twodimensional conformal quantum field theories. A new class of CTPS’s is constructed some of which are associated with certain modular invariants, thereby establishing the expected exi ..."
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Cited by 21 (6 self)
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Canonical tensor product subfactors (CTPS’s) describe, among other things, the embedding of chiral observables in twodimensional conformal quantum field theories. A new class of CTPS’s is constructed some of which are associated with certain modular invariants, thereby establishing the expected existence of the corresponding twodimensional theories. 1 Introduction and
Duality and defects in rational conformal field theory
, 2006
"... We study topological defect lines in twodimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We sh ..."
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Cited by 20 (7 self)
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We study topological defect lines in twodimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting onedimensional phase boundaries can be used to extract symmetries and orderdisorder dualities of the CFT. The case of central charge c = 4/5, for which there are two inequivalent world sheet phases corresponding to the tetracritical Ising model and the critical threestates
Conformal field theories, graphs and quantum algebras
 In MathPhys odyssey
, 2002
"... This article reviews some recent progress in our understanding of the structure of Rational Conformal Field Theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various ty ..."
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Cited by 14 (0 self)
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This article reviews some recent progress in our understanding of the structure of Rational Conformal Field Theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various types of boundary conditions –open, twisted – are encoded in a system of integer multiplicities that form matrix representations of fusionlike algebras. These multiplicities are also the combinatorial data that enable one to construct an abstract “quantum” algebra, whose 6j and 3jsymbols contain essential information on the Operator Product Algebra of the RCFT and are part of a cell system, subject to pentagonal identities. It looks quite plausible that a complete classification of RCFT amounts to a classification of “Weak C ∗ Hopf algebras”. 1