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52
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
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Cited by 52 (6 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = FVect, where F is a field. An object X ∈ A with twosided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1
CONFORMAL CORRELATION FUNCTIONS, FROBENIUS ALGEBRAS AND TRIANGULATIONS
, 2001
"... We formulate twodimensional rational conformal field theory as a natural generalization of twodimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A ..."
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Cited by 35 (18 self)
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We formulate twodimensional rational conformal field theory as a natural generalization of twodimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the MooreSeiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of Tduality. We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIMreps of the fusion rules, respectively.
TFT CONSTRUCTION OF RCFT CORRELATORS IV: STRUCTURE CONSTANTS AND CORRELATION FUNCTIONS
, 2005
"... We compute the fundamental correlation functions in twodimensional rational conformal field theory, from which all other correlators can be obtained by sewing: the correlators of three bulk fields on the sphere, one bulk and one boundary field on the disk, three boundary fields on the disk, and one ..."
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Cited by 28 (13 self)
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We compute the fundamental correlation functions in twodimensional rational conformal field theory, from which all other correlators can be obtained by sewing: the correlators of three bulk fields on the sphere, one bulk and one boundary field on the disk, three boundary fields on the disk, and one bulk field on the cross cap. We also consider conformal defects and calculate the correlators of three defect fields on the sphere and of one defect field on the cross cap. Each of these correlators is presented as the product of a structure constant and the appropriate conformal two or threepoint block. The structure constants are expressed as invariants of ribbon graphs in threemanifolds.
TFT construction of RCFT correlators III: Simple currents
 Nucl. Phys. B
"... We use simple currents to construct symmetric special Frobenius algebras in modular tensor categories. We classify such simple current type algebras with the help of abelian group cohomology. We show that they lead to the modular invariant torus partition functions that have been studied by Kreuzer ..."
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Cited by 22 (10 self)
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We use simple currents to construct symmetric special Frobenius algebras in modular tensor categories. We classify such simple current type algebras with the help of abelian group cohomology. We show that they lead to the modular invariant torus partition functions that have been studied by Kreuzer and Schellekens. We also classify boundary conditions in the associated conformal field theories and show that the boundary states are given by the formula proposed in hepth/0007174. Finally, we investigate conformal defects in these
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
Correspondences of ribbon categories
, 2006
"... Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories ..."
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Cited by 13 (3 self)
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Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.
Full field algebra
 HKo3] [HKr] [HL1] [HL2] [HL3] [HL4] [HL5] [K1] Y.Z. Huang and
"... We introduce the notions of openclosed field algebra and openclosed field algebra over a vertex operator algebra V. In the case that V satisfies certain finiteness and reductivity conditions, we show that an openclosed field algebra over V canonically gives an algebra over a Cextension of Swiss ..."
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Cited by 11 (1 self)
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We introduce the notions of openclosed field algebra and openclosed field algebra over a vertex operator algebra V. In the case that V satisfies certain finiteness and reductivity conditions, we show that an openclosed field algebra over V canonically gives an algebra over a Cextension of Swisscheese partial operad. We also give a tensorcategorical formulation and constructions of openclosed field algebras over V. 0
Statesum construction of twodimensional openclosed TQFTs
 In preparation
"... We present a state sum construction of twodimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional twodimensional cobordisms to those of openclosed cobordisms, i.e. smoo ..."
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Cited by 11 (5 self)
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We present a state sum construction of twodimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional twodimensional cobordisms to those of openclosed cobordisms, i.e. smooth compact oriented 2manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a twodimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an openclosed TQFT with a finite set of Dbranes using the example of the groupoid algebra of a finite groupoid.