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35
D.E.: Modular Invariants, Graphs and αInduction for Nets of Subfactors II
 In preparation
"... We analyze the induction and restriction of sectors for nets of subfactors defined by Longo and Rehren. Picking a local subfactor we derive a formula which specifies the structure of the induced sectors in terms of the original DHR sectors of the smaller net and canonical endomorphisms. We also obta ..."
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Cited by 80 (8 self)
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We analyze the induction and restriction of sectors for nets of subfactors defined by Longo and Rehren. Picking a local subfactor we derive a formula which specifies the structure of the induced sectors in terms of the original DHR sectors of the smaller net and canonical endomorphisms. We also obtain a reciprocity formula for induction and restriction of sectors, and we prove a certain homomorphism property of the induction mapping. Developing further some ideas of F. Xu we will apply this theory in a forthcoming paper to nets of subfactors arising from conformal field theory, in particular those coming from conformal embeddings or orbifold inclusions of SU(n) WZW models. This will provide a better understanding of the labeling of modular invariants by certain graphs, in particular of the ADE classification of SU(2) modular invariants.
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
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 αinduction, chiral generators and modular invariants for subfactors
 Commun. Math. Phys
, 1999
"... We consider a type III subfactor N ⊂ M of finite index with a finite system of braided NN morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply αinduction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle ..."
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Cited by 38 (10 self)
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We consider a type III subfactor N ⊂ M of finite index with a finite system of braided NN morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply αinduction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the αinduced sectors. A matrix Z is defined and shown to commute with the S and Tmatrices arising from the braiding. If the braiding is nondegenerate, then Z is a “modular invariant mass matrix ” in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of MM morphisms is generated by the images of both kinds of αinduction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will
An analogue of the KacWakimoto formula and black hole conditional entropy
 Commun. Math. Phys
"... Abstract. A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac ..."
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Cited by 21 (10 self)
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Abstract. A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by KacWakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the ConnesStœrmer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function the logarithm of an integer once the initial state or the final state is taken fixed.
E.: A theory of dimension
 KTheory
, 1997
"... Dedicated to R.V. Kadison on the occasion of his seventieth birthday Abstract. In which a theory of dimension related to the Jones index and based on the notion of conjugation is developed. An elementary proof of the additivity and multiplicativity of the dimension is given and there is an associate ..."
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Cited by 17 (0 self)
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Dedicated to R.V. Kadison on the occasion of his seventieth birthday Abstract. In which a theory of dimension related to the Jones index and based on the notion of conjugation is developed. An elementary proof of the additivity and multiplicativity of the dimension is given and there is an associated trace. Applications are given to a class of endomorphisms of factors and to the theory of subfactors. An important role is played by a notion of amenability inspired by the work of Popa.
On conditional expectations of finite index
 J. Operator Theory
, 1998
"... For a conditional expectation E on a (unital) C*algebra A there exists a real number K ≥ 1 such that the mapping K · E − idA is positive if and only if there exists a real number L ≥ 1 such that the mapping L ·E −idA is completely positive, among other equivalent conditions. The estimate (min K) ≤ ..."
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Cited by 15 (4 self)
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For a conditional expectation E on a (unital) C*algebra A there exists a real number K ≥ 1 such that the mapping K · E − idA is positive if and only if there exists a real number L ≥ 1 such that the mapping L ·E −idA is completely positive, among other equivalent conditions. The estimate (min K) ≤ (min L) ≤ (min K)[min K] is valid, where [·] denotes the entire part of a real number. As a consequence the notion of a ”conditional expectation of finite index ” is identified with that class of conditional expectations, which extends and completes results of M. Pimsner, S. Popa [27,28], M. Baillet, Y. Denizeau and J.F. Havet [6] and Y. Watatani [35] and others. Situations for which the index value and the Jones ’ tower exist are described in the general setting. In particular, the Jones ’ tower always exists in the W*case and for Ind(E) ∈ E(A) in the C*case. Furthermore, normal conditional expectations of finite index commute with the (abstract) projections of W*algebras to their finite, infinite, discrete and continuous type I, type II1, type II ∞ and type III parts, i.e. they respect and preserve these W*decompositions in full.
Crossedproducts by finite index endomorphisms and KMS
, 2003
"... ABSTRACT. Given a unital C*algebra A, an injective endomorphism α: A → A preserving the unit, and a conditional expectation E from A to the range of α we consider the crossedproduct of A by α relative to the transfer operator L = α −1 E. When E is of indexfinite type we show that there exists a c ..."
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Cited by 14 (3 self)
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ABSTRACT. Given a unital C*algebra A, an injective endomorphism α: A → A preserving the unit, and a conditional expectation E from A to the range of α we consider the crossedproduct of A by α relative to the transfer operator L = α −1 E. When E is of indexfinite type we show that there exists a conditional expectation G from the crossedproduct to A which is unique under certain hypothesis. We define a “gauge action ” on the crossedproduct algebra in terms of a central positive element h and study its KMS states. The main result is: if h> 1 and E(ab) = E(ba) for all a, b ∈ A (e.g. when A is commutative) then the KMSβ states are precisely those of the form ψ = φ ◦ G, where φ is a trace on A satisfying the identity where ind(E) is the JonesKosakiWatatani index of E. 1. Introduction. φ(a) = φ(L(h −β ind(E)a)), In [E2] we have introduced the notion of the crossedproduct of a C*algebra A by a *endomorphism α, a construction which also depends on the choice of a transfer operator, that is a positive continuous linear map L: A → A such that L ( α(a)b) = aL(b), for all a, b ∈ A. In the present work we treat the case in which
Notes for a quantum index theorem
 Commun. Math. Phys
"... We view DHR superselection sectors with finite statistics as Quantum Field Theory analogs of elliptic operators where KMS functionals play the role of the trace composed with the heat kernel regularization. We extend our local holomorphic dimension formula and prove an analogue of the index theorem ..."
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Cited by 14 (8 self)
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We view DHR superselection sectors with finite statistics as Quantum Field Theory analogs of elliptic operators where KMS functionals play the role of the trace composed with the heat kernel regularization. We extend our local holomorphic dimension formula and prove an analogue of the index theorem in the Quantum Field Theory context. The analytic index is the Jones index, more precisely the minimal dimension, and, on a 4dimensional spacetime, the DHR theorem gives the integrality of the index. We introduce the notion of holomorphic dimension; the geometric dimension is then defined as the part of the holomorphic dimension which is symmetric under charge conjugation. We apply the AHKT theory of chemical potential and we extend it to the low dimensional case, by using conformal field theory. Concerning Quantum Field Theory on curved spacetime, the geometry of the manifold enters in the expression for the dimension. If a quantum black hole is described by a spacetime with bifurcate Killing horizon and sectors are localizable on the horizon, the logarithm of the holomorphic dimension is proportional to the incremental free energy, due to the addition of the charge, and to the inverse temperature, hence to the surface gravity in the HartleHawking KMS state. For this analysis we consider a conformal net obtained by restricting the field to the horizon (“holography”). Compared with our previous work on Rindler spacetime, this result differs inasmuch as it concerns true black hole spacetimes, like the SchwarzschildKruskal manifold, and pertains to the entropy of the black hole itself, rather than of the outside system. An outlook concerns a possible relation with supersymmetry and noncommutative geometry.
Characterizing Invariants for Local Extensions of Current Algebras
, 1995
"... . Pairs A ae B of local quantum field theories are studied, where A is a chiral conformal quantum field theory and B is a local extension, either chiral or twodimensional. The local correlation functions of fields from B have an expansion with respect to A into conformal blocks, which are nonlocal ..."
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Cited by 13 (7 self)
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. Pairs A ae B of local quantum field theories are studied, where A is a chiral conformal quantum field theory and B is a local extension, either chiral or twodimensional. The local correlation functions of fields from B have an expansion with respect to A into conformal blocks, which are nonlocal in general. Two methods of computing characteristic invariant ratios of structure constants in these expansions are compared: (a) by constructing the monodromy representation of the braid group in the space of solutions of the KnizhnikZamolodchikov differential equation, and (b) by an analysis of the local subfactors associated with the extension with methods from operator algebra (Jones theory) and algebraic quantum field theory. Both approaches apply also to the reverse problem: the characterization and (in principle) classification of local extensions of a given theory. 1. Introduction The relevance of V. Jones' theory of (von Neumann) subfactors [1] for 2dimensional (2D) models of cr...