Intertwining operator algebras and vertex tensor categories for affine Lie algebras

Huang, Yi‐Zhi; Lepowsky, James

doi:10.1215/s0012-7094-99-09905-2

Cited by 43 publications

(74 citation statements)

References 34 publications

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“…Finkelberg [F1] [F2] transported these braided tensor category structures to the category of integrable highest weight modules of positive integral levels (with a few exceptions) for the same affine Lie algebra. Direct constructions of these braided tensor category structures were also given by Lepowsky and the author [HL5] based on the results in [HL1]- [HL4] [H1] and by Bakalov and Kirillov [BK]. In the general case, for a vertex operator algebra V satisfying suitable conditions (weaker than the conditions in the present paper), the braided tensor category structure on the category of V -modules was constructed by Lepowsky and the author and by the author in a series of papers [HL1]- [HL4] [H1] [H5].…”

Section: Introductionmentioning

confidence: 99%

Rigidity and Modularity of Vertex Tensor Categories

Huang

2008

Commun. Contemp. Math.

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Let V be a simple vertex operator algebra satisfying the following conditions: (i) V (n) = 0 for n < 0, V (0) = C1 and V ′ is isomorphic to V as a V -module. (ii) Every N-gradable weak V -module is completely reducible. (iii) V is C 2 -cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V -module is completely reducible.) Using the results obtained by the author in the formulation and proof of the general version of the Verlinde conjecture and in the proof of the Verlinde formula, we prove that the braided tensor category structure on the category of V -modules is rigid, balanced and nondegenerate. In particular, the category of V -modules has a natural structure of modular tensor category. We also prove that the tensor-categorical dimension of an irreducible V -module is the reciprocal of a suitable matrix element of the fusing isomorphism under a suitable basis. IntroductionIn the present paper, we prove the rigidity and modularity of the braided tensor category of modules for a vertex operator algebra satisfying certain natural conditions (see below). Finding proofs of these properties has been an open problem for many years. Our proofs in this paper are based on the results obatined by the author [H7] in the formulation and proof of the general version of the Verlinde conjecture and in the proof of the Verlinde formula.In MS2] also demonstrated that the theory of these polynomial equations is actually a conformal-field-theoretic analogue of the theory of the tensor categories. This work of Moore and Seiberg greatly advanced our understanding of the structure of conformal field theories and the name "modular tensor category" was suggested by I. Frenkel for the theory of these Moore-Seiberg equations. Later, the precise notion of modular tensor category was introduced to summarize the properties of these polynomial equations and has played a central role in the development of conformal field theories and three-dimensional topological field theories. See for example [T] and [BK] for the theory of modular tensor categories, their applications and references to many important works of mathematicians and physicists.Mathematically To prove that a semisimple braided tensor category actually carries a modular tensor category structure, we need to prove that it is rigid, balanced and nondegenerate. The balancing isomorphisms or twists in these braided tensor categories are actually trivial to construct and the balancing axioms are easy to prove. On the other hand, the rigidity has been an open problem for many years after the braided tensor category structure on the category of modules for a vertex operator algebra satisfying the conditions in [HL1]-[HL4] [H1] was constructed. The main difficulty is that from the construction, 2 it is not clear why the numbers determining the module maps given by the sequences in the axioms for the rigidity are not 0. The nondegeneracy of the semisimple braided tensor category of modules for a...

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Section: Introductionmentioning

confidence: 99%

Rigidity and Modularity of Vertex Tensor Categories

Huang

2008

Commun. Contemp. Math.

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“…Many important results for these and related theories, including the constructions of braided tensor category structures and the study of properties of correlation functions, are obtained using these equations (see, for example, [TK], [KazL], [V], [H3], [HL5] and [EFK]). …”

Section: Introductionmentioning

confidence: 99%

“…In the work [TK], Tsuchiya and Kanie used the Knizhnik-Zamolodchikov equations to show the convergence of the correlation functions. For all the concrete examples (see [H2], [H3], [HL5], [HM1] and [HM2]), the convergence and extension property was proved using the particular differential equations of regular singular points associated to the examples, including the Knizhnik-Zamolodchikov equations and the Belavin-Polyakov-Zamolodchikov equations mentioned above. Also, since braided tensor categories and intertwining operator algebras give representations of the braid groups, from the solution to the Riemann-Hilbert problem, we know that there must be some differential equations such that the monodromies of the differential equations give these representations of the braid groups.…”

Section: Introductionmentioning

confidence: 99%

Differential Equations and Intertwining Operators

Huang

2005

Commun. Contemp. Math.

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We show that if every module W for a vertex operator algebra V = n∈Z V (n) satisfies the condition dim W/C 1 (W ) < ∞, where C 1 (W ) is the subspace of W spanned by elements of the form u −1 w for u ∈ V + = n>0 V (n) and w ∈ W , then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations. Moreover, for prescribed singular points, there exist such systems of differential equations such that the prescribed singular points are regular. The finiteness of the fusion rules is an immediate consequence of a result used to establish the existence of such systems. Using these systems of differential equations and some additional reductivity conditions, we prove that products of intertwining operators for V satisfy the convergence and extension property needed in the tensor product theory for V -modules. Consequently, when a vertex operator algebra V satisfies all the conditions mentioned above, we obtain a natural structure of vertex tensor category (consequently braided tensor category) on the category of V -modules and a natural structure of intertwining operator algebra on the direct sum of all (inequivalent) irreducible V -modules. IntroductionIn the present paper, we show that for a vertex operator algebra satisfying certain finiteness and reductivity conditions, matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations of regular singular points. Similar results are also obtained by Nagatomo and Tsuchiya in [NT] . In the construction of these structures from representations of vertex operator algebras, one of the most important steps is to prove the associativity of intertwining operators, or a weaker version which, in physicists' terminology, is called the (nonmeromorphic) operator product expansion of chiral vertex operators. It was proved in [H1] that if a vertex operator algebra V is rational in the sense of [HL1], every finitely-generated lower-truncated generalized V -module is a V -module and products of intertwining operators for V have a convergence and extension property (see Definition 3.2 for the precise description of the property), then the associativity of intertwining operators holds. Consequently the category of V -modules has a natural structure of vertex tensor category (and braided tensor category) and the direct sum of all (inequivalent) irreducible V -modules has a natural structure of intertwining operator algebra.The results above reduce the construction of vertex tensor categories and intertwining operator algebras (in particular, the proof of the associativity of intertwining operators) to the proofs of the rationality of vertex operator algebras (in the sense of [HL1]), the condition on finitely-generated lowertruncated generalized V -modules and the convergence and extension property. Note that this rationality and the condition on finitely-generated lowertruncated generalized V -modules are both purely representation-theoretic properties. These and other relat...

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“…We normalize the intertwining operator by the condition: 25) and one can write the explicit formula for Φ ν µλ (· ⊗ v 0,λ ) in the polynomial realization. Namely,…”

Section: Polynomial Realization For Mmentioning

confidence: 99%

“…Eventually, this relation has been accomplished in the precise form of equivalence of certain tensor categories of representations [28], [17], [25]. The nonstandard tensor product of the representations of affine Lie algebras of the same level, motivated by two dimensional conformal field theory, becomes natural in the context of vertex operator algebras (VOA) [21].…”

Section: Introductionmentioning

confidence: 99%

Quantum Group as Semi-infinite Cohomology

Frenkel

Zeitlin

2010

Commun. Math. Phys.

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We obtain the quantum group SLq(2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c +c = 26. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SLq(2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.

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Intertwining operator algebras and vertex tensor categories for affine Lie algebras

Cited by 43 publications

References 34 publications

Rigidity and Modularity of Vertex Tensor Categories

Rigidity and Modularity of Vertex Tensor Categories

Differential Equations and Intertwining Operators

Quantum Group as Semi-infinite Cohomology

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