A theory of tensor products for module categories for a vertex operator algebra, I

Huang, Yi‐Zhi; Lepowsky, James

doi:10.1007/bf01587908

Cited by 146 publications

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“…Similar results are also obtained by Nagatomo and Tsuchiya in [NT]. Using these differential equations together with the results obtained by Lepowsky and the author in [HL1]- [HL4] [H1] and by the author in [H1] [H3]- [H7], we construct braided tensor categories and intertwining operator algebras.…”

Section: Introductionsupporting

confidence: 79%

“…More generally, the tensor product theory for the category of modules for a vertex operator algebra was developed by Lepowsky and the author [HL1]- [HL4] [H1] and the theory of intertwining operator algebras was developed by the author [H1] [H3]- [H7]. These structures are essentially equivalent to chiral genus-zero weakly conformal field theories [S1] [S2] [H5] [H6].…”

Section: Introductionmentioning

confidence: 99%

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Differential Equations and Intertwining Operators

Huang

2005

Commun. Contemp. Math.

124

112

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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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Section: Introductionsupporting

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Differential Equations and Intertwining Operators

Huang

2005

Commun. Contemp. Math.

124

112

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“…In fact, the theory in [HL1]- [HL6] and [H1]- [H2], based on this notion, automatically incorporates the corresponding algebraic constraints and the subtlety of "nuclear democracy" at every stage, and the theory works in considerable generality. This notion has led us to consider iterates of intertwining operators, to formulate the associativity of intertwining operators (in terms of such iterates) and to formulate a "compatibility condition," which was used to construct our tensor products in [HL3] and [HL4]. On the other hand, it was shown by Li [L1] [L3] that the "nuclear democracy theorem" generalizes to any g, and thus for any g, the notions of intertwining operator in the sense of [TK] and in the sense of [FHL] are indeed equivalent.…”

Section: Introductionmentioning

confidence: 99%

“…In [H1], the associativity of intertwining operators was proved using our theory of tensor products of modules for a vertex operator algebra developed in [HL3], [HL4] and [HL5] when the vertex operator algebra is rational in the sense of [HL3] and satisfies certain additional technical conditions. Combined with the results of Moore and Seiberg [MS], the result of [H1] served to construct a natural braided tensor category structure on the category of modules for such a vertex operator algebra.…”

Section: Introductionmentioning

confidence: 99%

Intertwining operator algebras and vertex tensor categories for affine Lie algebras

Huang¹,

Lepowsky²

1999

Duke Math. J.

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The category of finite direct sums of standard (integrable highest weight) modules of a fixed positive integral level k for an affine Lie algebraĝ is particularly important from the viewpoint of conformal field theory and related mathematics. Here we call this the category generated by the standardĝ-modules of level k. A central theme is a braided tensor category structure (in the sense of Joyal and Street [JS]) on this category, a structure explicitly discovered by Moore and Seiberg [MS] in their important study of conformal field theories. In [MS], Moore and Seiberg constructed this structure based on the assumption that there exists a suitable operator product expansion for chiral vertex operators; this is essentially equivalent to assuming the associativity of intertwining operators, in the language of vertex operator algebra theory. Actually, Belavin, Polyakov and Zamolodchikov [BPZ] had already formalized the relation between the operator product expansion and representation theory in the context of conformal field theory, especially for the Virasoro algebra, and Knizhnik and Zamolodchikov [KZ] had established the relation between conformal field theory and the representation theory of affine Lie algebras.In [KL1]-[KL5], Kazhdan and Lusztig achieved a breakthrough by indeed constructing a natural braided tensor category structure, with the additional property of rigidity, on a certain category ofĝ-modules of level k, when k is sufficiently negative, or more generally, when k is in a certain large subset of C excluding the positive integers, and, particularly, proving that 1 this braided tensor category is equivalent to a tensor category of modules for a quantum group constructed from the same finite-dimensional Lie algebra. The method used by Kazhdan and Lusztig, especially in their construction of the associativity isomorphisms, is algebro-geometric and is closely related to the algebro-geometric formulation and study of conformal-field-theoretic structures in the influential works of Tsuchiya-Ueno-Yamada [TUY], Drinfeld [Dr] and Beilinson-Feigin-Mazur [BFM]. (The work [BFM] discusses the case of the minimal models for the Virasoro algebra, and Beilinson informs us that a similar argument works for the case of the category generated by the standardĝ-modules.)In the important work [F1] and [F2], Finkelberg proved that the braided tensor category at positive level is tensor equivalent to a certain "subquotient" of the braided tensor category constructed by Kazhdan and Lusztig at negative level and is thus equivalent to a certain "subquotient" braided tensor category of quantum group representations. (Cf. also [Va].) We have been informed by Finkelberg that the arguments in his paper [F2] can in fact be reinterpreted to actually give a proof of the coherence relations for the braided tensor category structure on the category generated by the standard g-modules of positive integral level k; this reasoning uses Kazhdan-Lusztig's result mentioned above constructing rigid braided tensor category structure ...

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“…On the one hand, it is much easier to rigorously construct many aspects of (super)conformal field theories and study many of their properties using the algebraic formulation of vertex operator (super)algebra (or equivalently, chiral algebra) (e.g., [Wi], [BPZ], [Za], [FLM], [KT], [FZ], [FFR], [DL], [FF], [DMZ], [Wa], [Z1], [Z2]). On the other hand, the geometry of (super)conformal field theory can give insight and provide tools useful for studying the algebraic aspects of the theory, for example: giving rise to general results in Lie theory [BHL1]; giving the necessary insight for developing a theory of tensor products for vertex operator algebras [HL1], [HL4]- [HL7], [H3]; giving rise to change of variables formulas for vertex operator algebras [H2] and N = 1 Neveu-Schwarz vertex operator superalgebras [B6]; and giving rise to constructions in orbifold conformal field theory [BDM], [H4], [BHL2]. But one of the most important applications arises from the fact that this rigorous development of the differential geometric foundations of (super)conformal field theory (in particular an analytic development of the moduli space of (super-)Riemann surfaces and a sewing operation) is necessary for the construction of (super)conformal field theory in the sense of Segal [Se] and Kontsevich. In fact the work of Huang in [H1], [H2] along with [H3], [H6]- [H10] solves the problem of constructing holomorphic genus-zero (weakly) conformal field theories from certain representations of vertex operator algebras.…”

Section: Introductionmentioning

confidence: 99%

The Moduli Space of N = 2 Super-Riemann Spheres With Tubes

Barron

2007

Commun. Contemp. Math.

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Abstract. Within the framework of complex supergeometry and motivated by two-dimensional genus-zero holomorphic N = 2 superconformal field theory, we define the moduli space of N = 2 super-Riemann spheres with oriented and ordered half-infinite tubes (or equivalently, oriented and ordered punctures, and local superconformal coordinates vanishing at the punctures), modulo N = 2 superconformal equivalence. We develop a formal theory of infinitesimal N = 2 superconformal transformations based on a representation of the N = 2 Neveu-Schwarz algebra in terms of superderivations. In particular, via these infinitesimals we present the Lie supergroup of N = 2 superprojective transformations of the N = 2 super-Riemann sphere. We give a reformulation of the moduli space in terms of these infinitesimals. We introduce generalized N = 2 super-Riemann spheres with tubes and discuss some group structures associated to certain moduli spaces of both generalized and non-generalized N = 2 super-Riemann spheres. We define an action of the symmetric groups on the moduli space. Lastly we discuss the nonhomogeneous (versus homogeneous) coordinate system associated to N = 2 superconformal structures and the corresponding results in this coordinate system.

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A theory of tensor products for module categories for a vertex operator algebra, I

Abstract: Contents 1 Introduction 2 Review of basic concepts 3 Affinizations of vertex operator algebras and the * -operation 4 The notions of P (z)-and Q(z)-tensor product of two modules 5 First construction of Q(z)-tensor product 6 Second construction of Q(z)-tensor product

Cited by 146 publications

References 41 publications

Differential Equations and Intertwining Operators

Differential Equations and Intertwining Operators

Intertwining operator algebras and vertex tensor categories for affine Lie algebras

The Moduli Space of N = 2 Super-Riemann Spheres With Tubes

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