Yi-Zhi Huang

On axiomatic approaches to vertex operator algebras and modules

Introduction Vertex operator algebras Duality for vertex operator algebras Modules Duality for modules References.

Two-dimensional conformal geometry and vertex operator algebras

The focus of this volume is to formulate and prove one main theorem, the equivalance between the algebraic and geometric formulations of the notion of vertex operator algebra. The author introduces a geomatric notion of vertex operator algebra in terms of complex powers of the determinant line bundles over certain moduli spaces (parameter spaces) of spheres (genus-zero Riemann surfaces) with punctures and local analytic co-ordinates, and seeks to prove that this notion is precisely equivalent to the algebraic notion of vertex operator algebra. In particular, a detailed algebraic and analytic study of the sewing operation in the moduli space is presented.

VERTEX OPERATOR ALGEBRAS AND THE VERLINDE CONJECTURE

We prove the Verlinde conjecture in the following general form: 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 C2-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.) Then the matrices formed by the fusion rules among the irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation τ ↦ -1/τ on the space of characters of irreducible V-modules. Using this result, we obtain the Verlinde formula for the fusion rules. We also prove that the matrix associated to the modular transformation τ ↦ -1/τ is symmetric.

RIGIDITY AND MODULARITY OF VERTEX TENSOR CATEGORIES

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 C2-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.

Tensor Products of Modules for a Vertex Operator Algebra and Vertex Tensor Categories

In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announcement has also appeared [HL1]. The theory is based on both the formal-calculus approach to vertex operator algebra theory developed in [FLM2] and [FHL] and the precise geometric interpretation of the notion of vertex operator algebra established in [H1].

DIFFERENTIAL EQUATIONS AND INTERTWINING OPERATORS

We show that if every module W for a vertex operator algebra V satisfies the condition that the dimension of W/C_1(W) is less than infinity, where C_1(W) is the subspace of W spanned by elements of the form u_{-1}w for u in V of positive weight and w in 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 reducibility 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.

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

This is the first part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a “vertex tensor category” structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a “complex analogue” of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. The theory applies in particular to many familiar “rational” vertex operator algebras, including those associated with WZNW models, minimal models and the moonshine module. In this paper (Part I), we introduce the notions ofP(z)- andQ(z)-tensor product, whereP(z) andQ(z) are two special elements of the moduli space of spheres with punctures and local coordinates, and we present the fundamental properties and constructions ofQ(z)-tensor products.

Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I: Introduction and Strongly Graded Algebras and Their Generalized Modules

This is the first part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. This theory generalizes the tensor category theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a “conformal vertex algebra” or even more generally, for a “Mobius vertex algebra.” We do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. As in the earlier series of papers, our tensor product functors depend on a complex variable, but in the present generality, the logarithm of the complex variable is required; the general representation theory of vertex operator algebras requires logarithmic structure. This work includes the complete proofs in the present generality and can be read independently of the earlier series of papers. Since this is a new theory, we present it in detail, including the necessary new foundational material. In addition, with a view toward anticipated applications, we develop and present the various stages of the theory in the natural, general settings in which the proofs hold, settings that are sometimes more general than what we need for the main conclusions. In this paper (Part I), we give a detailed overview of our theory, state our main results and introduce the basic objects that we shall study in this work. We include a brief discussion of some of the recent applications of this theory, and also a discussion of some recent literature.

DIFFERENTIAL EQUATIONS, DUALITY AND MODULAR INVARIANCE

We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0 and V(0) = ℂ1, (ii) every ℕ-gradable weak V-module is completely reducible and (iii) V is C2-cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is completely different from the one previously used by Zhu and others. In particular, we show that the q-traces of products of certain geometrically-modified intertwining operators satisfy modular invariant systems of differential equations which, for any fixed modular parameter, reduce to doubly-periodic systems with only regular singular points. Together with the results obtained by the author in the genus-zero case, the results of the present paper solves essentially the problem of constructing chiral genus-one weakly conformal field theories from the representations of a vertex operator algebra satisfying the conditions above.

Braided Tensor Categories and Extensions of Vertex Operator Algebras

Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.