Differential Equations and Intertwining Operators

Abstract

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...

Cardy Condition for Open-Closed Field Algebras

“…We define an action of SL(2, Z) on the map (3.59) as follows: ∈ SL(2, Z) and w a ∈ W a . The following theorem is proved in [Mi2,H7].…”

Vertex operator algebras, the Verlinde conjecture, and modular tensor categories

“…The main obstructions were the duality and modular invariance properties for genus-zero and genus-one multipoint correlation functions constructed from intertwining operators for a vertex operator algebra satisfying the conditions mentioned above. These properties were proven recently (22,23).In this article, we announce a proof of the general version of the Verlinde conjecture above. Our theorem assumes only that the vertex operator algebra that we consider satisfies certain natural grading, finiteness, and reductivity properties (see Section 2).…”

“…The main obstructions were the duality and modular invariance properties for genus-zero and genus-one multipoint correlation functions constructed from intertwining operators for a vertex operator algebra satisfying the conditions mentioned above. These properties were proven recently (22,23).…”

“…2.3) heavily uses the results obtained in ref. 23 on the convergence and analytic extensions of the q traces of products of what we call ''geometrically modified intertwining operators,'' the genus-one associativity, and the modular invariance of these analytic extensions of the q traces, where q ϭ e 2 i . These results allow us to (rigorously) establish a formula that corresponds to the fact that the modular transformation ‫ۋ‬ Ϫ1͞ changes one basic Dehn twist on the Teichmüller space of genus-one Riemann surfaces to the other.…”

“…We also discuss some consequences of our theorem, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of ‫ۋ‬ Ϫ1͞ , and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of modules for such a vertex operator algebra constructed by Huang and Lepowsky (14)(15)(16)(17)(18)22) when V satisfies in addition the condition that irreducible modules not equivalent to the algebra (as a module) have no nonzero elements of weight 0. In particular, the category of modules for such a vertex operator algebra has a natural structure of modular tensor category.…”