Differential equations and intertwining operators

By Yi-Zhi Huang. 2002

160 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/303b06264c545db31833c482ff0cf6840cf2c179

How researchers see this paper

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In Algebra and Number Theory

1. Lin Zhang
2. James Lepowsky
3. Yi-Zhi Huang
4. C. Lam
5. H. Yamauchi
6. T. Creutzig
7. Haisheng Li
8. T. Arakawa

In Geometry and Topology

1. James Lepowsky
2. Lin Zhang
3. C. Lam
4. H. Yamauchi
5. Yi-Zhi Huang
6. T. Creutzig
7. T. Arakawa
8. Scott Carnahan

In Statistical and Nonlinear Physics

1. James Lepowsky
2. Lin Zhang
3. C. Lam
4. H. Yamauchi
5. Yi-Zhi Huang

Citations in the corpus

• Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I: Introduction and Strongly Graded Algebras and their Generalized Modules (2010, methodology): “In fact, because for a general vertex (operator) algebra satisfying those assumptions in the present work or in [65] no explicit form of the differential equations such as the form of the Knizhnik-Zamolodchikov equations exists, it was crucial that in the work [53, 59, 72–74], the present work and [”
• On the Structure of Framed Vertex Operator Algebras and Their Pointwise Frame Stabilizers (2006, background): “[FHL]). Moreover, it is shown in [ H2 ] that if V is rational, C2-cofinite and of CFT-type, then V(V ) is associative.”
• RIGIDITY AND MODULARITY OF VERTEX TENSOR CATEGORIES (2005, methodology): “Theorem 4.5 Let V be a simple vertex operator algebra satisfying the conditions in Section 1. Then the ribbon category structure on the category of V -modules constructed in [HL1]–[HL4], [H1] and [ H5 ] is nondegenerate.”