Differential equations and intertwining operators

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Abstract

We show that if every module W for a vertex operator algebra V = ∐ n∈z V (n)satis-fies the condition dimW/C 1(W) < ∞, where C 1(W) is the subspace of W spanned by elements of the form u-1w for u ∈ V + = ∐ n>o 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.

Original languageEnglish (US)
Pages (from-to)375-400
Number of pages26
JournalCommunications in Contemporary Mathematics
Volume7
Issue number3
DOIs
StatePublished - Jun 2005

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics

Keywords

  • Convergence and extension properties
  • Differential equations
  • Genus-zero conformal field theories
  • Intertwining operators

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