Analytic Conformal Blocks of C₂-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products

2023/05/17 by Gui, Bin, Zhang, Hao · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2305.10180

Abstract

This is the first paper of a three-part series in which we develop a theory of conformal blocks for C₂-cofinite vertex operator algebras (VOAs) that are not necessarily rational. The ultimate goal of this series is to prove a sewing-factorization theorem (and in particular, a factorization formula) for conformal blocks over holomorphic families of compact Riemann surfaces, associated to grading-restricted (generalized) modules of C₂-cofinite VOAs. In this paper, we prove that if \mathbb V is a C₂-cofinite VOA, if \mathfrak X is a compact Riemann surface with N incoming marked points and M outgoing ones, each equipped with a local coordinate, and if \mathbb W is a grading-restricted \mathbb V^{⊗ N}-modules, then the ``dual fusion product" exists as a grading-restricted \mathbb V^{⊗ M}-module. Indeed, we prove a more general version of this result without assuming \mathbb V to be C₂-cofinite. Our main method is a generalization of the propagation of conformal blocks.

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