One-Point Restricted Conformal Blocks and the Fusion Tensor Product

TL;DR

This paper introduces a new approach to compute fusion rules in vertex operator algebras by restricting conformal blocks, leading to a fusion tensor product construction on module categories for strongly rational VOAs.

Contribution

It provides a novel formula for fusion rules via a restriction of conformal blocks and establishes a fusion tensor product on module categories for strongly rational VOAs.

Findings

Derived a new formula for fusion rules using restricted conformal blocks.

Constructed a fusion tensor product on module categories of $A(V)$ for strongly rational VOAs.

Connected conformal block restrictions to the algebraic structure of module categories.

Abstract

We investigate a one-point restriction of conformal blocks on $(\mathbb{P}^1,\infty,1,0)$ associated with modules over a vertex operator algebra. By restricting the module attached to the point $\infty$ to its bottom degree, we obtain a new formula for computing fusion rules in terms of a left $A(V)$-module $M^1\odot M^2$ over the Zhu algebra $A(V)$. As a consequence, for strongly rational VOAs, the construction of $M^1\odot M^2$ induces the fusion tensor product on the module category $\mathsf{Mod}(A(V))$.