One-Point Restricted Conformal Blocks and the Fusion Tensor Product

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 $\mathsf{Mod}(A(V))$.