One-point restricted conformal blocks and the fusion tensor product

Abstract We investigate a one-point restriction of conformal blocks on $$(\mathbb {P}^1,\infty ,1,0)$$ ( P 1 , ∞ , 1 , 0 ) associated with modules over a vertex operator algebra V . 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 new left A ( V )-module $$M^1 \odot M^2$$ M 1 ⊙ M 2 over the Zhu algebra A ( V ), constructed from two V -modules $$M^1$$ M 1 and $$M^2$$ M 2 . As a consequence, for strongly rational vertex operator algebras, the construction of $$M^1 \odot M^2$$ M 1 ⊙ M 2 induces the fusion tensor product on the module category $$\textsf{Mod}(A(V))$$ Mod ( A ( V ) ) .