Restricted conformal blocks and the braiding on Zhu algebra modules

Abstract

Let $V$ be a vertex operator algebra (VOA), not necessarily rational or $C_2$-cofinite, such that the category $\mathcal C_1(V)$ of $C_1$-cofinite grading-restricted generalized $V$-modules is semisimple and contains the generalized Verma modules associated to finite-dimensional modules over the Zhu algebra $A(V)$. Such VOAs include affine VOAs at admissible and generic levels. We prove that the bottom degree of Huang-Lepowsky's $P(w)$-tensor product module $M^1\boxtimes_{P(w)}M^2$ is isomorphic to the left $A(V)$-module $M^1\odot_{w,0}M^2$ arising from the restricted conformal blocks on $(¶^1,\infty,w,0)$. By shifting the marked points and using the parallel transport of the restricted conformal blocks, we construct an explicit braiding isomorphism on the category of finite-dimensional $A(V)$-modules, which is the restriction of the braiding isomorphism of Huang-Lepowsky-Zhang to the bottom degrees. For the affine VOA $L_k(\mathfrak g)$ at a positive integral level, it is given by the Casimir tensor of $\mathfrak g$ on certain quotients of tensor products of finite-dimensional $\mathfrak g$-modules, and at generic levels it recovers Drinfeld's commutativity isomorphism.