Vertex operator algebra and parenthesized braid operad

Abstract

We study conformal blocks of vertex operator algebras on configuration spaces from the viewpoint of the parenthesized braid operad, a combinatorial model of the fundamental groupoid of the little 2-disk operad. For each binary tree we introduce coordinates and a simply connected domain in the configuration space, and show that conformal blocks admit convergent expansions on these domains. Inserting one binary tree into a leaf of another gives gluing maps for the corresponding conformal blocks, while analytic continuation along paths in configuration spaces gives isomorphisms between conformal blocks associated with different trees. We prove that these operations are compatible with the operadic composition in the parenthesized braid operad. As a consequence, the category of $C_1$-cofinite modules whose contragredient modules are finitely generated carries a canonical unital pseudo-braided category structure, without assuming rationality or $C_2$-cofiniteness of the vertex operator algebra. In the rational $C_2$-cofinite case, this structure is represented by tensor products and recovers the balanced braided tensor category structure with twist $\exp(2πiL(0))$.