Vertex operator algebra and parenthesized braid operad (opens in new tab)

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 give...