Analytic conformal blocks of C2-cofinite vertex operator algebras III: The sewing-factorization theorems

Abstract

Let (Formula presented.) be a (Formula presented.) -cofinite vertex operator algebra, not necessarily rational or self-dual. In this paper, we establish various versions of the sewing-factorization (SF) theorems for conformal blocks associated to grading-restricted generalized modules of (Formula presented.) (where (Formula presented.)). In addition to the versions announced in the Introduction of the first part of this series of papers by Gui and Zhang, we prove the following coend version of the SF theorem. Let (Formula presented.) be a compact Riemann surface with (Formula presented.) incoming and (Formula presented.) outgoing marked points, and let (Formula presented.) be another compact Riemann surface with (Formula presented.) incoming and (Formula presented.) outgoing marked points. Assign (Formula presented.) and (Formula presented.) to the incoming marked points of (Formula presented.) and (Formula presented.), respectively. For each (Formula presented.), assign (Formula presented.) and its contragredient (Formula presented.) to the outgoing marked points of (Formula presented.) and (Formula presented.), respectively. Denote the corresponding spaces of conformal blocks by (Formula presented.) and (Formula presented.). Let (Formula presented.) be the (Formula presented.) -pointed surface obtained by sewing (Formula presented.) along their outgoing marked points. Then, the sewing of conformal blocks — proved to be convergent in the second part of this series of papers by Gui and Zhang — yields an isomorphism of vector spaces (Formula presented.). We also discuss the relationship between conformal blocks and the modular functors defined using Lyubashenko's coend/construction.