Analytic conformal blocks of C2C₂‐cofinite vertex operator algebras III: The sewing‐factorization theorems

Let be a ‐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 (where ). 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 be a compact Riemann surface with incoming and outgoing marked points, and let be another compact Riemann surface with incoming and outgoing marked points. Assign and to the incoming marked points of and , respectively. For each , assign and its contragredient to the outgoing marked points of and , respectively. Denote the corresponding spaces of conformal blocks by and . Let be the ‐pointed surface obtained by sewing 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 We also discuss the relationship between conformal blocks and the modular functors defined using Lyubashenko's coend/construction.