Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras II: Convergence of Sewing and Higher Genus Pseudo-$q$-traces

Abstract: Let $\mathbb V=\bigoplus_{n\in\mathbb N}\mathbb V(n)$ be a $C_2$-cofinite vertex operator algebra. We prove the convergence of Segal's sewing of conformal blocks associated to analytic families of pointed compact Riemann surfaces and grading-restricted generalized $\mathbb V^{\otimes N}$-modules (where $N=1,2,\dots$) that are not necessarily tensor products of $\mathbb V$-modules, generalizing significantly the results on convergence in [Gui23]. We show that ``higher genus pseudo-$q$-traces" (called pseudo-sewing in this article) can be recovered from Segal's sewing. Therefore, our result on the convergence of Segal's sewing implies the convergence of pseudo-sewing, and hence covers both the convergence of genus-$0$ sewing in [HLZ12] and the convergence of pseudo-$q$-traces in [Miy04] and [Fio16]. Using a similar method, we also prove the convergence of Virasoro uniformization, i.e., the convergence of conformal blocks deformed by non-automomous meromorphic vector fields near the marked points. The local freeness of the analytic sheaves of conformal blocks is a consequence of this convergence. It will be used in the third paper of this series to prove the sewing-factorization theorem.