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

TL;DR

This paper proves the convergence of Segal's sewing of conformal blocks for $C_2$-cofinite vertex operator algebras, extending previous results to more general modules and higher genus surfaces, and relates this to pseudo-$q$-traces and Virasoro uniformization.

Contribution

It generalizes the convergence of Segal's sewing to analytic families of modules not limited to tensor products, and connects this to higher genus pseudo-$q$-traces and conformal block deformation.

Findings

Proved convergence of Segal's sewing for generalized modules.

Established convergence of higher genus pseudo-$q$-traces.

Demonstrated convergence of Virasoro uniformization.

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 [Gui24]. We show that ``higher genus pseudo-$q$-traces" (called pseudo-sewing in this article) can be recovered from the above generalization of Segal's sewing to $\mathbb{V}^{\otimes N}$-modules. Therefore, our result on the convergence of the generalized Segal's sewing implies the convergence of pseudo-sewing, and hence covers both the convergence of genus-$0$ sewing in [Hua05a,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.