Analytic conformal blocks of C2-cofinite vertex operator algebras II: Convergence of sewing and higher genus pseudo-q-traces

Let [Formula: see text] be a [Formula: see text]-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 [Formula: see text]-modules (where [Formula: see text]) that are not necessarily tensor products of [Formula: see text]-modules, generalizing significantly the results on convergence in [B. Gui, Convergence of sewing conformal blocks, Commun. Contemp. Math. 26(3) (2024) 2350007]. We show that “higher genus pseudo-[Formula: see text]-traces” (called pseudo-sewing in this paper) can be recovered from the above generalization of Segal’s sewing to [Formula: see text]-modules. Therefore, our result on the convergence of the generalized Segal sewing implies the convergence of pseudo-sewing, and hence covers both the convergence of genus-[Formula: see text] sewing in [Y.-Z. Huang, Differential equations and intertwining operators, Commun. Contemp. Math. 7(3) (2005) 375–400; Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2012), arXiv:1110.1929] and the convergence of pseudo-[Formula: see text]-traces in [M. Miyamoto, Modular invariance of vertex operator algebras satisfying [Formula: see text]-cofiniteness, Duke Math. J. 122(1) (2004) 51–91; F. Fiordalisi, Logarithmic intertwining operators and genus-one correlation functions, Commun. Contemp. Math. 18(6) (2016) 1650026]. Using a similar method, we also prove the convergence of Virasoro uniformization, i.e. the convergence of conformal blocks deformed by non-autonomous 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.