C2-cofinite analytic sewing convergence and pseudo-sewing applications

Let $V$ be a C2-cofinite conformal vertex algebra of CFT type over $\mathbb{C}$. For every proper analytic family of compact pointed Riemann surfaces over a complex manifold $B$ with $N\ge 1$ incoming sections and $R\ge 1$ disjoint paired coordinate neighborhoods, assume every component of each fiber before sewing meets an incoming or paired section and every component of every smooth sewn fiber meets an incoming section. Give the incoming points a grading-restricted generalized joint $V^{\otimes N}$-module $W$, and give the plus sewing points a grading-restricted generalized joint $V^{\otimes R}$-module $M$ and the minus sewing points its restricted contragredient $M^{\mathrm{\prime }}$, formed from the finite joint generalized-weight duals with the stated opposite-field action. The coordinate disks have positive radii $r_j,{\rho }_j$.

Fix a holomorphic family conformal block $\psi :W\otimes M\otimes M^{\mathrm{\prime }}\to \mathcal{O}(B)$ on the unsewn family: this means a $\mathbb{C}$-linear map whose evaluation at each $b\in B$ annihilates every global residue action on that fiber. For joint weight bases $e_{\lambda ,\alpha }$ of $M$ with dual bases $e_{\lambda ,\alpha }^{\vee }$ in $M^{\mathrm{\prime }}$, the formal sewing tensor and series are

Here $L_j(0)$ is the conformal operator of the $j$th joint action, and on a joint generalized-weight block the exponential means $e^{{\mathrm{\ell }}_j{\lambda }_j}{\sum }_{k\ge 0}{\mathrm{\ell }}_j^k(L_j(0)-{\lambda }_{j}I{)}^k\mathrm{/}k!$, a finite nilpotent sum. For $0<\mathrm{\mid }{q}_j\mathrm{\mid }<r_j{\rho }_j$, with the logarithms fixed on the universal cover, $S_{\psi }(w)$ converges absolutely and locally uniformly on compact subsets of the base and punctured sewing-parameter cover for every $w\in W$, and defines a holomorphic conformal block of the sewn smooth family.

For one sewing pair, fix also a holomorphic unsewn family conformal block $\varphi :W\otimes M^{\mathrm{\prime }}\otimes M\to \mathcal{O}(B)$, with the two paired labels enumerated in that order, and the following additional data: a grading-restricted generalized $V$-module $M$ with finite-dimensional ${\mathrm{E}\mathrm{n}\mathrm{d}}_V(M)$, a unital subalgebra $A\subseteq {\mathrm{E}\mathrm{n}\mathrm{d}}_V(M{)}^{\mathrm{o}\mathrm{p}}$ over which $M$ and the finite-dimensional lower truncations used in the series are projective, a symmetric functional $\omega :A\to \mathbb{C}$ satisfying $\omega (ab)=\omega (ba)$, and A-linearity of every finite-block map $T_{\lambda ,\mu ;b,w}:M_{[\mu ]}\to M_{[\lambda ]}$ defined by

Then the pseudo-sewing series $S_{\varphi }^{\omega }(w)={\sum }_{\lambda }{\mathrm{T}\mathrm{r}}_{\omega }(q^{L(0)}{\mathrm{\mid }}_{M_{[\lambda ]}}{T}_{\lambda ,\lambda ;b,w})$ has the same absolute locally uniform convergence and sewn-block conclusion. Here ${\mathrm{T}\mathrm{r}}_{\omega }$ is the finite-projective pseudo-trace, applied after extending a diagonal block map by zero to a projective lower truncation containing it, as in the pseudo-sewing supplier. These block maps need not assemble to an algebraic endomorphism of $M$. No rationality or self-duality assumption is added in either conclusion.

Analytic convergence of Segal sewing for C2-cofinite VOA blocks

Convergence of higher-genus pseudo-sewing

Proper family from disjoint analytic plumbing

Pointed analytic worldsheet and family

Plumbing family for punctured Riemann surfaces

VOA conformal block and dual fusion product

Hattori–Stallings pseudo-trace

Both analytic suppliers now have local proofs. Ordinary convergence uses the actual nodal positive-pole lifting and finite logarithmic Euler system; pseudo-sewing uses the finite-projective cyclic trace and two-pair sphere plumbing reduction. The deductions above therefore establish the stated combined conclusion. No fixed-surface sewing inverse, coend factorization or modular-functor assertion follows from convergence alone.