Analytic convergence of Segal sewing for C2-cofinite VOA blocks

First suppose every component of the unsewn family meets an incoming mark, and shrink its base to a coordinate polydisc B. Choose any strictly smaller sewing radii $r_j^0<r_j$, ${\rho }_j^0<{\rho }_j$, and put $Q={\prod }_j{D}_{r_j^0{\rho }_j^0}$. All constructions below are on the entire $P=Q\times B$, including $q_j=0$. F1 applies with D the incoming divisor. It constructs the finite current bundles $F_n=V_{\le n,C}\otimes {\omega }_{C\mathrm{/}P}$ and a common sufficiently positive incoming twist for each n. Its normalization-filtration degree argument is uniform on all partial-smoothing strata. In particular its global finite-generation assertion gives one fixed finite-dimensional $G\subset W$ generating modulo the global nodal Ward span on P. This assertion does not claim nodal local freeness.

We construct the projective coefficients used below. On the smooth normalization choose local vertical coordinates t. A projective coefficient is a collection $p_t$ satisfying $p_t=({\mathrm{\partial }}_{t}s{)}^2{p}_s+S_{t}s$, where $S_{t}s=s^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{s}^{\mathrm{\prime }}-(3\mathrm{/}2)(s^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{s}^{\mathrm{\prime }}{)}^2$. The chain rule, obtained by differentiating a composite three times, gives the affine cocycle identity for these transitions. A smooth partition of unity gives a smooth affine section. Allow poles of order at most k at D, with k sufficiently large that $K^2(kD)$ has zero fiber H1 on every normalization component. Subtract the vertical Green solution from F4 to its vertical dbar; this gives a smooth base family of fiber-holomorphic projective coefficients, meromorphic only at D. The remaining base dbar is a closed $(0,1)$ form with values in the finite bundle $H={\tilde{\pi }}_{\ast }{K}^2(kD)$. More explicitly, its canonical base $(0,1)$ connection is integrable by total dbar squared equal to zero, as in F4; use F4's explicit finite-dimensional gauge to take a holomorphic frame for H. In that frame the affine obstruction form is closed. F4 and the finite-bundle vanishing in F1 solve this base dbar equation. Subtracting that solution produces projective coefficients holomorphic on the total normalization family away from D and meromorphic of finite order at D. They are holomorphic on all paired sewing disks. A projective atlas or a holomorphic projective connection on the sewn family is not being assumed.

Let $\mathrm{\Omega }={\mathrm{\Omega }}_{\mathbf{q}}(\log \mathbf{q})$. For the jth pair write $x={\xi }_j$, $y={\varpi }_j$, $xy=q_j$, and $J=e^{L(1)}(-1{)}^{L(0)}$. The opposite-module identity gives the coefficientwise tensor equality $(Y_{M,j}(x^{L(0)}v,x)\otimes I)\mathrm{\Omega }=(I\otimes Y_{M^{\mathrm{\prime }},j}(y^{L(0)}Jv,y))\mathrm{\Omega }\mathrm{.}$ Indeed transposing the right field produces $Y_M(y^{-L(0)}v,y^{-1})$: $e^{yL(1)}(-y^{-1}{)}^{L(0)}{e}^{L(1)}(-1{)}^{L(0)}=y^{-L(0)}$. Moving this field through $q_j^{L_j(0)}$ changes it to the displayed left field by conformal conjugation. These are the genuine $L_j(0)$, with their nilpotent parts. Pairing finite generalized-weight blocks proves the identity without choosing a basis globally; the identity coevaluation on each finite block is basis independent. Multiplying by the node generator $dx\mathrm{/}x=-dy\mathrm{/}y$ cancels the two branch residues.

The coordinate action in F3 gives $U(\gamma )\omega ={\gamma }^{\mathrm{\prime }}(0{)}^2\omega +\frac{c}{12}S\gamma (0)\mathbf{1.}$ For verification, write $\gamma (t)=a\exp ({\sum }_{n\ge 1}{c}_n{t}^{n+1}{\mathrm{\partial }}_t)t$. Only $L(2)\omega =(c\mathrm{/}2)\mathbf{1}$ can contribute a vacuum component; $L(1)\omega =0$, so it is $c{c}_2\mathrm{/}2$. Direct expansion gives $S\gamma (0)=6c_2$. Therefore for any global vertical meromorphic vector field $v=h_t{\mathrm{\partial }}_t$ on the normalization, the local currents $(h_t\omega -(c\mathrm{/}12)h_t{p}_t\mathbf{1})dt$ agree on overlaps. Their poles are allowed at the incoming and paired marks. Their total Ward residue annihilates an unsewn block. Since $Y(\mathbf{1},t)=I$, the raw conformal residues consequently satisfy ${\sum }_i{\mathrm{Res}}_{t_i=0}{h}_i{Y}_i(\omega ,t_i)d{t}_i\psi =\frac{c}{12}{\sum }_i{\mathrm{Res}}_{t_i=0}{p}_{t_i}{h}_{i}d{t}_i\psi ,$ meaning equality after evaluating the block on the respective residue action on its argument. This proves the scalar identity, with its sign and factor; the meromorphic poles of p at D are permitted in the corrected current.

In particular sewing annihilates the entire nodal Ward span formally. To check this assertion, take a global finite-state finite-pole nodal current and expand its coefficients in q on the unchanged exterior. In the node frame of F1 it is a convergent sum of $x^a{y}^b$ times states. Substitution $y=q_j\mathrm{/}x$ gives, at each fixed q coefficient, a meromorphic current on the normalization with finite poles at the paired marks and D. Apply the unsewn Ward identity coefficientwise. Its two paired contributions cancel by step 1.3 and the opposite signs of the dualizing generator. All coefficient operations are finite: q exponents of the sewing module lie in finitely many lattice cosets bounded below, a fixed node Taylor coefficient has a finite pole cutoff, and sufficiently high positive modes kill every vector in a fixed finite generalized-weight block. Thus the formal result annihilates $J_{\mathrm{p}\mathrm{r}\mathrm{e}}(P)$; there is no product of two uncontrolled bilateral Laurent series.

We now remove the extra component hypothesis in step 1.1. Locally on B add one disjoint incoming section on each normalization component having no incoming mark, with a product coordinate and label V. Such sections exist in local submersion charts away from the finitely many old marks; smooth proper local triviality fixes the finite component list. Denote their number by s. The original unsewn family and the family with these additional labels both meet the marked-component hypotheses of F5, since every original component already has an incoming or paired mark. Their block sheaves are therefore finite locally free. Evaluation at ${\mathbf{1}}^{\otimes s}$ is a holomorphic bundle map and is an isomorphism on every fiber by fixed-curve vacuum propagation. A nonzero determinant minor gives its holomorphic inverse locally on B. Hence the given family block has an actual holomorphic propagated family $\tilde{\psi }$ with labels $W\otimes V^{\otimes s}$. This proves the family propagation used here; it is not assumed from a fixed-fiber statement alone.

Fix k. By F1 choose an actual logarithmic lift ${\tilde{y}}_k$ of $q_k{\mathrm{\partial }}_{q_k}$ with finite poles only at D. At incoming $t_i$ write it as $q_k{\mathrm{\partial }}_{q_k}+h_i(q,t_i,b){\mathrm{\partial }}_{t_i}$ and define $A_k={\sum }_i\mathrm{Res}{h}_i{Y}_i(\omega ,t_i)d{t}_i$ and ${\mathrm{\nabla }}_k=q_k{\mathrm{\partial }}_{q_k}-A_k$. On the jth fixture it is $a_j(x,y,q_{\mathrm{\ne }j},b)x{\mathrm{\partial }}_x+b_j(x,y,q_{\mathrm{\ne }j},b)y{\mathrm{\partial }}_y+(1-{\delta }_{jk})q_k{\mathrm{\partial }}_{q_k},a_j+b_j={\delta }_{jk}\mathrm{.}$ The a and b functions are holomorphic at the node. On the exterior expand its vertical part as ${\sum }_{\mathbf{n}\ge 0}{q}^{\mathbf{n}}{v}_{k,\mathbf{n}}$. Every $v_{k,\mathbf{n}}$ is a global meromorphic vertical field on the normalization. On the plus branch the coefficient of $q_j^{n_j}$ in $a_j(x,q_j\mathrm{/}x)x{\mathrm{\partial }}_x$ has powers $x^{l+1}{\mathrm{\partial }}_x$ with $l\ge -n_j$; the minus branch has the analogous bound. This follows directly from a monomial $x^a{y}^b=q_j^b{x}^{a-b}$, and proves the finite pole cutoff needed in step 2.2.

Apply step 2.1 to each $v_{k,\mathbf{n}}$ on the unsewn family, then apply the block to $w\otimes \mathrm{\Omega }$. The paired raw Virasoro residues, summed with coefficients $q^{\mathbf{n}}$, equal $L_k(0)\mathrm{\Omega }$. To see the sign and the constant term, step 1.3 applied to $v=\omega$ (for which $J\omega =\omega$) identifies the two actions after substitution $y=q_j\mathrm{/}x$. Their coefficients add to $a_j+b_j={\delta }_{jk}$, so their sum is ${\delta }_{jk}\mathrm{Res}x{Y}_{M,j}(\omega ,x)dx\mathrm{\Omega }={\delta }_{jk}{L}_j(0)\mathrm{\Omega }$. Also $q_k{\mathrm{\partial }}_{q_k}\mathrm{\Omega }=L_k(0)\mathrm{\Omega }$, including the logarithmic polynomial derivative. The incoming raw action is $A_{k}w$. It follows that $q_k{\mathrm{\partial }}_{q_k}{S}_{\psi }(w)=S_{\psi }({\mathrm{\nabla }}_{k}w)+g_k{S}_{\psi }(w),g_k=\frac{c}{12}{\sum }_{\mathbf{n}}{q}^{\mathbf{n}}{\sum }_{p\text{ all unsewn marks}}{\mathrm{Res}}_p{p}_t(v_{k,\mathbf{n}}{)}_{t}dt\mathrm{.}$ Here w is constant; the derivative in ${\mathrm{\nabla }}_k$ acts on holomorphic coefficients when w varies. This derives the formal sewing differential equation instead of assuming a Virasoro connection with unspecified scalar.

The scalar $g_k$ is holomorphic on the entire P. At incoming marks its contribution is $(c\mathrm{/}12)\mathrm{Res}{p}_{t_i}{h}_{i}d{t}_i$, a holomorphic function because both pole cutoffs are finite. For the paired marks its contributions are $\frac{c}{12}\frac{1}{2\pi i}{\oint }_{\mathrm{\mid }x\mathrm{\mid }=ϵ}{p}_x(x,b)a_j(x,q_j\mathrm{/}x,q_{\mathrm{\ne }j},b)xdx,$ and the analogous integral of $p_y(y,b)b_j(q_j\mathrm{/}y,y,q_{\mathrm{\ne }j},b)ydy$. Take $\mathrm{\mid }{q}_j\mathrm{\mid }\mathrm{/}{\rho }_j^0<ϵ<r_j^0$ for the first integral, and the reversed inequality for the second. On compact q subsets such contours exist, the integrands are holomorphic in every parameter, and Cauchy's theorem makes the result independent of the admissible radius. Their Taylor coefficients are exactly the residue coefficients in step 4.1, by normally convergent geometric substitution on the contours. As the contours may approach the disk radii, these integrals define the same holomorphic scalar on all Q, not merely near $q=0$.

Choose a basis $e_1,\dots ,e_d$ of the finite generating window G of step 1.1. The finite marked pole bounds of $h_i$ imply that ${\mathrm{\nabla }}_k{e}_i$ lies in one fixed finite-dimensional W-span with holomorphic coefficients on P: there are only finitely many negative conformal modes and sufficiently positive modes annihilate $e_i$. The global finite reduction in F1 therefore gives holomorphic functions ${\mathrm{\Omega }}_{k,ij}$ on the entire P with ${\mathrm{\nabla }}_k{e}_i-{\sum }_j{\mathrm{\Omega }}_{k,ij}{e}_j\in J_{\mathrm{p}\mathrm{r}\mathrm{e}}(P)\mathrm{.}$ By step 2.2 this relation may be sewn formally. Thus the column $F=(S_{\psi }(e_i){)}_i$ satisfies the actual finite system $q_k{\mathrm{\partial }}_{q_k}F=({\mathrm{\Omega }}_k+g_{k}I)F\mathrm{.}$ All its coefficients are holomorphic across every coordinate divisor, on all P, by the lift and scalar constructions and F1. In particular they are not postulated matrices on a punctured parameter space.

F2 gives finitely many exponent cosets and uniform finite logarithmic bounds for M. Each coefficient is holomorphic in b, because it pairs the original holomorphic block with one finite-dimensional weight coevaluation and finitely many nilpotent powers. The formal vector F therefore has exactly F2's prescribed exponent-polynomial-logarithm shape. Apply its convergence theorem to the actual system in step 6.1. It converges absolutely and locally uniformly on the universal cover of $Q^{\times }\times B$, coefficientwise in all logarithmic monomials. For any other $w\in W$, F1 gives a relation $w={\sum }_j{f}_j(q,b)e_j$ modulo the global Ward span, with holomorphic $f_j$ on all P. Step 2.2 gives the same formal relation after sewing. Cauchy coefficient bounds for each $f_j$ show its product with a convergent generalized-power-log series converges absolutely on every smaller polydisc, and formal convolution agrees with that product. Thus $S_{\psi }(w)$ has the same convergence property. There are finitely many logarithmic monomials, so taking the maximum of their coefficient bounds gives precisely the uniformity in the Statement.

The limit is a sewn block throughout the smooth sewing locus. Here it is necessary to test all Ward sections, not only those with a particular global frame. For every state cutoff n and sufficiently positive incoming pole order l, F1 gives a finite holomorphic bundle $H_{n,l}={\pi }_{\ast }{F}_n(lD)$ with its actual fiber identification. For fixed w its residue action takes values in a fixed finite-dimensional W-span: the state cutoff and the negative Laurent cutoff are fixed, and positive modes truncate on w. Evaluation by the holomorphic sewing limit is consequently a holomorphic bundle map $H_{n,l}\to \mathcal{O}$ on the connected covering domain. Near the sewing origin, a local frame of $H_{n,l}$ is represented by actual total holomorphic nodal sections; step 2.2 makes every Taylor-logarithm coefficient of this bundle map zero. Absolute convergence in step 7.1 therefore makes it zero on a nonempty open set. The identity theorem for local matrix coefficients extends zero over the entire connected domain. Taking every n,l,w exhausts the actual Ward span, by F1's positive-twist base change. This proves the block condition. It does not infer analytic sewing merely from formal Ward identities without convergence.

Apply the strengthened-component argument above to $\tilde{\psi }$, since the enlarged incoming divisor meets every original normalization component. Sewing commutes coefficientwise with evaluating the additional arguments at ${\mathbf{1}}^{\otimes s}$; they are not sewing factors. The resulting series is exactly the original $S_{\psi }(w)$, so it inherits the absolute locally uniform convergence. On every smooth sewn fiber, each component meets an original incoming mark by the original hypothesis. Fixed-curve propagation in F5 removes the added vacuum labels and makes the evaluated limit an original sewn block. Evaluation also preserves holomorphicity. Finally every compact subset of the original punctured sewing polydisc is contained in one product of strictly smaller radii chosen in step 1.1. The formal series is identical for every such choice, so the convergent limits agree on overlaps. Exhausting the original radii and covering compact base subsets proves all assertions of the unchanged Statement on its full domain. No rationality, semisimplicity, module tensor-factorization or cylinder-factorization hypothesis was introduced.