Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras III: The Sewing-Factorization Theorems

TL;DR

This work extends sewing-factorization (SF) theory to $C_2$-cofinite VOAs that are not necessarily rational, establishing analytic SF theorems for conformal blocks under both disjoint and self-sewing. Central to the results is a coend perspective for disjoint sewing, yielding the isomorphism $\int^{\mathbb M} \mathscr T^*_\mathfrak F(\mathbb M\otimes\mathbb W) \otimes \mathscr T^*_\mathfrak G(\mathbb M'\otimes\mathbb X) \simeq \mathscr T^*_{\mathfrak X_{p_\bullet}}(\mathbb W\otimes\mathbb X)$, and a dual-fusion-product formulation that encodes conformal blocks via canonical blocks and dinatural pairings. The authors develop a key special-case SF theorem, then lift to the general setting by analyzing parallel sections and the logarithmic $q$-expansion of conformal blocks, proving surjectivity and injectivity of the sewing map. The paper also relates this analytic SF theory to Lyubashenko’s topological modular functors, highlighting the role of left-exact coends and pseudo-$q$-traces in non-semisimple contexts, and shows how transitivity of fusion products recovers associativity isomorphisms in Huang–Lepowsky–Zhang’s framework. Overall, the results provide a robust, genus-zero to higher-genus, analytic SF program for irrational VOAs and connect VOA conformal blocks with Lyubashenko-style modular functors.

Abstract

Let $\mathbb V=\bigoplus_{n\in\mathbb N}\mathbb V(n)$ be a $C_2$-cofinite VOA, not necessarily rational or self-dual. In this paper, we establish various versions of the sewing-factorization (SF) theorems for conformal blocks associated to grading-restricted generalized modules of $\mathbb V^{\otimes N}$ (where $N\in\mathbb N$). In addition to the versions announced in the Introduction of [GZ23], we prove the following coend version of the SF theorem: Let $\mathfrak F$ be a compact Riemann surface with $N$ incoming and $R$ outgoing marked points, and let $\mathfrak G$ be another compact Riemann surface with $K$ incoming and $R$ outgoing marked points. Assign $\mathbb W\in\mathrm{Mod}(\mathbb V^{\otimes N})$ and $\mathbb X\in\mathrm{Mod}(\mathbb V^{\otimes K})$ to the incoming marked points of $\mathfrak F$ and $\mathfrak G$ respectively. For each $\mathbb{M} \in \mathrm{Mod}(\mathbb{V}^{\otimes R})$, assign $\mathbb{M}$ and its contragredient $\mathbb M'$ to the outgoing marked points of $\mathfrak F$ and $\mathfrak G$ respectively. Denote the corresponding spaces of conformal blocks by $\mathscr T_{\mathfrak F}^*(\mathbb M\otimes\mathbb W)$ and $\mathscr T_{\mathfrak{G}}^*(\mathbb M'\otimes\mathbb X)$. Let the $\mathfrak X$ be the $(N+K)$-pointed surface obtained by sewing $\mathfrak F$, $\mathfrak G$ along their outgoing marked points. Then the sewing of conformal blocks-proved to be convergent in [GZ25a]-yields an isomorphism of vector spaces $$\int^{\mathbb{M}\in\mathrm{Mod}(\mathbb V^{\otimes R})}\mathscr T_{\mathfrak F}^*(\mathbb M\otimes\mathbb{W})\otimes_{\mathbb C} \mathscr T_{\mathfrak G}^*(\mathbb M'\otimes \mathbb X)\simeq\mathscr T_{\mathfrak X}^*(\mathbb W\otimes \mathbb X)$$ We also discuss the relationship between conformal blocks and the modular functors defined using Lyubashenko's coend/construction.

Figures (5)

• Figure 1: . The pictorial illustration of Thm. \ref{['lb54']}.
• Figure 2: . Sewing $\mathfrak{F}$ with $\mathfrak{G}$.
• Figure 3: . The transitivity of fusion products $\boxtimes_{\mathfrak T}\mathbb V\simeq\boxtimes_\mathfrak{P}(\boxtimes_{\mathfrak Q}\mathbb V)$.
• Figure 1.1: . The exact sequence of chain complexes in Thm. \ref{['lb11']}.
• Figure 1.2: . The commutative diagram induced by \ref{['eq32']}.

Theorems & Definitions (81)

• Theorem 1: \ref{['lb54']}
• Theorem 2: \ref{['lb43']}
• Theorem 1
• proof
• Definition 1.1
• Proposition 1.2
• proof
• Definition 1.5
• Definition 1.6
• Remark 1.7