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Finite local freeness of the dual-fusion Hom sheaf

Statement

Let $V$ be a C2-cofinite conformal vertex algebra of CFT type, let $B$ be a nonempty open polydisk, and let $F, G \to B$ be analytic families of compact pointed worldsheets such that every component of every fiber of each family meets an incoming marked section. Give $F$ and $G$ respectively $N$ and $K$ incoming sections and the same number $R \geq 0$ of ordered outgoing sections. Assign fixed admitted generalized joint $V^{\otimes N}$ - and $V^{\otimes K}$ -modules $W, X$ to the incoming sections. By a dual fusion product over $B$ for $H = F$ or $G$ with incoming module $Z = W$ or $X$ , mean a fixed grading-restricted generalized joint $V^{\otimes R}$ -module $D_{H} (Z)$ and a holomorphic family conformal block $γ_{H}$ on $Z \otimes D_{H} (Z)$ . Holomorphic means that its evaluation on every fixed tensor is a holomorphic function on $B$ . Require, for every $b \in B$ and every admitted $V^{\otimes R}$ -module $M$ , that $T \mapsto γ_{H, b} \circ ({i d}_{Z} \otimes T)$ be the representing bijection ${Hom}_{V^{\otimes R}} (M, D_{H} (Z)) \to {C B}_{H_{b}} (Z \otimes M)$ . Suppose these two family products are supplied; their existence and a choice of holomorphic trivialization of arbitrary varying fiber modules are not conclusions of this theorem. Then the constant sheaf

H : = {Hom}_{V^{\otimes R}} ⁣ (D_{G} (X)^{'}, D_{F} (W)) \otimes_{C} O_{B}

is finite locally free over $B$ . Here $D_{G} (X)^{'}$ is the contragredient of $D_{G} (X)$ ; “finite locally free” means locally isomorphic to $O_{B}^{n}$ for one finite integer $n$ on each neighborhood.

Statement references

VOA conformal block and dual fusion product

Restricted contragredient of a generalized VOA module

Contragredient construction preserves generalized VOA modules

Facts & Assumptions

[F1]

Finite generation of C2-cofinite generalized joint modules supplies a finite set $E \subset C$ , depending only on $V$ , for which $G (A) = ⨁_{λ \in E^{R}} A_{[λ]}$ is a finite-dimensional generating subspace of every admitted generalized joint module $A$ .

[F2]

VOA conformal block and dual fusion product requires each assumed dual fusion product to be a grading-restricted generalized module; existence is a hypothesis here. Contragredient construction preserves generalized VOA modules makes the restricted contragredient another module in this same joint class.

[F3]

A component module map intertwines every mode, and therefore each $L_{i} (0)$ . Hence it preserves joint generalized weights: $(L_{i} (0) - λ_{i})^{n} f (a) = f ((L_{i} (0) - λ_{i})^{n} a)$ for every integer $n \geq 0$ .

Proof

Given: The C2-cofinite algebra, families, nonempty base, and indicated dual fusion products in the Statement.

1.1givenF1F2algebra

Put $A = D_{G} (X)^{'}$ and $C = D_{F} (W)$ . By [F2] these are grading-restricted generalized $V^{\otimes R}$ -modules. By [F1], $G (A) = ⨁_{λ \in E^{R}} A_{[λ]}$ is finite dimensional and generates $A$ under the component modes. Also $C_{E} = ⨁_{λ \in E^{R}} C_{[λ]}$ is finite dimensional, since it is a finite sum of finite-dimensional joint generalized-weight spaces. If $R = 0$ , the empty-tuple joint-space convention makes both modules finite dimensional and the same assertions hold with $G (A) = A$ and $C_{E} = C$ .

2.1F3step 1.1algebra

By [F3], restriction sends ${Hom}_{V^{\otimes R}} (A, C)$ into ${Hom}_{C} (G (A), C_{E})$ . This restriction map is injective: a module map zero on $G (A)$ is zero on every finite string of component modes applied to those generators, and such strings span $A$ . The target is a finite-dimensional vector space, so the indicated module Hom space has some finite dimension $d$ . No auxiliary grading-preservation condition is imposed on these maps.

3.1step 2.1choose∎

Choose a basis of this $d$ -dimensional Hom space. Tensoring its basis isomorphism with $O_{B}$ gives an $O_{B}$ -linear isomorphism $H ≅ O_{B}^{d}$ , proving finite local freeness of exactly the constant tensor-product sheaf in the Statement. This uses neither an identification with a varying conformal-block sheaf nor a base-change assertion about such a sheaf.

Remarks

This proof is conditional only on the existence and module-class interfaces expressly assumed for the displayed dual fusion products. It does not construct those products or prove the dual-fusion representing theorem. Identifying this constant Hom sheaf with a family conformal-block sheaf and proving compatibility with holomorphic base change remain separate obligations in consumers that require them.

Depends on

Dependency tree · 2 levels

Sources

Bin Gui and Hao Zhang, Analytic Conformal Blocks of C2-cofinite Vertex Operator Algebras III: The Sewing-Factorization Theorems