Coinvariants and conformal blocks on a pointed curve

Assume full AC as carried by the pointed-curve definition. Let $V$ be a conformal VOA of CFT type on a pointed curve $(X,p_i,t_i)$, and let $M_i$ be integrally graded conformal $V$-modules. No finite-dimensionality of their $L_0$-eigenspaces is assumed. Let ${\mathfrak{g}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ be the image and action defined above. On the algebraic tensor product $M_{\bullet }:=M_1{\otimes }_{\mathbb{C}}\cdots {\otimes }_{\mathbb{C}}{M}_n$, set $C_X(M_{\bullet }):=M_{\bullet }\mathrm{/}{\mathrm{span}}_{\mathbb{C}}\{x\cdot m:x\in {\mathfrak{g}}_{\mathrm{o}\mathrm{u}\mathrm{t}},m\in M_{\bullet }\}\mathrm{.}$ The algebraic conformal-block space is $B_X(M_{\bullet }):={\mathrm{Hom}}_{\mathbb{C}}(C_X(M_{\bullet }),\mathbb{C})$, equivalently the linear functionals on $M_{\bullet }$ that annihilate every ${\mathfrak{g}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$-action. The quotient-dual equivalence is proved in the following lemma. For a pointed family $\pi :\mathcal{X}\to S$ with the disjoint sections and relative formal coordinates specified in the pointed-curve definition, the relative construction requires the supplied relative chiral bundle, relative connection and finite-energy, finite-pole sheaves of the outgoing-mode definition. Explicitly, write $D_i=s_i(S)$ and let $V_{\le N,\mathcal{X}\mathrm{/}S}^r$ denote its supplied finite-stage relative bundle, tensored with the relative one-form line; put where all indices are nonnegative integers, ${\mathcal{T}}_{\mathcal{X}\mathrm{/}S}$ is the relative tangent line, and ${\pi }_{\ast }\mathcal{A}(W):=\mathcal{A}({\pi }^{-1}W)$ for open $W\subseteq S$. The direct limits are in sheaves of ${\mathcal{O}}_S$-modules. Require the relative right derivative $d^r:\mathcal{E}\to \mathcal{F}$, compatible Laurent expansion maps at the sections, and their induced outgoing Lie-algebra sheaf ${\mathcal{G}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$, defined as the image of $\mathrm{coker}(d^r)$ in the product of relative local mode quotients. For the chosen $V$-modules $M_i$, put ${\mathcal{M}}_i:={\mathcal{O}}_S{\otimes }_{\mathbb{C}}{M}_i$ (sheaf tensor product) and ${\mathcal{M}}_{\bullet }:={⨂}_{{\mathcal{O}}_S,i}{\mathcal{M}}_i$, with ${\mathcal{M}}_{\bullet }={\mathcal{O}}_S$ when $n=0$. Require the compatible ${\mathcal{O}}_S$-bilinear summed residue action $\alpha :{\mathcal{G}}_{\mathrm{o}\mathrm{u}\mathrm{t}}{\otimes }_{{\mathcal{O}}_S}{\mathcal{M}}_{\bullet }⟶{\mathcal{M}}_{\bullet }\mathrm{.}$ For these supplied relative data, define the relative coinvariant sheaf to be $\mathcal{C}:={\mathrm{coker}}_{{\mathcal{O}}_S}(\alpha )$, namely the sheaf associated to the presheaf $W\mapsto {\mathcal{M}}_{\bullet }(W)\mathrm{/}\mathrm{im}(\alpha (W))$. The existence of the required relative structures is not inferred merely from the definition of a family. No finite rank, local freeness, or automatic base-change claim follows from this definition.

This definition is algebraic. It does not assert equivalence to the analytic conformal-block definition via the strong residue theorem, nor finite dimensionality, local freeness, convergence, or a moduli-stack construction.

Pointed curve with formal coordinates

Coordinate-independent chiral algebra attached to a VOA

Global outgoing mode algebra and residue action

The outgoing mode algebra is coordinate independent and acts

Integrally graded conformal module

Module for an ordinary vertex algebra

The Axiom of Choice