Fixed-coordinate sphere Virasoro block family

Fix $c\in \mathbb{C}$, $n\ge 1$, and the configuration space ${\mathrm{Conf}}_n(\mathbb{C})=\{\mathbf{z}=(z_1,\dots ,z_n)\in {\mathbb{C}}^n:z_i\mathrm{\ne }{z}_j\text{ for }i\mathrm{\ne }j\}\mathrm{.}$ For each $j$, let $U_j$ be a representation of the Virasoro algebra in which the central element acts by $c{\mathrm{id}}_{U_j}$. At every $\mathbf{z}\in {\mathrm{Conf}}_n(\mathbb{C})$, specify a linear functional ${\mathrm{\Phi }}_{\mathbf{z}}\in {(U_1{\otimes }_{\mathbb{C}}\cdots {\otimes }_{\mathbb{C}}{U}_n)}^{\ast },$ where the tensor product is algebraic and $E^{\ast }:={\mathrm{Hom}}_{\mathbb{C}}(E,\mathbb{C})$ denotes the algebraic dual. The family is weakly holomorphic: for every fixed $u_j\in U_j$, the scalar function $\mathbf{z}\mapsto {\mathrm{\Phi }}_{\mathbf{z}}(u_1\otimes \cdots \otimes u_n)$ is holomorphic on ${\mathrm{Conf}}_n(\mathbb{C})$. Choose primary states $v_j\in U_j$ of weights $h_j\in \mathbb{C}$, meaning $L_0{v}_j=h_j{v}_j$ and $L_m{v}_j=0$ for every $m>0$, and define $F(\mathbf{z}):={\mathrm{\Phi }}_{\mathbf{z}}(v_1\otimes \cdots \otimes v_n)\mathrm{.}$

Translation covariance is required for every fixed tuple $u_j\in U_j$: ${\mathrm{\partial }}_{z_j}{\mathrm{\Phi }}_{\mathbf{z}}(u_1\otimes \cdots \otimes u_n)={\mathrm{\Phi }}_{\mathbf{z}}(u_1\otimes \cdots \otimes L_{-1}{u}_j\otimes \cdots \otimes u_n)\mathrm{.}$ In particular, the $L_{-1}$ insertion differentiates $F$. Use the fixed local coordinates $t_j=w-z_j$ at the marked points and $1\mathrm{/}w$ at infinity. As part of the family data, specify a meromorphic function $G_{\mathbf{z}}$ on ${\mathbb{P}}^1$ whose Laurent expansion at each $z_j$ is $G_{\mathbf{z}}(z_j+t_j)={\sum }_{m\in \mathbb{Z}}{\mathrm{\Phi }}_{\mathbf{z}}(v_1\otimes \cdots \otimes L_m{v}_j\otimes \cdots \otimes v_n)t_j^{-m-2}\mathrm{.}$ Thus $G_{\mathbf{z}}$ is the stress-tensor matrix element on the chosen primary insertions; the Laurent expansion above defines its mode indexing locally. Require that it have no finite poles other than the $z_j$, depend holomorphically on $(\mathbf{z},w)$ away from the incidence divisors $w=z_j$, and be $O(w^{-4})$ as $w\to \mathrm{\infty }$. The primary and translation conditions give its principal part at $z_j$ as $h_{j}F\mathrm{/}(w-z_j{)}^2+({\mathrm{\partial }}_{z_j}F)\mathrm{/}(w-z_j)$. Finally require the global sphere Ward identities: ${\sum }_j{\mathrm{\partial }}_{z_j}F=0,{\sum }_j(z_j{\mathrm{\partial }}_{z_j}+h_j)F=0,{\sum }_j(z_j^2{\mathrm{\partial }}_{z_j}+2h_j{z}_j)F=0.$

This is a conditional local interface, not an existence theorem for nonzero functionals or blocks. Coordinates are fixed, so fractional $L_0$-weights are allowed; no exponentiation to $\mathrm{Aut}\mathcal{O}$, generic coinvariant equivalence, or full CFT is asserted.

Virasoro algebra in the $L_m$ convention

The generic integral-$L_0$ block carrier Coinvariants and conformal blocks on a pointed curve is a separate interface; no equivalence is asserted here.