# Conformal blocks in genus zero

In section 10.4 of "Vertex Algebras and Algebraic Curves", Ben-Zvi & Frenkel (second edition), the authors claim that for any vertex algebra V, the space of one-pointed conformal blocks with insertions in $V$ $C(\mathbb{P}^1,0,V)$ is one-dimensional. They justify this by saying that the Lie algebra $U_{\mathbb{P}^1/0}(\mathcal{V}_0)$ is generated by the Fourier coefficients $A_{(n)}$ for $n<0.$ ($U_{\mathbb{P}^1/0}(\mathcal{V}_0)$ is such that $\mathcal{V}_0/U_{\mathbb{P}^1/0}(\mathcal{V}_0)\mathcal{V}_0$ is the space of coinvariants.) I don't see why this is true (and the later calculations in the section don't seem to justify this.) Also the statement as they wrote it seems false, as I believe $A_{(-1)}$ for $A$ the vacuum is the identity map?

First of all, we can't define conformal blocks for arbitrary vertex algebras - we need Möbius structure for genus zero, and quasiconformal structure for higher genera. Since we are working on $\mathbb{P}^1$ the bundle $\mathcal{V}$ splits as a sum of line bundles $\bigoplus_{n \in \mathbb{Z}} V_n \otimes \Omega^{-n}$, so the Lie algebra is given by global sections of $$(\bigoplus_n V_n \otimes \Omega^{1-n})/(T \otimes 1 + 1 \otimes d)(\bigoplus_n V_n \otimes \Omega^{-n}).$$ Choosing a global coordinate $t$, we find that in order for a polydifferential of the form $A \otimes t^k(dt)^{1-n}$ to be regular at infinity, it is necessary and sufficient that $2n-2-k \geq 0$. That is, we only get an action of the coefficient $A_{(k)}$ for $A$ of degree $n$ when $k \leq 2n-2$, so in particular, the vacuum cannot act as identity.
On the other hand, any positive degree vector $A$ can act on the vacuum by $A_{-1}$, so the vacuum axiom says that $A$ lies in the image. We conclude that if $V$ is non-negatively graded and the vacuum spans the degree 0 subspace (i.e., $V$ is of "CFT type"), then we get a 1-dimensional space of conformal blocks.
In general, we can identify the 2-point conformal block space with the space of invariant bilinear forms on $V$, and by a 1994 theorem of Li, this is dual to $V_0/L_1V_1$. Using the "propagation of vacuum" property (Theorem 10.3.1), this is also isomorphic to the 1-point conformal block space.