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?

## 1 Answer

Well, the claim is bogus, so you can't expect the proof to hold much water. On the other hand, it may be instructive to try filling in details to see why it fails.

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.