Vector bundle structure of conformal block bundle

Question

My question is about the conformal block bundle, which (following Kohno's "Conformal Field Theory and Topology") is constructed as follows: Consider the projection map onto the first $n$ coordinates \begin{equation} \pi: (\mathbb{C}P^1)^{n+1} \to (\mathbb{C}P^1)^n. \end{equation} Let $D_i$ denote the hyperplane of $(\mathbb{C}P^1)^{n+1}$ given by setting $z_i=z_{n+1}$ where $(z_1,\dots, z_{n+1})$ are coordinates for $(\mathbb{C}P^1)^{n+1}$. Let $\mathcal{M}_{D_1, \dots, D_n}(U)$ denote the space of meromorphic functions on $\pi^{-1}(U)$ with poles only on the hyperplanes $D_1,\dots, D_n$. Consider $\mathfrak{g} \otimes \mathcal{M}_{D_1, \dots, D_n}(U)$ as a Lie algebra with bracket defined by \begin{equation} [X \otimes f, Y \otimes g]=[X,Y] \otimes fg. \end{equation}

Given an element $f \in \mathcal{M}_{D_1, \dots, D_n}(U)$, we can expand $f$ as a Laurent series on the hyperplane $D_i$ \begin{equation} f_{D_i}(z)=\sum_{k=-N}^\infty a_n(z_1, \dots, z_n) (z-z_i)^k \end{equation} where $a_n(z_1, \dots, z_n)$ is holomorphic in $z_1, \dots, z_n$. Fixing $(z_1, \dots, z_n) = (p_1, \dots, p_n)$, we can view $f_{D_i}$ as an element of $\mathbb{C}((t_i))$ where we define $t_i=z - z_i$. This gives us a canonical inclusion map \begin{equation} \tau_i : \mathfrak{g} \otimes \mathcal{M}_{D_1, \dots, D_n}(U) \hookrightarrow \hat{\mathfrak{g}}_i \cong \mathfrak{g} \otimes \mathbb{C}((t_i)) \oplus \mathbb{C}c. \end{equation} To each $t_i$, associate an integrable highest weight $\hat{\mathfrak{g}}_i$-module $V_{\lambda_i}$ of highest weight $\lambda_i$. We can define an action of $\hat{\mathfrak{g}}_i$ on $\bigotimes_{i=1}^n V_{\lambda_i}$ by \begin{equation} X \cdot (v_1 \otimes \cdots \otimes v_n) = \sum_{i=1}^n v_1 \otimes \cdots \otimes \tau_i (X) v_i \otimes \cdots \otimes v_n. \end{equation}

Now, consider the trivial bundle \begin{equation} E:= \text{Conf}_n(\mathbb{C}P^1) \times \text{Hom}_{\mathbb{C}}\left(\bigotimes_{i=1}^n V_{\lambda_i},\mathbb{C}\right) \to \text{Conf}_n(\mathbb{C}P^1). \end{equation} For each open set $U \subseteq \text{Conf}_n(\mathbb{C}P^1)$, denote by $\mathcal{E}_{\lambda_1, \dots, \lambda_n}(U)$ the space of smooth sections $\sigma: U \to E$ satisfying \begin{equation} \sum_{i=1}^n (\sigma(p_1, \dots, p_n))(v_1 \otimes \cdots, \otimes \tau_i(f)v_i \otimes \cdots \otimes v_n)=0 \end{equation} for all $f \in \mathfrak{g} \otimes \mathcal{M}_{D_1, \dots, D_n}(U)$ and $v_1 \otimes \cdots \otimes v_n \in \bigotimes_{i=1}^n V_{\lambda_i}$.

Question: Why is the sheaf of smooth sections $\mathcal{E}_{\lambda_1, \dots, \lambda_n}(U)$ locally-free?

Ultimately, I'd like to understand why we can realise the disjoint union of the space of conformal blocks as a vector bundle over $\text{Conf}_n(\mathbb{C}P^1)$. Any help is greatly appreciated.

Answer 1

I also had the same question when I was reading Kohno's book. My opinion is that that book is only an introduction to the topic of conformal blocks and is not rigorous. For rigorous treatment you need to read the classical paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries by Tsuchiya, Ueno, Yamada. There are also many articles and monographs explaining their work. You may try Mukhopadhyay, or Ueno. As a survey of TUY's work I also like the last chapter of Bakalov-Kirillov.

The existence of a vector bundle structure is to me a very deep result. The argument of TUY could be summarized as follows. Basically you are considering a family of Riemann surfaces, i.e. a proper submersion $\pi:\mathcal C\rightarrow\mathcal B$ such that each fiber is a compact Riemann surface. In your case $\mathcal B=\text{Conf}_n(\mathbb CP^1)$ and $\mathcal C=\mathbb CP^1\times\mathcal B$. You want to introduce a vector bundle structure on the collection of conformal blocks (also called spaces of vacua). It is even not trivial to show that the dimensions of conformal blocks are independent of the fibers. To prove all these things one considers the sheaf of vacua (sheaf of conformal blocks) and its dual sheaf: sheaf of covacua. These are sheaves of $\mathcal O_{\mathcal B}$-modules where $\mathcal O_{\mathcal B}$ is the structure sheaf of $\mathcal B$. One then shows that the sheaf of covacua $\mathcal V$ is a coherent sheaf, and that there exists a connection on $\mathcal V$. There is a standard and elementary result in the literature of D-modules saying that any coherent sheaf with a connection is locally free, i.e. it behaves like a holomorphic vector bundle. (This is also proved in the above mentioned references.) This proves the existence of vector bundle structure and in particular, the constance of the dimensions of conformal blocks.