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.