Convergence of Sewing Conformal Blocks

In recent work, Damiolini-Gibney-Tarasca showed that for a $C_2$-cofinite rational CFT-type vertex operator algebra $\mathbb V$, sheaves of conformal blocks are locally free and satisfy the factorization property. In this article, we use analytic methods to prove that sewing conformal blocks is convergent, solving a conjecture proposed by Zhu and Huang.