Sewing and Propagation of Conformal Blocks

Abstract

We clarify the relations between sewing and propagating conformal blocks. In particular, we show that sewing and propagation and commuting procedures. As an application, we give a geometric construction of permutation-twisted modules for tensor product VOAs. Their first (algebraic) construction is due to Barron-Dong-Mason. The results and the point of view in this article are crucial for relating the (genus-0) permutation-twisted conformal blocks associated to a tensor product VOA $V^{\otimes k}$ and the untwisted conformal blocks (of possibly higher genera) associated to $V$, which will be discussed in an upcoming work.