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Sewing and Propagation of Conformal Blocks

Bin Gui

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.

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 and the untwisted conformal blocks (of possibly higher genera) associated to , which will be discussed in an upcoming work.

Paper Structure

This paper contains 11 sections, 14 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. The geometric setting
  3. Sheaves of VOA
  4. Conformal blocks
  5. Sewing conformal blocks
  6. An equivalence of sheaves
  7. Propagation of conformal blocks
  8. Multi-propagation
  9. Sewing and multi-propagation
  10. A geometric construction of permutation-twisted $\mathbb V^{\otimes k}$-modules
  11. Strong residue theorem for analytic families of curves

Key Result

Proposition 4.2

Let $\upphi:\mathscr W_{\mathfrak X}(\mathbb W_\bullet)\rightarrow\mathscr O_{\mathcal{B}}$ be an $\mathscr O_{\mathcal{B}}$-module morphism. Suppose that each connected component of $\mathcal{B}$ contains a non-empty open subset $V$ such that the restriction of $\upphi$ to $\mathscr W_{\mathfrak X_

Theorems & Definitions (38)

  • Definition 3.1
  • Definition 4.1
  • Proposition 4.2
  • Definition 5.2
  • Definition 5.3
  • Theorem 5.5: Gui23, Thm. 11.3
  • Example 5.6
  • Lemma 6.1
  • proof
  • Proposition 6.2
  • ...and 28 more