Genus-zero Permutation-twisted Conformal Blocks for Tensor Product Vertex Operator Algebras: The Tensor-factorizable Case

TL;DR

This paper establishes an explicit isomorphism between genus-0 conformal blocks for permutation-twisted modules of tensor product vertex operator algebras and untwisted modules, enabling determination of fusion rules and understanding of sewing and factorization properties.

Contribution

It constructs an explicit isomorphism linking permutation-twisted and untwisted conformal blocks, advancing the understanding of their structure and fusion rules.

Findings

Explicit isomorphism between twisted and untwisted conformal blocks

Fusion rules for permutation-twisted modules determined under certain conditions

Relations established between sewing and factorization of conformal blocks

Abstract

For a vertex operator algebra $V$, we construct an explicit isomorphism between the space of genus-0 conformal blocks associated to permutation-twisted $V^{\otimes n}$-modules and the space of conformal blocks associated to untwisted $V$-modules and a branched covering C of the Riemann sphere. As a consequence, when V is CFT-type, rational, and C2 cofinite, the fusion rules for permutation-twisted modules are determined. We also relate the sewing and factorization of permutation-twisted $V^{\otimes n}$-conformal blocks and untwisted $V$-conformal blocks. Various applications are discussed. Note the differences in theorem and equation numbering between the arXiv version and the published version. Some terminology also varies: See Def. 2.2.1 (Def. 2.20 of the published version) for a slight difference in the meanings of $\mathbb U$. The term "Analytic Jacobi identity" in the arXiv version is called the "duality property" in the published version.