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

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.

71 pages, 2 figures. The numbering of theorems, equations, and figures has been changed to align with the published version. To appear in Commun. Contemp. Math