Genus-zero permutation-twisted conformal blocks for tensor product vertex operator algebras: The tensor-factorizable case

Let [Formula: see text] be a vertex operator algebra (VOA), let [Formula: see text] be a finite set, and let [Formula: see text] be a subgroup of the permutation group [Formula: see text] which acts on [Formula: see text] in a natural way. For each [Formula: see text], the [Formula: see text]-twisted [Formula: see text]-modules were first constructed and characterized in [K. Barron, C. Dong and G. Mason, Twisted sectors for tensor product vertex operator algebras associated to permutation groups, Comm. Math. Phys. 227(2) (2002) 349–384] when [Formula: see text] has only one orbit, i.e. [Formula: see text] for some [Formula: see text]. In general, if [Formula: see text] is a disjoint union of several [Formula: see text]-orbits [Formula: see text], and if for each orbit [Formula: see text] one chooses a [Formula: see text]-twisted [Formula: see text]-module [Formula: see text], then [Formula: see text] is a [Formula: see text]-twisted [Formula: see text]-module. A direct sum of such [Formula: see text] is called a ⊗-factorizable [Formula: see text]-twisted [Formula: see text]-module. It is known that all [Formula: see text]-twisted modules are ⊗-factorizable if [Formula: see text] is rational [K. Barron, C. Dong and G. Mason, Twisted sectors for tensor product vertex operator algebras associated to permutation groups, Comm. Math. Phys. 227(2) (2002) 349–384]. In this paper, we use the main result of [B. Gui, Sewing and propagation of conformal blocks, New York J. Math. 30 (2024) 187–230] to construct an explicit isomorphism from the space of genus-[Formula: see text] conformal blocks associated to the [Formula: see text]-twisted [Formula: see text]-modules (i.e. [Formula: see text]-twisted [Formula: see text]-modules for some [Formula: see text]) that are ⊗-factorizable to the space of conformal blocks associated to the untwisted [Formula: see text]-modules and a branched covering [Formula: see text] of the Riemann sphere [Formula: see text]. When [Formula: see text] is CFT-type, [Formula: see text]-cofinite, and rational, we use the above result, the (untwisted) factorization property [C. Damiolini, A. Gibney and N. Tarasca, On factorization and vector bundles of conformal blocks from vertex algebras, Ann. Sci. École Norm. Sup. (2022)], and the Riemann–Hurwitz formula to completely determine the fusion rules among [Formula: see text]-twisted [Formula: see text]-modules. Furthermore, assuming [Formula: see text] is as above, we prove that the sewing/factorization of genus-[Formula: see text] [Formula: see text]-twisted [Formula: see text]-conformal blocks holds, and corresponds to the sewing/factorization of untwisted [Formula: see text]-conformal blocks associated to the branched coverings of [Formula: see text]. This proves, in particular, the operator product expansion (i.e. associativity) of [Formula: see text]-twisted [Formula: see text]-intertwining operators (a key ingredient of the [Formula: see text]-crossed braided tensor category [Formula: see text] of the [Formula: see text]-twisted [Formula: see text]-modules) without assuming that the fixed point subalgebra [Formula: see text] is [Formula: see text]-cofinite (and rational), a condition known so far only when [Formula: see text] is solvable and remains a conjecture in the general case. More importantly, this result implies that besides the fusion rules, the associativity isomorphism of [Formula: see text] is also characterized by the higher genus data of untwisted [Formula: see text]-conformal blocks, which gives a new insight into the category [Formula: see text]. We also discuss the applications to conformal nets, which are indeed the original motivations for the author to study the subject of this paper.