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Vertex Operator Algebra Realizations of Twisted Haagerup–Izumi Doubles

A mix of human-written and AI-generated textHuman understanding: some partsmath.QA — Quantum Algebra

Contributed by Andrew Riesen ↗, David E. Evans

Version 1 / Oct 23, 2026 / CC BY 4.0

Abstract

We construct vertex operator algebras realizing an infinite family of twisted, nonunitary Haagerup--Izumi doubles. For every odd integer N≥3N\geq 3, we construct a spherical fusion category FN\mathcal{F}_N with the Haagerup--Izumi fusion rules for Z/NZ\mathbb{Z}/N\mathbb{Z} and pointed associator of exact order NN, and realize its Drinfeld center as the ribbon category of modules of an explicit vertex operator algebra VN\mathscr{V}_N. The construction begins with a uniform family of strongly rational Virasoro extensions, whose irreducible modules and theta-function characters we determine. A simple-current extension involving a conformally shifted E7E_7 lattice vertex operator algebra removes the additional semion factor. The resulting VN\mathscr{V}_N is simple, regular, self-contragredient, and C2C_2-cofinite, with central charge −6N2−10-6N^2-10, effective central charge eight, and N2+3N^2+3 irreducible modules with explicit characters. Its weight-zero subspace has dimension 5656, so it is not of CFT type; we establish modularity and the ribbon equivalence without relying on the CFT-type hypothesis.

Provenance statement

This paper combines the authors’ mathematical work with contributions developed using generative AI under their direction. AI assisted in developing proof arguments and drafting substantial portions of the text, which the authors have revised and refined. The AI-assisted arguments have not yet undergone complete independent human verification. Comments welcome!

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Version history

  1. v1Submitted by Andrew RiesenInitial depositCurrentOct 23, 2026