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Vertex operator algebra realizations of untwisted Haagerup--Izumi centers

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 26, 2026 / CC BY 4.0

Abstract

For every odd integer N≥3N\geq3, we construct an untwisted Haagerup--Izumi fusion category HN\mathcal H_N on the negative spherical branch and a simple, self-contragredient, regular, C2C_2-cofinite vertex operator algebra ENE_N of central charge −16N-16N, together with an equivalence of modular fusion categories: \[ \operatorname{Rep}(E_N)\simeq Z(\mathcal H_N). \] The VOA ENE_N is rational, has exactly N2+3N^2+3 irreducible modules, and is lower bounded and integer graded, with least conformal degree −(N−1)/2-(N-1)/2. The proof begins with a CFT-type cyclic Virasoro extension of central charge −5N-5N, whose regularity we establish. Two categorical condensations recover HN\mathcal H_N, and a finite extension by shifted lattice modules removes the auxiliary pointed factor. We identify the complete ribbon modular data of Z(HN)Z(\mathcal H_N) with MD−(Z/N;Z/(N2+4),b↦b2/(N2+4))\mathrm{MD}_{-}(\mathbb Z/N;\mathbb Z/(N^2+4),b\mapsto b^2/(N^2+4)) in the notation of Gannon--Schopieray--Yadav, for every odd NN.

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