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Haagerup symmetry in a rank-six lattice conformal field theory

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

Contributed by Cédric ARHANCET ↗

Version 1 / Nov 10, 2026 / CC BY-SA 4.0

Abstract

Let UU be a positive strongly rational vertex operator algebra of central charge eight with Rep(U)≅Z(H3)\mathrm{Rep}(U) \cong Z(\cal H_3). We show that its Cartan Heisenberg commutant Ω\Omega embeds conformally into a lattice algebra VKV_K, where KK has rank six, determinant 3939 and root system A2⊕A2⊕A1A_2\oplus A_2\oplus A_1. The diagonal theory of VKV_K has a unique vacuum and carries each of the three Haagerup fusion categories through three nonconjugate embeddings of Ω\Omega. We show that, up to isomorphism fixing Ω\Omega, there are exactly five conformal extensions of Ω\Omega with positive strongly rational target and global representation-category dimension 3939, and that their images in VKV_K form four conjugacy classes. We also determine the common invariant relevant spectrum of the three canonical actions, their normalized cubic constants and their invariant genus-zero field algebra. We work in the TFT framework of rational full conformal field theory.

Provenance statement

The author acknowledges the use of AI tools for language polishing, exploratory mathematical discussions, drafting parts of the arguments, and preparing and checking exact calculations. The author takes responsibility for the mathematical content.

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Version history

  1. v1 · 38 pagesSubmitted by Cédric ARHANCETInitial depositCurrentNov 08, 2026