Recent developments around Haagerup

1. Three breakthroughs + 1

• Huang, arXiv:2609.15986, Cyclic Haagerup–Izumi fusion categories at every odd order.
• Radchenko, Wheeler, arXiv:2609.21892, Real quadratic fields and finite quantum dilogarithms I.
• Gannon, Schopieray, Yadav, arXiv:2609.25185, On Haagerup–Izumi fusion categories.
• He, Liu, Lu, Palcoux, Ren, arXiv:2610.11652, Triangular prism equations for near-group categories: algebraic solutions and a unitary conjecture.

2. Connections to quantum information and number theory

• Kopp, arXiv:1807.05877, SIC-POVMs and the Stark conjectures (IMRN 2021).
• Appleby, Flammia, Kopp, arXiv:2501.03970, A constructive approach to Zauner's conjecture via the Stark conjectures.
• Appleby, Flammia, Kopp, arXiv:2609.39192, The twisted convolution identity and ghost $r$-SICs from finite quantum dilogarithms.
• Radchenko, Wheeler, hexagon:2610.00186, Stark units for real quadratic fields and reciprocity laws. Heavy AI use.

3. VOA constructions

Prehistory

• Evans, Gannon, arXiv:1006.1326, The exoticness and realisability of twisted Haagerup–Izumi modular data (2010).
• Albert, Honda, Kaidi, Zheng, arXiv:2512.08225, Haagerup symmetry in $(E_8)_1$? (2025).
• Gannon, Rayhaun, arXiv:2606.05279, Hypergroup symmetry in relative quantum field theories and chiral algebras (2026).

The 2026 constructions

• Riesen, arXiv:2610.13933, The Haagerup VOA at $c=8$; full version hexagon:2610.00175. Heavy AI use.
• Huang, arXiv:2610.14441, Haagerup conformal field theory at central charge eight.
• Huang, hexagon:2610.00213, An order-five Haagerup–Izumi vertex operator algebra at central charge eight. Heavy AI use.
• Evans, arXiv:2610.26341, A vertex operator algebra for the S-fold realisation of a non-unitary Haagerup–Izumi modular fusion category. Heavy AI use.
• Evans, Riesen, hexagon:2610.00490, Vertex operator algebra realizations of twisted Haagerup–Izumi doubles. Heavy AI use.
• Evans, Riesen, arXiv:2610.32928, Vertex operator algebra realizations of untwisted Haagerup–Izumi centers; full version hexagon:2610.00508. Heavy AI use.

4. Connections to supersymmetric field theory

• Gang, Kim, Lee, arXiv:2310.14877, A non-unitary bulk-boundary correspondence: Non-unitary Haagerup RCFTs from S-fold SCFTs (2023), SciPost Phys. 17 (2024).
• Jeong, Sohn, arXiv:2608.11946, Non-unitary Haagerup-like TQFTs and RCFTs from generalized S-fold SCFTs (2026).
• Evans, arXiv:2610.18512, Haagerup–Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua (2026). Heavy AI use.

5. Exotic fusion categories known before September 2026

Fusion categories coming neither from finite groups nor from representations of affine Lie algebras (up to Morita equivalence = generalized gauging), in order of discovery

• 1994. Haagerup, Principal graphs of subfactors in the index range $4<[M:N]<3+\sqrt2$, in Subfactors (Taniguchi Symposium, Kyuzeso 1993), World Scientific. The list of candidate principal graphs, including the Haagerup and extended Haagerup ones.
• 1999. Asaeda, Haagerup, arXiv:math/9803044, Commun. Math. Phys. 202. The Haagerup and Asaeda–Haagerup subfactors.
• 2001. Izumi, The structure of sectors associated with Longo–Rehren inclusions II, Rev. Math. Phys. 13. The Haagerup–Izumi category for $\mathbb{Z}/5$.
• 2009. Bigelow, Morrison, Peters, Snyder, arXiv:0909.4099, Acta Math. 209 (2012). The extended Haagerup subfactor.
• 2010. Evans, Gannon, arXiv:1006.1326, Commun. Math. Phys. 307 (2011). Haagerup–Izumi categories for $\mathbb{Z}/7$ and $\mathbb{Z}/9$, with numerical evidence up to $\mathbb{Z}/19$.
• 2012. Morrison, Penneys, arXiv:1208.3637, Trans. AMS 367 (2015). The 4442 subfactor.
• 2012. Evans, Gannon, arXiv:1208.1500, Adv. Math. 255 (2014). Near-group categories $G+|G|$, over 40 new subfactors.
• 2014. Morrison, Penneys, arXiv:1406.3401. The 2D2 subfactor.
• 2015. Grossman, Izumi, Snyder, arXiv:1501.07324. A $\mathbb{Z}/4\times\mathbb{Z}/2$ generalized Haagerup category.
• 2015. Morrison, Peters, Snyder, arXiv:1501.06869, Selecta Math. 23 (2017). Ostrik's twisted Haagerup categories, from the $\mathbb{Z}/9+9$ near-groups.
• 2015. Evans, Gannon, arXiv:1506.03546, Adv. Math. 310 (2017). Non-unitary analogues of Haagerup–Izumi categories.
• 2016. Izumi, arXiv:1609.07604. The $3^n$ subfactors, including $3^{\mathbb{Z}/4}$ and $3^{\mathbb{Z}/2\times\mathbb{Z}/2}$.
• 2020. Huang, Lin, arXiv:2007.00670. Transparent Haagerup–Izumi categories up to $\mathbb{Z}/9$, and tetrahedrally symmetric ones up to $\mathbb{Z}/15$.
• 2022. Grossman, Izumi, Snyder, arXiv:2201.11901. Graded extensions of generalized Haagerup and Asaeda–Haagerup categories.
• September 2026. See §1.