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Haagerup Symmetry in $(E_{8})_{1}$ ?

Jan Albert ¹, Yamato Honda ², Justin Kaidi ^2,3,4, and Yunqin Zheng ⁵

¹Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
²Department of Physics, Kyushu University, Fukuoka 819-0395, Japan
³Institute for Advanced Study, Kyushu University, Fukuoka 819-0395, Japan
⁴Quantum and Spacetime Research Institute (QuaSR), Kyushu University, Fukuoka 819-0395, Japan
⁵Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

Phys. Rev. Lett. 136, 091603 – Published 2 March, 2026

DOI: https://doi.org/10.1103/6tzz-tvp7

Abstract

We suggest that the chiral $(e_{8})_{1}$ theory—in many senses the simplest vertex operator algebra—may have Haagerup symmetry $H_{i}$ for $i = 1$ , 2, 3. Likewise, we suggest that the nonchiral $(E_{8})_{1}$ Wess-Zumino-Witten model may have $H_{i} \times H_{i}^{op}$ symmetry, and that gauging the diagonal symmetry gives a $c = 8$ theory with $Z (H_{3})$ symmetry, which is the theory predicted in Evans and Gannon [Commun. Math. Phys. 307, 463 (2011)]. Along the way, we show that $(E_{8})_{1}$ also has a $Fib \times {Fib}^{op}$ symmetry, and that gauging the diagonal symmetry gives the $(G_{2})_{1} \times (F_{4})_{1}$ Wess-Zumino-Witten model, explaining the well-known conformal embedding $(G_{2})_{1} \times (F_{4})_{1} \subset (E_{8})_{1}$ . Finally, we suggest a relation to theories with $H_{3}$ symmetry at $c = 2$ , 6, complimenting the discussion with new modular bootstrap results.

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References (59)

U. Haagerup, in Subfactors (Proceedings of the Taniguchi Symposium, Kyuzeso, 1993) (World Scientific, River Edge, NJ, 1994), pp. 1–38.
P. Grossman and N. Snyder, Commun. Math. Phys. 311, 617 (2012).
See Supplemental Material at http://link.aps.org/supplemental/10.1103/6tzz-tvp7, which includes Refs. [4–30], for a brief review of the Haagerup fusion categories and their Drinfeld center, a more detailed discussion of line operators which do not commute with the chiral algebra, and a summary of some numerical modular bootstrap techniques.
Y.-H. Lin, M. Okada, S. Seifnashri, and Y. Tachikawa, J. High Energy Phys. 03 (2023) 094.
Y.-H. Lin and S.-H. Shao, Phys. Rev. D 107, 125025 (2023).
M. Izumi, Rev. Math. Phys. 13, 603 (2001).
D. E. Evans and P. R. Pinto, Advances in Operator Algebras and Mathematical Physics. (Sinaia 2003), edited by F.-P. Boca, O. Bratteli, R. Longo, and H. Siedentop, p. 67 (The Theta Foundation, Bucharest, 2006), https://bookstore.ams.org/THETA/8.
Y. Choi, B. C. Rayhaun, and Y. Zheng, arXiv:2409.02159.
J. Bockenhauer and D. E. Evans, Commun. Math. Phys. 200, 57 (1999).
J. A. Damia, G. Galati, O. Hulik, and S. Mancani, J. High Energy Phys. 04 (2024) 028.
D. E. Evans and T. Gannon, arXiv:1208.1500.
M. Izumi, arXiv:1512.04288.
M. Izumi, An operator algebra approach to the classification of certain fusion categories III, https://www.ms.u-tokyo.ac.jp/yasuyuki/izumi2016-3.pdf (2016).
B. C. Rayhaun, J. Math. Phys. (N.Y.) 65, 052301 (2024).
S. Möller and B. C. Rayhaun, arXiv:2410.18166.
R. Coquereaux, R. Rais, and E. H. Tahri, J. Math. Phys. (N.Y.) 51, 092302 (2010).
R. Han, A Construction of the “2221” Planar Algebra (University of California, Riverside, 2010).
F. Calegari, S. Morrison, and N. Snyder, Commun. Math. Phys. 303, 845 (2011).
C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, J. High Energy Phys. 01 (2019) 026.
D. Simmons-Duffin, J. High Energy Phys. 06 (2015) 174.
P. Grossman and N. Snyder, Commun. Math. Phys. 311, 617 (2012).
C. Dong and G. Mason, arXiv:math/0203005.
R. Thorngren and Y. Wang, J. High Energy Phys. 04 (2024) 132.
R. Thorngren and Y. Wang, J. High Energy Phys. 07 (2024) 051.
J. Kaidi, K. Ohmori, and Y. Zheng, Phys. Rev. Lett. 128, 111601 (2022).
J. Kaidi, G. Zafrir, and Y. Zheng, J. High Energy Phys. 08 (2022) 053.
Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, Phys. Rev. D 105, 125016 (2022).
Y. Choi, D.-C. Lu, and Z. Sun, J. High Energy Phys. 01 (2024) 142.
X. Yu and H. Y. Zhang, SciPost Phys. 19, 154 (2025).
S. Seifnashri, S.-H. Shao, and X. Yang, SciPost Phys. 19, 063 (2025).
L. E. Bottini and S. Schafer-Nameki, Phys. Rev. Lett. 134, 191602 (2025).
A gapped theory with Haagerup symmetry was constructed in [33]. Nonunitary candidates for RCFTs with Haagerup symmetry have also appeared in [34].
T.-C. Huang and Y.-H. Lin, J. Math. Phys. (N.Y.) 63, 042306 (2022).
D. Gang, D. Kim, and S. Lee, SciPost Phys. 17, 064 (2024).
R. Wolf, Microscopic models for fusion categories, Ph.D. thesis, Leibniz University, Hannover, 2020, arXiv:2101.04154.
T.-C. Huang, Y.-H. Lin, K. Ohmori, Y. Tachikawa, and M. Tezuka, Phys. Rev. Lett. 128, 231603 (2022).
R. Vanhove, L. Lootens, M. Van Damme, R. Wolf, T. J. Osborne, J. Haegeman, and F. Verstraete, Phys. Rev. Lett. 128, 231602 (2022).
Y. Liu, Y. Zou, and S. Ryu, Phys. Rev. B 107, 155124 (2023).
L.-Y. Hung, K. Ji, C. Shen, Y. Wan, and Y. Zhao, arXiv:2506.05324.
D. E. Evans and T. Gannon, Commun. Math. Phys. 307, 463 (2011).
Here, $C^{op}$ represents the opposite category to $C$ , which has the same objects and fusion rules but with opposite morphisms.
J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert, in Proceedings of the 16th International Congress on Mathematical Physics (2010), pp. 608–613, arXiv:0909.5013
O. Diatlyk, C. Luo, Y. Wang, and Q. Weller, J. High Energy Phys. 03 (2024) 127.
A. Perez-Lona, D. Robbins, E. Sharpe, T. Vandermeulen, and X. Yu, J. High Energy Phys. 02 (2024) 154.
A. Perez-Lona, D. Robbins, E. Sharpe, T. Vandermeulen, and X. Yu, J. High Energy Phys. 05 (2025) 066.
J. Albert, K. Hirasaki, Y. Honda, J. Kaidi, and Y. Zheng (to be published).
E. Verlinde, Nucl. Phys. B300, 360 (1988).
V. B. Petkova and J. B. Zuber, Phys. Lett. B 504, 157 (2001).
Previous discussions of symmetries of $(E_{8})_{1}$ that do not commute with the chiral algebra can be found in [14, 15, 50, 51].
S. Hegde and D. P. Jatkar, Mod. Phys. Lett. A 37, 2250193 (2022).
I. M. Burbano, J. Kulp, and J. Neuser, J. High Energy Phys. 10 (2022) 186.
V. G. Kac and I. T. Todorov, Commun. Math. Phys. 190, 57 (1997).
T. Gannon, in Operator Algebras and Mathematical Physics (Mathematical Society of Japan, 2019), Vol. 80, pp. 71–108.
We will assume that this $c = 8$ theory actually exists—to prove that $(E_{8})_{1}$ actually has Haagerup symmetry, it would be necessary to prove the existence of this theory. However, let us note that our computations relating $Z (H_{3})$ and $H_{3} \times H_{3}^{op}$ symmetries hold more generally, relating any theory with $Z (H_{3})$ symmetry to one with $H_{3} \times H_{3}^{op}$ symmetry.
A similar suggestion has been made previously in [15].
As a more general fact, note that for any fusion category $C$ , the Drinfeld center $Z (C)$ and $C \times C^{op}$ are Morita equivalent— $Z (C)$ can be obtained from $C \times C^{op}$ by gauging the Frobenius algebra $A = \oplus_{c \in M C} c ⊠ c^{op}$ [57].
M. Mueger, J. Pure Appl. Algebra 180, 159 (2001).
T.-C. Huang and Y.-H. Lin, arXiv:2007.00670.
Y.-H. Lin and S.-H. Shao, Phys. Rev. D 100, 025013 (2019).

Phys. Rev. Lett. 136, 091603– Published 2 March, 2026

Vol. 136, Iss. 9 — 6 March 2026

Received 18 December 2025
Accepted 13 February 2026

DOI: https://doi.org/10.1103/6tzz-tvp7

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Haagerup Symmetry in $(E_{8})_{1}$ ?

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Haagerup Symmetry in (E8)1?

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Haagerup Symmetry in $(E_{8})_{1}$ ?