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Haagerup symmetries of the E8E_8 theory and conformal nets at central charge eight

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

Contributed by Cédric ARHANCET

Version 2 / Nov 03, 2026 / CC BY-SA 4.0

Abstract

A Haagerup algebra is a positive strongly rational vertex operator algebra UU of central charge eight with Rep(U)\mathrm{Rep}(U) braided equivalent to the Drinfeld center Z(H3)Z(\cal H_3) of the Haagerup fusion category. We prove that UU has exactly three holomorphic extensions, each isomorphic to VE8V_{E_8}, and that the corresponding Lagrangian algebras have module categories H1\cal H_1, H2\cal H_2 and H3\cal H_3. The extensions attached to H2\cal H_2 and H3\cal H_3 are the two lattice extensions of VE6⊕A2V_{E_6 \oplus A_2}. We prove a recognition theorem for H2\cal H_2 and H3\cal H_3 among unitary fusion categories of rank six with three invertible objects, a noncommutative Grothendieck ring and trivial pointed associator. For simple regular unitary vertex operator algebras with bounded localized vertex operators, we construct the transport module of two ordinary unitary modules when the second integrates in Tener's annular sense. For a unitary conformal Haagerup subalgebra of VE6⊕A2V_{E_6 \oplus A_2}, these results give a completely rational net with representation category Z(H3)Z(\cal H_3), a realization of the Haagerup subfactor and the generalized orbifold of Bischoff. In particular, they apply to the strongly local explicit algebra SS of levels (3,13)(3,13), whose Haagerup property is established in \cite[Theorem 6.1]{ArhIII}. In \cite{ArhIII}, the net of SS is recognized from strong rationality and lattice branching, and a diagonal extension of the net is then used to identify Rep(S)\mathrm{Rep}(S). Here an exact construction of the compatible unitary structure on the diagonal lattice algebra VE8⊕E8V_{E_8 \oplus E_8}, followed by passage to the twin algebra, proves complete unitarity and Condition I+ for every Haagerup algebra. This also gives the VOA--net comparison through smeared vertex operators for unitary conformal embeddings in VE6⊕A2V_{E_6 \oplus A_2}.

Author comments: 66 pages

Provenance statement

The author acknowledges the use of AI tools for language polishing, exploratory mathematical discussions and feedback, for drafting parts of the arguments. The author takes full responsibility for the content of the manuscript.

References

  1. T. Abe, G. Buhl and C. Dong. Rationality, regularity, and C2C_2-cofiniteness. Trans. Amer. Math. Soc. 356 (2004), no. 8, 3391--3402.
  2. J. Albert, Y. Honda, J. Kaidi and Y. Zheng. Haagerup symmetry in (E8)1(E_8)_1? Phys. Rev. Lett. 136 (2026), 091603. Page and equation references are to arXiv:2512.08225v2.arXiv
  3. C. Arhancet. Haagerup vertex operator algebras at central charge eight. Preprint, 2026. arXiv:2611.01618 (https://arxiv.org/abs/2611.01618).arXiv
  4. C. Arhancet. The Haagerup center at central charge eight via conformal nets. Preprint, 2026.
  5. M. Asaeda and U. Haagerup. Exotic subfactors of finite depth with Jones indices (5+13)/2(5+\sqrt{13})/2 and (5+17)/2(5+\sqrt{17})/2. Comm. Math. Phys. 202 (1999), no. 1, 1--63.
  6. M. Bischoff. A remark on CFT realization of quantum doubles of subfactors: case index <4< 4. Lett. Math. Phys. 106 (2016), no. 3, 341--363.
  7. M. Bischoff. The relation between subfactors arising from conformal nets and the realization of quantum doubles. In Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones' 60th birthday, 15--24, Proc. Centre Math. Appl. Austral. Nat. Univ. 46, Austral. Nat. Univ., Canberra, 2017. Page references are to this published version of arXiv:1511.08931.arXiv
  8. M. Bischoff. Generalized orbifold construction for conformal nets. Rev. Math. Phys. 29 (2017), no. 1, 1750002. Page references are to arXiv:1608.00253.arXiv
  9. M. Bischoff and C. Jones. Computing fusion rules for spherical GG-extensions of fusion categories. Selecta Math. (N.S.) 28 (2022), no. 2, Paper No. 26. Page references are to arXiv:1909.02816v2.arXiv
  10. M. Bischoff, Y. Kawahigashi and R. Longo. Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case. Doc. Math. 20 (2015), 1137--1184. Page references are to arXiv:1410.8848v2.arXiv
Show all 81 referencesShow fewer references
  1. M. Bischoff, Y. Kawahigashi, R. Longo and K.-H. Rehren. Tensor categories and endomorphisms of von Neumann algebras. With applications to quantum field theory. SpringerBriefs in Mathematical Physics 3, Springer, Cham, 2015. Page references are to arXiv:1407.4793v3.arXiv
  2. M. Bischoff, Y. Kawahigashi, R. Longo and K.-H. Rehren. Phase boundaries in algebraic conformal QFT. Comm. Math. Phys. 342 (2016), no. 1, 1--45.
  3. L. E. Bottini and S. Schäfer-Nameki. Construction of a gapless phase with Haagerup symmetry. Phys. Rev. Lett. 134 (2025), 191602. arXiv version: arXiv:2410.19040v4.arXiv
  4. N. Bourbaki. Groupes et algèbres de Lie, Chapitres IV, V et VI. Hermann, Paris, 1968.
  5. A. Brothier. On Jones' connections between subfactors, conformal field theory, Thompson's groups and knots. Celebratio Mathematica, Vaughan F. R. Jones volume, Mathematical Sciences Publishers, 2020. Page references are to arXiv:1912.07140v2.arXiv
  6. S. Carpi, T. Gaudio, L. Giorgetti and R. Hillier. Haploid algebras in C∗C^*-tensor categories and the Schellekens list. Comm. Math. Phys. 402 (2023), no. 1, 169--212. Page references are to arXiv:2211.12790v3.arXiv
  7. S. Carpi, Y. Kawahigashi, R. Longo and M. Weiner. From vertex operator algebras to conformal nets and back. Mem. Amer. Math. Soc. 254 (2018), no. 1213. Page and equation references are to arXiv:1503.01260v4.arXiv
  8. T. Creutzig, S. Kanade and R. McRae. Gluing vertex algebras. Adv. Math. 396 (2022), Paper No. 108174. Page references are to arXiv:1906.00119v2.arXiv
  9. T. Creutzig, S. Kanade and R. McRae. Tensor categories for vertex operator superalgebra extensions. Mem. Amer. Math. Soc. 295 (2024), no. 1472. Page references are to arXiv:1705.05017v2.arXiv
  10. D. H. Collingwood and W. M. McGovern. Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold, New York, 1993.
  11. C. Dong, H. Li and G. Mason. Modular-invariance of trace functions in orbifold theory and generalized Moonshine. Comm. Math. Phys. 214 (2000), no. 1, 1--56. Page references are to arXiv:q-alg/9703016.arXiv
  12. A. Davydov, M. Müger, D. Nikshych and V. Ostrik. The Witt group of non-degenerate braided fusion categories. J. Reine Angew. Math. 677 (2013), 135--177. Page references are to arXiv:1009.2117v2.arXiv
  13. C. Dong. Vertex algebras associated with even lattices. J. Algebra 161 (1993), no. 1, 245--265.
  14. C. Dong, X. Jiao and F. Xu. Quantum dimensions and quantum Galois theory. Trans. Amer. Math. Soc. 365 (2013), no. 12, 6441--6469. Page references are to arXiv:1201.2738v1.arXiv
  15. C. Dong and X. Lin. Unitary vertex operator algebras. J. Algebra 397 (2014), 252--277. Page references are to arXiv:1308.2361v1.arXiv
  16. C. Dong and G. Mason. Holomorphic vertex operator algebras of small central charge. Pacific J. Math. 213 (2004), no. 2, 253--266. Page references are to arXiv:math/0203005.arXiv
  17. C. Dong and F. Xu. Conformal nets associated with lattices and their orbifolds. Adv. Math. 206 (2006), no. 1, 279--306. Page references are to arXiv:math/0411499v2.arXiv
  18. P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik. Tensor categories. Mathematical Surveys and Monographs 205, American Mathematical Society, Providence, RI, 2015.
  19. P. Etingof, D. Nikshych and V. Ostrik. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581--642. Page references are to arXiv:math/0203060v11.arXiv
  20. P. Etingof, D. Nikshych and V. Ostrik. Fusion categories and homotopy theory. Quantum Topol. 1 (2010), no. 3, 209--273. Page references are to arXiv:0909.3140v2.arXiv
  21. P. Etingof, D. Nikshych and V. Ostrik. Weakly group-theoretical and solvable fusion categories. Adv. Math. 226 (2011), no. 1, 176--205. Page references are to arXiv:0809.3031v2.arXiv
  22. D. E. Evans and T. Gannon. The exoticness and realisability of twisted Haagerup--Izumi modular data. Comm. Math. Phys. 307 (2011), no. 2, 463--512. Page and equation references are to arXiv:1006.1326v1.arXiv
  23. I. B. Frenkel, Y.-Z. Huang and J. Lepowsky. On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104 (1993), no. 494.
  24. I. Frenkel, J. Lepowsky and A. Meurman. Vertex operator algebras and the Monster. Pure and Applied Mathematics 134, Academic Press, Boston, 1988.
  25. T. Gannon and B. C. Rayhaun. Hypergroup symmetry in relative quantum field theories and chiral algebras. Preprint, arXiv:2606.05279v1. Page references are to this version.arXiv
  26. T. Gannon, A. Schopieray and H. Yadav. On Haagerup--Izumi fusion categories. Preprint, arXiv:2609.25185.arXiv
  27. P. Grossman and N. Snyder. Quantum subgroups of the Haagerup fusion categories. Comm. Math. Phys. 311 (2012), no. 3, 617--643. Page references are to arXiv:1102.2631v2.arXiv
  28. B. Gui. Unitarity of the modular tensor categories associated to unitary vertex operator algebras, II. Comm. Math. Phys. 372 (2019), no. 3, 893--950. Page references are to arXiv:1712.04931v6.arXiv
  29. B. Gui. Categorical extensions of conformal nets. Comm. Math. Phys. 383 (2021), no. 2, 763--839. Page references are to arXiv:1812.04470v8.arXiv
  30. B. Gui. Q-systems and extensions of completely unitary vertex operator algebras. Int. Math. Res. Not. IMRN 2022, no. 10, 7550--7614. Page references are to arXiv:1908.03484v3.arXiv
  31. B. Gui. Geometric positivity of the fusion products of unitary vertex operator algebra modules. Comm. Math. Phys. 405 (2024), Paper No. 72. Page references are to arXiv:2306.11856v2.arXiv
  32. B. Gui. Comparison of extensions of unitary vertex operator algebras and conformal nets. Preprint, arXiv:2505.03235v3.arXiv
  33. B. Gui. Unbounded field operators in categorical extensions of conformal nets. Invent. Math., published online 2026. Page references are to arXiv:2001.03095v5.arXiv
  34. S. Helgason. Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics 80. Academic Press, New York--London, 1978.
  35. A. G. Henriques and J. E. Tener. Every conformal net has an associated unitary VOA. Preprint, arXiv:2507.20735v3.arXiv
  36. T.-C. Huang. Cyclic Haagerup--Izumi fusion categories at every odd order. Preprint, arXiv:2609.15986.arXiv
  37. T.-C. Huang. Haagerup conformal field theory at central charge eight. Preprint, arXiv:2610.14441v2. The results are announced, with an outline of proof.arXiv
  38. T.-C. Huang, Y.-H. Lin, K. Ohmori, Y. Tachikawa and M. Tezuka. Numerical evidence for a Haagerup conformal field theory. Phys. Rev. Lett. 128 (2022), 231603. arXiv:2110.03008.arXiv
  39. Y.-Z. Huang. Differential equations and intertwining operators. Commun. Contemp. Math. 7 (2005), no. 3, 375--400.
  40. Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, II: Logarithmic formal calculus and properties of logarithmic intertwining operators . arXiv:1012.4196v2 (https://arxiv.org/abs/1012.4196v2) (2012).arXiv
  41. Y.-Z. Huang. Rigidity and modularity of vertex tensor categories. Commun. Contemp. Math. 10 (2008), suppl. 1, 871--911.
  42. Y.-Z. Huang, A. Kirillov Jr. and J. Lepowsky. Braided tensor categories and extensions of vertex operator algebras. Comm. Math. Phys. 337 (2015), no. 3, 1143--1159. Page references are to arXiv:1406.3420v1.arXiv
  43. M. Izumi. The structure of sectors associated with Longo--Rehren inclusions. II. Examples. Rev. Math. Phys. 13 (2001), no. 5, 603--674.
  44. M. Izumi. The classification of 3n3^n subfactors and related fusion categories. Quantum Topol. 9 (2018), no. 3, 473--562. Page references are to arXiv:1609.07604v2.arXiv
  45. X. Jiao. An explicit lattice construction for the Haagerup center. Preprint, October 4, 2026, 54 pages, https://www.researchgate.net/publication/415561299_An_explicit_lattice_construction_for_the_Haagerup_center.link
  46. V. F. R. Jones. A no-go theorem for the continuum limit of a periodic quantum spin chain. Comm. Math. Phys. 357 (2018), no. 1, 295--317.
  47. V. F. R. Jones, S. Morrison and N. Snyder. The classification of subfactors of index at most 5. Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 2, 277--327. Page references are to arXiv:1304.6141.arXiv
  48. V. G. Kac. Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990.
  49. Y. Kawahigashi. Conformal field theory, vertex operator algebras and operator algebras. Proceedings of the International Congress of Mathematicians, Rio de Janeiro 2018. Vol. III, 2597--2616, World Sci. Publ., Hackensack, NJ, 2018. Page references are to arXiv:1711.11349v1.arXiv
  50. Y. Kawahigashi and R. Longo. Classification of local conformal nets. Case c<1c < 1. Ann. of Math. (2) 160 (2004), no. 2, 493--522. Page references are to arXiv:math-ph/0201015v3.arXiv
  51. Y. Kawahigashi, R. Longo and M. Müger. Multi-interval subfactors and modularity of representations in conformal field theory. Comm. Math. Phys. 219 (2001), no. 3, 631--669. Page references are to arXiv:math/9903104v2.arXiv
  52. A. Kirillov Jr. and V. Ostrik. On a qq-analogue of the McKay correspondence and the ADE classification of sl^2\widehat{\mathfrak{sl}}_2 conformal field theories. Adv. Math. 171 (2002), no. 2, 183--227. Page references are to arXiv:math/0101219v3.arXiv
  53. S. Köster. Local nature of coset models. Rev. Math. Phys. 16 (2004), no. 3, 353--382.
  54. H. K. Larson. Pseudo-unitary non-self-dual fusion categories of rank 4. J. Algebra 415 (2014), 184--213. Page references are to arXiv:1401.1879v2.arXiv
  55. H. Li. Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra 96 (1994), no. 3, 279--297.
  56. R. Longo. Conformal subnets and intermediate subfactors. Comm. Math. Phys. 237 (2003), no. 1-2, 7--30. Page references are to arXiv:math/0102196v2.arXiv
  57. R. Longo and K.-H. Rehren. Nets of subfactors. Rev. Math. Phys. 7 (1995), no. 4, 567--597. Page references are to arXiv:hep-th/9411077v2.arXiv
  58. R. Longo and F. Xu. Topological sectors and a dichotomy in conformal field theory. Comm. Math. Phys. 251 (2004), no. 2, 321--364.
  59. V. Morinelli, Y. Tanimoto and M. Weiner. Conformal covariance and the split property. Comm. Math. Phys. 357 (2018), no. 1, 379--406.
  60. M. Müger. On superselection theory of quantum fields in low dimensions. In XVIth International Congress on Mathematical Physics, 496--503. World Sci. Publ., Hackensack, NJ, 2010. Page references are to arXiv:0909.2537v1.arXiv
  61. S.-H. Ng and P. Schauenburg. Frobenius--Schur indicators and exponents of spherical categories. Adv. Math. 211 (2007), no. 1, 34--71. Page references are to arXiv:math/0601012v3.arXiv
  62. S.-H. Ng and P. Schauenburg. Higher Frobenius--Schur indicators for pivotal categories. In Hopf algebras and generalizations, 63--90. Contemp. Math. 441. Amer. Math. Soc., Providence, RI, 2007. Page references are to arXiv:math/0503167v2.arXiv
  63. V. Ostrik. Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8 (2003), no. 2, 177--206. Page references are to arXiv:math/0111139v1.arXiv
  64. V. Ostrik. Pivotal fusion categories of rank 3. Mosc. Math. J. 15 (2015), no. 2, 373--396. Page references are to arXiv:1309.4822v1.arXiv
  65. D. Reutter. Uniqueness of unitary structure for unitarizable fusion categories. Comm. Math. Phys. 397 (2023), no. 1, 37--52. Page references are to arXiv:1906.09710v2.arXiv
  66. A. Riesen. The Haagerup VOA at c=8c = 8. Research announcement, arXiv:2610.13933v1 (October 9, 2026).arXiv
  67. A. Riesen. The Haagerup VOA at central charge eight. Working manuscript haagerup_readable_proof.pdf in the package Haagerup_VOA_POST.zip , release snapshot-2026-10-09 , https://github.com/Doctorfm98/Haagerup_VOA/releases/tag/snapshot-2026-10-09. Section, page and statement numbers refer to this release.link
  68. J. E. Tener. Geometric realization of algebraic conformal field theories. Adv. Math. 349 (2019), 488--563. Page references are to arXiv:1611.01176v2.arXiv
  69. J. E. Tener. Representation theory in chiral conformal field theory: from fields to observables. Selecta Math. (N.S.) 25 (2019), no. 5, Paper No. 76. Page references are to arXiv:1810.08168v2.arXiv
  70. J. E. Tener. Fusion and positivity in chiral conformal field theory. Geom. Funct. Anal. 34 (2024), no. 4, 1226--1296. Page references are to arXiv:1910.08257v3.arXiv
  71. R. Vanhove, L. Lootens, M. Van Damme, R. Wolf, T. J. Osborne, J. Haegeman and F. Verstraete. Critical lattice model for a Haagerup conformal field theory. Phys. Rev. Lett. 128 (2022), 231602. arXiv:2110.03532.arXiv

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  1. v1 · 66 pagesSubmitted by Cédric ARHANCETInitial depositSupersededNov 02, 2026
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