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The Haagerup center at central charge eight via conformal nets

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

Contributed by Cédric ARHANCET ↗

Version 1 / Nov 06, 2026 / CC BY-SA 4.0

Abstract

Let S⊂VE6⊕A2S\subset V_{E_6\oplus A_2} be the unitary vertex operator algebra of central charge eight with currents at levels (3,13)(3,13) constructed in our previous work. An exact presentation of a full corner of its Zhu algebra gives strong rationality, the twelve simple modules and the lattice branching rules. Using these results, we identify the representation category of its conformal net with the Haagerup center Z(H3)Z(\mathcal{H}_3). We construct a holomorphic diagonal extension of AS⊗AS\mathcal {A}_S\otimes \mathcal{A}_S and apply the vacuum trace class theorem of Henriques and Tener to prove trace class heat operators in every sector of AS\mathcal{A}_S. Reconstruction gives a unitary holomorphic vertex operator algebra containing S⊗SS\otimes S as mutual commutants. The coset theorems of Gui then give the complete unitarity of SS and a braided equivalence Rep(S)≅Z(H3)\mathrm{Rep}(S) \cong Z(\mathcal{H}_3), so that SS is a Haagerup algebra. Using results of Dong, Jiao and Xu, we also identify the Huang--Riesen algebra with the twin of SS and describe their common fixed point algebra using the intersection of the two binary lattices of discriminant −39-39.

Author comments: 73 pages

Provenance statement

The author acknowledges the use of AI tools for language polishing, LaTeX editing, exploratory mathematical discussions and feedback, for drafting parts of the arguments, for reading and checking the statements of the cited literature, and for writing and running the computer programs of Section \ref{sec:certificates}. 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. Haagerup symmetries of the E8E_8 theory and conformal nets at central charge eight. Preprint, 2026, arXiv:2611.03560 (https://arxiv.org/abs/2611.03560).arXiv
  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. Baillet, Y. Denizeau and J.-F. Havet. Indice d'une espérance conditionnelle. Compos. Math. 66 (1988), no. 2, 199--236. Numdam (https://www.numdam.org/item/CM_1988__66_2_199_0/).link
  7. G. M. Bergman. The diamond lemma for ring theory. Adv. in Math. 29 (1978), no. 2, 178--218.
  8. 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
  9. M. Bischoff. Generalized orbifold construction for conformal nets. Rev. Math. Phys. 29 (2017), no. 1, 1750002. Page references are to arXiv:1608.00253.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 82 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. G. Buhl. A spanning set for VOA modules. J. Algebra 254 (2002), no. 1, 125--151.
  3. S. Carnahan and M. Miyamoto. Regularity of fixed-point vertex operator subalgebras. Preprint, arXiv:1603.05645v4 (2018).arXiv
  4. S. Carpi. On the representation theory of Virasoro nets. Comm. Math. Phys. 244 (2004), no. 2, 261--284. Numbering of arXiv:math/0306425.arXiv
  5. 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
  6. T. Creutzig, S. Kanade, A. R. Linshaw and D. Ridout. Schur--Weyl duality for Heisenberg cosets. Transform. Groups 24 (2019), no. 2, 301--354.
  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. Statement numbers refer 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, 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
  12. C. Dong, X. Jiao and F. Xu. On the Haagerup vertex operator algebras. Preprint, arXiv:2611.04778v1 (2026).arXiv
  13. C. Dong, X. Lin and S.-H. Ng. Congruence property in conformal field theory. Algebra Number Theory 9 (2015), no. 9, 2121--2166.
  14. 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
  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 J. Lepowsky. Generalized vertex algebras and relative vertex operators. Progress in Mathematics 112, Birkhäuser, Boston, 1993.
  17. C. Dong and G. Mason. Rational vertex operator algebras and the effective central charge. Int. Math. Res. Not. 2004, no. 56, 2989--3008. Page references are to arXiv:math/0201318.arXiv
  18. 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
  19. C. Dong and G. Mason. On quantum Galois theory. Duke Math. J. 86 (1997), no. 2, 305--321. Numbering of arXiv:hep-th/9412037.arXiv
  20. C. Dong. Vertex algebras associated with even lattices. J. Algebra 161 (1993), no. 1, 245--265.
  21. 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
  22. P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik. Tensor categories. Mathematical Surveys and Monographs 205, American Mathematical Society, Providence, RI, 2015.
  23. 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
  24. 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
  25. 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
  26. 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
  27. D. E. Evans and T. Gannon. Non-unitary fusion categories and their doubles via endomorphisms. Adv. Math. 310 (2017), 1--43.
  28. 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.
  29. 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
  30. 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
  31. B. Gui. Categorical extensions of conformal nets. Comm. Math. Phys. 383 (2021), no. 2, 763--839. Page references are to arXiv:1812.04470v8.arXiv
  32. 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
  33. B. Gui. Comparison of extensions of unitary vertex operator algebras and conformal nets. Preprint, arXiv:2505.03235v3.arXiv
  34. B. Gui. Unbounded field operators in categorical extensions of conformal nets. Invent. Math., published online 2026. Page references are to arXiv:2001.03095v5.arXiv
  35. D. Guido and R. Longo. The conformal spin and statistics theorem. Comm. Math. Phys. 181 (1996), no. 1, 11--35.
  36. U. Haagerup. Principal graphs of subfactors in the index range 4<[M:N]<3+24 < [M:N] < 3+\sqrt2. In: Subfactors (Kyuzeso, 1993), World Sci. Publ., River Edge, NJ, 1994, 1--38.
  37. S. Helgason. Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics 80. Academic Press, New York--London, 1978.
  38. A. Henriques. Loop groups and diffeomorphism groups of the circle as colimits. Comm. Math. Phys. 366 (2019), no. 2, 537--565.
  39. B. Gui. On a Connes fusion approach to finite index extensions of conformal nets. Preprint, arXiv:2112.15396v2, 2025.arXiv
  40. A. G. Henriques and J. E. Tener. Every conformal net has an associated unitary VOA. Preprint, arXiv:2507.20735v3.arXiv
  41. A. G. Henriques and J. E. Tener. A genus zero functorial CFT associated to the vacuum sector of a conformal net. Preprint, arXiv:2608.24437v1, 2026.arXiv
  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. Y.-Z. Huang. Differential equations and intertwining operators. Commun. Contemp. Math. 7 (2005), no. 3, 375--400.
  44. Y.-Z. Huang. Rigidity and modularity of vertex tensor categories. Commun. Contemp. Math. 10 (2008), suppl. 1, 871--911.
  45. Y.-Z. Huang. Vertex operator algebras and the Verlinde conjecture. Commun. Contemp. Math. 10 (2008), no. 1, 103--154.
  46. 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
  47. Y.-Z. Huang and J. Lepowsky. A theory of tensor products for module categories for a vertex operator algebra, I, II. Selecta Math. (N.S.) 1 (1995), no. 4, 699--756, 757--786.
  48. M. Izumi. The structure of sectors associated with Longo--Rehren inclusions. II. Examples. Rev. Math. Phys. 13 (2001), no. 5, 603--674.
  49. 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
  50. 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
  51. 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
  52. V. G. Kac. Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990.
  53. Y. Kawahigashi and R. Longo. Noncommutative spectral invariants and black hole entropy. Comm. Math. Phys. 257 (2005), no. 1, 193--225.
  54. 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
  55. 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
  56. H. Li. Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra 96 (1994), no. 3, 279--297.
  57. 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
  58. 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
  59. G. Mason. Lattice subalgebras of strongly regular vertex operator algebras. In: Conformal field theory, automorphic forms and related topics, Contrib. Math. Comput. Sci. 8, Springer, Heidelberg, 2014, 31--53. Page references are to arXiv:1110.0544.arXiv
  60. R. McRae. On rationality for C2C_2-cofinite vertex operator algebras. Preprint, arXiv:2108.01898v3.arXiv
  61. R. McRae. On semisimplicity of module categories for finite non-zero index vertex operator subalgebras. Lett. Math. Phys. 112 (2022), Paper No. 25.
  62. S. Möller and B. C. Rayhaun. Equivalence relations on vertex operator algebras, I: Genus. Preprint, arXiv:2408.07117v1 (2024).arXiv
  63. 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
  64. 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
  65. A. Riesen. The Haagerup VOA at c=8c = 8. Research announcement, arXiv:2610.13933v1 (October 9, 2026).arXiv
  66. A. Riesen. The Haagerup VOA at central charge eight. Working manuscript haagerup_readable_proof.pdf , 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
  67. C. Raymond, Y. Tanimoto and J. E. Tener. Unitary vertex algebras and Wightman conformal field theories. Comm. Math. Phys. 395 (2022), no. 1, 299--330. Page references are to arXiv:2203.10795v1.arXiv
  68. 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
  69. 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
  70. J. H. M. Wedderburn. A theorem on finite algebras. Trans. Amer. Math. Soc. 6 (1905), no. 3, 349--352.
  71. M. Weiner. Conformal covariance and positivity of energy in charged sectors. Comm. Math. Phys. 265 (2006), no. 2, 493--506.
  72. Y. Zhu. Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9 (1996), no. 1, 237--302.

Version history

  1. v1 · 73 pagesSubmitted by Cédric ARHANCET73 pagesCurrentNov 06, 2026