Communications in Mathematical Physics

, Volume 356, Issue 3, pp 981–1015 | Cite as

Modular Data for the Extended Haagerup Subfactor

  • Terry Gannon
  • Scott Morrison


We compute the modular data (that is, the S and T matrices) for the centre of the extended Haagerup subfactor [BMPS12]. The full structure (i.e., the associativity data, also known as 6-j symbols or F matrices) still appears to be inaccessible. Nevertheless, starting with just the number of simple objects and their dimensions (obtained by a combinatorial argument in [MW14]) we find that it is surprisingly easy to leverage knowledge of the representation theory of \({SL (2, \mathbb{Z})}\) into a complete description of the modular data. We also investigate the possible character vectors associated with this modular data. This is the published version of arXiv:1606.07165.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AH99.
    Asaeda M., Haagerup U.: Exotic subfactors of finite depth with Jones indices \({(5+\sqrt{13})/2}\) and \({(5+\sqrt{17})/2}\) . Commun. Math. Phys. 202(1), 1–63 (1999). doi: 10.1007/s002200050574 arXiv:math.OA/9803044 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. ASD71.
    Atkin, A.O.L., Swinnerton-Dyer, H.P.F.: Modular forms on noncongruence subgroups. In: Combinatorics (Proceedings of Symposia in Pure Mathematics, vol. 19. Univ. California, Los Angeles, Calif., 1968), pp. 1–25. American Mathematical Society, Providence (1971)Google Scholar
  3. Ban03.
    Bantay P.: The kernel of the modular representation and the Galois action in RCFT. Commun. Math. Phys. 233(3), 423–438 (2003). doi: 10.1007/s00220-002-0760-x arXiv:math/0102149 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. BG07.
    Bantay P., Gannon T.: Vector-valued modular functions for the modular group and the hypergeometric equation. Commun. Number Theory Phys. 1(4), 651–680 (2007). doi: 10.4310/CNTP.2007.v1.n4.a2 arXiv:0705.2467 CrossRefzbMATHMathSciNetGoogle Scholar
  5. BMPS12.
    Bigelow S., Morrison S., Peters E., Snyder N.: Constructing the extended Haagerup planar algebra. Acta Math. 209(1), 29–82 (2012). doi: 10.1007/s11511-012-0081-7 arXiv:0909.4099 CrossRefzbMATHMathSciNetGoogle Scholar
  6. BNRW13.
    Bruillard, P., Ng, S.-H., Rowell, E.C., Wang, Z.: Rank-finiteness for modular categories. J. Am. Math. Soc. (2013). doi: 10.1090/jams/842. arXiv:1310.7050
  7. BNRW15.
    Bruillard, P., Ng, S.-H., Rowell, E.C., Wang, Z.: On classification of modular categories by rank (2015). arXiv:1507.05139
  8. CG99.
    Coste, A., Gannon, T.: Congruence subgroups and rational conformal field theory (1999). arXiv:math/9909080
  9. CS99.
    Conway J.H., Sloane N.J.A.: Sphere Packings, Lattices and Groups. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  10. DLN12.
    Dong C., Lin X., Ng S.-H.: Congruence property in conformal field theory. Algebra Number Theory 9(9), 2121–2166 (2015) arXiv:1201.6644 CrossRefzbMATHMathSciNetGoogle Scholar
  11. EG11.
    Evans D.E., Gannon T.: The exoticness and realisability of twisted Haagerup–Izumi modular data. Commun. Math. Phys. 307(2), 463–512 (2011). doi: 10.1007/s00220-011-1329-3 arXiv:1006.1326 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. EG14.
    Evans D.E., Gannon T.: Near-group fusion categories and their doubles. Adv. Math. 255, 586–640 (2014). doi: 10.1016/j.aim.2013.12.014 arXiv:1208.1500 CrossRefzbMATHMathSciNetGoogle Scholar
  13. Gan12.
    Gannon, T.: Much ado about Mathieu. Adv. Math. 301, 322–358 (2016). arXiv:1211.5531
  14. Gan14.
    Gannon T.: The theory of vector-valued modular forms for the modular group. In: Kohnen, W., Weissauer, R. (eds.) Conformal Field Theory, Automorphic Forms and Related Topics, pp. 247–286. Springer, Berlin (2014). arXiv:1310.4458
  15. GIS15.
    Grossman, P., Izumi, M., Snyder, N.: The Asaeda–Haagerup fusion categories (2015). arXiv:1501.07324
  16. Haa94.
    Haagerup, U.: Principal graphs of subfactors in the index range \({4 < [M : N] < 3 + \sqrt2}\). In: Subfactors (Kyuzeso, 1993), pp. 1–38. World Scientific Publishing, River Edge (1994)Google Scholar
  17. Hua05.
    Huang, Y.-Z.: Vertex operator algebras, the Verlinde conjecture, and modular tensor categories. Proc. Natl. Acad. Sci. USA, 102(15), 5352–5356 (2005). doi: 10.1073/pnas.0409901102. arXiv:math/0412261. (electronic)
  18. Izu01.
    Izumi M.: The structure of sectors associated with Longo–Rehren inclusions. II. Examples. Rev. Math. Phys. 13(5), 603–674 (2001). doi: 10.1142/S0129055X01000818 CrossRefzbMATHMathSciNetGoogle Scholar
  19. Izu15.
    Izumi, M.: A Cuntz algebra approach to the classification of near-group categories. In: Proceeding of the 2014 Mavi and 2015 Qinhuvangdao Conferences in Honour of Vaughan F.R. Jones’ 60th Birthday, pp. 222–343. Center for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra Aus (2017). arXiv:1512.04288
  20. LL04.
    Lepowsky, J., Li, H.: Introduction to Vertex Operator Algebras and Their Representations, Volume 227 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston (2004). doi: 10.1007/978-0-8176-8186-9
  21. MS12.
    Morrison S., Snyder N.: Non-cyclotomic fusion categories. Trans. Am. Math. Soc. 364(9), 4713–4733 (2012). doi: 10.1090/S0002-9947-2012-05498-5 arXiv:1002.0168 CrossRefzbMATHMathSciNetGoogle Scholar
  22. Mug00.
    Müger M.: Galois theory for braided tensor categories and the modular closure. Adv. Math. 150(2), 151–201 (2000). doi: 10.1006/aima.1999.1860 arXiv:math.CT/9812040 CrossRefzbMATHMathSciNetGoogle Scholar
  23. MW14.
    Morrison S., Walker K.: The centre of the extended Haagerup subfactor has 22 simple objects. Int. J. Math. 28, 1750009 (2017) arXiv:1404.3955 CrossRefzbMATHGoogle Scholar
  24. NS10.
    Ng S.-H., Schauenburg P.: Congruence subgroups and generalized Frobenius–Schur indicators. Commun. Math. Phys. 300(1), 1–46 (2010). doi: 10.1007/s00220-010-1096-6 arXiv:0806.2493 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. Ost03.
    Ostrik V.: Fusion categories of rank 2. Math. Res. Lett. 10(2–3), 177–183 (2003) arXiv:math.QA/0203255 CrossRefzbMATHMathSciNetGoogle Scholar
  26. Ost13.
    Ostrik, V.: Pivotal fusion categories of rank 3. (with an Appendix written jointly with Dmitri Nikshych) (2013). arXiv:1309.4822
  27. Zhu96.
    Zhu Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9(1), 237–302 (1996). doi: 10.1090/S0894-0347-96-00182-8 CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.University of AlbertaCalgaryCanada
  2. 2.Australian National UniversityCanberraAustralia

Personalised recommendations