Communications in Mathematical Physics

, Volume 320, Issue 3, pp 761–781 | Cite as

On Examples of Intermediate Subfactors from Conformal Field Theory



Motivated by our subfactor generalization of Wall’s conjecture, in this paper we determine all intermediate subfactors for conformal subnets corresponding to four infinite series of conformal inclusions, and as a consequence we verify that these series of subfactors verify our conjecture. Our results can be stated in the framework of Vertex Operator Algebras. We also verify our conjecture for Jones-Wassermann subfactors from representations of Loop groups extending our earlier results.


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  1. 1.
    Altschüler D., Bauer M., Itzykson C.: The branching rules of conformal embeddings. Commun. Math. Phys. 132, 349–364 (1990)ADSCrossRefGoogle Scholar
  2. 2.
    Böckenhauer J., Evans D.E.: Modular invariants, graphs and α-induction for nets of subfactors. I. Commun. Math. Phys. 197, 361–386 (1998)ADSMATHCrossRefGoogle Scholar
  3. 3.
    Böckenhauer J., Evans D.E., Kawahigashi Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210, 733–784 (2000)ADSMATHCrossRefGoogle Scholar
  4. 4.
    Bisch D., Jones V.F.R.: Algebras associated to intermediate subfactors. Invent. Math. 128(1), 89–157 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Dong, C.: Introduction to vertex operator algebras I. Sūrikaisekikenkyūsho Kōkyūroku, No. 904 (1995), pp. 1–25. Also see, 1995
  6. 6.
    Dong, C., Lepowsky, J.: Generalized vertex algebras and relative vertex operators. Progress in Mathematics, 112, Basel-Bosten: Birkhäuser, 1993Google Scholar
  7. 7.
    Fröhlich J., Gabbiani F.: Operator algebras and Conformal field theory. Commun. Math. Phys. 155, 569–640 (1993)ADSMATHCrossRefGoogle Scholar
  8. 8.
    Frenkel, I.B., Lepowsky, J., Ries, J.: Vertex operator algebras and the Monster. New York: Academic, 1988Google Scholar
  9. 9.
    Frenkel I., Zhu Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66(1), 123–168 (1992)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gannon T., Ruelle P., Walton M.A.: Automorphism modular invariants of current algebras. Commun. Math. Phys. 179(1), 121–156 (1996)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Goodman F., Wenzl H.: Littlewood-Richardson coefficients for Hecke algebras at roots of unity. Adv. Math. 82(2), 244–265 (1990)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Grossman P., Jones V.F.R.: Intermediate subfactors with no extra structure. J. Amer. Math. Soc. 20(1), 219–265 (2007)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Guido D., Longo R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11–35 (1996)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Guralnick R., Xu F.: On a subfactor generalization of Wall’s conjecture. J. Algebra 332, 457–468 (2011)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Jones, V.F.R.: Fusion en algFbres de von Neumann et groupes de lacets (d’aprFs A. Wassermann). (French) [Fusion in von Neumann algebras and loop groups (after A. Wassermann)] Seminaire Bourbaki, Vol. 1994/95. Asterisque No. 237 (1996), Exp. No. 800, 5, 251–273Google Scholar
  16. 16.
    Kac V.G.: Vertex algebras for beginners. Providence, RI: Amer. Math. Soc., 1997Google Scholar
  17. 17.
    Kac, V.G.: “Infinite Dimensional Lie Algebras”, 3rd Edition, Cambridge: Cambridge University Press, 1990Google Scholar
  18. 18.
    Kac V.G., Longo R., Xu F.: Solitons in affine and permutation orbifolds. Commun. Math. Phys. 253(3), 723–764 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    Tsuchiya A., Kanie Y.: Vertex Operators in conformal field theory on P 1 and monodromy representations of braid group. Adv. Studies in Pure Math. 16(88), 297–372 (1988)MathSciNetGoogle Scholar
  20. 20.
    Kac V.G., Wakimoto M.: Modular and conformal invariance constraints in representation theory of affine algebras. Adv. in Math. 70, 156–234 (1988)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Levstein F., Liberati J.I.: Branching rules for conformal embeddings. Commun. Math. Phys. 173, 1–16 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Longo R.: Conformal subnets and intermediate subfactors. Commun. Math. Phys. 237(1–2), 7–30 (2003)MathSciNetADSMATHGoogle Scholar
  23. 23.
    Longo R., Rehren K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Goddard P., Nahm W., Olive D.: Symmetric spaces, Sugawara’s energy momentum tensor in two dimensions and free fermions. Phys. Lett. B 160(1–3), 111–116 (1985)MathSciNetADSMATHGoogle Scholar
  25. 25.
    Izumi M., Longo R., Popa S.: A Galois correspondence for compact groups of automorphisms of von Neumann Algebras with a generalization to Kac algebras. J. Funct. Anal. 155, 25–63 (1998)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Jones V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Jones V.F.R., Xu F.: Intersections of finite families of finite index subfactors. Internat. J. Math. 15(7), 717–733 (2004)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Liebeck M.W., Pyber L., Shalev A.: On a conjecture of G. E. Wall. J. Algebra 317(1), 184–197 (2007)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Pimsner M., Popa S.: Entropy and index for subfactors. Ann. Scient. Ec. Norm. Sup. 19, 57–106 (1986)MathSciNetMATHGoogle Scholar
  30. 30.
    Pressley, A., Segal, G.: “Loop Groups”. Oxford: Oxford University Press, 1986Google Scholar
  31. 31.
    Wall, G.E.: Some applications of the Eulerian functions of a finite group. J. Austral. Math. Soc. 2, 35–59 (1961/1962)Google Scholar
  32. 32.
    Wassermann A.: Operator algebras and Conformal field theories III. Invent. Math. 133, 467–538 (1998)MathSciNetADSMATHCrossRefGoogle Scholar
  33. 33.
    Xu F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys. 192, 347–403 (1998)ADSCrossRefGoogle Scholar
  34. 34.
    Xu F.: Algebraic coset conformal field theories. Commun. Math. Phys. 211(1), 1–43 (2000)ADSMATHCrossRefGoogle Scholar
  35. 35.
    Xu F.: Mirror extensions of local nets. Commun. Math. Phys. 270(3), 835–847 (2007)ADSMATHCrossRefGoogle Scholar
  36. 36.
    Xu F.: An application of mirror extensions. Commun. Math. Phys. 290(1), 83–103 (2009)ADSMATHCrossRefGoogle Scholar
  37. 37.
    Xu F.: On intermediate subfactors of Goodman-de la Harpe-Jones subfactors. Commun. Math. Phys. 298(3), 707–739 (2010)ADSMATHCrossRefGoogle Scholar
  38. 38.
    Xu F.: Some computations in the cyclic permutations of completely rational nets. Commun. Math. Phys. 267(3), 757–782 (2006)ADSMATHCrossRefGoogle Scholar
  39. 39.
    Xu, F.: On representing some lattices as lattices of intermediate subfactors of finite index. Adv. Math. 220(5), 1317–1356; corrections in the proof of Cor. 5.23 in [math.OA], 2009

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at RiversideRiversideUSA

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