Skip to main content
SpringerLink
Log in

Congruence Subgroups and Generalized Frobenius-Schur Indicators

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category \({\mathcal {C}}\) , an equivariant indicator of an object in \({\mathcal {C}}\) is defined as a functional on the Grothendieck algebra of the quantum double \({Z(\mathcal {C})}\) via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholzer. In addition, we obtain two formulae for the generalized indicators, one of them a generalization of Bantay’s second indicator formula for a rational conformal field theory. This formula implies a conjecture of Pradisi-Sagnotti-Stanev, as well as a conjecture of Borisov-Halpern-Schweigert.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Altschüler D., Coste A.: Quasi-quantum groups, knots, three-manifolds, and topological field theory. Commun. Math. Phys. 150(1), 83–107 (1992)

    Article  ADS  Google Scholar 

  2. Bantay P.: The Frobenius-Schur indicator in conformal field theory. Phys. Lett. B 394(1–2), 87–88 (1997)

    MATH  MathSciNet  ADS  Google Scholar 

  3. Bantay P.: The kernel of the modular representation and the Galois action in RCFT. Commun. Math. Phys. 233(3), 423–438 (2003)

    MATH  MathSciNet  ADS  Google Scholar 

  4. Bauer M., Coste A., Itzykson C., Ruelle P.: Comments on the links between su(3) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards. J. Geom. Phys. 22(2), 134–189 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Borisov L., Halpern M.B., Schweigert C.: Systematic approach to cyclic orbifolds. Internat. J. Mod. Phys. A 13(1), 125–168 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Bakalov, B., Kirillov, A. Jr.: Lectures on tensor categories and modular functors. University Lecture Series, Vol. 21, Providence, RI: Amer. Math. Soc., 2001

  7. Brenner J.L.: The linear homogeneous group. III. Ann. of Math. 71(2), 210–223 (1960)

    Article  MathSciNet  Google Scholar 

  8. Cardy J.L.: Operator content of two-dimensional conformally invariant theories. Nucl. Phys. B 270(2), 186–204 (1986)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Coste A., Gannon T.: Remarks on Galois symmetry in rational conformal field theories. Phys. Lett. B 323(3–4), 316–321 (1994)

    MathSciNet  ADS  Google Scholar 

  10. Coste, A., Gannon, T.: Congruence Subgroups and Rational Conformal Field Theory. math.QA/9909080

  11. de Boer J., Goeree J.: Markov traces and II1 factors in conformal field theory. Commun. Math. Phys. 139(2), 267–304 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Dong, C., Mason, G.: Vertex operator algebras and Moonshine: a survey. In: Progress in algebraic combinatorics (Fukuoka, 1993), Adv. Stud. Pure Math., Vol. 24, Tokyo: Math. Soc. Japan, 1996, pp. 101–136

  13. Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi-Hopf algebras, group cohomology and orbifold models. Integrable systems and quantum groups (Pavia, 1990), River Edge, NJ: World Sci. Publishing, 1992, pp. 75–98

  14. Eholzer W.: On the classification of modular fusion algebras. Commun. Math. Phys. 172(3), 623–659 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Etingof P., Nikshych D., Ostrik V.: On fusion categories. Ann. of Math. (2) 162(2), 581–642 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Eholzer W., Skoruppa N.-P.: Modular invariance and uniqueness of conformal characters. Commun. Math. Phys. 174(1), 117–136 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Fuchs J., Ganchev A.Ch., Szlachányi K., Vecsernyés P.: S4 symmetry of 6 j symbols and Frobenius-Schur indicators in rigid monoidal C* categories. J. Math. Phys. 40(1), 408–426 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Fuchs, J., Schweigert, C.: Category theory for conformal boundary conditions. In: Vertex operator algebras in mathematics and physics (Toronto, ON, 2000), Fields Inst. Commun., Vol. 39, Providence, RI: Amer. Math. Soc., 2003, pp. 25–70

  19. Gannon T.: Integers in the open string. Phys. Lett. B 473(1–2), 80–85 (2000)

    MATH  MathSciNet  ADS  Google Scholar 

  20. Gelfand S., Kazhdan D.: Invariants of three-dimensional manifolds. Geom. Funct. Anal. 6(2), 268–300 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Goff C., Mason G., Ng S.-H.: On the gauge equivalence of twisted quantum doubles of elementary abelian and extra-special 2-groups. J. Algebra 312(2), 849–875 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Huang Y.-Z.: Vertex operator algebras, the Verlinde conjecture, and modular tensor categories. Proc. Natl. Acad. Sci. USA 102(15), 5352–5356 (2005) (electronic)

    Article  MathSciNet  ADS  Google Scholar 

  23. Kassel C.: Quantum groups. Springer-Verlag, New York (1995)

    MATH  Google Scholar 

  24. Kurth C.A., Long L.: On modular forms for some noncongruence arithmetic subgroups. J. Number Theory 128(7), 1989–2009 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kac V.G., Longo R., Xu F.: Solitons in affine and permutation orbifolds. Commun. Math. Phys. 253(3), 723–764 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Kashina, Y., Sommerhäuser, Y., Zhu, Y.: On higher Frobenius-Schur indicators. Mem. Amer. Math. Soc. 181, no. 855, viii+65. (2006)

  27. Lepowsky J.: From the representation theory of vertex operator algebras to modular tensor categories in conformal field theory. Proc. Natl. Acad. Sci. USA 102(15), 5304–5305 (2005) (electronic)

    Article  MathSciNet  ADS  Google Scholar 

  28. Linchenko V., Montgomery S.: A Frobenius-Schur theorem for Hopf algebras. Algebr. Represent. Theory 3(4), 347–355 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Larson R.G., Radford D.E.: Semisimple cosemisimple Hopf algebras. Amer. J. Math. 109(1), 187–195 (1987)

    MathSciNet  Google Scholar 

  30. Larson R.G., Radford D.E.: Finite-dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple. J. Algebra 117(2), 267–289 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mason G., Ng S.-H.: Group cohomology and gauge equivalence of some twisted quantum doubles. Trans. Amer. Math. Soc. 353(9), 3465–3509 (2001) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  32. Mason G., Ng S.-H.: Central invariants and Frobenius-Schur indicators for semisimple quasi-Hopf algebras. Adv. Math. 190(1), 161–195 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  33. Majid S.: Quantum double for quasi-Hopf algebras. Lett. Math. Phys. 45(1), 1–9 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  34. Montgomery, S.: Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, Vol. 82, Washington, DC: Conference Board of the Mathematical Sciences, 1993

  35. Moore G.: Atkin-Lehner symmetry. Nucl. Phys. B 293(1), 139–188 (1987)

    Article  ADS  Google Scholar 

  36. Moore G., Seiberg N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123(2), 177–254 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. Moore, G., Seiberg, N.: Lectures on RCFT. In: Physics, geometry, and topology (Banff, AB, 1989), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 238, New York: Plenum, 1990, pp. 263–361

  38. Müger M.: From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180(1–2), 159–219 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Ng S.-H., Schauenburg P.: Frobenius-Schur indicators and exponents of spherical categories. Adv. Math. 211(1), 34–71 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  40. Ng, S.-H., Schauenburg, P.: Higher Frobenius-Schur Indicators for Pivotal Categories. In: Hopf Algebras and Generalizations. Contemp. Math., Vol. 441, Providence, RI: Amer. Math. Soc., 2007, pp. 63–90

  41. Ng S.-H., Schauenburg P.: Central invariants and higher indicators for semisimple quasi-Hopf algebras. Trans. Amer. Math. Soc. 360(4), 1839–1860 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  42. Nikshych, D., Turaev, V., Vainerman, L.: Invariants of knots and 3-manifolds from quantum groupoids. In: Proceedings of the Pacific Institute for the Mathematical Sciences Workshop “Invariants of Three-Manifolds” (Calgary, AB, 1999), Vol. 127, 2003, pp. 91–123.

  43. Pradisi G., Sagnotti A., Stanev Ya.S.: Planar duality in SU(2) WZW models. Phys. Lett. B 354(3–4), 279–286 (1995)

    MathSciNet  ADS  Google Scholar 

  44. Radford D.E.: The trace function and Hopf algebras. J. Algebra 163(3), 583–622 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  45. Rowell E., Stong R., Wang Z.: On Classification of Modular Tensor Categories. Commun. Math. Phys. 292(2), 343–389 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  46. Schauenburg P.: On the Frobenius-Schur indicators for quasi-Hopf algebras. J. Alg. 282(1), 129–139 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  47. Sommerhäuser, Y., Zhu, Y.: Hopf algebras and congruence subgroups. http://arxiv.org/abs/0710.0705v2[math.RA], 2008

  48. Szlachányi, K.: Finite quantum groupoids and inclusions of finite type. In: Mathematical physics in mathematics and physics (Siena, 2000), Fields Inst. Commun., Vol. 30, Providence, RI: Amer. Math. Soc., 2001, pp. 393–407

  49. Vafa C.: Toward classification of conformal theories. Phys. Lett. B 206(3), 421–426 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  50. Verlinde E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300(3), 360–376 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  51. Xu F.: Some computations in the cyclic permutations of completely rational nets. Commun. Math. Phys. 267(3), 757–782 (2006)

    Article  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Siu-Hung Ng

  2. Mathematisches Institut der Universität München, Theresienstr. 39, 80333, München, Germany

    Peter Schauenburg

Authors
  1. Siu-Hung Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Schauenburg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siu-Hung Ng.

Additional information

Communicated by Y. Kawahigashi

The first author is supported by the NSA grant H98230-08-1-0078.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ng, SH., Schauenburg, P. Congruence Subgroups and Generalized Frobenius-Schur Indicators. Commun. Math. Phys. 300, 1–46 (2010). https://doi.org/10.1007/s00220-010-1096-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-010-1096-6

Keywords

Search

Navigation