Communications in Mathematical Physics

, Volume 265, Issue 1, pp 47–93 | Cite as

Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center

  • B.L. Feigin
  • A.M. Gainutdinov
  • A.M. Semikhatov
  • I.Yu. Tipunin
Article

Abstract

The SL(2, ℤ)-representation π on the center of the restricted quantum group Open image in new window at the primitive 2pth root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the Open image in new window ribbon element determines the decomposition of π into a ``pointwise'' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of Open image in new window at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of Open image in new window.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras. I. J. Amer. Math. Soc. 6, 905–947 (1993); II. J. Amer. Math. Soc. 6, 949–1011 (1993); III. J. Amer. Math. Soc. 7, 335–381 (1994); IV. J. Amer. Math. Soc. 7, 383–453 (1994)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Moore, G., Seiberg, N.: Lectures on RCFT. In: Physics, Geometry, and Topology (Trieste Spring School 1989), New York: Plenum, 1990, p. 263Google Scholar
  3. 3.
    Finkelberg, M.: An equivalence of fusion categories. Geometric and Functional Analysis (GAFA) 6, 249–267 (1996)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. Berlin–New York: Walter de Gruyter, 1994Google Scholar
  5. 5.
    Lyubashenko, V.: Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity. Commun. Math. Phys. 172, 467–516 (1995); Modular properties of ribbon abelian categories. In: Symposia Gaussiana, Proc. of the 2nd Gauss Symposium, Munich, 1993, Conf. A , Berlin-New York: Walter de Gruyter, 1995, pp. 529–579; Modular Transformations for Tensor Categories. J. Pure Applied Algebra 98, 279–327 (1995)CrossRefADSMATHMathSciNetGoogle Scholar
  6. 6.
    Lyubashenko, V., Majid, S.: Braided groups and quantum Fourier transform. J. Algebra 166, 506–528 (1994)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Kausch, H.G.: Extended conformal algebras generated by a multiplet of primary fields. Phys. Lett. B 259, 448 (1991)CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Gaberdiel, M.R., Kausch, H.G.: A rational logarithmic conformal field theory. Phys. Lett. B 386, 131 (1996)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Flohr, M.A.I.: On modular invariant partition functions of conformal field theories with logarithmic operators. Int. J. Mod. Phys. A11, 4147 (1996)Google Scholar
  10. 10.
    Flohr, M.: On Fusion Rules in Logarithmic Conformal Field Theories. Int. J. Mod. Phys. A12, 1943–1958 (1997)Google Scholar
  11. 11.
    Kerler, T.: Mapping class group action on quantum doubles. Commun. Math. Phys. 168, 353–388 (1995)CrossRefADSMATHMathSciNetGoogle Scholar
  12. 12.
    Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge: Cambridge University Press, 1994Google Scholar
  13. 13.
    Gaberdiel, M.R., Kausch, H.G.: Indecomposable fusion products. Nucl. Phys. B 477, 293 (1996)CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360 (1988)CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Fuchs, J., Hwang, S., Semikhatov, A.M., Tipunin, I.Yu.: Nonsemisimple fusion algebras and the Verlinde formula. Commun. Math. Phys. 247, 713–742 (2004)CrossRefADSMATHGoogle Scholar
  16. 16.
    Gurarie, V., Ludwig, A.W.W.: Conformal field theory at central charge c=0 and two-dimensional critical systems with quenched disorder. http://arxiv.org/list/hep-th/0409105, 2004
  17. 17.
    Feigin, B.L., Gainutdinov, A.M., Semikhatov, A.M., Tipunin, I.Yu.: Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT, math.QA/0512621Google Scholar
  18. 18.
    Reshetikhin, N.Yu., Turaev, V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys., 127, 1–26 (1990)CrossRefADSMATHMathSciNetGoogle Scholar
  19. 19.
    Lachowska, A.: On the center of the small quantum group. http://arxiv.org/list/math.QA/0107098, 2001
  20. 20.
    Ostrik, V.: Decomposition of the adjoint representation of the small quantum sl2. Commun. Math. Phys. 186, 253–264 (1997)CrossRefADSMATHMathSciNetGoogle Scholar
  21. 21.
    Gluschenkov, D.V., Lyakhovskaya, A.V.: Regular representation of the quantum Heisenberg double {Uq (sl(2)), Funq(SL(2))} (q is a root of unity). http://arxiv.org/list/hep-th/9311075, 1993
  22. 22.
    Jimbo, M., Miwa, T., Takeyama, Y.: Counting minimal form factors of the restricted sine-Gordon model http://arxiv.org/list/math-ph/0303059, 2003
  23. 23.
    Gaberdiel, M.R.: An algebraic approach to logarithmic conformal field theory. Int. J. Mod. Phys. A18, 4593–4638 (2003)Google Scholar
  24. 24.
    Flohr, M.: Bits and Pieces in Logarithmic Conformal Field Theory. Int. J. Mod. Phys. A18, 4497–4592 (2003)Google Scholar
  25. 25.
    Gurarie, V.; Logarithmic operators in conformal field theory. Nucl. Phys. B410, 535 (1993)Google Scholar
  26. 26.
    Rohsiepe, F.: Nichtunitäre Darstellungen der Virasoro-Algebra mit nichttrivialen Jordanblöcken. Diploma Thesis, Bonn, (1996) [BONN-IB-96-19]Google Scholar
  27. 27.
    Fjelstad, J., Fuchs, J., Hwang, S., Semikhatov, A.M., Tipunin, I.Yu.: Logarithmic conformal field theories via logarithmic deformations. Nucl. Phys. B633, 379 (2002)Google Scholar
  28. 28.
    Semikhatov, A.M., Taormina, A., Tipunin, I.Yu.: Higher-level Appell functions, modular transformations, and characters. http://arxiv.org/list/math.QA/0311314, 2003
  29. 29.
    Kač, V.G.: Infinite Dimensional Lie Algebras. Cambridge: Cambridge University Press, 1990Google Scholar
  30. 30.
    Fuchs, J.: Affine Lie algebras and quantum groups. Cambridge: Cambridge University Press, 1992Google Scholar
  31. 31.
    Reshetikhin, N.Yu., Semenov-Tian-Shansky, M.A.: Quantum R-matrices and factorization problems. J. Geom. Phys. 5, 533–550 (1988)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Bakalov, B., Kirillov, A.A.: Lectures on Tensor Categories and Modular Functors. Providence, RI: AMS, 2001Google Scholar
  33. 33.
    Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators I: Partition functions. Nucl. Phys. B 646, 353 (2002)CrossRefADSMATHGoogle Scholar
  34. 34.
    Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators II: Unoriented world sheets. Nucl. Phys. B678, 511–637 (2004)Google Scholar
  35. 35.
    Kerler, T., Lyubashenko, V.V.: Non-Semisimple Topological Quantum Field Theories for 3- Manifolds with Corners. Springer Lecture Notes in Mathematics 1765, Berlin-Heidelberg-New York: Springer Verlag, 2001Google Scholar
  36. 36.
    Larson, R.G., Sweedler, M.E.: An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91, 75–94 (1969)MATHMathSciNetGoogle Scholar
  37. 37.
    Radford, D.E.: The order of antipode of a finite-dimensional Hopf algebra is finite. Amer. J. Math 98, 333–335 (1976)MATHMathSciNetGoogle Scholar
  38. 38.
    Drinfeld, V.G.: On Almost Cocommutative Hopf Algebras. Leningrad Math. J. 1(2), 321–342 (1990)MathSciNetGoogle Scholar
  39. 39.
    Kassel, C.: Quantum Groups. New York: Springer-Verlag, 1995Google Scholar
  40. 40.
    Sweedler, M.E.: Hopf Algebras. New York: Benjamin, 1969Google Scholar
  41. 41.
    Radford, D.E.: The trace function and Hopf algebras. J. Alg. 163, 583–622 (1994)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Gantmakher, F.R.: Teoriya Matrits [in Russian]. Moscow: Nauka, 1988Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • B.L. Feigin
    • 1
  • A.M. Gainutdinov
    • 2
  • A.M. Semikhatov
    • 3
  • I.Yu. Tipunin
    • 3
  1. 1.Landau Institute for Theoretical PhysicsMoscowRussia
  2. 2.Physics DepartmentMoscow State UniversityMoscowRussia
  3. 3.Lebedev Physics InstituteMoscowRussia

Personalised recommendations