Communications in Mathematical Physics

, Volume 247, Issue 3, pp 713–742 | Cite as

Nonsemisimple Fusion Algebras and the Verlinde Formula

  • J. Fuchs
  • S. Hwang
  • A.M. Semikhatov
  • I.Yu. Tipunin


We find a nonsemisimple fusion algebra Open image in new window associated with each (1, p) Virasoro model. We present a nonsemisimple generalization of the Verlinde formula which allows us to derive Open image in new window from modular transformations of characters.


Verlinde Formula Fusion Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360 (1988)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Kawai, T.: On the structure of fusion algebras. Phys. Lett. B 217, 247 (1989)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Caselle, M., Ponzano, G., Ravanini, F.: Towards a classification of fusion rule algebras in rational conformal field theories. Int. J. Mod. Phys. B 6, 2075 (1992) [hep-th/9111027]MathSciNetzbMATHGoogle Scholar
  4. 4.
    Fuchs, J.: Fusion rules in conformal field theory. Fortschr. Phys. 42(1), (1994) [hep-th/9306162]Google Scholar
  5. 5.
    Eholzer, W.: On the classification of modular fusion algebras. Commun. Math. Phys. 172, 623 (1995) [hep-th/9408160]MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bannai, E., Ito, T.: Algebraic Combinatorics I: Association Schemes. New York: Benjamin–Cummings, 1984Google Scholar
  7. 7.
    Gurarie, V.: Logarithmic operators in conformal field theory. Nucl. Phys. B 410, 535 (1993) [hep-th/9303160]CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Gaberdiel, M.R., Kausch, H.G.: Indecomposable fusion products. Nucl. Phys. B 477, 293 (1996) [hep-th/9604026]CrossRefMathSciNetGoogle Scholar
  9. 9.
    Gaberdiel, M.R., Kausch, H.G.: A rational logarithmic conformal field theory. Phys. Lett. B 386, 131 (1996) [hep-th/9606050]CrossRefMathSciNetGoogle Scholar
  10. 10.
    Rohsiepe, F.: Nichtunitäre Darstellungen der Virasoro-Algebra mit nichttrivialen Jordanblöcken. Diploma Thesis, Bonn (1996) [BONN-IB-96-19]Google Scholar
  11. 11.
    Kogan, I.I., Mavromatos, N.E.: World-sheet logarithmic operators and target space symmetries in string theory. Phys. Lett. B 375, 111 (1996) [hep-th/9512210]CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Kogan, I.I., Lewis, A.: Origin of logarithmic operators in conformal field theories. Nucl. Phys. B 509, 687 (1998) [hep-th/9705240]CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Flohr, M.A.I.: On modular invariant partition functions of conformal field theories with logarithmic operators. Int. J. Mod. Phys. A 11, 4147 (1996) [hep-th/9509166]Google Scholar
  14. 14.
    Gaberdiel, M.R.: An algebraic approach to logarithmic conformal field theory. [hep-th/0111260]Google Scholar
  15. 15.
    Fjelstad, J., Fuchs, J., Hwang, S., Semikhatov, A.M., Tipunin, I.Yu.: Logarithmic conformal field theories via logarithmic deformations. Nucl. Phys. B 633, 379 (2002) [hep-th/0201091]CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Gannon, T.: Modular data: the algebraic combinatorics of conformal field theory. [math.QA/0103044]Google Scholar
  17. 17.
    Kausch, H.G.: Extended conformal algebras generated by a multiplet of primary fields. Phys. Lett. B 259, 448 (1991)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Eholzer, W., Skoruppa, N.-P.: Conformal characters and theta series. Lett. Math. Phys. 35, 197 (1995) [hep-th/9410077]MathSciNetzbMATHGoogle Scholar
  19. 19.
    Pierce, R.S.: Associative Algebras. Graduate Text in Mathematics 88, Berlin-Heidelberg-New York: Springer Verlag, 1982Google Scholar
  20. 20.
    Flohr, M.A.I.: Bits and pieces in logarithmic conformal field theory. hep-th/0111228Google Scholar
  21. 21.
    Mumford, D.: Tata Lectures on Theta. Basel-Boston: Birkhäuser, 1983, 1984Google Scholar
  22. 22.
    Moore, G., Seiberg, N.: Lectures on RCFT. In: Physics, Geometry, and Topology (Trieste Spring School 1989), London-New York: Plenum 1990, p. 263ffGoogle Scholar
  23. 23.
    Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves. Providence RI: AMS, 2001Google Scholar
  24. 24.
    Bakalov, B., Kirillov, A.A.: Lectures on Tensor Categories and Modular Functors. Providence RI: AMS 2001Google Scholar
  25. 25.
    Bredthauer, A., Flohr, M.: Boundary states in c = −2 logarithmic conformal field theory. Nucl. Phys. B 639, 450 (2002) [hep-th/0204154]CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Bredthauer, A.: Boundary states in logarithmic conformal field theory – A novel approach. Diploma Thesis, Hannover (2002). []Google Scholar
  27. 27.
    Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators I: Partition functions. Nucl. Phys. B 646, 353 (2002). [hep-th/0204148]CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators II: Unoriented world sheets. Nucl. Phys. B 678, 511 (2004). [hep-th/0306164]CrossRefGoogle Scholar
  29. 29.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • J. Fuchs
    • 1
  • S. Hwang
    • 1
  • A.M. Semikhatov
    • 2
  • I.Yu. Tipunin
    • 2
  1. 1.Karlstad UniversityKarlstadSweden
  2. 2.Lebedev Physics InstituteMoscowRussia

Personalised recommendations