Communications in Mathematical Physics

, Volume 180, Issue 3, pp 671–707 | Cite as

Simple currents and extensions of vertex operator algebras

  • Chongying Dong
  • Haisheng Li
  • Geoffrey Mason


We consider how a vertex operator algebra can be extended to an abelian interwining algebra by a family of weak twisted modules which aresimple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except forE8) and to get various extensions of the vertex operator algebras associated with integrable representations.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Integrable Representation 
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. [BPZ] Belavin, A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetries in two-dimensional quantum field theory. Nucl. Phys.B241, 333–380 (1984)Google Scholar
  2. [B] Borcherds, R.E.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA83, 3068–3071 (1986)Google Scholar
  3. [D1] Dong, C.: Vertex algebras associated with even lattices. J. Algebra161, 245–265 (1993)Google Scholar
  4. [D2] Dong, C.: Twisted modules for vertex operator algebras associated with even lattices. J. Algebra165, 91–112 (1993)Google Scholar
  5. [DL] Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators. Progress in Math.Vol. 112, Boston: Birkhauser, 1993Google Scholar
  6. [DLM1] Dong, C., Li, H., Mason, G.: Regularity of rational vertex operator algebras. Adv. Math., to appear, q-alg/9508018Google Scholar
  7. [DLM2] Dong, C., Li, H., Mason, G.: Twisted representations of vertex operator algebras. Preprint, q-alg/9509005Google Scholar
  8. [DM] Dong, C., Mason, G.: Nonabelian orbifolds and the boson-fermion correspondence. Commun. Math. Phys.163, 523–559 (1994)Google Scholar
  9. [FFR] Alex J. Feingold, Igor, B., Frenkel, John F.X. Ries: Spinor Construction of Vertex Operator Algebras, Triality, andE (1)8. Contemp. Math.121, 1991Google Scholar
  10. [FHL] Frenkel, I., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs Amer. Math. Soc.104, 1993Google Scholar
  11. [FLM] Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Appl. Math.Vol. 134. Boston: Academic Press, 1988Google Scholar
  12. [FZ] Frenkel, I., Zhu, Y.-C.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J.66, 123–168 (1992)Google Scholar
  13. [F] Fuchs, J.: Simple WZW currents. Commun. Math. Phys.136, 345–356 (1991)Google Scholar
  14. [FG] Fuchs, J., Gepner, D.: On the connection between WZW and free field theories. Nucl. Phys.B294, 30–42 (1998)Google Scholar
  15. [G] Guo, H.: On abelian intertwining algebras and modules. Ph.D. thesis, Rutgers University, 1994Google Scholar
  16. [Hua] Huang, Y.-Z.: A nonmeromorphic extension of the moonshine module vertex operator algebra. Contemp. Math.193, 123–148 (1995)Google Scholar
  17. [HL1] Huang, Y.-Z., Lepowsky, J.: Towards a theory of tensor product for representations for a vertex operator algebra. In: proc. 20th International Conference on Differential Geometric Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, Singapore: World Scientific, 1992,Vol. 1, pp. 344–354Google Scholar
  18. [HL2] Huang, Y.-Z., Lepowsky, J.: A theory of tensor product for module category of a vertex operator algebra, I, II, Selecta Mathematica, to appearGoogle Scholar
  19. [H] Humphreys, J.E.: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics9, New York: Springer-Verlag, 1984Google Scholar
  20. [K] Kac, V.G.: Infinite dimensional Lie algebras, 3rd ed., Cambridge: Cambridge Univ. Press, 1990Google Scholar
  21. [KW] Kac, V.G., Wang, W.-Q.: Vertex operator superalgebras and representations. Contemp. Math.Vol.175, 161–191 (1994)Google Scholar
  22. [Le] Lepowsky, J.: Calculus of twisted vertex operators. Proc. Natl. Acad. Sci. USA82, 8295–8299 (1985)Google Scholar
  23. [LP] Lepowsky, J., Prime, M.: Standard modules for type one affine Lie algebras In: Number Theory, New York, 1982, Lectures Notes in Math.1052, New York: Springer-Verlag, 1984, pp. 194–251Google Scholar
  24. [Li1] Li, H.-S.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure and Appl. Algebra109, 143–195 (1996)Google Scholar
  25. [Li2] Li, H.-S.: Local systems of twisted vertex operators, vertex superalgebras and twisted modules. Contemp. Math.193, 203–236 (1995)Google Scholar
  26. [Li3] Li, H.-S.: Representation theory and tensor product theory for vertex operator algebras. Ph.D. thesis, Rutgers University, 1994Google Scholar
  27. [Li4] Li, H.-S.: The theory of physical super selection sectors in terms of vertex operator algebra language. PreprintGoogle Scholar
  28. [LX] Li, H.-S., Xu, X.-P.: A characterization of vertex algebras associated to even lattices. J. Algebra173, 253–270 (1995)Google Scholar
  29. [MaS] Mack, G., Schomerus, V.: Conformal field algebras with quantum symmetry from the theory of superselection sectors. Commun. Math. Phys.134, 139–196 (1990)Google Scholar
  30. [MP] Meurman, A., Primc, M.: Annihilating fields of standard modules of\(\tilde sl\left( {2, \mathbb{C}} \right)\) and combinatorial identities. Preprint (1994)Google Scholar
  31. [MoS] Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys.123, 177–254 (1989)Google Scholar
  32. [M] Mossberg, G.: Axiomatic vertex algebras and the Jacobi identity. Ph.D dissertation, University of Lund, Sweden, 1993Google Scholar
  33. [SY] Schellekens, A.N., Yankielowicz, S.: Extended chiral algebras and modular invariant partition functions. Nucl. Phys.327, 673–703 (1989)Google Scholar
  34. [X] Xu, X.: Intertwining operators for twisted modules of a colored vertex operator superalgebra. PreprintGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Chongying Dong
    • 1
  • Haisheng Li
    • 1
  • Geoffrey Mason
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaSanta CruzUSA

Personalised recommendations