Abstract
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 forE 8) and to get various extensions of the vertex operator algebras associated with integrable representations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[BPZ] Belavin, A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetries in two-dimensional quantum field theory. Nucl. Phys.B241, 333–380 (1984)
[B] Borcherds, R.E.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA83, 3068–3071 (1986)
[D1] Dong, C.: Vertex algebras associated with even lattices. J. Algebra161, 245–265 (1993)
[D2] Dong, C.: Twisted modules for vertex operator algebras associated with even lattices. J. Algebra165, 91–112 (1993)
[DL] Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators. Progress in Math.Vol. 112, Boston: Birkhauser, 1993
[DLM1] Dong, C., Li, H., Mason, G.: Regularity of rational vertex operator algebras. Adv. Math., to appear, q-alg/9508018
[DLM2] Dong, C., Li, H., Mason, G.: Twisted representations of vertex operator algebras. Preprint, q-alg/9509005
[DM] Dong, C., Mason, G.: Nonabelian orbifolds and the boson-fermion correspondence. Commun. Math. Phys.163, 523–559 (1994)
[FFR] Alex J. Feingold, Igor, B., Frenkel, John F.X. Ries: Spinor Construction of Vertex Operator Algebras, Triality, andE 8(1) . Contemp. Math.121, 1991
[FHL] Frenkel, I., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs Amer. Math. Soc.104, 1993
[FLM] Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Appl. Math.Vol. 134. Boston: Academic Press, 1988
[FZ] Frenkel, I., Zhu, Y.-C.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J.66, 123–168 (1992)
[F] Fuchs, J.: Simple WZW currents. Commun. Math. Phys.136, 345–356 (1991)
[FG] Fuchs, J., Gepner, D.: On the connection between WZW and free field theories. Nucl. Phys.B294, 30–42 (1998)
[G] Guo, H.: On abelian intertwining algebras and modules. Ph.D. thesis, Rutgers University, 1994
[Hua] Huang, Y.-Z.: A nonmeromorphic extension of the moonshine module vertex operator algebra. Contemp. Math.193, 123–148 (1995)
[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. 344–354
[HL2] Huang, Y.-Z., Lepowsky, J.: A theory of tensor product for module category of a vertex operator algebra, I, II, Selecta Mathematica, to appear
[H] Humphreys, J.E.: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics9, New York: Springer-Verlag, 1984
[K] Kac, V.G.: Infinite dimensional Lie algebras, 3rd ed., Cambridge: Cambridge Univ. Press, 1990
[KW] Kac, V.G., Wang, W.-Q.: Vertex operator superalgebras and representations. Contemp. Math.Vol.175, 161–191 (1994)
[Le] Lepowsky, J.: Calculus of twisted vertex operators. Proc. Natl. Acad. Sci. USA82, 8295–8299 (1985)
[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. 194–251
[Li1] Li, H.-S.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure and Appl. Algebra109, 143–195 (1996)
[Li2] Li, H.-S.: Local systems of twisted vertex operators, vertex superalgebras and twisted modules. Contemp. Math.193, 203–236 (1995)
[Li3] Li, H.-S.: Representation theory and tensor product theory for vertex operator algebras. Ph.D. thesis, Rutgers University, 1994
[Li4] Li, H.-S.: The theory of physical super selection sectors in terms of vertex operator algebra language. Preprint
[LX] Li, H.-S., Xu, X.-P.: A characterization of vertex algebras associated to even lattices. J. Algebra173, 253–270 (1995)
[MaS] Mack, G., Schomerus, V.: Conformal field algebras with quantum symmetry from the theory of superselection sectors. Commun. Math. Phys.134, 139–196 (1990)
[MP] Meurman, A., Primc, M.: Annihilating fields of standard modules of\(\tilde sl\left( {2, \mathbb{C}} \right)\) and combinatorial identities. Preprint (1994)
[MoS] Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys.123, 177–254 (1989)
[M] Mossberg, G.: Axiomatic vertex algebras and the Jacobi identity. Ph.D dissertation, University of Lund, Sweden, 1993
[SY] Schellekens, A.N., Yankielowicz, S.: Extended chiral algebras and modular invariant partition functions. Nucl. Phys.327, 673–703 (1989)
[X] Xu, X.: Intertwining operators for twisted modules of a colored vertex operator superalgebra. Preprint
Author information
Authors and Affiliations
Additional information
Communicated by G. Felder
Supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.
Supported by NSF grant DMS-9401272 and a research grant from the Committee on Research, UC Santa Cruz.
Rights and permissions
About this article
Cite this article
Dong, C., Li, H. & Mason, G. Simple currents and extensions of vertex operator algebras. Commun.Math. Phys. 180, 671–707 (1996). https://doi.org/10.1007/BF02099628
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02099628