Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 008, 13 pages      math.QA/0610322
Contribution to the Proceedings of the O'Raifeartaigh Symposium

The Virasoro Algebra and Some Exceptional Lie and Finite Groups

Michael P. Tuite
Department of Mathematical Physics, National University of Ireland, Galway, Ireland

Received October 09, 2006, in final form December 16, 2006; Published online January 08, 2007

We describe a number of relationships between properties of the vacuum Verma module of a Virasoro algebra and the automorphism group of certain vertex operator algebras. These groups include the Deligne exceptional series of simple Lie groups and some exceptional finite simple groups including the Monster and Baby Monster.

Key words: vertex operator algebras; Virasoro algebras; Deligne exceptional series; Monster group.

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  1. Borcherds R., Vertex algebras, Kac-Moody algebras and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068-3071.
  2. Deligne P., La série exceptionnelle de groupes de Lie (The exceptional series of Lie groups), C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 321-326.
  3. Dong C., Mason G., Holomorphic vertex operator algebras of small central charge, Pacific J. Math. 213 (2004), 253-266, math.QA/0203005.
  4. Frenkel I., Huang Y-Z., Lepowsky J., On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494.
  5. Frenkel I., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Academic Press, New York, 1988.
  6. Griess R.L., The vertex operator algebra related to E8 with automorphism group O+(10,2), in The Monster and Lie algebras (1996, Columbus, Ohio State University), Math. Res. Inst. Public., Vol. 7, de Gruyter, Berlin, 1998.
  7. Hoehn G., Selbstduale Vertexoperatorsuperalgebren und das Babymonster, PhD Thesis, Bonn. Math. Sch. 286 (1996), 1-85.
  8. Kac V., Vertex operator algebras for beginners, University Lecture Series, Vol. 10, AMS, Boston, 1998.
  9. Kac V., Raina A.K., Bombay lectures on highest weight representations of infinite dimensional Lie algebras, World Scientific, Singapore, 1987.
  10. Li H., Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 (1994), 279-297.
  11. Matsuo A., Norton's trace formula for the Griess algebra of a vertex operator algebra with large symmetry, Comm. Math. Phys. 224 (2001), 565-591, math.QA/0007169.
  12. Maruoka H., Matsuo A., Shimakura H., Trace formulas for representations of simple Lie algebras via vertex operator algebras, 2005, unpublished preprint.
  13. Matsuo A., Nagatomo K., Axioms for a vertex algebra and the locality of quantum fields, MSJ Memoirs, Vol. 4, Tokyo, Mathematical Society of Japan, 1999, hep-th/9706118.
  14. Schellekens A.N., Meromorphic C = 24 conformal field theories, Comm. Math. Phys. 153 (1993), 159-185, hep-th/9205072.
  15. Tuite M.P., to appear.

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