On Exceptional Vertex Operator (Super) Algebras

  • Michael P. TuiteEmail author
  • Hoang Dinh Van
Part of the Developments in Mathematics book series (DEVM, volume 38)


We consider exceptional vertex operator algebras and vertex operator superalgebras with the property that particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendents of the vacuum. We show that the genus one partition function and characters for simple ordinary modules must satisfy modular linear differential equations. We show the rationality of the central charge and lowest weights of modules, modularity of solutions, the dimension of each graded space is a rational function of the central charge and that the lowest weight primaries generate the algebra. We also discuss conditions on the reducibility of the lowest weight primary vectors as a module for the automorphism group. Finally we analyse solutions for exceptional vertex operator algebras with primary vectors of lowest weight up to 9 and for vertex operator superalgebras with primary vectors of lowest weight up to 17/2. Most solutions can be identified with simple ordinary modules for known algebras but there are also four conjectured algebras generated by weight two primaries and three conjectured extremal vertex operator algebras generated by primaries of weight 3, 4 and 6, respectively.

Key words

Vertex operator algebras Vertex operator super algebras Virasoro algebra Group theory 

Mathematics Subject Classification (2010):

Primary 17B69 17B68 17B25 17B67 Secondary 20E32. 



The authors are particularly grateful to Geoffrey Mason for very useful comments over many discussions about this work. The authors also thank Terry Gannon, Atsushi Matsuo and Ching Hung Lam for their comments.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Applied MathematicsNational University of Ireland GalwayGalwayIreland

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