On Exceptional Vertex Operator (Super) Algebras

Abstract

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 module lowest weights, 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.

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“…Note that C2 2 m is called the quadratic Casimir vector in [Tuite 2009;Tuite and Van 2014]. We also note that if…”

“…More precisely, SVOAs with minimal conformal weight 1 are isomorphic to lattice VOAs associated to the root lattices of type A 1 and E 8 if the conformal weight-1 space forms a conformal 6-design. Also, it was proved in [Tuite 2009] that a VOA whose 4th Casimir vector belongs to V ! is isomorphic to one of the simple affine VOA associated to the Deligne exceptional series of the simple Lie algebras A 1 ; A 2 ; G 2 ; D 4 ; F 4 ; E 6 ; E 7 , and E 8 at level 1 if the minimal conformal weight is 1.…”

“…This result is actually an analogue of the result in [Venkov 2001], as already mentioned. Due to the classification in [Tuite 2009], one can consider the classification problem in the case of SVOAs with minimal conformal weight 3 2 and t D 4. The commutant superalgebra (see [Yamauchi 2005 [Miyamoto 1996a] for the definition of code SVOAs).…”

Classification of vertex operator algebras of class ${\mathcal S}^4$ with minimal conformal weight one

“…Conversely, if a VOA V of central charge c with minimal conformal weight 2 is of class S 8 , then c = 24 and dim V 2 = 196884 ( [Mat01,Hö08]), which are satisfied by the moonshine VOA. For VOAs of class S 6 , there are constraints on the central charge and the dimension of the minimal conformal weight space (see [Mat01, Table 3.3], [Hö08, Theorem 4.1] and [TV14,§6]). In particular, VOAs of class S 6 with minimal conformal weight one are lattice VOAs associated to the root lattices of type A 1 and E 8 ( [Hö08,Tu09]).…”

Conformal designs and minimal conformal weight spaces of vertex operator superalgebras

“…This result is actually an analogue of the result in [Venkov 2001], as already mentioned. Due to the classification in [Tuite 2009], one can consider the classification problem in the case of SVOAs with minimal conformal weight 3 2 and t D 4. The commutant superalgebra (see [Yamauchi 2005]) of an Ising vector in the lattice type VOA…”