Self-Dual Vertex Operator Superalgebras of Large Minimal Weight

The new general upper bound μ ≤ [ c 24 ] + 1 for the minimal weight μ of a selfdual vertex operator superalgebra of central charge c 6= 23 1 2 is proven. For central charges c ≤ 48, further improved estimates are given and examples of vertex operator superalgebras with large minimal weight are discussed. We also study the case of vertex operator superalgebras with N=1 supersymmetry which was first considered by Witten in connection with three-dimensional quantum gravity. The upper bound μ ≤ 1 2 [ c 12 ] + 1 2 for the minimal superconformal weight is obtained for c 6= 23 1 2 . In addition, we show that it is impossible that the monster sporadic group acts on an extremal self-dual N=1 supersymmetric vertex operator superalgebra of central charge 48 in a way proposed by Witten if certain standard assumptions about orbifold constructions hold. The same statement holds for extremal self-dual vertex operator algebras of central charge 48.