Supersymmetry and the Suzuki chain

Abstract

We classify $N=1$ SVOAs with no free fermions and with bosonic subalgebra a simply connected WZW algebra which is not of type $E$. The latter restriction makes the classification tractable; the former restriction implies that the $N=1$ automorphism groups of the resulting SVOAs are finite. We discover two infinite families and nine exceptional examples. The exceptions are all related to the Leech lattice: their automorphism groups are the larger groups in the Suzuki chain (${Co}_1$, $Suz:2$, $G_2\left(4\right):2$, $J_2:2$, $U_3\left(3\right):2$) and certain large centralizers therein ($2^{10}:M_{12}:2$, $M_{12}:2$, $U_4\left(3\right):D_8$, $M_{21}:2^2$). Along the way, we elucidate fermionic versions of a number of VOA operations, including simple current extensions, orbifolds, and ’t Hooft anomalies.