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 $\mathrm{E}$. 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 ($\mathrm{Co}_1$, $\mathrm{Suz}{:}2$, $\mathrm{G}_2(4){:}2$, $\mathrm{J}_2{:}2$, $\mathrm{U}_3(3){:}2$) and certain large centralizers therein ($2^{10}{:}\mathrm{M}_{12}{:}2$, $\mathrm{M}_{12}{:}2$, $\mathrm{U}_4(3){:}D_8$, $\mathrm{M}_{21}{:}2^2$). Along the way, we elucidate fermionic versions of a number of VOA operations, including simple current extensions, orbifolds, and 't Hooft anomalies.