  We construct equivalences among four notions associated to a reductive group
scheme $G$: factorization super central extensions of the loop group of $G$ by
$\mathbb G_m$ subject to a condition on the commutator, factorization super
line bundles on the affine Grassmannian of $G$, rigidified sections of a
quotient of $2$-truncated K-theory over the Zariski classifying stack of $G$,
and combinatorial data defined by Brylinski and Deligne in a conjectural
extension of their classification theorem.
