Half-integral levels

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.