Central extensions by $${\mathbf {K}}_2$$ and factorization line bundles

Abstract

Let X be a smooth, geometrically connected curve over a perfect field k. Given a connected, reductive group G, we prove that central extensions of G by the sheaf \({\mathbf {K}}_2\) on the big Zariski site of X, studied in Brylinski–Deligne [5], are equivalent to factorization line bundles on the Beilinson–Drinfeld affine Grassmannian \(\text{ Gr}_G\). Our result affirms a conjecture of Gaitsgory–Lysenko [13] and classifies factorization line bundles on \(\text{ Gr}_G\).