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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 by J.-L.Brylinski and P. Deligne, are equivalent to factorization line bundles on theBeilinson-Drinfeld affine Grassmannian $\operatorname{Gr}_G$. Our resultaffirms a conjecture of D. Gaitsgory and S. Lysenko and classifiesfactorization line bundles on $\operatorname{Gr}_G$.

Extensions by $\mathbf K_2$ and factorization line bundles