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 \(\text{ Gr}_G\).

中文翻译：

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$$ {\ mathbf {K}} _ 2 $$ K 2的中央扩展和分解线束

令X为在理想场k上的平滑几何连接曲线。给定一个连通的归约群G，我们证明在Brylinski–Deligne [5]中研究的X的大Zariski位置上的捆\（{\ mathbf {K}} _ 2 \）的G中心扩展与Beilinson–Drinfeld仿射Grassmannian \（\ text {Gr} _G \）上的分解线束。我们的结果肯定了Gaitsgory–Lysenko [13]的一个猜想，并且对\（\ text {Gr} _G \）上的分解线束进行了分类。