Universal O-modules and representations of the automorphism group of a formal disk

Universal O- and D-modules (introduced by Beilinson and Drinfeld) arise in the study of the equivalence between certain categories of vertex algebras and chiral algebras. A universal O-module of dimension d is an assignment of a quasi-coherent sheaf to each smooth variety of dimension d, in a way compatible with étale morphisms between the varieties. This seems like a lot of data, but it turns out that it is equivalent to the data of a single representation of a group, the group of automorphisms of the formal d-dimensional disk. This equivalence of categories can be proved via an intermediate category, the category of quasi-coherent sheaves on a stack of étale germs of d-dimensional varieties. For those who have not seen stacks in action, this is a nice and approachable example of how they can be used to organize complicated structures and answer questions which otherwise seem quite intractable. No prior knowledge of stacks will be assumed.

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