Universal D-modules, and factorisation structures on Hilbert schemes of points - Cliff, E (2015)
This thesis concerns the study of chiral algebras over schemes of arbitrary dimension 𝑛. In Chapter I, we construct a chiral algebra over each smooth variety 𝑋 of dimension 𝑛. We do this via the Hilbert scheme of points of 𝑋, which we use to build a factorisation space over 𝑋. Linearising this space produces a factorisation algebra over 𝑋, and hence, by Koszul duality, the desired chiral algebra. We begin the chapter with an overview of the theory of factorisation and chiral algebras, before introducing our main constructions. We compute the chiral homology of our factorisation algebra, and show that the 𝒟-modules underlying the corresponding chiral algebras form a universal 𝒟-module of dimension 𝑛. In Chapter II, we discuss the theory of universal 𝒟-modules and 𝒪𝒪- modules more generally. We show that universal modules are equivalent to sheaves on certain stacks of étale germs of 𝑛-dimensional varieties. Furthermore, we identify these stacks with the classifying stacks of groups of automorphisms of the 𝑛-dimensional disc, and hence obtain an equivalence between the categories of universal modules and the representation categories of these groups. We also define categories of convergent universal modules and study them from the perspectives of the stacks of étale germs and the representation theory of the automorphism groups.