Abstract
We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld (American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI, 2004), to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen’s homotopy theory of differential graded Lie algebras. We prove the equivalence of higher-dimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these techniques to rederive some fundamental results of Beilinson and Drinfeld (American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI, 2004) on chiral enveloping algebras of \({\star}\) -Lie algebras.
Highlights
We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in [BD1], to higher-dimensional varieties
Beilinson and Drinfeld developed the theory of chiral and factorizationalgebras on curves in their seminal work, [BD1], as a geometric counterpart of the algebraic theory of vertex algebras
In order to sensibly extend the theory to varieties, they observed the necessity of developing the homotopy theory of chiral Lie algebras, a problem of independent interest.[1]. We develop just such a homotopy theory of chiral and factorization structures and apply it to prove a generalization of the above theorem of [BD1], to establish an equivalence between chiral Lie algebras and factorization coalgebras on higher-dimensional varieties
Summary
Beilinson and Drinfeld developed the theory of chiral and factorization (co)algebras on curves in their seminal work, [BD1], as a geometric counterpart of the algebraic theory of vertex algebras. (and such that this system forms a D-module as S ranges over Ran X), and a system of homotopy equivalences (1.1) that satisfy the natural compatibility conditions under further partitions of finite sets into disjoint unions When written in this form, the notion of factorization D-module looks symmetric from the algebra/coalgebra perspective. For a ground field k, we shall denote by Vectk the commutative algebra object of -Catsptres,cont given by the -category associated to the simplicial category of chain complexes of k-vector spaces. Let Schcorr denote the 1, 1 -category whose objects are schemes È Ô Õ of finite type, and morphisms are correspondences, i.e., for Y1, Y2 Schcorr, HomSchcorr Y1, Y2 is the groupoid of diagrams, an element f in which is of the form (1.6). -Catsptres,cont, obtained from D! Ô Õ Schproper op by taking left adjoints
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