Chiral Koszul Duality

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in [BD1], 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 higherdimensional 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 [BD1] on chiral enveloping algebras of -Lie algebras.