# Chiral Koszul Duality

 Title: Chiral Koszul Duality Author: Francis, John; Gaitsgory, Dennis Note: Order does not necessarily reflect citation order of authors. Citation: Francis, John, and Dennis Gaitsgory. 2012. Chiral Koszul duality. Selecta Mathematica, n.s., 18(1): 27-87. Full Text & Related Files: 1103.5803v4.pdf (532.7Kb; PDF) 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. Published Version: doi:10.1007/s00029-011-0065-z Other Sources: http://arxiv.org/abs/1103.5803 Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10043337 Downloads of this work: