Selecta Mathematica

, Volume 18, Issue 1, pp 27–87 | Cite as

Chiral Koszul duality

  • John FrancisEmail author
  • Dennis GaitsgoryEmail author


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.


Chiral algebras Chiral homology Factorization algebras Conformal field theory Koszul duality Operads ∞-Categories 

Mathematics Subject Classification (2010)

Primary 81R99 Secondary 14H81 18G55 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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