TY - JOUR
T1 - Chiral Koszul duality
AU - Francis, John
AU - Gaitsgory, Dennis
N1 - Funding Information:
Acknowledgments This work originated in the course of Mike Hopkins’ seminar on chiral algebras at Harvard during the fall of 2007, and we thank all the seminar participants for their collaboration and fellowship. We especially thank Jacob Lurie for many conversations, during which many of the present ideas were jointly conceived, and for numerous explanations on various topics addressed in this paper. (Unfortunately, Jacob has declined to sign this work as a coauthor). We warmly thank Sasha Beilinson for conversations on chiral algebras and for his generous encouragement on first hearing of this work. DG thanks Nick Ro-zenblyum for many helpful discussions. JF thanks Reimundo Heluani for first explaining chiral algebras to him as a first-year graduate student, which led him to [1] and the comments of 3.3.13 therein. Finally, we wish to thank the anonymous referee, whose remarks have helped to improve the paper. JF is supported by an NSF postdoctoral fellowship. JF’s travel to Cambridge, during which a draft of this paper was written, was supported by the Midwest Topology Network. DG is supported by NSF grant DMS-0600903.
PY - 2012/3
Y1 - 2012/3
N2 - 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 {black star}-Lie algebras.
AB - 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 {black star}-Lie algebras.
KW - Chiral algebras
KW - Chiral homology
KW - Conformal field theory
KW - Factorization algebras
KW - Koszul duality
KW - Operads
KW - ∞-Categories
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U2 - 10.1007/s00029-011-0065-z
DO - 10.1007/s00029-011-0065-z
M3 - Article
AN - SCOPUS:84857798497
SN - 1022-1824
VL - 18
SP - 27
EP - 87
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
IS - 1
ER -