Chiral Koszul duality

TL;DR

This paper generalizes the theory of chiral and factorization algebras from curves to higher-dimensional varieties, establishing a homotopy-theoretic framework and a form of Koszul duality.

Contribution

It develops the homotopy theory for higher-dimensional chiral and factorization structures and proves their equivalence via a new chiral Koszul duality.

Findings

Established homotopy theory for higher-dimensional chiral algebras

Proved equivalence of higher-dimensional chiral and factorization algebras

Rederived fundamental results on chiral enveloping algebras

Abstract

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in \cite{bd}, 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 \cite{bd} on chiral enveloping algebras of $\star$-Lie algebras.