Deformations of chiral algebras

Abstract

We start studying chiral algebras (as defined by A. Beilinson and V. Drinfeld) from the point of view of deformation theory. First, we define the notion of deformation of a chiral algebra on a smooth curve $X$ over a bundle of local artinian commutative algebras on $X$ equipped with a flat connection (whereas `usual' algebraic structures are deformed over a local artinian algebra) and we show that such deformations are controlled by a certain *-Lie algebra $\mathfrak g$. Then we try to contemplate a possible additional structure on $\mathfrak g$ and we conjecture that this structure up to homotopy is a chiral analogue of Gerstenhaber algebra, i.e. a coisson algebra with odd coisson bracket (in the terminology of Beilinson-Drinfeld). Finally, we discuss possible applications of this structure to the problem of quantization of coisson algebras.