Chiral algebras are objects living on algebraic curves. They are ‘quantizations’ of structures given by local Poisson brackets on a space of ‘classical’ fields. In the special case, where the underlying curve is the affine line Spec(C[t]), chiral algebras invariant under translations are the same as vertex algebras. Requiring invariance under the whole affine group and the Virasoro algebra leads essentially to conformal vertex algebras. Taking the disc Spec(C[[t]]) instead of the affine line gives quasi-conformal vertex algebras. In this sense, chiral algebras comprise the basic features of conformal field theory. The aim of the book under review is to set up rudiments of chiral algebras theory. The first chapter of the book gives some necessary algebraic foundations of pseudo-tensor categories and the related ‘compound’ geometry. The second chapter is devoted to ‘coisson’ algebras, which is the authors’ name for local Poisson algebras (an abbreviation for ‘compound Poisson’). The third chapter deals with the chiral algebras proper. In the final chapter the global theory in the formalism of chiral homology is developed. The book is aimed at students and researchers interested in vertex algebras, quantization, and applications of geometry to mathematical physics in general. It assumes a solid preliminary knowledge of algebraic geometry and homological algebra.

Reviewer:

mm