xii PREFACE TO THE FIRST EDITION

the interplay between vertex algebras and various geometric structures related to

algebraic curves, bundles and moduli spaces.

In particular, we attach to each vertex algebra and any pointed algebraic curve

the spaces of coinvariants and conformal blocks. When we vary the curve X and

other data on X (such as G-bundles), these spaces combine into a sheaf on the

relevant moduli space. One can gain new insights into the structure of moduli

spaces from the study of these sheaves.

The language of the book gradually changes from that of formal power series as

in [FLM, Kac3] to that of bundles, sheaves, and connections on algebraic curves.

Our goal however is to avoid using sophisticated techniques, such as the theory of

D-modules, as much as possible. In particular, we present most of the material

without mentioning the "CD-word". Only at the end of the book do we use rudiments

of D-module theory when describing the relationship between vertex algebras and

the Beilinson-Drinfeld chiral algebras, and the sheaves of coinvariants. Ultimately,

the formalism developed in this book will enable us to relate the algebraic theory of

vertex algebras to the geometric theory of factorization algebras and factorization

spaces.

The first five chapters of this book contain a self-contained elementary intro-

duction to the algebraic theory of vertex algebras and modules over them. We

motivate all definitions and results, give detailed proofs and consider numerous ex-

amples. This part of the book is addressed mainly to beginners. No prerequisites

beyond standard college algebra are needed to understand it.

In Chapters 6-10 we develop the geometric approach to vertex algebras. Here

some familiarity with basic notions of algebraic geometry should be helpful. We

have tried to make the exposition as self-contained as possible, so as to make it

accessible for non-experts.

Next, we review in Chapters 11-16 various constructions and applications of

vertex algebras, such as, the free field realization of affine Kac-Moody algebras,

solutions of the Knizhnik-Zamolodchikov equations, and the Drinfeld-Sokolov re-

duction. We also study quasi-classical analogues of vertex algebras, called vertex

Poisson algebras.

The last four chapters of the book are more algebro-geometrically oriented.

Here we construct the sheaves of coinvariants on the moduli spaces of curves

and bundles and introduce the chiral algebras and factorization algebras following

Beilinson and Drinfeld. In particular, we show how to attach to any quasi-conformal

vertex algebra a chiral algebra on an arbitrary smooth algebraic curve. We discuss

various examples of factorization algebras and factorization spaces, including the

Beilinson-Drinfeld Grassmannians. We also give a brief overview of the geometric

Langlands correspondence.

This book may be used by the beginners as an entry point to the modern theory

of vertex algebras and its geometric incarnations, and by more experienced readers

as a guide to advanced studies in this beautiful and exciting field.

Acknowledgments

We are grateful to Ivan Mirkovic and Matthew Szczesny for their careful reading

of drafts of this book and detailed comments which helped us improve the exposi-

tion and correct errors. We owe thanks to Matthew Emerton, George Glauberman,

Namhoon Kim, Mark Kisin, Manfred Lehn, Evgeny Mukhin, Markus Rosellen,