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
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
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
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.
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,