Introduction

Some history and motivation

Vertex operators appeared in the early days of string theory as local operators

describing propagation of string states. Mathematical analogues of these operators

wrere discovered in the representation theory of affine Kac-Moody algebras in the

works of J. Lepowsky-R. Wilson [LW] and I. Frenkel-V. Kac [FK]. In order to

formalize the emerging structure, and motivated in particular by the I. Frenkel-

J. Lepowsky-A. Meurman construction of the Moonshine Module of the Monster

group, R. Borcherds gave the definition of vertex algebra in [Bo2]. The foundations

of the theory were subsequently laid down in [FLM, FHL]; in particular, it was

shown in [FLM] that the Moonshine Module indeed possessed a vertex algebra

structure.

In the meantime, A. Belavin, A. Polyakov and A. Zamolodchikov [BPZ] initi-

ated the study of two-dimensional conformal field theory (CFT). Vertex algebras

can be seen in retrospect as the mathematical equivalent of the chiral symmetry

algebras of CFT. Moreover, the key property of associativity of vertex algebras is

equivalent to the property of operator product expansion in CFT, which goes back

to the pioneering works of A. Polyakov and K. Wilson. Thus, vertex algebras may

be thought of as the mathematical language of two-dimensional conformal field

theory.

In recent years, in the course of their study of conformal field theories and string

theories, physicists have come up with astonishing conjectures predicting and relat-

ing to each other various geometric invariants: mirror symmetry, Gromov-Witten

invariants, Seiberg-Witten theory, etc. While many of these conjectures have been

proved rigorously afterwards, the process of making these predictions remains for

the most part a mystery for mathematicians. It is based on the usage of tools, such

as the path integral, that have so far resisted rigorous mathematical formulation.

The theory of vertex algebras, on the other hand, provides a rigorous mathematical

foundation for two-dimensional conformal field theory and string theory from the

Hamiltonian (i.e., operator) point of view. Namely, the space of states of the chiral

sector of a CFT may be described as a representation of a vertex algebra, and chi-

ral correlation functions may be considered as sections of various vector bundles on

the moduli spaces of pointed curves. Thus, vertex algebras provide a natural point

of entry for a mathematician into the world of conformal field theory and string

theory.

The interaction between conformal field theory and algebraic geometry has

already produced remarkable results, such as the computation of the dimensions

of the spaces of "non-abelian theta-functions" by means of the Verlinde formula.

These results came from examining particular examples of conformal field theories,

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