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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