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Book Review: Vertex algebras and algebraic curves

Yi-Zhi Huang

2002
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Bulletin of the American Mathematical Society
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Vertex algebras and algebraic curves by E. Frenkel and D. Ben-Zvi is an excellent introduction to the theory of vertex algebras and their connection with the geometry of algebraic curves. This superb book contains the first published systematic exposition of the algebro-geometric approach to vertex algebras and its relation with the algebraic approach. To provide the reader with a broad background, I will first review the mathematical development of vertex operator algebras and conformal field
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... heories, in relation to the main theme of the book. The book under review has fine historical discussions in its own style, including a section of bibliographic notes at the end of each chapter. Vertex algebras are a class of algebras underlying a number of recent constructions in mathematics and physics. A vertex algebra is an "algebra" of vertex operators, or equivalently, an "algebra" whose operations are given by vertex operators. Vertex operators, operators describing the interaction of particles at a vertex, were first introduced and studied in the early 1970's by physicists in dual resonance theory, which led physicists to string theory. In the representation theory of affine Lie algebras, examples of vertex operators were rediscovered in the late 1970's by J. Lepowsky and R. Wilson and vertex operators were further exploited and developed by I. Frenkel and V. Kac. In a breakthrough in 1983 by I. Frenkel, Lepowsky and A. Meurman, vertex operators were used to give a construction, incorporating a vertex operator realization of the Griess algebra, of what they called the "moonshine module", an infinite-dimensional representation of the Fischer-Griess Monster finite simple group. This work in fact proved a conjecture of J. McKay and J. Thompson, a part of the monstrous moonshine conjecture of J. Conway and S. Norton. In 1986, motivated partly by this construction of the moonshine module, R. Borcherds initiated a general theory of vertex operators and, in particular, axiomatized the notion of vertex algebra. He constructed vertex algebras associated to even lattices and stated that the moonshine module carries a vertex algebra structure. The latter statement was proved by I. Frenkel, Lepowsky and Meurman based on their earlier construction of and their results on the moonshine module. Since then, the theory of vertex (operator) algebras has been rapidly developed and has found a number of applications in a wide range of branches of mathematics, including Borcherds' beautiful proof of the remaining parts of the full monstrous moonshine conjecture. Meanwhile, in physics, A. Belavin, A. Polyakov and A. Zamolodchikov systematized the basic properties of the operator product structure of (two-dimensional) conformal field theory, a physical theory arising in both condensed matter physics and string theory. In the study of these algebraic structures, physicists arrived at a physical notion of "chiral algebra". Physically, the underlying vector space of such a chiral algebra is the space of "meromorphic fields" in a conformal field theory,

doi:10.1090/s0273-0979-02-00955-2
fatcat:lrutac4r6rc2jefhy7luzukfbi