Introduction to Vertex Operator Superalgebras and Their Modules

ISBN-13: 9780792352426 / Angielski / Twarda / 1998 / 360 str.

Xiaoping Xu; Xu Xiaoping Xu

Opis

Vertex algebra was introduced by Boreherds, and the slightly revised notion "vertex oper ator algebra" was formulated by Frenkel, Lepowsky and Meurman, in order to solve the problem of the moonshine representation of the Monster group - the largest sporadie group. On the one hand, vertex operator algebras ean be viewed as extensions of eertain infinite-dimensional Lie algebras such as affine Lie algebras and the Virasoro algebra. On the other hand, they are natural one-variable generalizations of commutative associative algebras with an identity element. In a certain sense, Lie algebras and commutative asso ciative algebras are reconciled in vertex operator algebras. Moreover, some other algebraie structures, such as integral linear lattiees, Jordan algebras and noncommutative associa tive algebras, also appear as subalgebraic structures of vertex operator algebras. The axioms of vertex operator algebra have geometrie interpretations in terms of Riemman spheres with punctures. The trace functions of a certain component of vertex operators enjoy the modular invariant properties. Vertex operator algebras appeared in physies as the fundamental algebraic structures of eonformal field theory, whieh plays an important role in string theory and statistieal meehanies. Moreover, eonformalfieldtheoryreveals animportantmathematiealproperty, the so called "mirror symmetry" among Calabi-Yau manifolds. The general correspondence between vertex operator algebras and Calabi-Yau manifolds still remains mysterious. Ever since the first book on vertex operator algebras by Frenkel, Lepowsky and Meur man was published in 1988, there has been a rapid development in vertex operator su peralgebras, which are slight generalizations of vertex operator algebras.

Recenzje

`... well-written, with many improvements of known results and existing proofs. The researchers and the graduate students will use this book both as a graduate textbook and as a useful reference...'
Zentralblatt MATH, 929 (2000)

Spis treści

Preface. Introduction: Notational Conventions.I. Self-Dual Lattices and Codes. 1. Self-Dual Codes. 2. Self-Dual Lattices. II. Vertex Operator Superalgebras and Their Modules. 3. Definitions and General Properties. 4. Conformal Superalgebras, Affine Kac-Moody Algebras and KZ Equations. 5. Analogue of the Highest-Weight Theory. 6. Lattice Vertex Operator Superalgebras. 7. VOSAs Generated by Their Subspaces of Small Weights. Bibliography. Index.

O autorze

Xiaoping Xu received his Ph.D. from Rutgers University, USA in 1992. He is currently a research professor at the Chinese Academy of Sciences Institute of Mathematics, and has been working on representation theory and applied partial differential equations for twenty years, during which he has published over fifty substantial research papers and two monographs on mathematics.