Two-dimensional conformal geometry and vertex operator algebras
type: Item
instance of: scholarly article
title: Two-dimensional conformal geometry and vertex operator algebras
zbMATH Open document ID: 0884.17021
author: Yi-Zhi Huang
published in: Progress in Mathematics
publication date: 1997-07-28T00:00:00Z
review text: This book is an extension of the author's Ph.D. thesis in 1990, whose main results were announced in his article ``Geometric interpretation of vertex operator algebras'' [Proc. Natl. Acad. Sci. USA 88, 9964-9968 (1991; Zbl 0810.17019)]. The notion of vertex algebra was introduced by \textit{R. E. Borcherds} [Proc. Natl. Acad. Sci. USA 83, 3068-3071 (1986; Zbl 0613.17012)]. Vertex operator algebras are vertex algebras with a certain Virasoro algebra structure, which were formulated by \textit{I. Frenkel}, \textit{J. Lepowsky} and \textit{A. Meurman} [Vertex operator algebras and the Monster, Pure Appl. Math. 134 (Academic Press, New York, 1988; Zbl 0674.17001)]. Vertex operator algebras are fundamental algebraic structures in conformal field theory. Based on work of \textit{G. B. Segal} [Proc. IXth international congress on mathematical physics, Swansea, 1988 (Hilger, Bristol), 22-37 (1989)], \textit{C. Vafa} [Phys. Lett. B 199, 195-202 (1987)] and Frenkel (see Appendices in the above FLM's book), the author gives a geometric interpretation of the axioms of vertex operator algebra in terms of so-called ``partial operads'' of complex powers of the determinant line bundles over certain moduli spaces of spheres with punctures and local analytic coordinates. The first axiom restricts vertex operator algebras to be positive energy modules for what turns out eventually to be the Virasoro algebra. The second axiom corresponds to the eigenspace decomposition with respect to the operator of rescaling local coordinates. The third axiom states that the correlation functions of vertex operators are meromorphic as functions of any one of the punctures in a sphere and with the other punctures as the only possible poles. The fourth axiom requires that the ordering of the punctures of spheres with tubes does not enter the structure of a geometric vertex operator algebra. The fifth axiom requires that the image of an element sew from two elements in the moduli space is equal to the c…
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 17-02
Mathematics Subject Classification ID: 81T40
Mathematics Subject Classification ID: 81R10
Mathematics Subject Classification ID: 17B68
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 1040033
zbMATH Keywords: sewing operation
zbMATH Keywords: vertex partial operads
zbMATH Keywords: vertex operator algebra
zbMATH Keywords: Virasoro algebra
zbMATH Keywords: correlation functions
zbMATH Keywords: punctures of spheres with tubes
reviewed by: Xiaoping Xu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1040033