Virasoro-type algebras, Riemann surfaces and strings in Minkowski space
type: Item
instance of: scholarly article
title: Virasoro-type algebras, Riemann surfaces and strings in Minkowski space
zbMATH Open document ID: 0659.17012
DOI: 10.1007/BF01077803
author: Igor Krichever
author: S. P. Novikov
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Gamma\) denote a compact Riemann surface of genus \(g\) with two distinguished points \(P_+\), \(P_-\). The generalized Heisenberg algebra connected with \(\Gamma\) is defined by the authors as an algebra with generators \(\alpha_ n\) and a central element t, satisfying the relations \([\alpha_ n,\alpha_ m]=\gamma_{nm}\cdot t\), \([\alpha_ n,t]=0\), where \(\alpha_{mn}=(1/2\pi i)\oint A_ m dA_ n\). The \(A_ n\)'s are suitable meromorphic functions on \(\Gamma\), holomorphic away from \(P_+\), \(P_-\) and characterized by their behavior in the neighbourhoods of these points. An analog of the Virasoro algebra connected with \(\Gamma\) is defined as an algebra with generators \(E_ n\) and \(t\) and the relations \[ [E_ n,t]=0,\quad [E_ n,E_ m]=\sum c^ k_{nm} E_{n+m-k}+\chi_{nm} t. \] The \(c^ k_{nm}\) are the structural constants of the algebra \({\mathcal L}^{\Gamma}\) of meromorphic vector fields on \(\Gamma\) with respect to a suitable basis and \(\chi_{nm}\) is a 2-cocycle on \({\mathcal L}^{\Gamma}\). With this structure the authors give an algebro-geometric model for strings, the ideas of an earlier paper [Funct. Anal. Appl. 21, 126-142 (1987); translation from Funkts. Anal. Prilozh. 21, No.2, 46-63 (1987; Zbl 0634.17010)] are modified.
Mathematics Subject Classification ID: 17B68
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 81T30
zbMATH DE Number: 4077503
zbMATH Keywords: compact Riemann surface
zbMATH Keywords: generalized Heisenberg algebra
zbMATH Keywords: strings
reviewed by: Günter Czichowski
MaRDI profile type: Publication
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4077503