Algebras of Virasoro type, energy-momentum tensor, and decomposition operators on Riemann surfaces
type: Item
instance of: scholarly article
title: Algebras of Virasoro type, energy-momentum tensor, and decomposition operators on Riemann surfaces
zbMATH Open document ID: 0684.17012
DOI: 10.1007/BF01078570
author: Igor Krichever
author: S. P. Novikov
publication date: 1989-01-01T00:00:00Z
review text: This is the third in a series of papers by the same authors [see ibid. 21, No.2, 46-63 (1987; Zbl 0634.17010)] and 21, No.4, 47-61 (1987; Zbl 0659.17012)] developing a program of the operator quantization of multiloop diagrams in the bosonic string theory. The approach departs from a twice pointed non-singular Riemannian surface \(\Gamma\) as an algebro-geometric model of a bosonic string; the fixed points \(P_{\pm}\) correspond to the conformal compactification of the string world sheet at \(t\to \pm \infty\) in the Minkowski space. The so-called `almost graded' central extensions of certain tensor algebras on \(\Gamma\) play a crucial role in the operator theory of interacting strings; they are analogues of the Virasoro and Heisenberg algebras. The{\S} 1 contains a reminder of the basic ideas in a `more appropriate for the sequel' form. Operator realization of a bosonic string in the Fock space \({\mathcal H}^{\pm}\) of Dirac fermions on \(\Gamma\) is discussed in the {\S}2. In the case of genus \(g>0\) the energy-impulse tensor proves to be ill- defined, and the {\S}3 is devoted to the introduction of its proper substitution, the energy-impulse `pseudotensor' on \(\Gamma\), which is defined invariantly and depends on the triple \(\Gamma\), \(P_+\), \(P_-\) only. The concluding {\S}4 sketches a program of extending the results presented beyond the bosonic sector of the closed string, via the BRST techniques.
reviewed by: Vladimir G. Pestov
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 81Q30
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 81T60
Mathematics Subject Classification ID: 30F99
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 4120365
zbMATH Keywords: normal ordering
zbMATH Keywords: almost-graded algebras
zbMATH Keywords: operator quantization
zbMATH Keywords: bosonic string
zbMATH Keywords: Riemannian surface
zbMATH Keywords: Fock space
zbMATH Keywords: Dirac fermions
MaRDI profile type: Publication
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4120365