A triangulation of moduli space from light-cone string theory
type: Item
instance of: scholarly article
title: A triangulation of moduli space from light-cone string theory
zbMATH Open document ID: 0636.32012
DOI: 10.1007/BF01215219
author: Steven B. Giddings
author: Scott A. Wolpert
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: The authors prove that the light-cone diagrams for the closed strings provide a single cover of the moduli space of Riemann surfaces. A diagram consists of flat tubes corresponding to free propagation of strings; at the interaction points there are curvature singularities. A diagram with n external states and g loops is characterized by \(6g+2n-6\) parameters: the internal momenta \(\alpha_ I\), \(I=1,...,g\), (the external ones \(\alpha_ i\), \(i=1,...,n\) are fixed), the twist angles \(\theta_{\beta}\), \(\beta =1,...,3g+n-3\), and the interaction times \(\tau_ a\), \(a=1,...,2g+n-3\) (the first interaction time is taken \(\tau=0)\). The variables \((\alpha,\theta,\tau)\) are the modular parameters labelling the conformal structure on a Riemann surface of genus g with n punctures. The main idea in proving that this correspondence is bijective is that the light-cone diagram is equivalent to the existence of an abelian differential \(\omega\) having only simple poles with specified real residues whose sum is zero, and pure imaginary periods. Namely the natural abelian differential dw on the world sheet is the restriction of an abelian differential defined on a planar diagram from which the light-cone diagram can be obtained by identification of boundary segements. In this way a light-cone diagram defines a Riemann surface with such an abelian differential. Conversely, given an abelian differential \(\omega\) on an arbitrary Riemann surface, let \(w=\int^{z}_{z_ 0}\omega\) in a neighbourhood U excluding a zero or pole of \(\omega\), where z is a local coordinate and \(z_ 0\) some basepoint in U. Then \(dw=\omega\) defines an almost globally flat metric on the surface \(dwd\bar w=| \omega |^ 2\). A simple (higher order) zero of \(\omega\) corresponds to a simple (higher order) interaction point, a simple pole corresponds to an incoming or outcoming string, depending on the sign of the residue at the pole. On a Riemann surface R of genus g, with n specified points \(P_…
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 32B25
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 81T60
Mathematics Subject Classification ID: 30F30
zbMATH DE Number: 4036181
zbMATH Keywords: abelian differentials
zbMATH Keywords: light-cone string theory
zbMATH Keywords: light-cone diagrams
zbMATH Keywords: moduli space of Riemann surfaces
reviewed by: Cabiria Andreian Cazacu
MaRDI profile type: Publication
cites work: The Euler characteristic of the moduli space of curves
cites work: Q3931656
cites work: Q5565313
cites work: Q4771484
description: scientific article; zbMATH DE number 4036181