A triangulation of moduli space from light-cone string theory
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 \(P_ i\), \(i=1,...,n\), there exists a unique abelian differential \(\omega\) with pure imaginary periods, and whose singularities are simple poles at the points \(P_ i\) with given real residues \(\alpha_ i\), \(\sum^{n}_{i=1}\alpha_ i=0\). Further, such a differential determines uniquely a light-cone diagram. (Imaginary trajectories of \(\omega\) close.) A globally defined time \(\tau(z)= Re\int^{z}_{z_ 0}\omega\), \(\omega(z_ 0)\neq 0\), is discussed in connection with its critical set. Thus light-cone diagrams over moduli space, the parameters (\(\alpha\),\(\theta\),\(\tau)\) act as a global set of coordinates and a new cell decomposition of moduli space is obtained. The top dimensional cell is \({\mathcal S}\) (the set of diagrams with only simple interactions) with some boundary identifications; \({\mathcal S}\) is connected. Overcounting of configurations, when \((\alpha,\theta,\tau)\) ranges freely, is eliminated dividing by discrete symmetry factors. The coordinates \((\alpha,\theta,\tau)\) are closely related to the complex structure of moduli space. This is explicitly given by theta functions for the torus and the extension to higher genus is proposed. Conclusion underlines the importance of the work in physics. The proof that the light-cone diagram give a 1-1 cover of moduli space is a part of the proof of the equivalence between this light-cone formalism and the Polyakov approach. A note added in proof acknowledges that D. E. D'Hoker and S. Giddings have completed this proof establishing the equivalence.
scientific article; zbMATH DE number 4036181