Teichmüller spaces of string theory
The author gives a rigorous mathematical analysis of string theory. In Section 2 he shows for a Lorentzian world sheet swept by a string propagation in \(\mathbb{R}^{I,D}\) that a one-parameter family of conformally equivalent Riemann structures on the sheet forms an infinite geodesic in the appropriate Teichmüller space, the so called \(P\)-line. In Section 3 the author investigates the harmonic \(P\)-lines which are induced by harmonic homothetic Gauss maps. Considering Bers embedding of the Teichmüller space to the Banach space of bounded quadratic differentials the author shows in Section 4 that if the \(P\)-line relates to a quadratic differential of Jenkins-Strebel type (i.e. holomorphic quadratic differential that has closed horizontal and vertical trajectories) then endpoints of the \(P\)-line are represented by boundary points of that embedding and that leads to the decomposition of the world sheet onto one or more Riemann surfaces. In physical language it means that the \(P\)-lines corresponding to Jenkins-Strebel differentials describes world sheet which has to be created and to decay. In Section 5 the author proves that if the Jenkins-Strebel differential determining the \(P\)-line has only one horizontal and vertical cylinder of the same heights and circumferences then the space-time has to be discrete. At last in Section 6 the author introduces in Teichmüller space the global Fenchel-Nielsen coordinates and shows that the \(P\)-line corresponding to the Jenkins-Strebel differentials with \(3p-3\) cylinders \((p\) is a genus) can be related to a Hamiltonian system with respect to the Weil- Petersson-Kähler symplectic form.
scientific article; zbMATH DE number 445105
scientific article