Integrable classical and quantum gravity

The lectures are divided into four parts. We first describe how the dimensional reduction of 3d gravity-coupled non-linear #sigma#-models gives rise to the particular 2d models studied in these lectures, of which pure Einstein gravity with two commuting Killing vectors is the simplest example. Chapter 3 deals with the classical integrability of the model, that is manifest in the existence of a linear system. The spectral parameter current is introduced as new fundamental quantity and the dynamics studied in these terms. Restriction to an somonodromic ansatz results in a set of decoupled differential equations completely describing the dynamics; the relaxation of these restrictions is briefly sketched in the end. In Chapter 4 we set up a Hamiltonian framework for the model, such that translations along the light-cone are governed by two independent Hamiltonians and the phase space is essentially attached to one point in space-time. We compare the new scheme to the conventional Hamiltonian approach and discuss some open questions. Exploiting the new Hamiltonian structure, we develop the quantization of the model in Chapter 5. The Wheeler-DeWitt equations that identify physical states in the quantum theory are explicitly stated. Physical states solving these equations are built from solutions of a modified version of the Knizhnik-Zamolodchikov system. Finally, the appendix sketches the extension of the formalism to supersymmetric models. (orig.)Available from TIB Hannover: RA 2999(96-249) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman