Gravity in Two Spacetime Dimensions

TL;DR

This thesis explores exactly solvable models of gravity in two dimensions, detailing their classical solutions, quantization, and relations to Poisson sigma models, advancing understanding of quantum gravity in simplified settings.

Contribution

It provides a comprehensive introduction to 2D gravitational models, including new insights into their solvability, classical solutions, and quantization methods using Poisson sigma models.

Findings

Exact solutions for 2D gravity-Yang-Mills systems

Quantization of models yields all physical quantum states

Relations among different 2D gravitational actions clarified

Abstract

In this habilitation thesis we provide an introduction to gravitational models in two spacetime dimensions. Focus is put on exactly solvable models. We begin by introducing and motivating different possible gravitational actions, including those of generalized dilaton theories as well as of purely geometrical, higher derivative theories with and without torsion. The relation among them as well as to Poisson sigma models is worked out in some detail. In the exactly solvable cases, such as pure gravity-Yang-Mills systems, the general solution to the field equations on a global level is reviewed. Quantization of such models is performed in the Dirac approach, where, by use of the formulation as Poisson sigma models, all admissible physical quantum states are obtained. Table of contents: 1. Introduction, 2. 2d geometry and gravitational actions, 3. Generalized dilaton theories and matter actions, 4. 2d gravity-Yang-Mills systems in terms of Poisson sigma models, 5. Classical solutions on a local level, 6. Classical solutions on a global level, 7. Towards quantum gravity. (In part this work contains/summarizes previous joint work with T. Kloesch and P. Schaller).