A class of two dimensional field theories, based on (generically degenerate) Poisson structures and generalizing gravity-Yang-Mills systems, is presented. Locally, the solutions of the classical equations of motion are given. A general scheme for the quantization of the models in a Hamiltonian formulation is found.

We introduce a new topological sigma model, whose fields are bundle maps from the tangent bundle of a 2-dimensional world-sheet to a Dirac subbundle of an exact Courant algebroid over a target manifold. It generalizes simultaneously the (twisted) Poisson sigma model as well as the G/G-WZW model. The equations of motion are satisfied, iff the corresponding classical field is a Lie algebroid morphism. The Dirac Sigma Model has an inherently topological part as well as a kinetic term which uses a metric on worldsheet and target. The latter contribution serves as a kind of regulator for the theory, while at least classically the gauge invariant content turns out to be independent of any additional structure. In the (twisted) Poisson case one may drop the kinetic term altogether, obtaining the WZ-Poisson sigma model; in general, however, it is compulsory for establishing the morphism property.

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We consider the topological gauged WZW model in the generalized momentum representation. The chiral field g is interpreted as a counterpart of the electric field E of conventional gauge theories. The gauge dependence of wave functionals Ψ(g) is governed by a new gauge cocycle φ GW ZW . We evaluate this cocycle explicitly using the machinery of Poisson σ-models. In this approach the GWZW model is reformulated as a Schwarz type topological theory so that the action does not depend on the worldsheet metric. The equivalence of this new formulation to the original one is proved for genus one and conjectured for an arbitrary genus Riemann surface. As a by-product we discover a new way to explain the appearance of Quantum Groups in the WZW model. ETH-TH/95-14TUW-95-07 PITHA -95/9 ESI 225 (1995) hep-th/9505012 * On leave of absence from Steklov Mathematical Institute,

For a 1+1 dimensional theory of gravity with torsion different approaches to the formulation of a quantum theory are presented. They are shown to lead to the same finite dimensional quantum system. Conceptual questions of quantum gravity like e.g. the problem of time are discussed in this framework.