Extended 2d generalized dilaton gravity theories

We show that an anomaly-free description of matter in (1+1) dimensions requires a deformation of the 2d relativity principle, which introduces a non-trivial center in the 2d Poincare algebra. Then we work out the reduced phase-space of the anomaly-free 2d relativistic particle, in order to show that it lives in a noncommutative 2d Minkowski space. Moreover, we build a Gaussian wave packet to show that a Planck length is well-defined in two dimensions. In order to provide a gravitational interpretation for this noncommutativity, we propose to extend the usual 2d generalized dilaton gravity models by a specific Maxwell component, which gauges the extra symmetry associated with the center of the 2d Poincare algebra. In addition, we show that this extension is a high energy correction to the unextended dilaton theories that can affect the topology of space-time. Further, we couple a test particle to the general extended dilaton models with the purpose of showing that they predict a noncommutativity in curved space-time, which is locally described by a Moyal star product in the low energy limit. We also conjecture a probable generalization of this result, which provides a strong evidence that the noncommutativity is described by a certain star product which is not of the Moyal type at high energies. Finally, we prove that the extended dilaton theories can be formulated as Poisson-Sigma models based on a nonlinear deformation of the extended Poincare algebra.