Poisson-sigma model for 2D gravity with non-metricity

We present a Poisson-sigma model describing general 2D dilaton gravity with non-metricity, torsion and curvature. It involves three arbitrary functions of the dilaton field, two of which are well-known from metric compatible theories, while the third one characterizes the local strength of non-metricity. As an example we show that alpha corrections in 2D string theory can generate (target space) non-metricity.