Confinement and screening of the Schwinger model on the Poincare half plane

We discuss the confining features of the Schwinger model on the Poincare ha lf plane. We show that despite the fact that the expectation value of the l arge Wilson loop of massless Schwinger model displays perimeter behavior, t he system can be in confining phase due eo the singularity of the metric at horizontal axis. It is also shown that in the quenched Schwinger model, th e area dependence of the Wilson loop, in contrast to the flat case, is a no t a sign of confinement and the model has a finite energy even for large ex ternal charges separation. The presence of dynamical fermions cannot modify the screening or the confining behavior of the system. Finally we show tha t in the massive Schwinger model, the system is again in screening phase. T he zero curvature limit of the solutions is also discussed.