Field Theories on the Poincar\'e Disk

The massive scalar field theory and the chiral Schwinger model are quantized on a Poincar\'e disk of radius $\rho$. The amplitudes are derived in terms of hypergeometric functions. The behavior at long distances and near the boundary of some of the relevant correlation functions is studied. The exact computation of the chiral determinant appearing in the Schwinger model is obtained exploiting perturbation theory. This calculation poses interesting mathematical problems, as the Poincar\'e disk is a noncompact manifold with a metric tensor which diverges approaching the boundary. The results presented in this paper are very useful in view of possible extensions to general Riemann surfaces. Moreover, they could also shed some light in the quantization of field theories on manifolds with constant curvature scalars in higher dimensions.