Two-dimensional effective action for matter fields coupled to the dilaton

We revise the calculation of the one-loop effective action for scaler and spinor fields coupled to the dilaton in two dimensions. Applying the method of covariant perturbation theory for the heat kernel we derive the effective action in an explicitly covariant form that produces both the conformally invariant and the conformally anomalous terms. For scalar fields the conformally invariant part of the action is nonlocal. The obtained effective action is proved to be infrared finite. We also compute the one-loop effective action for scalar fields at finite temperature.