  We investigate the quantum effects of the non-minimal matter-gravity
couplings derived by Cangemi and Jackiw in the realm of a specific fermionic
theory, namely the abelian Thirring model on a Riemann surface of genus zero
and one. The structure and the strength of the new interactions are seen to be
highly constrained, when the topology of the underlying manifold is taken into
account. As a matter of fact, by requiring to have a well-defined action, we
are led both to quantization rules for the coupling constants and to selection
rules for the correlation functions. Explicit quantum computations are carried
out in genus one (torus). In particular the two-point function and the chiral
condensate are carefully derived for this case. Finally the effective
gravitational action, coming from integrating out the fermionic degrees of
freedoom, is presented. It is different from the standard Liouville one: a new
non-local functional of the conformal factor arises and the central charge is
improved, depending also on the Thirring coupling constant. This last feature
opens the possibility of giving a new explicit representation of the minimal
series in terms of a fermionic interacting model.
