  We study a new two-dimensional quantum gravity theory, based on a
gravitational action containing both the familiar Liouville term and the
Mabuchi functional, which has been shown to be related to the coupling of
non-conformal matter to gravity. We compute the one-loop string susceptibility
from a first-principle, path integral approach in the Kahler parameterization
of the metrics and discuss the particularities that arise in the case of the
pure Mabuchi theory. While we mainly use the most convenient spectral cutoff
regularization to perform our calculations, we also discuss the interesting
subtleties associated with the multiplicative anomaly in the familiar
zeta-function scheme, which turns out to have a genuine physical effect for our
calculations. In particular, we derive and use a general multiplicative anomaly
formula for Laplace-type operators on arbitrary compact Riemann surfaces.
