2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions

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.