Abstract
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 Kähler 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 ζ-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.
Highlights
Two-dimensional gravity on Riemann surfaces has been studied since long by various methods
We formulate and study twodimensional quantum gravity in the Kähler formalism on an arbitrary Riemann surface of genus h, defining the integration measure over the space of metrics and the gravitational action which is an arbitrary combination of the Liouville and Mabuchi functionals
We have studied various two-dimensional quantum gravity partition functions in the path integral approach with various integration measures, the gravitational action being a combination of the Liouville and Mabuchi functionals
Summary
Two-dimensional gravity on Riemann surfaces has been studied since long by various methods. We formulate and study twodimensional quantum gravity in the Kähler formalism on an arbitrary Riemann surface of genus h, defining the integration measure over the space of metrics and the gravitational action which is an arbitrary combination of the Liouville and Mabuchi functionals. Unlike the known quantum field theoretic examples [17], where this anomaly is unphysical and can be absorbed in the local counterterms [18], it does play a non-trivial physical role in the present quantum gravity context This is explained in details in Appendix A where, in particular, we derive a new general multiplicative anomaly formula for Laplace-type operators on arbitrary compact Riemann surfaces. We have included a brief discussion of the sharp cutoff method for the round sphere in Appendix B
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