Abstract
It was recently shown that other functionals contribute to the effective action for the Liouville field when considering massive matter coupled to two-dimensional gravity in the conformal gauge. The most important of these new contributions corresponds to the Mabuchi functional. We propose a minisuperspace action that reproduces the main features of the Mabuchi action in order to describe the dynamics of the zero-mode. We show that the associated Hamiltonian coincides with the (quantum mechanical) Liouville Hamiltonian. As a consequence the Liouville theory and our model of the Mabuchi theory both share the same spectrum, eigenfunctions and - in this approximation - correlation functions.
Highlights
As a first step towards understanding four-dimensional quantum gravity, one may consider studying twodimensional gravity as a toy model since many computations can be carried out exactly
The latter is described by an effective action Sgrav[g0, φ] that arises from quantum effects and which provides dynamics to the metric which otherwise
We have shown that it coincides with the spectrum of Liouville theory
Summary
As a first step towards understanding four-dimensional quantum gravity, one may consider studying twodimensional gravity as a toy model since many computations can be carried out exactly. In this letter we ignore the terms denoted by the dots and focus on the Mabuchi action SM [g0, φ] For example it arises as the leading term in a small mass expansion in the case where the matter is a massive scalar field with a coupling to the curvature [6]. This action has been largely studied in differential geometry, starting with the seminal work [7], but it did not appear in physics until recently [5, 6]. Additional considerations on the Mabuchi theory (including the derivation of our model) and the coupling of massive matter to 2d gravity will be presented elsewhere [16]
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