For the TZ metric on the moduli space bcc9b6" title="Click to view the MathML source">M0,n of n A0;-pointed rational curves, we construct a Kähler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space e4f36b606c622a5" title="Click to view the MathML source">Sg,n as holomorphic fibration bc7c05f764baad31d315e4" title="Click to view the MathML source">Sg,n→Sg over the Schottky space Sg of compact Riemann surfaces of genus g, where the fibers are configuration spaces of n points. For the tautological line bundles Li over e4f36b606c622a5" title="Click to view the MathML source">Sg,n, we define Hermitian metrics e6237c51f5e31429f4e8181c4cca" title="Click to view the MathML source">hi in terms of Fourier coefficients of a covering map J of the Schottky domain. We define the regularized classical Liouville action S and show that 842087c5a55780dd6e3" title="Click to view the MathML source">exp{S/π} is a Hermitian metric in the line bundle bc83484836b060081b10fc5db7c461a"> over e4f36b606c622a5" title="Click to view the MathML source">Sg,n. We explicitly compute the Chern forms of these Hermitian line bundles
We prove that a smooth real-valued function on e4f36b606c622a5" title="Click to view the MathML source">Sg,n, a potential for this special difference of WP and TZ metrics, coincides with the renormalized hyperbolic volume of a corresponding Schottky 3-manifold. We extend these results to the quasi-Fuchsian groups of type e582d856" title="Click to view the MathML source">(g,n).