Potentials and Chern forms for Weil–Petersson and Takhtajan–Zograf metrics on moduli spaces

For the TZ metric on the moduli space M_0,n of n-pointed rational curves, we construct a Kähler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space S_g,n as holomorphic fibration S_g,n→S_g over the Schottky space S_g of compact Riemann surfaces of genus g, where the fibers are configuration spaces of n points. For the tautological line bundles L_i over S_g,n, we define Hermitian metrics h_i 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 exp⁡{S/π} is a Hermitian metric in the line bundle L=⊗_i=1ⁿL_i over S_g,n. We explicitly compute the Chern forms of these Hermitian line [formula ommitted] We prove that a smooth real-valued function −S=−S+π∑_i=1ⁿlog⁡h_i on S_g,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 (g,n).