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

For the TZ metric on the moduli space M-0,M-n of n-pointed rational curves, we construct a Kahler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space S-g,S-n as holomorphic fibration S-g,S-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,S-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/pi} is a Hermitian metric in the line bundle L = circle times(n)(i=1) L-i over S-g,S-n We explicitly compute the Chern forms of these Hermitian line bundles c(1)(L-i, h(i)) = 4/3 omega Tz,i, c(1) (L, exp{S/pi}) = 1/pi(2)omega WP. We prove that a smooth real-valued function -S = -S + pi Sigma(n)(i=1) log h(i) on S-g,S-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). (C) 2016 Elsevier Inc. All rights reserved.