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

Abstract

For the TZ metric on the moduli space M0,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 Sg,n as holomorphic fibration 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 Sg,n, we define Hermitian metrics 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 exp⁡{S/π} is a Hermitian metric in the line bundle L=⊗i=1nLi over Sg,n. We explicitly compute the Chern forms of these Hermitian line bundlesc1(Li,hi)=43ωTZ,i,c1(L,exp⁡{S/π})=1π2ωWP. We prove that a smooth real-valued function −S=−S+π∑i=1nlog⁡hi on 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 (g,n).