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

Abstract

For the TZ metric on the moduli space ${\mathcal{M}}_{0,n}$ 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 ${\mathfrak{S}}_{g,n}$ as holomorphic fibration ${\mathfrak{S}}_{g,n}\to {\mathfrak{S}}_g$ over the Schottky space ${\mathfrak{S}}_g$ of compact Riemann surfaces of genus $g$, where the fibers are configuration spaces of $n$ points. For the tautological line bundles ${\mathcal{L}}_i$ over ${\mathfrak{S}}_{g,n}$ 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 $\exp \{S/\pi \}$ is a Hermitian metric in the line bundle $\mathcal{L}={\otimes }_{i=1}^n{\mathcal{L}}_i$ over ${\mathfrak{S}}_{g,n}$. We explicitly compute the Chern forms of these Hermitian line bundles $$c_1({\mathcal{L}}_i,h_i)=\frac{4}{3}{\omega }_{\mathrm{TZ},i},c_1(\mathcal{L},\exp \{S/\pi \})=\frac{1}{{\pi }^2}{\omega }_{\mathrm{WP}}.$$ We prove that a smooth real-valued function $-\mathcal{S}=-S+\pi {\sum }_{i=1}^n\log h_i$ on ${\mathfrak{S}}_{g,n}$, 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 $(g,n)$.

arXiv comment36 pages, Typos corrected and presentation improved