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

TL;DR

This paper constructs explicit potentials and computes Chern forms for Weil-Petersson and Takhtajan-Zograf metrics on moduli spaces, linking them to hyperbolic volume and extending results to quasi-Fuchsian groups.

Contribution

It introduces new explicit formulas for potentials and Chern forms of these metrics, connecting geometric analysis with hyperbolic volume in moduli space theory.

Findings

Explicit Kähler potential for TZ metric on moduli space.

Chern forms of line bundles expressed in terms of Weil-Petersson and TZ forms.

Potential function matches renormalized hyperbolic volume.

Abstract

For the TZ metric on the moduli space $\mathscr{M}_{0,n}$ of $n$-pointed rational curves, we construct a K\"ahler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space $\mathfrak{S}_{g,n}$ as holomorphic fibration $\mathfrak{S}_{g,n}\rightarrow\mathfrak{S}_{g}$ 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 $\mathscr{L}_{i}$ 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 $\mathscr{L}=\otimes_{i=1}^{n}\mathscr{L}_{i}$ over $\mathfrak{S}_{g,n}$. We explicitly compute the Chern forms of these Hermitian line bundles $$c_{1}(\mathscr{L}_{i},h_{i})=\frac{4}{3}\omega_{\mathrm{TZ},i},\quad c_{1}(\mathscr{L},\exp\{S/\pi\})=\frac{1}{\pi^{2}}\omega_{\mathrm{WP}}.$$ We prove that a smooth real-valued function $-\mathscr{S}=-S+\pi\sum_{i=1}^{n}\log h_{i}$ 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 $(g,n)$.