Hyperbolic 2-spheres with conical singularities, accessory parameters and Kaehler metrics on $\mathcal{M}_{0,n}$

TL;DR

This work addresses how a hyperbolic metric with conical singularities on the sphere yields both the accessory parameters of the associated Fuchsian equation and a family of Kaehler metrics on $\mathcal{M}_{0,n}$. It proves, for conical orders $α$ with $α_i\in(0,1)$ and $\sum α_i>2$, that the critical value $S_α$ of the Liouville action satisfies $c_i=-\frac{1}{2\pi}\frac{\partial S_α}{\partial z_i}$ and that $\frac{\partial c_i}{\partial \bar{z}_k}=\frac{1}{2\pi}\langle \partial/\partial z_i, \partial/\partial z_k\rangle_α$, where the right-hand side defines a Weil-Petersson-like metric. Moreover, the second derivatives of $S_α$ yield a Kaehler structure on $\mathcal{M}_{0,n}$ with $-S_α$ as its potential, achieved via a novel $α$-dependent kernel construction. The results generalize previous Z-T2 findings to general conical singularities and are obtained through elementary, analytic methods that avoid Teichmüller theory, highlighting direct links between Liouville theory, accessory parameters, and Kaehler geometry on moduli spaces.

Abstract

We show that the real-valued function $S_α$ on the moduli space $\mathcal{M}_{0,n}$ of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic 2-sphere with $n\geq 3$ conical singularities of arbitrary orders $α=\{α_1,...,α_n\}$, generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kaehler metrics on $\mathcal{M}_{0,n}$ parameterized by the set of orders $α$, explictly relate accessory parameters to these metrics, and prove that the functions $S_α$ are their Kaehler potentials.

Theorems & Definitions (20)

• Remark 1
• Lemma 1
• Remark 2
• Lemma 2
• proof
• Remark 3
• Lemma 3
• Lemma 4
• proof
• Remark 4