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

Abstract

We show that the real-valued function $S_\alpha$ 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 $\alpha=\{\alpha_1,...,\alpha_n\}$, 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 $\alpha$, explictly relate accessory parameters to these metrics, and prove that the functions $S_\alpha$ are their Kaehler potentials.