Hyperbolic $2$-spheres with conical singularities, accessory parameters and Kähler 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 $\alpha =\{\alpha _1,\dots , \alpha _n\}$, generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on ${\mathcal {M}}_{0,n}$ parameterized by the set of orders $\alpha$, explicitly relate accessory parameters to these metrics, and prove that the functions $S_\alpha$ are their Kähler potentials.