Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler metrics on ℳ_{0,𝓃}

Source abstract

We show that the real-valued function S α S_\alpha on the moduli space M 0 , n {\mathcal {M}}_{0,n} of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic 2 2 -sphere with n ≥ 3 n\geq 3 conical singularities of arbitrary orders α = { α 1 , … , α n } \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 M 0 , n {\mathcal {M}}_{0,n} parameterized by the set of orders α \alpha , explicitly relate accessory parameters to these metrics, and prove that the functions S α S_\alpha are their Kähler potentials.