Solutions to SU(n + 1) Toda system generated by spherical metrics

Abstract

Following A.B. Givental (1989) [5], we refer to an n-tuple $({\omega }_1,\dots ,{\omega }_n)$ of Kähler forms on a Riemann surface S as a solution to the SU$(n+1)$ Toda system if and only if$\left(\mathrm{Ric}\right({\omega }_1),\dots ,\mathrm{Ric}({\omega }_n\left)\right)=(2{\omega }_1,\dots ,2{\omega }_n)C_n,$ where $C_n$ is the Cartan matrix of type $A_n$. In particular, when $n=1$, this solution corresponds to a spherical metric. Using the correspondence between solutions and totally unramified unitary curves, we show that a spherical metric ω generates a family of solutions, including ${\left(i\right(n+1-i\left)\omega \right)}_{i=1}^n$. Moreover, we characterize this family in terms of the monodromy group of the spherical metric. As a consequence, we obtain a new solution class to the SU$(n+1)$ Toda system with cone singularities on compact Riemann surfaces, complementing the existence results of Lin et al. (2020) [9].