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

Following A.B. Givental (1989) [5] , we refer to an n -tuple ( ω 1 , … , ω n ) of Kähler forms on a Riemann surface S as a solution to the SU ( n + 1 ) Toda system if and only if ( Ric ( ω 1 ) , … , Ric ( ω n ) ) = ( 2 ω 1 , … , 2 ω 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 ( i ( n + 1 − i ) ω ) 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] .