Solutions to SU (n+1) Toda system with cone singularities via toric curves on compact Riemann surfaces

On a compact Riemann surface (X) with finite punctures (P₁, , Pₖ), we define toric curves as multi-valued, totally unramified holomorphic maps to (Pⁿ) with monodromy in a maximal torus of (PSU (n+1) ). Toric solutions for the (SU (n+1) ) system on X\P₁, , Pₖ\ are recognized by their associated toric curves in (Pⁿ). We introduce a character n-ensemble as an (n) -tuple of meromorphic one-forms with simple poles and purely imaginary periods, generating toric curves on (X) minus finitely many points. We establish on X a correspondence between character n-ensembles and toric solutions to the (SU (n+1) ) system with finitely many cone singularities. Our approach not only broadens seminal solutions for up to two cone singularities on the Riemann sphere, as classified by Jost-Wang (Int. Math. Res. Not. , (6): 277-290, 2002) and Lin-Wei-Ye (Invent. Math. , 190 (1): 169-207, 2012), but also advances beyond the limits of Lin-Yang-Zhong's existence theorems (J. Differential Geom. , 114 (2): 337-391, 2020) by introducing a new solution class.