Solutions to $\mathrm{S}\mathrm{U}(n+1)$ Toda system with cone singularities via toric curves on compact Riemann surfaces

On a compact Riemann surface (X) with finite punctures (P_1, \ldots, P_k), we define toric curves as multi-valued, totally unramified holomorphic maps to ($\mathbb{P}$^n) with monodromy in a maximal torus of ({\rm PSU}(n+1)). $\mathit{\text{Toric}}\mathit{\text{solutions}}$ for the ({\rm SU}(n+1)) system on $X\setminus \{P_1,\dots ,P_k\}$ are recognized by their associated {\it toric} curves in ($\mathbb{P}$^n). 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 ({\rm 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.