Global forms of $\mathcal{N}=4$ theories and non-minimal Seiberg-Witten solutions

Abstract: To each four dimensional $\mathcal{N}\geq 2$ supersymmetric quantum field theory, one can associate an algebraic completely integrable (ACI) system that encodes the low energy dynamics of theory. In this paper we explicitly derive the appropriate ACI systems for the global forms of $\mathcal{N}=4$ super Yang-Mills (sYM) using isogenies of polarised abelian varieties. In doing so, we relate the complex moduli of the resulting varieties to the exactly marginal coupling of the theory, thus allowing us to probe the $S$-duality groups of the global forms. Finally, we comment on whether the resulting varieties are the Jacobians of a minimal genus Riemann surface, coming to the conclusion that many global forms of $\mathcal{N}=4$ sYM do not admit a minimal genus Seiberg-Witten curve that correctly reproduces the global form.