Global variants of four-dimensional gauge theories are specified by their spectrum of genuine Wilson-’t Hooft line operators. The choice of global variant has significant consequences when spacetime is taken to be ℝ3 × S1. We focus on \( \mathcal{N} \) = 1* theories, which are closely connected to twisted elliptic Calogero-Moser systems. We establish, on general grounds, how this gauge-theoretic topological data manifests itself on the integrable system side by introducing a notion of global variants for complex many-body integrable systems associated with Lie algebras. Focusing on \( \mathcal{N} \) = 1* theories of type A and B2, we elucidate the implications for the structure of gapped vacua, the emergent (generalized) symmetries realized in each vacuum, and the action of spontaneously broken modular invariance.
