Global variants of N = 1* theories and Calogero-Moser systems

A bstract 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 ℝ 3 × S 1 . We focus on N 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 N N = 1 * theories of type A and B 2 , 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.