Global variants of $\mathcal{N}=1^{\ast }$ theories and Calogero-Moser systems

Abstract

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 ${\mathbb{R}}^3\times S^1$. We focus on $\mathcal{N}=1^{\ast }$ 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 $\mathcal{N}=1^{\ast }$ 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.

arXiv commentv1: 69 pages, 27 figures; v2: 74 pages, 27 figures, final version

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