Motivated by the study of spontaneously broken non-invertible symmetries, I will examine the connection between N=1* gauge theories and elliptic Calogero-Moser (CM) integrable systems, with a focus on the well-studied type A case. N=1* gauge theories are mass-deformations of N=4 super Yang-Mills theories, while elliptic Calogero-Moser systems describe integrable systems of particles on a torus associated with compact or complex simple Lie algebras. A puzzling feature of this correspondence is the fact that there seems to be a one-to-one mapping between the isolated extrema of CM systems and the massive vacua of N=1* theories on R4. These extrema, however, should rather be in one-to-one correspondence with the massive vacua of N=1∗ theories compactified on a circle, which are typically more numerous. I will explain how this apparent discrepancy is resolved by distinguishing global variants of N=1* theories, leading to an association of CM systems with global variants of Lie groups rather than Lie algebras.
Sibasish Banerjee
