Dualities and trialities in $\mathcal{N}=2$ supersymmetric gauged quantum mechanics

We study new Seiberg-like dualities for 1d $\mathcal{N}=2$ supersymmetric gauge theories – that is, supersymmetric gauged quantum mechanics – with unitary gauge group and (anti)fundamental matter in chiral and fermi multiplets, and with non-zero Fayet–Iliopoulos parameter. Similarly to its higher-dimensional analogues, this 1d Seiberg duality is an infrared duality: the supersymmetric ground states of the dual gauge theories match exactly. We provide strong evidence for the dualities, including the matching of the flavoured Witten indices, a Higgs-branch derivation in terms of dual Grassmannian manifolds, and a detailed study of the Coulomb-branch ground states in the abelian case. We study how the supersymmetric ground states, in either dual description, depend on the sign of the Fayet–Iliopoulos parameter, and we explore the corresponding wall-crossing phenomenon. For some special values of the discrete parameters defining the unitary gauge theory, the dualities, combined with trivial wall-crossing, enhances to a triality. This includes, as a special case, the dimensional reduction to 1d of the 2d $\mathcal{N}=(0,2)$ Gadde–Gukov–Putrov triality.