Dualities and trialities in N = 2 supersymmetric gauged quantum mechanics

We study new Seiberg-like dualities for 1d N$$ \mathcal{N} $$ = 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 N$$ \mathcal{N} $$ = (0, 2) Gadde-Gukov-Putrov triality.