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

TL;DR

This paper introduces new Seiberg-like dualities for 1d $ abla=2$ supersymmetric gauge theories, demonstrating their equivalence through ground state matching, index calculations, and exploring wall-crossing and triality phenomena.

Contribution

It presents the first detailed study of dualities and trialities in 1d $ abla=2$ supersymmetric gauged quantum mechanics, extending higher-dimensional dualities to quantum mechanics.

Findings

Matching of flavoured Witten indices confirms dualities.

Higgs-branch derivation via dual Grassmannians supports duality.

Wall-crossing behavior depends on FI parameter sign.

Abstract

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.