Gauge/Bethe correspondence from quiver BPS algebras

TL;DR

This paper explores the Gauge/Bethe correspondence in 2D supersymmetric quiver gauge theories linked to toric Calabi-Yau three-folds, revealing conditions under which the correspondence holds or encounters obstructions.

Contribution

It constructs integrable models from quiver Yangian representations and clarifies when the Gauge/Bethe correspondence is valid or obstructed in chiral quivers.

Findings

Bethe ansatz equations match vacuum equations for non-chiral quivers

Obstructions arise in chiral quivers preventing the correspondence

Trigonometric versions relate to 3D $ ext{N}=2$ gauge theories

Abstract

We study the Gauge/Bethe correspondence for two-dimensional $\mathcal{N}=(2,2)$ supersymmetric quiver gauge theories associated with toric Calabi-Yau three-folds, whose BPS algebras have recently been identified as the quiver Yangians. We start with the crystal representations of the quiver Yangian, which are placed at each site of the spin chain. We then construct integrable models by combining the single-site crystals into crystal chains by a coproduct of the algebra, which we determine by a combination of representation-theoretical and gauge-theoretical arguments. For non-chiral quivers, we find that the Bethe ansatz equations for the crystal chain coincide with the vacuum equation of the quiver gauge theory, thus confirming the corresponding Gauge/Bethe correspondence. For more general chiral quivers, however, we find obstructions to the $R$-matrices satisfying the Yang-Baxter equations and the unitarity conditions, and hence to their corresponding Gauge/Bethe correspondence. We also discuss trigonometric (quantum toroidal) versions of the quiver BPS algebras, which correspond to three-dimensional $\mathcal{N}=2$ gauge theories and arrive at similar conclusions. Our findings demonstrate that there are important subtleties in the Gauge/Bethe correspondence, often overlooked in the literature.