Relating Gauge Theories via Gauge/Bethe Correspondence

TL;DR

This paper leverages the Gauge/Bethe correspondence to relate three seemingly distinct ${\mathcal N}=(2,2)$ quiver gauge theories in two dimensions by identifying their supersymmetric vacua with the Bethe spectra of integrable spin chains. By exploiting the fact that integrable models can admit multiple, equivalent Bethe equation formulations, the authors show that three different quiver theories—Case A, Case B, and Case C—have the same chiral ring and hence identical supersymmetric ground states, despite differing gauge groups and matter content. The analysis centers on the XXX and $tJ$ spin chains, with $sl(1|2)$ symmetry for the latter, and constructs explicit mappings between the twisted superpotentials $\widetilde{W}_{\text{eff}}(\sigma)$ and Yang–Yang functions $Y(\lambda)$, establishing a precise dictionary between gauge-theory data and integrable-system data. Additionally, the work embeds these correspondences in a Type IIA brane setup, showing that the three quiver theories can be connected via brane motions in the massless limit. Overall, the approach offers a framework for deriving nontrivial relations among gauge theories through integrable models and suggests broader applicability beyond the ground-state sector.

Abstract

In this note, we use techniques from integrable systems to study relations between gauge theories. The Gauge/Bethe correspondence, introduced by Nekrasov and Shatashvili, identifies the supersymmetric ground states of an N=(2,2) supersymmetric gauge theory in two dimensions with the Bethe states of a quantum integrable system. We make use of this correspondence to relate three different quiver gauge theories which correspond to three different formulations of the Bethe equations of an integrable spin chain called the tJ model.

Figures (4)

• Figure 1: Quiver diagrams for the three example theories. The twisted masses for the chiral fields are given in red.
• Figure 2: Example quiver diagram for the Gauge/Bethe correspondence. Gauge groups are labeled in black, matter fields in blue, the corresponding twisted masses in red.
• Figure 3: Brane transitions connecting the quiver gauge theories of cases A, B, C
• Figure 4: Equivalent integration contours around the poles of $p(z)$.