# Nested coordinate Bethe wavefunctions from the Bethe/gauge correspondence

## TL;DR

This paper extends the Bethe/gauge correspondence to derive nested coordinate Bethe wavefunctions for higher-rank spin chains using $A_M$ quiver gauge theories, including Higgsing procedures and parameter specializations.

## Contribution

It generalizes the Bethe/gauge correspondence to $A_M$ quivers, producing nested wavefunctions and interpreting Higgsing and Hanany-Witten moves in spin-chain terms.

## Key findings

- Derived $rak{su}(M+1)$ nested Bethe wavefunctions from $A_M$ quiver gauge theories.
- Extended the construction to XXZ spin chains.
- Connected Higgsing procedures to Izergin-Korepin parameter specializations.

## Abstract

In [1, 2], Nekrasov applied the Bethe/gauge correspondence to derive the $\mathfrak{su}\, (2)$ XXX spin-chain coordinate Bethe wavefunction from the IR limit of a 2D $\mathcal{N}=(2, 2)$ supersymmetric $A_1$ quiver gauge theory with an orbifold-type codimension-2 defect. Later, Bullimore, Kim and Lukowski implemented Nekrasov's construction at the level of the UV $A_1$ quiver gauge theory, recovered his result, and obtained further extensions of the Bethe/gauge correspondence [3]. In this work, we extend the construction of the defect to $A_M$ quiver gauge theories to obtain the $\mathfrak{su} \, ( M + 1 )$ XXX spin-chain nested coordinate Bethe wavefunctions. The extension to XXZ spin-chain is straightforward. Further, we apply a Higgsing procedure to obtain more general $A_M$ quivers and the corresponding wavefunctions, and interpret this procedure (and the Hanany-Witten moves that it involves) on the spin-chain side in terms of Izergin-Korepin-type specializations (and re-assignments) of the parameters of the coordinate Bethe wavefunctions.