Spin Chain Integrability as a Supersymmetric Gauge Duality

TL;DR

This paper establishes a new exact duality between 4D supersymmetric gauge theories and open XYZ spin chains with elliptic R-matrices, extending the Bethe/gauge duality to higher dimensions and boundary conditions.

Contribution

It constructs the first exact duality linking 4D BCD-type gauge theories with general-boundary XYZ spin chains, broadening the scope of gauge-spin chain correspondences.

Findings

Unified duality framework across dimensions via the $oldsymbol{ ext{Omega}}$-deformation

Exact correspondence between 4D $oldsymbol{ ext{N=1}}$ gauge theories and XYZ spin chains

Extension of Bethe/gauge duality to boundary conditions and elliptic R-matrices

Abstract

We establish a novel correspondence between 4D $\mathcal N=1$ supersymmetric gauge theories on $D^2\times T^2$ and open XYZ spin chains with generalized boundary conditions, extending beyond previous 3D Bethe/gauge duality frameworks. Our primary contribution is the rigorous construction of the first exact duality between 4D BCD-type gauge theories and general-boundary XYZ spin chains governed by elliptic $R$-matrices. This framework provides a universal mechanism for resolving supersymmetric gauge theory/spin-chain duality across dimensional hierarchies: 2D $\mathcal{N}=(2,2)$ $\longleftrightarrow$ XXX spin chain, 3D $\mathcal{N}=2$ $\longleftrightarrow$ XXZ spin chain, 4D $\mathcal{N}=1$ $\longleftrightarrow$ XYZ spin chain, mediated through the $\Omega$-deformation parameter $\epsilon$.