A New 2d/4d Duality via Integrability

TL;DR

The paper proves a 2d/4d duality between four-dimensional ${\cal N}=2$ theories in an ${\Omega}$-background and two-dimensional ${\cal N}=(2,2)$ theories by performing a saddle-point analysis of the Nekrasov partition function in the Nekrasov-Shatashvili limit. The key step is deriving the Bethe Ansatz Equations of the $SL(2,\mathbb{R})$ spin chain from the instanton density in the NS limit, and showing that under a quantization condition the system truncates to a finite Baxter equation whose solutions match the 2d theory’s Yang-Yang functional, thereby equating the on-shell twisted superpotentials. The authors extend the duality to linear quiver gauge theories, identifying 2d duals with corresponding quivers and demonstrating perturbative matching of the twisted superpotentials. This work connects protected holomorphic data in 4d ${\cal N}=2$ theories to quantum integrable systems and provides a unifying framework that intersects with integrability, Seiberg-Witten theory, and AGT-type correspondences.

Abstract

We prove a duality, recently conjectured in arXiv:1103.5726, which relates the F-terms of supersymmetric gauge theories defined in two and four dimensions respectively. The proof proceeds by a saddle point analysis of the four-dimensional partition function in the Nekrasov-Shatashvili limit. At special quantized values of the Coulomb branch moduli, the saddle point condition becomes the Bethe Ansatz Equation of the SL(2) Heisenberg spin chain which coincides with the F-term equation of the dual two-dimensional theory. The on-shell values of the superpotential in the two theories are shown to coincide in corresponding vacua. We also identify two-dimensional duals for a large set of quiver gauge theories in four dimensions and generalize our proof to these cases.

Figures (2)

• Figure 3.1: The IIA-brane construction for Theory I in the linear quiver case.
• Figure 3.2: The IIA-brane brane construction for Theory II in the linear quiver case.