Quantization of Integrable Systems and a 2d/4d Duality

TL;DR

This paper establishes a duality linking 4d N=2 supersymmetric gauge theories with 2d gauged linear sigma-models, revealing a deep connection between quantum integrable systems, surface operators, and string theory.

Contribution

It introduces a novel duality connecting 4d and 2d supersymmetric theories, relating their chiral rings and providing insights into quantization of integrable systems.

Findings

Coulomb branch quantizes the Heisenberg SL(2) spin chain

Agreement with Bethe Ansatz confirms the duality

Seiberg-Witten solution is captured by 2d one-loop computation

Abstract

We present a new duality between the F-terms of supersymmetric field theories defined in two- and four-dimensions respectively. The duality relates N=2 supersymmetric gauge theories in four dimensions, deformed by an Omega-background in one plane, to N=(2,2) gauged linear sigma-models in two dimensions. On the four dimensional side, our main example is N=2 SQCD with gauge group SU(L) and 2L fundamental flavours. Using ideas of Nekrasov and Shatashvili, we argue that the Coulomb branch of this theory provides a quantization of the classical Heisenberg SL(2) spin chain. Agreement with the standard quantization via the Algebraic Bethe Ansatz implies the existence of an isomorphism between the chiral ring of the 4d theory and that of a certain two-dimensional theory. The latter can be understood as the worldvolume theory on a surface operator/vortex string probing the Higgs branch of the same 4d theory. We check the proposed duality by explicit calculation at low orders in the instanton expansion. One striking consequence is that the Seiberg-Witten solution of the 4d theory is captured by a one-loop computation in two dimensions. The duality also has interesting connections with the AGT conjecture, matrix models and topological string theory where it corresponds to a refined version of the geometric transition.