Quantization of Integrable Systems and a 2d/4d Duality
Nick Dorey, Timothy J. Hollowood, Sungjay Lee
TL;DR
The authors establish a precise 2d/4d duality linking 4d ${\cal N}=2$ SQCD in an ${\Omega}$-background to a 2d ${\cal N}=(2,2)$ GLSM, via a shared quantization of an underlying classical integrable system. The Coulomb branch of the 4d theory yields the NS quantization of a classical $SL(2)$ spin chain, while the 2d theory on a surface operator encodes the same Bethe Ansatz structure as the spin chain, leading to a robust chiral-ring isomorphism. They perform detailed weak-coupling checks, matching classical, one-loop, and instanton contributions up to two instantons with corresponding 2d results, after a precise map of parameters including $N+L=\sum_l n_l$, $\hat{\tau}=\tau+\frac{1}{2}(N+1)$, and shifts in masses by $\epsilon$. The work connects to broader themes in supersymmetric gauge theory, including the AGT conjecture, matrix models, and refined topological strings, and suggests a refined geometric transition interpretation in the NS limit.
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.