A new 2d/4d duality via integrability

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.

On integrable structure and geometric transition in supersymmetric gauge theories

“…We can easily generalize the result proved in [2] by attaching to them a compactified S 1 of radius R, which can be identified with the M-theory cycle. As the result, .…”

“…Here we begin by discussing the simple K-theoretic generalization of the exact correspondence proposed in [1,2], and in the later sections we shall consider the realization of these gauge theories in the M-theory interpretation of the refined topological string, and how the exact correspondence discovered in [1] can be recast as a beautiful realization of refined geometric transition. We can easily generalize the result proved in [2] by attaching to them a compactified S 1 of radius R, which can be identified with the M-theory cycle.…”

“…To extend the exact correspondence in [1,2] which gives a one to one map between the chiral rings of certain four and two dimensional supersymmetric gauge theories, our starting point is the instanton partition function for Theory I computed by localization techniques in [19,20] . This amounts to topologically twisting the gauge theory on Omega-deformed background and summing over the fixed points of the action of cohomological charges.…”

“…is dominated by an infinite set of saddle points, as can be seen by the variation of density function ρ(x) or equivalently the end points {X li }. However it was shown in [2] that we can truncate such infinite set by imposing the quantization condition on the cycles corresponding to the undeformed Higgs branch root condition (2.1). The quantization condition is equivalent to truncate the Young diagram Y li to only n l columns, i.e.…”

“…As an initial step, in [1,2] a new exact correspondence between the quantum vacua of four dimensional N = 2 supersymmetric gauge theories and two dimensional N = (2, 2) gauged linear sigma models was proposed, as both sets of vacua can be identified with the eigenstates of the same quantum integrable Hamiltonian under specific conditions for the parameters. More explicitly, here we introduced the so-called Omega deformation in two of the four dimensions in the N = 2 gauge theory [3], and the resultant equivariant prepotential was proposed to be the Yang-Yang functional [4] for quantizing the classical integrable system from the Seiberg-Witten curve.…”