Quantization of integrable systems and a 2d/4d duality

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 Ω-background in one plane, to N = (2, 2) gauged linear σ-models in two dimensions. On the four dimensional side, our main example is N = 2 SQCD with gauge group G = SU(L) and N F = 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 SeibergWitten 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.

On integrable structure and geometric transition in supersymmetric gauge theories

“…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.…”

“…Here {a l } are the vevs of the adjoint scalar in the vector multiplet, and n l ∈ Z is the unit of quantized electromagnetic flux under l-th U (1) factor, and as discussed extensively in [1], the condition (2.1) can be interpreted as the quantization of a special locus in the moduli space of Theory I. In particular when n l = 0 (2.1) coincides precisely with the "root of baryonic Higgs branch" on the moduli space.…”

“…Here q = e 2πiτ = (−1) Nc+1q and {λ li } are the vevs of the adjoint scalar in the vector multiplet, and we have introduced the double index notation li, where {n l } satisfy Nc l=1 n l = K, this can be most easily understood from the D-brane picture that the theory is now in the Higgs phase and we are distributing K different vevs among N c different mass parameters {M l } [1], n l is the number of the vevs associated with M l .…”

“…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.…”

“…In this note we shall consider an alternative but equivalent string realization of the aforementioned field theories 1 . Using the refined topological strings [8][9][10] 2 it turns out that the theories on both sides of correspondence can be realized as open (2d) and closed (4d) refined topological string amplitudes respectively.…”