The Bethe/Gauge Correspondence

66 Chapter 4. Bethe/gaugethat the resulting vacuum equations correspond to the bae of, respectively, the anisotropic xyzand xxz spin chains [1, §2.2, §2.4].Graphically the situation can be represented as follows:BetheGaugexyz model ←→ T 2 × R 1,1|4 R 3,1|4R ′ → 0xxz model ←→ S 1 R × R1,1|4 R 2,1|4R → 0xxx model ←→ R 1,1|4 R 1,1|4Here the dotted lines denote compactification, and the squiggly lines stand for reduction. Thesquares commute provided the limits of vanishing radius are taken in such a way that the otherparameters stay finite [1]. On the Bethe side the limits of vanishing anisotropy parameters areindicated.The microscopically three and four dimensional theories can be viewed as lifts to higherdimension of the theory that we have studied.Matching vacuum structures. As a small ‘application’ of the Bethe/gauge correspondenceconsider the following two gauge theories [50, §2.3, §3.4]. In both theories we take L f = L¯f = Land L = 1, with corresponding twisted masses ˜m l f= ˜m l¯f = −isu and ˜m a = iu. The firsttheory has gauge groups G 1 = U(N) and the second one G 2 = U(L − N); the FI-parametersare related by r 1 = −r 2 . On the Bethe side it is clear that the resulting vacuum equations arephysically equivalent. Indeed, there are N overturned spins in one case and L − N in the other.Since the length of the spin chain is L in either case, the two systems are related by simplyinterchanging our labels ‘↑’ and ‘↓’, which doesn’t affect the physics. The conclusion is that thevacuum structure of the two gauge theories is the same. For a more involved application alongthese lines see §2.4, §3.5 and §3.6 of [50].Quantization. The Bethe/gauge correspondence is rather reminiscent of Seiberg-Witten theory[65]. In this theory one studies so-called ‘electric-magnetic duality’ in N = 2 supersymmetricnon-abelian gauge theories in four spacetime dimensions.In the low-energy limit we get an abelian theory; we are again interested in the theory onthe Coulomb branch. The analogues of our σ n are usually denoted by a i . Recall that an N = 2vector superfield contains an adjoint N = 1 chiral superfield. The a i are the eigenvalues of thescalar field component of this chiral superfield.The effective theory in the infrared is governed by the prepotential F(a). It is the analogue ofour effective twisted superpotential, and can again be computed exactly; the result also involvesa logarithm. Donagi and Witten showed that this low-energy theory gives rise to a classicalalgebraically integrable system, which is a complex generalization of ordinary classical integrablesystems [66, 67].Together with the Bethe/gauge correspondence, Seiberg-Witten theory can be used to quantizesuch algebraic integrable systems [3]. For this, the four-dimensional N = 2 theories have tobe related to our two-dimensional cN = (2, 2) theories. This is done via the Omega deformation(also known as ‘Nekrasov deformation’) in the so-called Nekrasov-Shatashvili (ns) limit asdescribed in §3.1 of [3].Again we summarize the idea of [3] in a diagram. Seiberg-Witten theory is displayed on theleft and the Bethe/gauge correspondence on the right. The dotted arrow on the bottom denotesthe quantization.