The Bethe/Gauge Correspondence

4.2. Further topics 654.1.3 DictionaryWe can summarize everything we have seen so far in the dictionary that is provided by theBethe/gauge correspondence:BetheGaugeN = (2, 2) sym withmassive matter,L a = 1, ˜m a = iu, andsuperpotential (4.7)low energy limitspin chain ←→vacuum structureon Coulomb brachbae ←→ vacuum equationsN-particle sector ←→ G = U(N)rapidities λ n ←→ susy vacua σ nlength L ←→ L f = L¯f = LY (λ) ←→ ˜Weff (σ)twist parameter ϑ ←→ τ = ir + ϑ/2πlocal spins s l andinhomogeneities ν l←→˜m l f = (ν l − is l ) u˜m l¯f = (−ν l − is l ) uWe can use this dictionary to translate things we know on one side to the other side. In thisway we may hope to obtaining interesting new statements and uncover new structures. We willmention some developments along these lines in Section 4.2.A still more elaborate version of the dictionary, including the Kirillov-Reshetikhin modulesthat we touched upon in Section 2.2.3, can be found in §3.3 of [50].4.2 Further topicsTo conclude this chapter we outline three further aspects of the Bethe/gauge correspondence.Anisotropy/higher dimensions. Consider an N = 1 supersymmetric gauge theory in fourdimensions. By dimensional reduction to two dimensions we get a N = (2, 2) theory thathas all the nice properties which we already listed in Section 1.3.3 and which we need for theBethe/gauge correspondence to work. However, instead of getting rid of the two dimensions atonce, we can proceed in smaller steps.We start with a superspace (isomorphic to) R 3,1|4 as described in Section 1.2.2. Instead ofreducing the spacetime we can compactify one of the spatial dimensions to a circle SR 1 withradius R, resulting in the superspace SR 1 × R2,1|4 . The component of the four-vector fieldcorresponding to the compactified dimension yields Kaluza-Klein modes with masses that areinversely proportional to the radius R. At low energy the resulting theory is effectively threedimensional.When we let R become very small, the kk modes become very massive, and canbe integrated out.We are on the right track: if we decrease the dimension by one more we get a theory thatis effectively two-dimensional and has N = (2, 2) supersymmetry at low energies, so that wecan apply the machinery of the Bethe/gauge correspondence. There are two ways to achievethis. Firstly we can compactify one more dimension and get T 2 × R 1,1|4 ∼ = SR 1 × S1 R × ′ R1,1|4 .Alternatively, we reduce by one dimension, giving SR 1 × R1,1|4 . In either case the low energylimit gives an effective two-dimensional theory with N = (2, 2) supersymmetry. It turns out