The Bethe/Gauge Correspondence

3.5. The nonabelian case 59the case L a = 0 we refer to §2.2 of [15]; see also §2.2 of [2]. As before, we are interested in thevacuum structure on the Coulomb branch in which all of the φ’s have to be integrated out.Vacuum equation. The ϑ-term (3.67) shows that the periodicity of ϑ can be accounted for inseveral ways: we can independently shift of each of the σ n in W tree (σ) by an amount −im n σ n ,for m n ∈ Z. This leads us to consider the shifted twisted superpotential ˜W ⃗m = ˜W − i ∑ m n σ n .The same steps as in the abelian case yield a system of N coupled vacuum equations:(exp 2π ∂ ˜W)eff∂σ n = 1 , 1 ≤ n ≤ N . (3.69)We see that the vacuum structure of our theory on the Coulomb branch is once more determinedby the effective twisted superpotential. By the decoupling and non-renormalization theorems,˜W eff can be found by integrating out the matter scalar fields, and we may assume that thesuperpotential W vanishes.Effective twisted superpotential. Let’s start with the fields in the fundamental representationand consider integrating out the field Φ l fwith twisted mass parameter ˜m l f. The multipletΦ l f contains N chiral superfields (Φl f )n whose mass terms contain σ n and are given by |σ n − ˜m l f |.We can find the resulting contribution to ˜W eff from (3.62):δ ˜W l eff,f = 12πN∑ ((σ n − ˜m l f ) log σn − ˜m l fµn=1Similarly, the result for ¯Φ¯f,k isδ ˜W k eff,¯f = 12πN∑((−σ n − ˜m k¯f ) log σn − ˜m k¯fµn=1The contribution due to Φ a turns out to beδ ˜W j eff,a = 12πm≠n)− 1 . (3.70))− 1 . (3.71)N∑((σ m − σ n − ˜m j a) log σm − σ n − ˜m j )a− 1 . (3.72)µSumming up all these contributions, we arrive at the following total effective twisted superpotentialis˜W eff (σ) = i τN∑n=1σ n + 12π+ 12π+ 12πN∑ ∑L f(σ n − ˜m l f )n=1 l=1N∑ ∑L¯f(−σ n − ˜m k¯f )n=1 k=1N∑ ∑L a(σ m − σ n − ˜m j a)m,n j=1m≠n(log σn − ˜m l fµ)− 1(log −σn − ˜m k¯fµ)− 1(log σm − σ n − ˜m j )a− 1µ(3.73)Conclusion: the vacuum equation. In conclusion, plugging (3.73) into the vacuum equations(3.69), we get the following set of equations:∏ Lfl=1 (σn − ˜m l f∏ )L¯fk=1 (σn + ˜m k¯f ) = ∏NµL f−L¯f (−1) L¯f−L ae 2πiτ∏L am≠n j=1σ n − σ m + ˜m j aσ n − σ m − ˜m j a, 1 ≤ n ≤ N (3.74)These are the equations that we were after, and conclude this chapter.