The Bethe/Gauge Correspondence

50 Chapter 3. GaugeVacuum manifolds. We can analyze the vacuum structure of the theory from the conditionof vanishing energy, so we have to find the zeros of (3.50). The allowed values of the φ l f , φk¯fand σ depend on the values of the FI parameter r and the twisted masses. Recall that in thesupersymmetric sigma model picture, the dynamical scalar fields are viewed as coordinates inthe target manifold M. Thus, for fixed values of the parameters, the requirement U = 0 definesa subspace of M: the vacuum manifold. This is the target space of the supersymmetric sigmamodel obtained in the low-energy limit of the theory.The solutions can roughly be divided into two classes. Firstly we can fix the value of σ,so that the U(1) G -invariance is spontaneously broken, and the φ l fand φ k¯f may acquire a massby the Higgs mechanism. This class of solutions is called the Higgs branch of the theory. Forvanishing twisted masses, we must have σ = 0, while the φ’s are restricted to∑L fl=1∑L¯f|φ l f | 2 − |φ k¯f | 2 = r .k=1There is a global symmetry group which rotates all of the scalars by the same phase. Thus, westill have to take the quotient by the diagonal action of U(1). In the special case L¯f = 0, whichonly a solution when r > 0, this implies that the resulting vacuum manifold is isomorphic tothe complex projective space CP Lf−1 . Similarly, when L f = 0 and r < 0 we find CP L¯f −1 . Fornonzero twisted masses the geometry of the classical Higgs branch is more complicated; see §3of [16] and §2 of [17].However, we are interested in gauge theories and therefore in the class of solutions where allφ l fand vanish. We can do this because the matter fields are massive so they are ‘frozen’ atφk¯flow energies. For zero twisted masses, this corresponds to large |σ|, but generically the valueof σ is not constrained. In particular, the gauge group is preserved, and we’re on the Coulombbranch of the theory. It is clear from (3.50) that this only happens for r = 0. The analysischanges when we take quantum effects into account. In the remainder we restrict ourselves tothe theory on the Coulomb branch.3.4.2 Quantum effectsThe set-up is like before: we have U(1) G gauge theory with abelian super field strength Σ withmassive matter, L f fundamental fields and L¯f anti-fundamentals, with corresponding twistedmasses. In general there might be a superpotential for the matter fields (compatible with thevalues of the twisted masses) and a twisted superpotential ˜W for Σ.Our goal is to integrate out the massive matter fields as well as the high-energy modes of Σ.Before we do this, however, we discuss some preliminaries. We will see that supersymmetrymakes our life easier once again, allowing us to derive exact results.Mass dimensions. As always, quantum effects can lead to divergences and we will have torenormalize the theory. From the classical Lagrangian (3.48) we can find the superficial degrees ofdivergence by looking at the mass dimensions of our parameters, the gauge coupling constant e,the complex coupling τ = i r + ϑ/2π and the twisted masses ˜m l f , ˜mk¯f . As always, the startingpoint is that [x µ ] = −1 and, in units where = 1, the action must have mass dimension zero.Since we work in two dimensions, this means that the Lagrangian density has mass dimension[L] = 2. Looking at L kin we find that[φ] = 0 , [ψ ± ] = 1 2 , [F ] = 1 .Furthermore, [θ ± ] = [¯θ ± ] = − 1 2, and the superfield Φ has mass dimension zero. Because weexponentiate the vector superfield, [V ] = 0 too. This implies [Σ] = [ ¯D + D − V ] = 1 2 + 1 2 + 0 = 1,so that[σ] = [A µ ] = 1 , [λ ± ] = [¯λ ± ] = 3 2 , [D] = 2 .