11.1 Semiclassical analysis

Let us consider SU(N) gauge theory with Nf hypermultiplets in the fundamental N-dimensional representation. The 𝒩=2 vector multiplet consists of the 𝒩=1 adjoint chiral multiplet Φ and the 𝒩=1 vector multiplet Wα, both N × N matrices. The hypermultiplets, in terms of 𝒩=1 chiral multiplets, can be written as

Qia,Q̃ai,a = 1,,N; i = 1,,Nf. (11.1.1)

One branch of the supersymmetric vacuum is given by the condition

[Φ, Φ] = 0. (11.1.2)

This means that Φ can be diagonalized. We denote it by

Φ = diag(a1,,aN), ai = 0. (11.1.3)

Let us consider a generic situation when aiaj for all ij. Then the gauge group is broken from SU(N) to U(1)N 1. The W-boson mass is given by

MW = |ai aj| (11.1.4)

for the W-boson coming from the entry (i,j) of the N × N matrix. As for the monopole, it is known that the ’t Hooft-Polyakov monopole solution for the breaking from SU(2) to U(1) can be directly regarded as a solution for the breaking from SU(N) to U(1), by choosing 2 × 2 submatrices of N × N matrices, given by picking the entries at the positions (i,i), (i,j), (j,i) and (j,j) for ij. The masses are then

Mmonopole = |τUV (ai aj)|. (11.1.5)

The one-loop running is

E d dEτ = 1 2π(2N Nf). (11.1.6)

Then the theory is asymptotically free when 0 Nf < 2N. The dynamical scale is then

Λ2N Nf := ΛUV 2N Nfe2πiτUV . (11.1.7)

When Nf = 2N, the theory is asymptotically conformal, and τUV is a dimensionless parameter in the quantum theory.

When there are flavors, the 𝒩=1 superpotential in this vacua is

i(QiΦQ̃i μiQiQ̃i) = i(Qi1,Qi2,,QiN) a1 μi a2 μi a N μi Q̃1i Q̃2i Q̃Ni .(11.1.8)

Then we have one massless charged hypermultiplet component whenever we have ai μs = 0 for some i and s.

In the strongly-coupled quantum theory, the definition of ai as the diagonal entries of the gauge-dependent quantity Φ does not make much sense. Instead, as we did in the case of SU(2) gauge theory, we define ai as the complex numbers entering in the BPS mass formula:

M |niai + miaDi + sfsμs| (11.1.9)

where (ni,mi) are the electric and the magnetic charges under U(1)N 1 infrared gauge group, and fs are the flavor charges. We can also consider gauge-invariant combinations of Φ defined as

xN + u2xN 2 + + uN1x + uN := det(x Φ) (11.1.10)

where x is a dummy variable. For N = 2, we had Φ diag(a,a) and therefore u2 = a2 up to quantum corrections. Similarly, for general N, uk is the degree k elementary symmetric polynomials of the variables a1, …, aN up to quantum corrections. Our task then is to determine the mapping between (u2,,uN) and (a1,,aN) including the quantum corrections.