Let us consider $SU\left(N\right)$ gauge theory with ${N}_{f}$ hypermultiplets in the fundamental $N$-dimensional representation. The $\mathcal{\mathcal{N}}=2$ vector multiplet consists of the $\mathcal{\mathcal{N}}=1$ adjoint chiral multiplet $\Phi $ and the $\mathcal{\mathcal{N}}=1$ vector multiplet ${W}_{\alpha}$, both $N\times N$ matrices. The hypermultiplets, in terms of $\mathcal{\mathcal{N}}=1$ chiral multiplets, can be written as

$${Q}_{i}^{a},\phantom{\rule{1em}{0ex}}{\stackrel{\u0303}{Q}}_{a}^{i},\phantom{\rule{2em}{0ex}}a=1,\dots ,N;\phantom{\rule{1em}{0ex}}i=1,\dots ,{N}_{f}.$$ | (11.1.1) |

One branch of the supersymmetric vacuum is given by the condition

$$\left[\Phi ,{\Phi}^{\u2020}\right]=0.$$ | (11.1.2) |

This means that $\Phi $ can be diagonalized. We denote it by

$$\Phi =diag\left({a}_{1},\dots ,{a}_{N}\right),\phantom{\rule{2em}{0ex}}\sum {a}_{i}=0.$$ | (11.1.3) |

Let us consider a generic situation when ${a}_{i}\ne {a}_{j}$ for all $i\ne j$. Then the gauge group is broken from $SU\left(N\right)$ to $U{\left(1\right)}^{N-1}$. The W-boson mass is given by

$${M}_{W}=|{a}_{i}-{a}_{j}|$$ | (11.1.4) |

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

$${M}_{\text{monopole}}=\left|{\tau}_{UV}\left({a}_{i}-{a}_{j}\right)\right|.$$ | (11.1.5) |

The one-loop running is

$$E\frac{d}{dE}\tau =\frac{1}{2\pi}\left(2N-{N}_{f}\right).$$ | (11.1.6) |

Then the theory is asymptotically free when $0\le {N}_{f}<2N$. The dynamical scale is then

$${\Lambda}^{2N-{N}_{f}}:={\Lambda}_{UV}^{2N-{N}_{f}}{e}^{2\pi i{\tau}_{UV}}.$$ | (11.1.7) |

When ${N}_{f}=2N$, the theory is asymptotically conformal, and ${\tau}_{UV}$ is a dimensionless parameter in the quantum theory.

When there are ﬂavors, the $\mathcal{\mathcal{N}}=1$ superpotential in this vacua is

$$\begin{array}{cc}& \sum _{i}\left({Q}_{i}\Phi {\stackrel{\u0303}{Q}}^{i}-{\mu}_{i}{Q}_{i}{\stackrel{\u0303}{Q}}^{i}\right)=\\ & \sum _{i}\left({Q}_{i}^{1},{Q}_{i}^{2},\dots ,{Q}_{i}^{N}\right)\left(\begin{array}{c}\hfill {a}_{1}-{\mu}_{i}\hfill \\ \hfill \hfill & \hfill {a}_{2}-{\mu}_{i}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \ddots \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {a}_{N}-{\mu}_{i}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\stackrel{\u0303}{Q}}_{1}^{i}\hfill \\ \hfill {\stackrel{\u0303}{Q}}_{2}^{i}\hfill \\ \hfill \vdots \hfill \\ \hfill {\stackrel{\u0303}{Q}}_{N}^{i}\hfill \end{array}\right).& \text{(11.1.8)}\end{array}$$Then we have one massless charged hypermultiplet component whenever we have ${a}_{i}-{\mu}_{s}=0$ for some $i$ and $s$.

In the strongly-coupled quantum theory, the deﬁnition of ${a}_{i}$ as the diagonal entries of the gauge-dependent quantity $\Phi $ does not make much sense. Instead, as we did in the case of $SU\left(2\right)$ gauge theory, we deﬁne ${a}_{i}$ as the complex numbers entering in the BPS mass formula:

$$M\ge |{n}^{i}{a}_{i}+{m}_{i}{a}_{D}^{i}+\sum _{s}{f}_{s}{\mu}_{s}|$$ | (11.1.9) |

where $\left({n}^{i},{m}_{i}\right)$ are the electric and the magnetic charges under $U{\left(1\right)}^{N-1}$ infrared gauge group, and ${f}_{s}$ are the ﬂavor charges. We can also consider gauge-invariant combinations of $\Phi $ deﬁned as

$${x}^{N}+{u}_{2}{x}^{N-2}+\cdots +{u}_{N-1}x+{u}_{N}:=\u27e8det\left(x-\Phi \right)\u27e9$$ | (11.1.10) |

where $x$ is a dummy variable. For $N=2$, we had $\Phi \sim diag\left(a,-a\right)$ and therefore ${u}_{2}=-{a}^{2}$ up to quantum corrections. Similarly, for general $N$, ${u}_{k}$ is the degree $k$ elementary symmetric polynomials of the variables ${a}_{1}$, …, ${a}_{N}$ up to quantum corrections. Our task then is to determine the mapping between $\left({u}_{2},\dots ,{u}_{N}\right)$ and $\left({a}_{1},\dots ,{a}_{N}\right)$ including the quantum corrections.