Next, consider the $SU\left(N\right)$ theory with one ﬂavor $\left(Q,\stackrel{\u0303}{Q}\right)$ of bare mass $\mu $. The curve is given by

$$\Sigma :\phantom{\rule{2em}{0ex}}\frac{{\Lambda}^{N-1}\left(x-\mu \right)}{z}+{\Lambda}^{N}z={x}^{N}+{u}_{2}{x}^{N-2}+\cdots +{u}_{N}.$$ | (11.3.1) |

Recall that in the semiclassical analysis we saw that a light charged hypermultiplet arises when ${a}_{i}\sim \mu $. Let us check that the curve written above reproduces this behavior.

First, we introduce ${\underline{a}}_{i}$ as before, and consider the semiclassical regime when all $\left|{\underline{a}}_{i}\right|$ is far larger than $\left|\Lambda \right|$. The A-cycle on the ultraviolet curve was $\left|z\right|=1$ as before. Then we ﬁnd ${a}_{i}\sim \underline{{a}_{i}}+O\left(\Lambda \right)$ just as was in the case of the pure theory.

To see additional singularities in the weakly-coupled region, deﬁne $\stackrel{\u0303}{z}=z\u2215{\Lambda}^{N-1}$. The curve is then

$$\frac{x-\mu}{\stackrel{\u0303}{z}}+{\Lambda}^{2N-1}\stackrel{\u0303}{z}={x}^{N}+{u}_{2}{x}^{N-2}+\cdots +{u}_{N},$$ | (11.3.2) |

which can be approximated by

$$\frac{x-\mu}{\stackrel{\u0303}{z}}={x}^{N}+{u}_{2}{x}^{N-2}+\cdots +{u}_{N}=\prod \left(x-{\underline{a}}_{i}\right)$$ | (11.3.3) |

in the extremely weakly coupled limit. The equation factorizes and the curve separates into two when ${\underline{a}}_{i}=\mu $; otherwise the curve is a smooth degree-$N$ covering of the $z$ sphere. This shows that when ${\underline{a}}_{i}=\mu $, a one-cycle on the Seiberg-Witten curve shrinks, and the membrane suspended there produces a massless hypermultiplet, see Fig. 5.9.

The one-loop running can also be checked. The branch points ${z}_{i}^{+}$ in the large $z$ region is unchanged, as the structure of the ${N}_{f}=1$ curve in the large $z$ region itself is unchanged from the pure curve. Then

$${z}_{i}^{+}\sim {\left(E\u2215\Lambda \right)}^{N}.$$ | (11.3.4) |

In the small $z$ region, the branch points are around where ${\Lambda}^{N-1}x\u2215z$ and $P\left(x\right)$ are of the same order. Assuming $\left|x\right|\sim \left|{\underline{a}}_{i}\right|\sim \left|E\right|$, we see

$${z}_{i}^{-}\sim {\left(\Lambda \u2215E\right)}^{N-1}.$$ | (11.3.5) |

Then the monopole has the mass

$$\begin{array}{lll}\hfill {M}_{\text{monopole}}& =\left|\frac{1}{2\pi i}{\int}_{{B}_{i}}\lambda \right|\phantom{\rule{2em}{0ex}}& \hfill \text{(11.3.6)}\\ \hfill & \sim \left|\left({a}_{i}-{a}_{i+1}\right)\frac{1}{2\pi i}{\int}_{{\Lambda}^{N-1}\u2215{E}^{N-1}}^{{E}^{N}\u2215{\Lambda}^{N}}\frac{dz}{z}\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(11.3.7)}\\ \hfill & \sim |\left({a}_{i}-{a}_{i+1}\right)\frac{2N-1}{2\pi i}log\frac{E}{\Lambda}|.\phantom{\rule{2em}{0ex}}& \hfill \text{(11.3.8)}\end{array}$$This gives

$$\tau \left(E\right)=\frac{2N-1}{2\pi i}log\frac{E}{\Lambda}$$ | (11.3.9) |

as it should be.

More generally, we can consider the curve given by

$$\begin{array}{cc}& \Sigma :\phantom{\rule{2em}{0ex}}\frac{{\Lambda}^{N-{N}_{L}}\prod _{i=1}^{{N}_{L}}\left(x-{\mu}_{i}\right)}{z}+z{\Lambda}^{N-{N}_{R}}\prod _{i=1}^{{N}_{R}}\left(x-{\mu}_{i}^{\prime}\right)\\ & ={x}^{N}+{u}_{2}{x}^{N-2}+\cdots +{u}_{N-1}x+{u}_{N}& \text{(11.3.10)}\end{array}$$where ${N}_{L},{N}_{R}\le N$. When ${N}_{L}={N}_{R}=N$, we need to introduce complex numbers $f$, ${f}^{\prime}$ as in the curve of $SU\left(2\right)$ with four ﬂavors, (9.1.5):

$$\begin{array}{cc}& \Sigma :\phantom{\rule{2em}{0ex}}f\cdot \frac{\prod _{i=1}^{N}\left(x-{\mu}_{i}\right)}{z}+{f}^{\prime}\cdot z\prod _{i=1}^{N}\left(x-{\mu}_{i}^{\prime}\right)\\ & ={x}^{N}+{u}_{2}{x}^{N-2}+\cdots +{u}_{N-1}x+{u}_{N}.& \text{(11.3.11)}\end{array}$$We also need to distinguish the mass parameters in the curve and the mass parameters in the BPS mass formula, carefully studied in Sec. 9.2 for $SU\left(2\right)$ with four ﬂavors. In the following we mainly discuss the case with less than $2N-1$ ﬂavors.

Consider the case when ${\mu}_{i}$ and ${\mu}_{i}^{\prime}$ are all small. Further, consider the regime where $\left|{\underline{a}}_{i}\right|\gg \Lambda $. As always we ﬁnd ${a}_{i}={\underline{a}}_{i}+O\left(\Lambda \right)$. The branch points are at

$$\left|{z}_{i}^{+}\right|\sim \frac{{E}^{N-{N}_{R}}}{{\Lambda}^{N-{N}_{R}}},\phantom{\rule{2em}{0ex}}\left|{z}_{i}^{-}\right|\sim \frac{{\Lambda}^{N-{N}_{L}}}{{E}^{N-{N}_{L}}}.$$ | (11.3.12) |

Then we ﬁnd

$$\begin{array}{lll}\hfill {M}_{\text{monopole}}& \sim \left|\left({a}_{i}-{a}_{i+1}\right)\frac{1}{2\pi i}{\int}_{{\Lambda}^{N-{N}_{L}}\u2215{E}^{N-{N}_{L}}}^{{E}^{N-{N}_{R}}\u2215{\Lambda}^{N-{N}_{R}}}\frac{dz}{z}\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(11.3.13)}\\ \hfill & \sim |\left({a}_{i}-{a}_{i+1}\right)\frac{2N-\left({N}_{L}+{N}_{R}\right)}{2\pi i}log\frac{E}{\Lambda}|,\phantom{\rule{2em}{0ex}}& \hfill \text{(11.3.14)}\end{array}$$and therefore the one-loop running is

$$\tau \left(E\right)=\frac{2N-\left({N}_{L}+{N}_{R}\right)}{2\pi i}log\frac{E}{\Lambda}.$$ | (11.3.15) |

In the other regime when $\left|{\mu}_{i}\right|,\left|{\underline{a}}_{i}\right|\gg \Lambda $, we can use the redeﬁning trick to ﬁnd singularities on the Coulomb branch. For example, deﬁning $\stackrel{\u0303}{z}=z\u2215{\Lambda}^{N-{N}_{L}}$, the curve is

Then the limit $\Lambda \to 0$ can be taken, which gives

$$\frac{\prod _{i=1}^{{N}_{L}}\left(x-{\mu}_{i}\right)}{\stackrel{\u0303}{z}}=\prod _{i=1}^{N}\left(x-{\underline{a}}_{i}\right).$$ | (11.3.17) |

This means that whenever ${\underline{a}}_{i}={\mu}_{s}$ for some $i$ and $s=1,\dots ,{N}_{L}$, the curve splits into two, because the equation can be factorized. The same can be done for the variable $w=1\u2215z$. Then we also ﬁnd singularities when ${\underline{a}}_{i}={\mu}_{s}^{\prime}$ for some $i$ and $s=1,\dots ,{N}_{R}$. In total, these reproduce the semiclassical, weakly-coupled physics of $SU\left(N\right)$ theory with ${N}_{f}={N}_{R}+{N}_{L}$ hypermultiplets in the fundamental representation. The situation is summarized in Fig. 11.5.

We have a sphere $C$ described by the coordinate $z$. The curve $\Sigma $ is an $N$-sheeted cover of $C$. We have one M5-brane wrapping $\Sigma $. We call the 6d theory living on $C$ the $\mathcal{\mathcal{N}}=\left(2,0\right)$ theory of type $SU\left(N\right)$. Roughly speaking, it arises from $N$ coincident M5-branes.

Consider $Arg\phantom{\rule{0.3em}{0ex}}z$ as the sixth direction ${x}_{6}$, and $log\left|z\right|$ as the ﬁfth direction ${x}_{5}$. Reducing along the ${x}_{6}$ direction, we have a 5d theory on a segment. The 5d theory is the maximally supersymmetric Yang-Mills theory with gauge group $SU\left(N\right)$. The term

$$\frac{{\Lambda}^{N-{N}_{L}}\prod _{i=1}^{{N}_{L}}\left(x-{\mu}_{i}\right)}{z}$$ | (11.3.18) |

in the curve can be thought of deﬁning a certain boundary condition on the left side of the ﬁfth direction. We regard it as giving ${N}_{L}$ hypermultiplets in the $SU\left(N\right)$ fundamental representation there. Similarly, the term

$$z{\Lambda}^{N-{N}_{R}}\prod _{i=1}^{{N}_{R}}\left(x-{\mu}_{i}^{\prime}\right)$$ | (11.3.19) |

is regarded as the boundary condition such that ${N}_{R}$ fundamental hypermultiplets there. By further reducing the theory along the ﬁfth direction, we have $SU\left(N\right)$ gauge theory with ${N}_{f}={N}_{L}+{N}_{R}$ fundamental hypermultiplets in total. We saw that the eﬀect of the boundary conditions becomes noticeable around when

$$log\left|{z}_{R}\right|\sim \left(N-{N}_{R}\right)log\frac{E}{\Lambda}<0,\phantom{\rule{2em}{0ex}}log\left|{z}_{L}\right|\sim \left(N-{N}_{L}\right)log\frac{\Lambda}{E}>0.$$ | (11.3.20) |

In the ﬁve dimensional Yang-Mills, we have monopole strings, which have ends around $\left|{z}_{R}\right|$ and $\left|{z}_{L}\right|$. From the four-dimensional point of view, $log\left|{z}_{L}\right|\u2215\left|{z}_{R}\right|$ then controlled the mass of the monopoles, which then gave the one-loop running of the theory.

Note that from the four-dimensional point of view, the split of ${N}_{f}$ into ${N}_{R}$ and ${N}_{L}$ is rather arbitrary. In fact, by redeﬁning $z$, we can easily come to the form of the curve given by

$$\frac{{\Lambda}^{2N-{N}_{f}}\prod _{i}^{{N}_{f}}\left(x-{\mu}_{i}\right)}{z}+z={x}^{N}+{u}_{2}{x}^{N-2}+\cdots {u}_{N}$$ | (11.3.21) |

where we deﬁned ${\mu}_{{N}_{L}+i}:={\mu}_{i}^{\prime}$. In this form the symmetry exchanging all ${N}_{f}$ mass parameters is manifest.

From the higher-dimensional perspective, it is however sometimes convenient to stick to the situation where the equation of the Seiberg-Witten curve $\Sigma $ is of degree $N$ regarded as a polynomial in $x$. This guarantees that $\Sigma $ is always an $N$-sheeted cover of the ultraviolet curve $C$. Numerically, this condition means that the boundary condition such as (11.3.18) and (11.3.19) also has degrees less than or equal to $N$. This imposes the constraint $N\ge {N}_{L,R}$, and therefore $2N\ge {N}_{f}$. This is the condition that the theory is asymptotically free or asymptotically conformal.