Let us start with the $SU\left(2\right)$ with ${N}_{f}=2$ ﬂavors. The Seiberg-Witten curve was guessed in (6.4.6), which we repeat here:

$$\Sigma :\phantom{\rule{2em}{0ex}}\frac{2\Lambda \left(x-{\mu}_{1}\right)}{z}+2\Lambda \left(x-{\mu}_{2}\right)z={x}^{2}-u$$ | (8.2.1) |

with the Seiberg-Witten diﬀerential $\lambda =xdz\u2215z$. The ultraviolet curve $C$ is still just an ${S}^{2}$, shown in Fig. 8.1.

When $\left|u\right|\gg \left|{\Lambda}^{2}\right|,|{\mu}_{i}{|}^{2}$, we can estimate the line integrals easily. First, we put the $A$-cycle at $\left|z\right|=1$. Using $x\simeq \sqrt{u}$ around there, we have

$$a=\frac{1}{2\pi i}{\oint}_{A}x\frac{dz}{z}\simeq \sqrt{u}.$$ | (8.2.2) |

The positions of the branch points of $x\left(z\right)$ on the curve $C$ can also be easily estimated: there are two around $z\simeq \sqrt{u}\u2215\Lambda $ and two more around $z\simeq \Lambda \u2215\sqrt{u}$. Then we see

$${a}_{D}=\frac{1}{2\pi i}{\oint}_{B}x\frac{dz}{z}\simeq \frac{2\cdot 2}{2\pi i}{\int}_{\sqrt{u}\u2215\Lambda}^{1}a\frac{dz}{z}\simeq -\frac{4}{2\pi i}alog\frac{a}{\Lambda}.$$ | (8.2.3) |

From this we can compute $\tau \left(a\right)=\partial {a}_{D}\u2215\partial a$, which reproduces the one-loop running (8.1.6).

Let us next study the structure of the singularities on the $u$-plane. When ${\mu}_{1,2}\gg \Lambda $, the gauge coupling is rather small around the energy scale ${\mu}_{1,2}$. Then we expect that when

$$u\simeq {\mu}_{i}^{2}\phantom{\rule{1em}{0ex}}\text{for}i=1,2\text{}$$ | (8.2.4) |

one component of $\left({Q}_{i},{\stackrel{\u0303}{Q}}^{i}\right)$ become very light, producing a singularity. Below the scale of ${\mu}_{1,2}$, the theory is eﬀectively equivalent to pure $SU\left(2\right)$ theory, which should have two singularities where either monopoles or dyons are very light. In total we expect that there are four singularities on the $u$-plane, see Fig. 8.2.

This structure can be checked starting from the curve (8.2.1) by studying its discriminant, which is left as an exercise to the reader. Here we study the massless case ${\mu}_{1}={\mu}_{2}=0$ in detail. The Seiberg-Witten curve for the massless case is simply

$${x}^{2}-2\Lambda \left(z+\frac{1}{z}\right)x-u=0.$$ | (8.2.5) |

Then we see that four branch points of $x\left(z\right)$ meet in pairs when $u=0$ or $u=-4{\Lambda}^{2}$ as depicted in Fig. 8.3.

Explicitly, when $u=0$ they meet at $z=\pm i$ and when $u=-{\Lambda}^{2}$ they meet at $z=\pm 1$. There are no other singularities on the $u$-plane, so we see that when ${\mu}_{1}={\mu}_{2}=0$ the $u$-plane has the structure shown in Fig. 8.4.

At each of $u=0$, $u=-{\Lambda}^{2}$, two pairs of branch points of $x\left(z\right)$ collide. This means that each of $u=0$, $u=-{\Lambda}^{2}$ should be considered as two singularities on the $u$-plane. This situation was shown in Fig. 8.4 by putting almost overlapping two blobs at $u=0,$ $-{\Lambda}^{2}$. In total there are four singularities, matching what we found above for ${\mu}_{1,2}\gg \Lambda $. Let us denote the monodromies around various closed paths as shown in Fig. 8.4.

The monodromy ${M}_{\infty}$ at inﬁnity can be found from the explicit form of $a$, ${a}_{D}$ found in (8.2.2), (8.2.3) to be

$${M}_{\infty}=\left(\begin{array}{cc}\hfill -1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right).$$ | (8.2.6) |

The monodromy ${M}_{+}$ around $u=0$ can be found by following the motion of the branch points when we make a slow change along the path $u=\mathit{\epsilon}{e}^{i\mathit{\theta}}$ for a very small $\mathit{\epsilon}$ from $\mathit{\theta}=0$ to $\mathit{\theta}=2\pi $.

The pair of branch points exchanges positions as shown in Fig. 8.5. We see that the $B$ cycle remains the same, while $A$ is sent to $A-2B$, thus generating

$${M}_{+}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 1\hfill \end{array}\right)=S{T}^{2}{S}^{-1}.$$ | (8.2.7) |

We see that

$${M}_{\infty}={M}_{+}{M}_{-}$$ | (8.2.8) |

with

$${M}_{-}=T{M}_{+}{T}^{-1}.$$ | (8.2.9) |

Before proceeding, it is instructive to use another description of the curve to ﬁnd the same $u$-plane structure. The curve was given in (6.4.7). When massless, this just becomes

$$\frac{{x}^{2}}{z}+{\Lambda}^{2}z={x}^{2}-u.$$ | (8.2.10) |

The branch points collide when $u=0$ or $u=-{\Lambda}^{2}$ as before, but it looks rather diﬀerent on the ultraviolet curve, as shown in Fig. 8.6.

Note that $\lambda $ diverges at $z=0$ and $z=1$ independent of $u$. One branch point of $x\left(z\right)$ on the ultraviolet curve moves as $u$ changes, and this point hits either $z=0$ or $z=1$ at $u=0$ or $u=-{\Lambda}^{2}$ respectively. It is left to the reader to recover the monodromies ${M}_{\pm}$ from this latter view point.