Let us now construct the holomorphic functions $a\left(u\right)$, ${a}_{D}\left(u\right)$ satisfying the monodromies determined above. It is again done by using the Seiberg-Witten curve, which is given in this case by

$$\Sigma :\phantom{\rule{1em}{0ex}}\frac{2\Lambda \left(x-\mu \right)}{z}+{\Lambda}^{2}z={x}^{2}-u$$ | (5.2.1) |

with auxiliary complex variables $z$ and $x$, together with the Seiberg-Witten diﬀerential

$$\lambda =x\frac{dz}{z}.$$ | (5.2.2) |

We dropped the subscript $1$ from $\Lambda $ to lighten the notation.

Again, we add a point $z=\infty $ and regard $z$ as a complex coordinate on the sphere $C$. This is the ultraviolet curve. The variable $x$ is now a function on it, see Fig. 5.6. Note that $z=0$ is no longer a branch point; indeed, the local behavior of $x$ there is now

$$\begin{array}{lll}\hfill {x}_{+}& \sim \frac{2\Lambda}{z}-\mu +O\left(z\right),\phantom{\rule{2em}{0ex}}& \hfill \text{(5.2.3)}\\ \hfill {x}_{-}& \sim +\mu +O\left(z\right).\phantom{\rule{2em}{0ex}}& \hfill \text{(5.2.4)}\end{array}$$Note also that $\lambda $ has a residue $\mp \mu $ at $z=0$. The curve $\Sigma $ is a two-sheeted cover of $C$ shown in Fig. 5.7.

We deﬁne cycles $A$ and $B$ as shown, and then the functions $a\left(u\right)$ and ${a}_{D}\left(u\right)$ are given by

$$a=\frac{1}{2\pi i}{\oint}_{A}\lambda ,\phantom{\rule{2em}{0ex}}{a}_{D}=\frac{1}{2\pi i}{\oint}_{B}\lambda .$$ | (5.2.5) |

The proof $Im\tau \left(a\right)>0$ goes exactly as in the pure case. The curve $\Sigma $ can be mapped to a parallelogram within a complex $t$ plane by ${\int}_{{P}_{0}}^{P}\partial \lambda \u2215\partial u={\int}_{{P}_{0}}^{P}dz\u2215\left(xz\right)$, see Fig. 5.8. The poles with residues $\pm \mu $ of $\lambda $ are denoted explicitly in the ﬁgure. When a closed cycle $L$ on the torus winds the $A$ cycles $n$ times, $B$ cycles $m$ times, and the poles $f$ times, the integral of $\lambda $ is then

$$\frac{1}{2\pi i}{\oint}_{L}\lambda =na+m{a}_{D}+f\mu ,$$ | (5.2.6) |

just as in the BPS mass formula (5.1.4).

Let us check that the curve correctly reproduces the running of the coupling in the weakly-coupled region. For simplicity, set $\mu =0$, and assume $\left|u\right|\gg \left|\Lambda \right|$. We put the $A$ cycle at $\left|z\right|=1$. We easily ﬁnd

$$\frac{1}{2\pi i}{\oint}_{A}x\frac{dz}{z}\sim \sqrt{u}$$ | (5.2.7) |

as before. As for the $B$ integral, two branch points are around $z\sim \Lambda \u2215\sqrt{u}$ and one branch point is around $z\sim u\u2215{\Lambda}^{2}$. The dominant contribution to the integral is then

$$\frac{1}{2\pi i}{\oint}_{B}x\frac{dz}{z}\sim \frac{2}{2\pi i}{\int}_{u\u2215{\Lambda}^{2}}^{\Lambda \u2215\sqrt{u}}a\frac{dz}{z}=-\frac{6}{2\pi i}alog\frac{a}{\Lambda}.$$ | (5.2.8) |

Then we ﬁnd

$$\tau \left(a\right)=\frac{\partial {a}_{D}}{\partial a}=-\frac{6}{2\pi i}log\frac{a}{\Lambda},$$ | (5.2.9) |

reproducing the running (5.1.6).

Let us next check that the curve correctly reproduces the singularity structure on the $u$-plane. The branch points of the function $x\left(z\right)$ can be determined by studying when the equation of $\Sigma $, given in (5.2.1), has double roots. The equation for the branch points is given by

$${z}^{3}+\frac{u{z}^{2}}{{\Lambda}^{2}}-\frac{2\mu z}{\Lambda}+1=0.$$ | (5.2.10) |

The singularity in the $u$-plane is caused by two of the branch points of $x\left(z\right)$ colliding in the ultraviolet curve $C$ with the coordinate $z$. This condition can be found by taking the discriminant of the equation of $z$ above, giving

$${u}^{3}-{\mu}^{2}{u}^{2}+9{\Lambda}^{3}\mu u+\frac{27}{4}{\Lambda}^{6}-8{\Lambda}^{3}{\mu}^{3}=0.$$ | (5.2.11) |

When $\mu =0$, this equation simpliﬁes to ${u}^{3}+\frac{27}{4}{\Lambda}^{6}=0$, giving the solutions

$$u=c{\Lambda}^{2},\phantom{\rule{1em}{0ex}}{e}^{2\pi i\u22153}c{\Lambda}^{2},\phantom{\rule{1em}{0ex}}{e}^{4\pi i\u22153}c{\Lambda}^{2}$$ | (5.2.12) |

for a constant $c$, reproducing Fig. 5.3.

When $\left|\mu \right|\gg \left|\Lambda \right|$, the equation (5.2.11) can be solved by making two separate approximations. Assuming $u$ is rather big, we can truncate the equation to just ${u}^{3}-{\mu}^{2}{u}^{2}\sim 0$, ﬁnding a singularity at

$$u\sim {\mu}^{2}.$$ | (5.2.13) |

Next, assuming $u$ is rather small, we ﬁnd $-{\mu}^{2}{u}^{2}-8{\Lambda}^{3}{\mu}^{3}\sim 0$ giving

$$u\sim \pm \sqrt{-8{\Lambda}^{3}\mu}.$$ | (5.2.14) |

Together, they reproduce Fig. 5.2. From this, we ﬁnd that the eﬀective pure $SU\left(2\right)$ theory in the region $\left|u\right|\ll \left|\mu \right|$ has the dynamical scale

$${\Lambda}_{0}^{2}\sim \sqrt{{\Lambda}^{3}\mu}.$$ | (5.2.15) |

This agrees with what we saw in (5.1.11).

It is instructive to study another way to derive the singularity at $u\sim {\mu}^{2}$ from the curve. We would like to take the approximation $\left|\Lambda \right|\ll \left|\mu \right|$. To facilitate to take the limit, we introduce $\stackrel{\u0303}{z}=z\u2215\Lambda $ in (5.2.1) and ﬁnd

$$\frac{2\left(x-\mu \right)}{\stackrel{\u0303}{z}}+{\Lambda}^{3}\stackrel{\u0303}{z}={x}^{2}-u.$$ | (5.2.16) |

Now the limit is easy to take: we just ﬁnd

$$\frac{2\left(x-\mu \right)}{\stackrel{\u0303}{z}}={x}^{2}-u.$$ | (5.2.17) |

Then it is clear that when $u={\mu}^{2}$, the equation can be factorized to

$$\left(x-\mu \right)\left(x+\mu -\frac{2}{\stackrel{\u0303}{z}}\right)=0,$$ | (5.2.18) |

therefore it represents two sheets intersecting at a point. When $u\ne {\mu}^{2}$, two sheets are connected smoothly. The change is schematically shown in Fig. 5.9. We learned that the singularity at $u\sim {\mu}^{2}$ arises essentially from the structure $2\Lambda \left(x-\mu \right)\u2215z$ in the curve.