5.2 The curve

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

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

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

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

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 ${\int \; <!--nolimits-->}_{P_0}^P\partial \lambda ∕\partial u={\int \; <!--nolimits-->}_{P_0}^{P}dz∕\left(xz\right)$, 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

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

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

Then we find

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

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

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

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

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

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

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 $\tilde{z}=z∕\Lambda$ in (5.2.1) and find

Now the limit is easy to take: we just find

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

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 $2\Lambda \left(x-\mu \right)∕z$ in the curve.