Let us consider $SU\left(2\right)$ gauge theory with four doublet hypermultiplets with masses ${\mu}_{1,2,3,4}$. In the very high energy region, the one-loop running is given by (8.1.6), which is just

$$\tau \left(a\right)={\tau}_{UV}.$$ | (9.1.1) |

From this we learn a distinguishing feature of the ${N}_{f}=4$ theory compared to the theories with less ﬂavors: it has a dimensionless parameter ${\tau}_{UV}$. When ${N}_{f}<4$, the bare coupling ${\tau}_{UV}$ was combined with the scale ${\Lambda}_{UV}$ to form the dynamical scale $\Lambda $, which just set the overall scale of the theory.

Now suppose the gauge coupling is small at the ultraviolet. Equivalently, suppose ${\tau}_{UV}$ has a large positive imaginary part. Further suppose ${\mu}_{1,2,3,4}$ are all of the same order, $\sim \mu $. Then the coupling at the energy scale $\sim \mu $ is small, and the semiclassical analysis is OK. We see that when $a\sim \pm {\mu}_{i}$, or equivalently when $u\sim {\mu}_{i}{\phantom{\rule{0.0pt}{0ex}}}^{2}$, the low-energy limit is described by $U\left(1\right)$ gauge theory with one charged hypermultiplet. Far below this scale, the theory is eﬀectively the pure $SU\left(2\right)$ theory, which has the monopole point and the dyon point. Then the $u$-plane schematically has the structure shown in Fig. 9.1.

When ${\mu}_{1,2,3,4}=0$, we can consider the R-symmetry with the charge assignment

This is not anomalous. Then the only sensible point to have a singularity in the $u$-plane is at the origin, where six singularities in the generic case collide, see Fig. 9.2.

The coupling is given by ${\tau}_{UV}$ everywhere,

$$a=\sqrt{u},\phantom{\rule{2em}{0ex}}{a}_{D}={\tau}_{UV}{a}_{D}.$$ | (9.1.3) |

Therefore the monodromy ${M}_{\infty}$ at inﬁnity is just

$${M}_{\infty}=\left(\begin{array}{cc}\hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right).$$ | (9.1.4) |

It looks relatively uninteresting. We will see however that there is a lot of interesting physics going on when we study the dependence on ${\tau}_{UV}$.

The curve of the ${N}_{f}=4$ theory is given by

where $f$ and ${f}^{\prime}$ are complex numbers, whose ratio will eventually be related to ${\tau}_{UV}$. The diﬀerential is $\stackrel{\u0303}{\lambda}=\stackrel{\u0303}{x}d\stackrel{\u0303}{z}\u2215\stackrel{\u0303}{z}$. The reason for additional tildes will become clear later.

The structure of the function $\stackrel{\u0303}{x}\left(\stackrel{\u0303}{z}\right)$ over the ultraviolet curve $C$ which is a sphere with coordinate $z$ is shown in Fig. 9.3. The diﬀerential $\stackrel{\u0303}{\lambda}$ always diverges at $\stackrel{\u0303}{z}=0$, $\infty $, and at the two solutions $\stackrel{\u0303}{z}={c}_{1,2}\left(f\right)$ of $f\u2215\stackrel{\u0303}{z}+{f}^{\prime}\stackrel{\u0303}{z}=1$. These points do not move when $u$ is changed, and shown in red blobs in the ﬁgure. There are four additional branch points where $\stackrel{\u0303}{x}\left(z\right)$ is ﬁnite, shown in black blobs.

Let us rewrite the curve in a more illuminating way. We ﬁrst rescale the coordinate $z$ to set ${f}^{\prime}=1$. We then collect terms with the same power of $\stackrel{\u0303}{x}$:

$$\left(1-\stackrel{\u0303}{z}-\frac{f}{\stackrel{\u0303}{z}}\right){\stackrel{\u0303}{x}}^{2}-\u2661\stackrel{\u0303}{x}-{\u2661}^{\prime}=0$$ | (9.1.6) |

where $\u2661,{\u2661}^{\prime}$ are some complicated expressions, which readers should ﬁll in. We divide the whole expression by $\left(1-\stackrel{\u0303}{z}-f\u2215\stackrel{\u0303}{z}\right)$, and ﬁnd

$${\stackrel{\u0303}{x}}^{2}-\u2663\stackrel{\u0303}{x}-{\u2663}^{\prime}=0.$$ | (9.1.7) |

We note that $\u2663$ and ${\u2663}^{\prime}$ have poles at the two solutions ${c}_{1,2}\left(f\right)$ of $1-z-f\u2215z=0$. Here it is instructive to spell out $\u2663$, which is given by

Deﬁning $x=\stackrel{\u0303}{x}-\u2663\u22152$, we have

$${x}^{2}-\u2662=0$$ | (9.1.9) |

where $\u2662$ now has double poles at ${c}_{1,2}\left(f\right)$ due to the completion of the square. Instead of $\stackrel{\u0303}{\lambda}=\stackrel{\u0303}{x}d\stackrel{\u0303}{z}\u2215\stackrel{\u0303}{z}$ we will use $\lambda =xd\stackrel{\u0303}{z}\u2215\stackrel{\u0303}{z}$ henceforth. Note that

This is independent of the Coulomb branch modulus $u$, and its residues are all linear combinations of ${\stackrel{\u0303}{\mu}}_{i}$. We encountered in (6.4.8) a similar shift of $\lambda $ by a one-form which is independent of $u$ and whose residues are given by the mass terms only. Such shift only amounts to a re-deﬁnition of the ﬂavor charge and the mass terms, and does not aﬀect the physics, as discussed there.

Now we deﬁne the coordinate $z=\stackrel{\u0303}{z}\u2215{c}_{1}\left(f\right)$ so that the double poles are at $z=q$ and $1$ for $\left|q\right|<1$, see Fig. 9.4. The ﬁnal form of the curve is then:

where $P\left(z\right)$ is a quartic polynomial, as can be seen by re-following the change of variables starting from (9.1.5). The explicit expression of $P\left(z\right)$ in terms of $u$, $f$ and ${\stackrel{\u0303}{\mu}}_{i}$ is not very important, however.

The quadratic diﬀerential ${\varphi}_{2}\left(z\right)$ has double poles at $z=0,q,1,\infty $. To see this for $z=\infty $, set $w=1\u2215z$. Then $d{z}^{2}\u2215{z}^{2}=d{w}^{2}\u2215{w}^{2}$. This has poles of order two when $w=0$, i.e. when $z=\infty $. We identify this dimensionless parameter $q$ as a function of the UV coupling ${\tau}_{UV}$.