9.1 The curve as ${\lambda }^2={\varphi }_2\left(z\right)$

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

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 $u\sim {\mu }_i{}^2$, 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,

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

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 $\tilde{\lambda }=\tilde{x}d\tilde{z}∕\tilde{z}$. The reason for additional tildes will become clear later.

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

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

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

Defining $x=\tilde{x}-♣∕2$, we have

where $♢$ now has double poles at $c_{1,2}\left(f\right)$ due to the completion of the square. Instead of $\tilde{\lambda }=\tilde{x}d\tilde{z}∕\tilde{z}$ we will use $\lambda =xd\tilde{z}∕\tilde{z}$ henceforth. Note that

This is independent of the Coulomb branch modulus $u$, and its residues are all linear combinations of ${\tilde{\mu }}_i$. 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 $z=\tilde{z}∕c_1\left(f\right)$ 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 ${\tilde{\mu }}_i$ 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 $w=1∕z$. Then $d{z}^2∕z^2=d{w}^2∕w^2$. 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 ${\tau }_{UV}$.