12.1 S-dual of $SU\left(N\right)$ with $N_f=2N$ flavors, part I

12.1.1 Rewriting of the curve

We learned in the last section that the curve of $SU\left(N\right)$ theory with $2N$ flavors is given by:

where $f$ is a complex number; the differential is $\tilde{\lambda }=\tilde{x}dz∕z$. This theory is superconformal, and $f$ is a function of the UV coupling constant ${\tau }_{UV}$. We would like to understand the strong-coupling limits of this theory.

As we did in Sec. 9, it is convenient to rewrite the curve in terms of the Seiberg-Witten differential $\lambda$, to the form

We start from (12.1.1). First we gather terms with the same power of $\tilde{x}$:

where

We divide the whole equation by $\left(1-1∕\tilde{z}-f\tilde{z}\right)$ and define $x=\tilde{x}+{♡}_1∕\left(1-1∕\tilde{z}-f\tilde{z}\right)∕N$. We now have

where ${♣}_k$ has poles of order $k$ at two zeros ${\tilde{z}}_{1,2}$ of $1-1∕\tilde{z}-f\tilde{z}=0$, due to the shift from $\tilde{x}$ to $x$. We set $z=\tilde{z}∕{\tilde{z}}_1$ so that one zero is now at 1, and another is at $q={\tilde{z}}_2∕{\tilde{z}}_1$.

Introducing $\lambda =xdz∕z$, we have an equation of the form (12.1.2); ${\varphi }_k\left(z\right)$ has poles of order at most $k$ at $z=0$, $q$, $1$ and $\infty$. Consider the case when all ${\tilde{\mu }}_i$ and ${\tilde{\mu }}_i^{\prime }$ are generic, and assume $q\ll 1$. Then it is straightforward to determine how $\lambda$ behaves close to each of the singularity. As we are solving a degree-$N$ equation, we have $N$ residues at each singularity. They are given by

Here

and similarly for the ${\mu }_i^{\prime }$, ${\mu }^{\prime }$. Note that ${\mu }_i$ and $\mu$ are the mass parameters which enter the BPS mass formula. We found that they are related to the parameters ${\tilde{\mu }}_i$ via a finite renormalization.

When $N=2$, the structure of the residues at all four punctures were of the same type, as they are all given by $\left(m,-m\right)$. For $N>2$, we see that the structure of the residues at $z=0,\infty$ and the structure at $z=q,1$ are different. The former is of the form $\left(m_1,\dots ,m_N\right)$ with $\sum m_i=0$, and the latter is of the form $m\left(1,1,\dots ,1-N\right)$.

It is also instructive to consider the completely massless case, when we have ${\tilde{\mu }}_i={\tilde{\mu }}_i^{\prime }=0$ for all $i$. The original curve is just

After the same manipulation as above, we find

Therefore,

We observe here again that the behavior of the poles are all the same when $N=2$, while the behavior at $z=0,\infty$ and the behavior at $z=1,q$ are distinct when $N>2$.

We have $2N$ mass terms in the system. First of all we split them into $N$ mass terms encoded in the region $z\sim 0$, and another $N$ mass terms in the region $z\sim \infty$. Correspondingly, we started from the flavor symmetry $U\left(2N\right)$ and decomposed it into $U\left(N\right)\times U\left(N\right)$. We further decompose each of $U\left(N\right)$ into $SU\left(N\right)$ and $U\left(1\right)$. Combined, we use the decomposition of the flavor symmetry of the form

The residues of $\lambda$ at the puncture $A$ at $z=0$ and those at the puncture $D$ at $z=\infty$ encode the mass terms for $SU{\left(N\right)}_{A,D}$ respectively, whereas those at the puncture $B$ at $z=q$ and those at the puncture $C$ at $z=1$ encode the mass terms for $U{\left(1\right)}_{B,C}$; compare (12.1.6).

We then say that the singularity at $z=0$ carry the $SU\left(N\right)$ symmetry, the one at $z=q$ carry the $U\left(1\right)$ symmetry, and similarly for those at $z=1$, $=\infty$. We can visualize the situation as in Fig. 12.1. We call the punctures at $z=0,\infty$ the full punctures, and those at $z=q,1$ the simple punctures. In the 6d viewpoint, these are four-dimensional defect objects extending along the Minkowski ${ℝ}^{3,1}$, and they carry respective flavor symmetries on them.

When $N=2$, the original symmetry is not just $U\left(2\right)$ but $SO\left(4\right)$. Accordingly, the split $U\left(2\right)\simeq SU\left(2\right)\times U\left(1\right)$ is enhanced to the following structure

and therefore the distinction of the types of punctures is gone.

12.1.2 Weak-coupling limit

Clearly $f\sim q\sim e^{2\pi i{\tau }_{UV}}$ in the weak coupling region, see Fig. 12.2. When the coupling is extremely weak, we can think that the four-punctured sphere on the left is composed of two three-punctured spheres. In the tube region connecting the two, the behavior of $\lambda$ is essentially given just by

Writing

we find that the residues of $\lambda$ in the tube region is given by $a_1,\dots ,a_N$. Therefore, we find full punctures after we split off two spheres.

The resulting three-punctured sphere has one simple puncture and two full punctures. Therefore it should carry $U\left(1\right)\times SU\left(N\right)\times SU\left(N\right)$ symmetry. The four-punctured sphere represents the $SU\left(N\right)$ theory with $2N$ flavors. The tube region carries the $SU\left(N\right)$ vector multiplet. Then each three-punctured sphere just represents $N$ flavors, i.e. hypermultiplets $\left(Q_i^a,{\tilde{Q}}_a^i\right)$ where $a,i=1,\dots ,N$. Then two $SU\left(N\right)$ symmetries can be identified with those acting on the index $a$ and $i$ respectively, and the $U\left(1\right)$ symmetry is such that $Q$ has charge $+1$ while $\tilde{Q}$ has charge $-1$.

The ultraviolet curve of the $SU\left(N\right)$ theory with $2N$ flavors, shown in Fig. 12.1, is composed of two copies of this three-punctured sphere. The $2N$ hypermultiplets are split into $N$ hypermultiplets $\left(Q_i^a,{\tilde{Q}}_a^i\right)$ charged under $SU{\left(N\right)}_A$ and $U{\left(1\right)}_B$, and another $N$ hypermultiplets $\left(Q^{\prime }{}_i^a,{\tilde{Q}}^{\prime }{}_a^i\right)$ charged under $SU{\left(N\right)}_D$ and $U{\left(1\right)}_C$.

12.1.3 A strong-coupling limit

Let us consider what happens when $q\to \infty$. As shown in Fig. 12.3, it just ends up exchanging the puncture $B$ and $C$, at the same time redefining the coupling $q$ via $q^{\prime }=1∕q$. This means that this strongly-coupled limit turns out to be another weakly-coupled $SU\left(N\right)$ gauge theory with $2N$ flavors. This time, the $2N$ hypermultiplets are split into $N$ hypermultiplets $\left(q_i^a,{\tilde{q}}_a^i\right)$ and another $N$ hypermultiplets $\left(q^{\prime }{}_i^a,{\tilde{q}}^{\prime }{}_a^i\right)$, but notice that the first $N$ are charged under $SU{\left(N\right)}_A$ and $U{\left(1\right)}_C$ while the second $N$ are charged under $SU{\left(N\right)}_D$ and $U{\left(1\right)}_B$. As we learned for the case of the $SU\left(2\right)$ theory with four flavors in Sec. 9.4, the new quarks are magnetic from the point of view of the original theory.

We would like to understand the limit $q\to 1$ too. We need to split the four-punctured sphere as shown in Fig. 12.4. But the configuration of punctures are not what we already know: we have two full punctures on one side, and two simple punctures on the other side. We need to study more about the 6d construction before answering what happens in the limit.