Now we have learned enough techniques to understand the S-dual of $SU\left(N\right)$ theory with $2N$ ﬂavors, see the ﬁrst row of Fig. 12.13. Originally, we have a sphere with four punctures: two at $z=0$, $\infty $ are full punctures, and two at $z=q$, $1$ are simple punctures. We would like to understand the limit $q\to 1$. We end up decoupling two simple punctures from the other two. We already learned what happens in this decoupling process.

The simple puncture is a puncture of type $\left(N-1,1\right)$. Decoupling two of them, we generate a puncture of type $\left(N-2,1,1\right)$. This puncture has a ﬂavor symmetry $SU\left(2\right)\times U\left(1\right)$ when $N>3$, and $SU\left(3\right)$ when $N=3$. The behavior of the duality when $N=3$ is somewhat more peculiar than the other cases. In any case, there is an $SU\left(2\right)$ symmetry exchanging the last two columns of height 2, and a weakly-coupled dynamical $SU\left(2\right)$ group gauges this $SU\left(2\right)$ symmetry. There is in addition one ﬂavor in the doublet representation for this $SU\left(2\right)$ gauge group coming from the almost decoupled sphere on the right, see the last row of Fig. 12.13.

The question is the nature of the sphere on the left hand side. It has three punctures: two are full punctures, and one is of type $\left(N-2,1,1\right)$. Assuming all the mass parameters are zero, we can determine the behavior of ﬁelds ${\varphi}_{k}\left(z\right)$ easily, as the pole structure at $z=\infty $ is $\left({p}_{2},{p}_{3},\dots ,{p}_{N}\right)=\left(1,2,\dots ,2\right)$. We see that

$${\varphi}_{2}\left(z\right)=0,\phantom{\rule{2em}{0ex}}{\varphi}_{k}\left(z\right)=\frac{{u}_{k}}{{\left(z-1\right)}^{k-1}{z}^{k-1}}d{z}^{k}.$$ | (12.3.1) |

This theory has one dimension-$k$ operator for each $k=3,4,\dots ,N$. The ﬂavor symmetry is at least $SU\left(N\right)\times SU\left(N\right)$ associated to the full punctures, and $SU\left(2\right)\times U\left(1\right)$ associated to the puncture of type $\left(N-2,1,1\right)$. Call this funny conformal ﬁeld theory ${R}_{N}$, for which we introduce a graphical notation as in Fig. 12.14. In the original theory, the symmetry $SU\left(N\right)\times SU\left(N\right)\times U\left(1\right)$ was part of the ﬂavor symmetry $SU\left(2N\right)$ rotating the whole $2N$ hypermultiplets in the fundamental representation. We then need to demand that this theory ${R}_{N}$ has a larger ﬂavor symmetry

$$SU\left(2N\right)\times SU\left(2\right)\supset \left[SU\left(N\right)\times SU\left(N\right)\times U\left(1\right)\right]\times SU\left(2\right).$$ | (12.3.2) |

We ﬁnally have the S-duality statement:

This general statement was found by Chacaltana and Distler in [72]. We know that the dual $SU\left(2\right)$ gauge coupling has zero beta function. Applying the analysis as in Sec. 10.5, we ﬁnd that the $SU\left(2\right)$ ﬂavor symmetry of the ${R}_{N}$ theory contributes to the running of the $SU\left(2\right)$ coupling as if it has eﬀectively three hypermultiplets in the doublet. Equivalently, we have

$${\u27e8{j}_{\mu}{j}_{\nu}\u27e9}_{{R}_{N}}=3{\u27e8{j}_{\mu}{j}_{\nu}\u27e9}_{\text{freehyperinadoubletof}SU\left(2\right)\text{}}$$ | (12.3.4) |

where ${j}_{\mu}$ is the $SU\left(2\right)$ ﬂavor symmetry current. See Fig. 12.15.

When $N=3$ we can say a little more about this duality. This was originally found by Argyres and Seiberg in [74]; the presentation here follows that given by Gaiotto in [8].

Now the puncture of type $\left(N-2,1,1\right)=\left(1,1,1\right)$ is a full puncture. Therefore the theory ${R}_{3}$ is given by a sphere with three full punctures, see Fig. 12.16. The structure of ${\varphi}_{k}\left(z\right)$ is already given in (12.3.1). Therefore, this theory has just one Coulomb branch operator, of dimension 3.

We know that there is an enhancement of the ﬂavor symmetry $SU\left(3\right)\times SU\left(3\right)$ associated to two full punctures to $SU\left(6\right)$, as in (12.3.2). We have three full punctures. Therefore, it should be that the ﬂavor symmetry $F$ of this theory should be such that we have the following diagram

for any choice of two out of three $SU\left(3\right)$s.
Fortunately, there is unique such $F$,
that is ${E}_{6}$,
see Fig. 12.17. There, on the left, we introduce a diagrammatic notation for this
theory. On the center and on the right, we have the extended Dynkin diagram of
${E}_{6}$ with one node
removed.^{16} We
clearly see subgroups $SU{\left(3\right)}^{3}$
and $SU\left(6\right)\times SU\left(2\right)$.
We already saw above that this theory has only one Coulomb branch operator, and
its dimension is three. This nicely ﬁts the feature of a rank-1 superconformal theory
announced to exist in Sec. 10.4. This is equivalent to Minahan-Nemeschansky’s theory
$MN\left({E}_{6}\right)$.

We conclude that we have the following duality:

We can give a few more checks to this duality. The ﬁrst one concerns the current two-point functions. Firstly, we computed the current two-point function for the $SU\left(2\right)$ ﬂavor symmetry in (12.3.4). Then the whole ${E}_{6}$ ﬂavor currents, which include the $SU\left(2\right)$ ones, should have the same coeﬃcient in front of the two-point function. Note that $SU\left(6\right)$ ﬂavor symmetry of the $SU\left(3\right)$ gauge theory with six ﬂavors is also a subgroup of this ${E}_{6}$ ﬂavor symmetry. Therefore, we should have

This is indeed the case, since the left hand side can be computed in the extreme weakly-coupled regime, where they just come from three hypermultiplets in the fundamental representation of $SU\left(6\right)$.

The second check is about the Higgs branch. The $SU\left(3\right)$ theory with six ﬂavors has a Higgs branch of quaternionic dimension

$$3\cdot 6-dimSU\left(3\right)=10.$$ | (12.3.8) |

Let us perform the computation in the dual side. The theory $MN\left({E}_{6}\right)$ has a Higgs branch of quaternionic dimension 11, as we tabulated in Table 10.1. We have a doublet of $SU\left(2\right)$ in addition, and we perform the hyperkähler quotient with respect to $SU\left(2\right)$ gauge group. Therefore the quaternionic dimension is

$$11+2-dimSU\left(2\right)=10,$$ | (12.3.9) |

which agrees with what we found above in the original gauge theory side. Here we only compared the dimensions, but they can be shown to be equivalent as hyperkähler manifolds, see [75].