We can now have some fun manipulating punctures. For example, consider a gauge theory with gauge group $SU{\left(N\right)}^{N-2}$, with bifundamental hypermultiplets between consecutive groups, together with $N$ additional fundamental hypermultiplets for the ﬁrst and the last one, see the ﬁrst row of Fig. 12.18. The Seiberg-Witten solution is easily given: it is given by a sphere of type $SU\left(N\right)$ theory, with two full punctures and $N-1$ simple punctures. We go to a duality frame where we decouple all of these $N-1$ simple punctures. Applying the decoupling procedure we learned in Sec. 12.2, we ﬁnd that we generate a quiver tail with gauge group

$$SU\left(N-1\right)\times SU\left(N-2\right)\times \cdots SU\left(2\right),$$ | (12.4.1) |

with bifundamental hypermultiplets between two consecutive groups and one doublet for the last $SU\left(2\right)$. The ﬁrst $SU\left(N-1\right)$ gauges an $SU\left(N-1\right)$ subgroup of the ﬂavor symmetry $SU\left(N\right)$ of the puncture of type $\left(1,1,\dots ,1\right)$, i.e. the full puncture.

In this way, we can construct a theory described by a sphere with three full punctures. This is called the ${T}_{N}$ theory, see Fig. 12.19. Note that ${R}_{3}={T}_{3}$. As we have three full punctures, the ﬂavor symmetry is at least $SU{\left(N\right)}^{3}$. When $N=3$, we saw above that this ﬂavor symmetry enhances to ${E}_{6}$. When $N\ge 4$, there are more than one gauge group in the original gauge theory. Therefore, we do not have an enhancement from $SU\left(N\right)\times SU\left(N\right)$ to any other group. This matches with the fact that there is no group containing $SU{\left(N\right)}^{3}$ such that $SU{\left(N\right)}^{2}$ enhances to $SU\left(2N\right)$ when $N\ge 4$. Putting the punctures at $z=0,1,\infty $, we see that ${\varphi}_{k}$ has the form

$${\varphi}_{k}=\frac{{u}_{k}^{\left(1\right)}+\cdots +{u}_{k}^{\left(k-2\right)}{z}^{k-3}}{{z}^{k-1}{\left(z-1\right)}^{k-1}}d{z}^{k}.$$ | (12.4.2) |

Therefore this theory has one Coulomb branch operator of dimension $3$, two Coulomb branch operators of dimension $4$, …, and $N-2$ Coulomb branch operators of dimension $N$.

Now we can take two copies of this ${T}_{N}$ theory and couple them by an $SU\left(N\right)$ gauge multiplet. In the 6d construction, we just have four full punctures on the sphere. Therefore, we have the S-duality structure exactly as in $SU\left(2\right)$ theory with four ﬂavors, exchanging all four punctures. In fact, ${T}_{2}$ theory is just the trifundamental hypermultiplet ${Q}_{ijk}$.

Next, consider the duality shown in Fig. 12.21. We end up with a three-punctured sphere with two full puncture and one puncture of type $\left(2,2\right)$. In the original gauge theory, we have six fundamental ﬂavors coupling to the $SU\left(4\right)$ gauge multiplet with $SU\left(6\right)$ ﬂavor symmetry. To construct the ultraviolet curve, we split these six ﬂavors into four ﬂavors and two ﬂavors, and applied the rule shown in the third row of Fig. 12.7. Therefore, we see that the theory represented by the three-punctured sphere have a ﬂavor symmetry $F$ of the form

Thankfully, there is a unique such group $F$, that is ${E}_{7}$, see Fig. 12.22. We can of course compute the number of Coulomb branch operators this theory has, by studying ${\varphi}_{k}\left(z\right)$. Here, let us try a diﬀerent procedure. Originally, we had the gauge group $SU\left(4\right)\times SU\left(2\right)$. Therefore, the numbers of the Coulomb branch operators of dimension 2,3,4 were respectively $2,1,1$. On the dual side, the quiver tail contains $SU\left(3\right)\times SU\left(2\right)$, which has two operators of dimension 2 and one operator of dimension 1. The theory represented by the three-punctured sphere should account for the diﬀerence. Therefore there is just one Coulomb branch operator, of dimension 4. This again ﬁts the feature of a rank-1 superconformal theory announced to exist in Sec. 10.4. This is equivalent to Minahan-Nemeschansky’s theory $MN\left({E}_{7}\right)$. We can also check the agreement of the current two-point functions and the dimensions of the Higgs branch, as we did at the end of Sec. 12.3.2.

Generalizing this to the ${E}_{8}$ symmetry is by now rather straightforward. We perform the duality as shown in Fig. 12.23. In the dual side, we have a three-punctured sphere with one full puncture, another of type $\left(2,2,2\right)$, and of type $\left(3,3\right)$. We see that the ﬂavor symmetry $F$ of the theory should satisfy

This nicely ﬁts the structure of Minahan-Nemeschansky’s theory $MN\left({E}_{8}\right)$, see Fig. 12.24. Checks of various properties are left as an exercise to the reader.

Finally, let us study a non-conformal example. Consider $SU\left(N\right)$ theory with ${N}_{f}=2n$ ﬂavors, with $N>n$. The curve is

with the diﬀerential $\lambda =xdz\u2215z$. Here we demanded ${\sum}_{i}{\mu}_{i}+{\stackrel{\u0303}{\mu}}_{i}=0$ and split the $U\left(1\right)$ mass term as $\mu $. Clearly something happens when ${u}_{N-n}=2{\Lambda}^{N-n}$ around $z\sim 1$. This point was ﬁrst considered in [76]. The correct physics was ﬁrst discussed in [77]. We will see below that the low-energy limit is an infrared-free $SU\left(2\right)$ gauge theory coupled to the theories ${R}_{n}$ and ${X}_{N-n+4}$.

To study the infrared behavior, we let

$${u}_{N-n,\text{old}}=2{\Lambda}^{N-n}+{u}_{N-n,\text{new}},\phantom{\rule{2em}{0ex}}z=1+\delta z$$ | (12.4.6) |

and assume the scaling

and

We then need to assume

$${\mathit{\epsilon}}^{\prime}{\phantom{\rule{0.0pt}{0ex}}}^{N-n+2}\sim {\mathit{\epsilon}}^{2}.$$ | (12.4.9) |

In particular we have

$$\mathit{\epsilon}\ll {\mathit{\epsilon}}^{\prime}\ll 1.$$ | (12.4.10) |

In the region $x\sim \mathit{\epsilon}$, we can approximate the curve (12.4.5) as

$$\begin{array}{cc}& \frac{{\Lambda}^{N-n}\prod _{i=1}^{n}\left(x+\mu +{\mu}_{i}\right)}{z}+{\Lambda}^{N-n}\prod _{i=1}^{n}\left(x+\mu +{\stackrel{\u0303}{\mu}}_{i}\right)z\\ & =\left(2{\Lambda}^{N-n}+{u}_{N-n}\right){x}^{n}+{u}_{N-n+2}{x}^{n-2}+\cdots +{u}_{N}& \text{(12.4.11)}\end{array}$$with the scaling (12.4.7). When this is written as a degree-$n$ equation for $x$, the coeﬃcient of the ${x}^{n}$ term is given by

$$\frac{{\Lambda}^{N-n}}{z}+{\Lambda}^{N-n}z-2{\Lambda}^{N-n}-{u}_{N-n}$$ | (12.4.12) |

In the limit $\mathit{\epsilon}\to 0$, two zeros of (12.4.12) collide at $z=1$. This is exactly the situation we studied in Sec. 12.3 for $SU\left(n\right)$ theory with $2n$ ﬂavors in the $q\to 1$ limit. We see that we generate the ${R}_{n}$ theory coupled to $SU\left(2\right)$ gauge group; the operator ${u}_{N-n+2}$ is now regarded as the Coulomb branch vev of this $SU\left(2\right)$. The parameters ${\mu}_{i}$ and ${\stackrel{\u0303}{\mu}}_{i}$ are now the mass parameters for the $SU\left(2n\right)$ symmetry of the ${R}_{n}$ theory.

In the region $x\sim {\mathit{\epsilon}}^{\prime}$, the curve (12.4.5) can be approximated as

$$c\phantom{\rule{0.3em}{0ex}}\delta {z}^{2}=\left({x}^{N-n}+{u}_{2}{x}^{N-n-2}+\cdots +\frac{{u}_{N-n+1}}{x}+\frac{{u}_{N-n+2}}{{x}^{2}}\right),$$ | (12.4.13) |

where the diﬀerential $\lambda =xd\delta z$ and $c$ is an unimportnat constant. We already encountered this in Sec. 11.5; this is the curve describing the Argyres-Douglas point of $SU\left(N-n+1\right)$ theory with 2 ﬂavors. Equivalently, we called this theory ${X}_{N-n+4}$ in Sec. 10.5.

Summarizing, we see that the limiting theory has the structure given in Fig. 12.25. Namely, there is a weakly-coupled $SU\left(2\right)$ gauge group, connecting the region $x\sim \mathit{\epsilon}$ given by a sphere of 6d theory of type $SU\left(n\right)$, representing the ${R}_{n}$ theory, to the region $x\sim {\mathit{\epsilon}}^{\prime}$, given by a sphere of 6d theory of type $SU\left(2\right)$, representing the theory ${X}_{N-n+4}$.

In the intermediate region ${\mathit{\epsilon}}^{\prime}\gg x\gg \mathit{\epsilon}$, the curve is just

$$\delta {z}^{2}\sim \frac{{u}_{N-n+2}}{{x}^{2}}$$ | (12.4.14) |

with $\lambda =\delta zdx\sim \sqrt{{u}_{N-n+2}}dx\u2215x$. We see that there is an $SU\left(2\right)$ gauge group, with

$$a\sim \frac{1}{2\pi i}\oint \delta z\frac{dx}{x}\sim \sqrt{{u}_{N-n+2}}.$$ | (12.4.15) |

The dual coordinate ${a}_{D}$ is then given roughly by

Using $a\sim \mathit{\epsilon}$ and the relation (12.4.9), we see

$${a}_{D}=\frac{2}{2\pi i}\frac{N-n}{N-n+2}aloga+\cdots \phantom{\rule{0.3em}{0ex}}.$$ | (12.4.17) |

Recall that the running is given by

$${a}_{D}\sim \frac{2}{2\pi i}\left(4-{N}_{f}\right)aloga+\cdots $$ | (12.4.18) |

for $SU\left(2\right)$ theory with ${N}_{f}$ ﬂavors. This system then eﬀectively has

$${N}_{f}=\frac{5N-5n+8}{N-n+2}>4.$$ | (12.4.19) |

The $SU\left(2\right)$ is now infrared free. Note that this is correctly the sum of the eﬀective number of ﬂavors of the ${R}_{N}$ theory and the ${X}_{N-n+4}$ theory, as computed already. Indeed, it is $3$ for the ${R}_{N}$ theory, and $2\left(N-n+1\right)\u2215\left(N-n+2\right)$ for the ${X}_{N-n+4}$ theory, see (12.3.4) and (10.5.8), respectively.