To this aim, we introduce a new diagrammatic notation for $\mathcal{\mathcal{N}}=2$ gauge theories. This notation is related to but distinct from the trivalent one introduced in Sec. 9.5.

A diagram is composed of squares and circles with integers written in them, and edges connecting squares and circles. A square with $N$ stands for a $U\left(N\right)$ ﬂavor symmetry, and a circle with $N$ an $SU\left(N\right)$ gauge symmetry. An edge connecting two objects with $N$ and $M$ written within them represents a hypermultiplet $\left({Q}_{i}^{a},{\stackrel{\u0303}{Q}}_{a}^{i}\right)$ where $i=1,\dots ,N$ and $j=1,\dots ,M$. They are in the tensor product of the fundamental representation of $SU\left(N\right)$ and $SU\left(M\right)$, and is called the bifundamental hypermultiplet. Such a diagram speciﬁes an $\mathcal{\mathcal{N}}=2$ gauge theory. This class of theories is often called quiver gauge theories.

The simplest cases are when all the squares and circles have the same number $N$ written in them, see Fig. 12.5. The ﬁrst one in the ﬁgure is just a bifundamental hypermultiplet. The second one is the $SU\left(N\right)$ theory with $2N$ ﬂavors. The last one is an $SU{\left(N\right)}_{1}\times SU{\left(N\right)}_{2}$ theory, so that

- there is a bifundamental hypermultiplet for $SU{\left(N\right)}_{1}\times SU{\left(N\right)}_{2}$, and
- there are $N$ fundamental hypermultiplets for $SU{\left(N\right)}_{1}$, and
- there are $N$ fundamental hypermultiplets for $SU{\left(N\right)}_{2}.$

Note that both $SU{\left(N\right)}_{1}$ and $SU{\left(N\right)}_{2}$ have zero beta function.

Their Seiberg-Witten solutions can be obtained by combining the knowledge we acquired so far. Namely, each edge corresponds to the bifundamental hypermultiplet of $SU\left(N\right)\times SU\left(N\right)$, which we know to come from a three punctured sphere of 6d theory of type $SU\left(N\right)$, with two full punctures and one simple puncture. All we have to do then is to prepare one such sphere for each edge, and connect pairs of full punctures by tubes. For example, the Seiberg-Witten solution for the third theory in Fig. 12.5 is given by

$${\lambda}^{N}+{\varphi}_{2}\left(z\right){\lambda}^{N-2}+\cdots +{\varphi}_{N}\left(z\right)=0$$ | (12.2.1) |

where ${\varphi}_{k}\left(z\right)$ has ﬁve singularities, such that two at $z=0$, $=\infty $ are full and the other three at $z=1,$ $q$ and $q{q}^{\prime}$ are simple.

For simplicity, let us assume that all the mass parameters are zero. Then, from the condition of the order of the poles of the singularities given in (12.1.11), the ﬁelds ${\varphi}_{k}\left(z\right)$ are uniquely ﬁxed to be

$${\varphi}_{k}\left(z\right)=\frac{{u}_{k}^{\left(1\right)}z+{u}_{k}^{\left(2\right)}}{\left(z-1\right)\left(z-q\right)\left(z-q{q}^{\prime}\right)}\frac{d{z}^{k}}{{z}^{k-1}}.$$ | (12.2.2) |

The reader should check that it has the correct behavior at $z=\infty $. This theory is superconformal, as both $SU{\left(1\right)}_{1}$ and $SU{\left(2\right)}_{2}$ have zero one-loop beta function. This is reﬂected by the fact that the variables appearing in the Seiberg-Witten curve (12.2.1) can be assigned scaling dimensions in a natural way. The diﬀerential $\lambda $ should have scaling dimension one, since its integral gives the BPS mass formula: $\left[\lambda \right]=1$. We then set $\left[z\right]=0$ and $\left[{\varphi}_{k}\right]=k$. This means that ${u}_{k}^{\left(i=1,2\right)}$ should be two Coulomb branch operators with scaling dimension $k$. Indeed, we are dealing with an $SU{\left(N\right)}^{2}$ gauge theory which is superconformal, and there are exactly one Coulomb branch operator of scaling dimension $k$ for $k=2,\dots ,N$.

A rather degenerate situation arises when we take just one bifundamental hypermultiplet $\left({Q}_{i}^{a},{\stackrel{\u0303}{Q}}_{i}^{a}\right)$ and couple one $SU\left(N\right)$ gauge multiplet to both indices, see Fig. 12.6. The $N\times N$ hypermultiplet components now behave as an adjoint representation plus a singlet. The singlet part is completely decoupled, and therefore the theory is essentially the $SU\left(N\right)$ gauge theory with an adjoint hypermultiplet. When massless this is the $\mathcal{\mathcal{N}}=4$ super Yang-Mills, whereas it is called $\mathcal{\mathcal{N}}={2}^{\ast}$ theory when massive. The Seiberg-Witten solution can then be obtained by taking a three-punctured sphere and connecting the two full punctures. We end up having a torus with one simple puncture. This solution was ﬁrst found in [70], to which the readers should refer for details.

So far we learned how to solve gauge theories shown in Fig. 12.5. They have the gauge group

$$SU\left(N\right)\times \cdots \times SU\left(N\right)\times \cdots \times SU\left(N\right)$$ | (12.2.3) |

with bifundamentals between adjacent $SU\left(N\right)$ groups, and additional $N$ ﬂavors each for the ﬁrst and the last $SU\left(N\right)$ groups. All $SU\left(N\right)$ groups have zero beta function.

Let us consider a slight generalization of this class of theories. The gauge group is of the following form

$$SU\left(N\right)\times \cdots \times SU\left(N\right)\times SU\left({N}_{k}\right)\times SU\left({N}_{k-1}\right)\times \cdots SU\left({N}_{2}\right)\times SU\left({N}_{1}\right).$$ | (12.2.4) |

We put the bifundamental hypermultiplets between adjacent $SU\left(N\right)$ and $SU\left({N}^{\prime}\right)$. Such gauge theories are often called linear quiver gauge theories, since the gauge factors are arranged in a linear fashion.

Here, we introduce additional ﬂavors for every $SU$ group, so that they all have zero beta functions. Deﬁne ${N}_{0}=0$ and ${N}_{k+1}=N$. Then the condition we need to impose is

$${N}_{i-1}+{N}_{i+1}+{n}_{i}=2{N}_{i},\phantom{\rule{2em}{0ex}}i=1,\dots ,k$$ | (12.2.5) |

where ${n}_{i}$ is the number of additional fundamental hypermultiplet for $SU\left({N}_{i}\right)$. Since ${n}_{i}\ge 0$, we have ${s}_{i}\ge {s}_{i+1}$ where ${s}_{i}={N}_{i}-{N}_{i-1}$. Clearly ${\sum}_{i=1}^{k+1}{s}_{i}=N$.

A decreasing sequence of integers ${s}_{1}\ge {s}_{2}\ge \cdots \ge {s}_{k+1}$ whose sum is $N$ is called a partition of $N$. Then we can phrase our ﬁnding here by saying that this type of gauge theory can be characterized by a partition of $N$. A partition can be graphically represented by a Young diagram. Here we draw it by arranging boxes so that the widths of the rows are given by ${s}_{i}$. Examples are shown for $N=4$ on the left hand side of Fig. 12.7. There, additional ${n}_{i}$ ﬂavors are shown by connecting a box ${n}_{i}$ to a circle ${N}_{i}$.

What is the Seiberg-Witten solutions of this class of theories? There are a few independent methods to arrive at the solutions. Originally they are obtained using a conﬁguration of branes in type IIA string theory and lifting it to M-theory [6]. We now also have a ﬁeld theoretical derivation in terms of instanton computation [71]. In this subsection, we just state the results, and give a few justiﬁcation. We will come back to this point in more details in Sec. 12.5.

The Seiberg-Witten solution is obtained by the following procedure. First, consider a sphere of 6d theory of type $SU\left(N\right)$, realizing the theory where all ${N}_{i}$ is equal to $N$. It was given by the Seiberg-Witten curve of the form

$${\lambda}^{N}+{\varphi}_{2}\left(z\right){\lambda}^{N-2}+\cdots +{\varphi}_{N}\left(z\right)=0.$$ | (12.2.6) |

As explained above, we have two full punctures and a number of simple punctures. We then replace one full puncture at $z=\infty $ with a new type of puncture labeled by the Young diagram, see the right hand side of Fig. 12.7. These new types of punctures, together with the simple and the full punctures introduced already, are called tame $SU\left(N\right)$ punctures.

The change of the type of the puncture is the change of the structure of the singularities of the ﬁelds ${\varphi}_{k}\left(z\right)$. We can also write the curve (12.2.6) as

$$det\left(\lambda -\phi \left(z\right)\right)=0$$ | (12.2.7) |

where $\phi \left(z\right)$ is a meromorphic one-form which is a traceless $N\times N$ matrix, as we did for the $SU\left(2\right)$ case in Sec. 9.5.5. Then ${\varphi}_{k}\left(z\right)$ is given by an elementary symmetric degree-$k$ polynomial combination of the eigenvalues of $\phi \left(z\right)$. Then the structure of the singularities of ${\varphi}_{k}\left(z\right)$ can be described also by the structure of the residue of $\phi \left(z\right)$.

We already saw a full puncture carries the ﬂavor symmetry $SU\left(N\right)$, and a simple puncture $U\left(1\right)$. To correctly reproduce the ﬂavor symmetry of the total theory, the singularity at $z=\infty $ labeled by the Young diagram ${s}_{1}\ge {s}_{2}\ge \cdots \ge {s}_{k+1}$ needs to be associated to the ﬂavor symmetry

$$S\left[U\left({n}_{1}\right)\times U\left({n}_{2}\right)\times \dots U\left({n}_{k}\right)\right]$$ | (12.2.8) |

where the $S\left[\cdots \phantom{\rule{0.3em}{0ex}}\right]$ means that we remove the diagonal $U\left(1\right)$ of the following unitary gauge groups.

The description becomes complete once we describe how the ﬁelds ${\varphi}_{k}\left(z\right)$ behave at this new puncture. When the hypermultiplets are all massless, the rule is given as follows. Given a Young diagram with row widths ${s}_{1}\ge {s}_{2}\ge \cdots \phantom{\rule{0.3em}{0ex}}$, deﬁne ${p}_{k}=k-{\nu}_{k}$ where

$$\left({\nu}_{1},{\nu}_{2},\dots ,{\nu}_{N}\right)=\left(\underset{{s}_{1}}{\underbrace{1,\dots ,1}},\underset{{s}_{2}}{\underbrace{2,\dots ,2}},\dots ,\right)$$ | (12.2.9) |

Then ${\varphi}_{k}\left(z\right)$ should have a pole of order ${p}_{k}$ at the puncture. For an example, see Fig. 12.8.

In terms of the $N\times N$ matrix-valued one-form $\phi \left(z\right)$ the statement is somewhat simpler. Namely, the residue of $\phi \left(z\right)$ at the puncture should be given by

$$Res\phi \left(z\right)\sim {J}_{{s}_{1}}\oplus {J}_{{s}_{2}}\oplus \cdots \oplus {J}_{{s}_{k}}$$ | (12.2.10) |

where ${J}_{s}$ is an $s\times s$ Jordan block:

It is a good exercise to check that the pole orders ${p}_{k}$ of ${\varphi}_{k}\left(z\right)$ can be reproduced by plugging in (12.2.10) into (12.2.7) and comparing it with (12.2.6).

When the hypermultiplets are massive, the rule goes instead as follows. Take the same Young diagram, but describe it with column heights ${t}_{1}\ge {t}_{2}\ge \cdots {t}_{x}$ where $x$ is the number of columns. Then $\lambda $ should have $N$ residues with following structure:

$$\left(\underset{{t}_{1}}{\underbrace{{\mu}_{1},\dots ,{\mu}_{1}}},\underset{{t}_{2}}{\underbrace{{\mu}_{2},\dots ,{\mu}_{2}}},\dots ,\right)$$ | (12.2.12) |

where we need to impose

$$\sum {t}_{i}{\mu}_{i}=0.$$ | (12.2.13) |

This is equivalent to say that the residue of the matrix-valued one-form $\phi \left(z\right)$ should be conjugate to a diagonal matrix with entries given by (12.2.12).

We identify these residues with the mass parameters associated to the ﬂavor symmetry (12.2.8). There are ${n}_{i}$ mass parameters ${\mu}_{a}^{\left(i\right)}$, $a=1,\dots ,{n}_{i}$ for each $U\left({n}_{i}\right)$. We then make the identiﬁcation

Note that $\sum {n}_{i}$ equals the number of columns $x$. The individual ${n}_{i}$ corresponds to the number of columns of a certain given height, say $h$, then there is an index $a$ such that

$${t}_{a}={t}_{a+1}=\cdots ={t}_{a+{n}_{i}-1}=h.$$ | (12.2.15) |

Then the Weyl group of the $U\left({n}_{i}\right)$ ﬂavor symmetry can be identiﬁed with the permutation of the columns of height $h$.

It is conventional in the $\mathcal{\mathcal{N}}=2$ literature to label the punctures using column heights $\left({t}_{i}\right)$. The full puncture is then associated to the Young diagram $\left(1,1,\dots ,1\right)$, and the simple puncture has the Young diagram $\left(N-1,1\right)$. We can indeed check that the general formulas (12.2.9) and (12.2.12) reproduce (12.1.6) and (12.1.11). Note also that the puncture of type $\left(N\right)$ does not have poles at all. This corresponds to an absence of the puncture.

Let us apply this general discussion to the particular case $N=2$ which we discussed extensively in Sec. 9. There, we introduced a diﬀerent diagrammatic notation using trivalent vertices, reﬂecting special properties of $SU\left(2\right)$, see Fig. 12.9. In the current approach, we see that both the full puncture and the simple puncture for $N=2$ have the Young diagram $\left(1,1\right)$, thus losing the distinction. The only other type of puncture is $\left(2\right)$, which corresponds to the absence of puncture in the ﬁrst place. Therefore the construction in this section does not give anything new for $N=2$.

Let us check that the prescription described above reproduces the expected number of Coulomb branch operators. Compare, for example, the ﬁrst and the fourth rows of Fig. 12.7. The Seiberg-Witten solutions are both given by

$${\lambda}^{4}+{\varphi}_{2}\left(z\right){\lambda}^{2}+{\varphi}_{3}\left(z\right)\lambda +{\varphi}_{4}\left(z\right)=0.$$ | (12.2.16) |

In both cases, ${\varphi}_{k}\left(z\right)$ has one full puncture at $z=0$ and ﬁve simple punctures at $z={z}_{i}$. The puncture at $z=\infty $ changes types. For the theory at the ﬁrst row, the puncture at $z=\infty $ is a full puncture, where ${\varphi}_{k}\left(z\right)$ has poles of order $k-1$. This determines the ﬁelds ${\varphi}_{k}\left(z\right)$ to be given by

Note that the degree of the polynomial in the numerator is ﬁxed by the order of the pole at $z=\infty $. We identify ${u}_{k}^{\left(i\right)}$ as the dimension-$k$ Coulomb branch operator of the $i$-th $SU\left(4\right)$ gauge group.

Now change the type of the puncture at $z=\infty $. The allowed order of the pole there is reduced by ${\nu}_{k}$ as given in (12.2.9). In this particular case, the orders of the poles for ${\varphi}_{2}\left(z\right)$, ${\varphi}_{3}\left(z\right)$, ${\varphi}_{4}\left(z\right)$ are reduced by $0$, $1$, $2$ respectively. This reduces the degree of the polynomials in the numerator of (12.2.17) by $0$, $1$, $2$ respectively, resulting in

$$\begin{array}{lll}\hfill {\varphi}_{2}\left(z\right)& =\frac{{u}_{2}^{\left(1\right)}+{u}_{2}^{\left(2\right)}z+{u}_{2}^{\left(3\right)}{z}^{2}+{u}_{2}^{\left(4\right)}{z}^{3}}{\prod _{i}^{5}\left(z-{z}_{i}\right)}\frac{d{z}^{2}}{z}\phantom{\rule{2em}{0ex}}& \hfill \text{(12.2.18)}\\ \hfill {\varphi}_{3}\left(z\right)& =\frac{{u}_{3}^{\left(1\right)}+{u}_{3}^{\left(2\right)}z+{u}_{3}^{\left(3\right)}{z}^{2}}{\prod _{i}^{5}\left(z-{z}_{i}\right)}\frac{d{z}^{3}}{{z}^{2}}\phantom{\rule{2em}{0ex}}& \hfill \text{(12.2.19)}\\ \hfill {\varphi}_{4}\left(z\right)& =\frac{{u}_{4}^{\left(1\right)}+{u}_{4}^{\left(2\right)}z}{\prod _{i}^{5}\left(z-{z}_{i}\right)}\frac{d{z}^{4}}{{z}^{3}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(12.2.20)}\end{array}$$We identify ${u}_{k}^{\left(i\right)}$ as a dimension $k$ Coulomb branch operator for the $i$-th gauge group. We see that the third gauge group now has the Coulomb branch operators of dimension 2 and of dimension 3, and that the fourth gauge group only has the Coulomb branch operator of dimension 2. This agrees with our claim that the gauge group is now $SU\left(4\right)\times SU\left(4\right)\times SU\left(3\right)\times SU\left(2\right)$.

This analysis of the number of the Coulomb branch operators can be extended to arbitrary $N$ and to arbitrary Young diagram. By a straightforward but somewhat cumbersome combinatorial computation we see that the pole structure (12.2.9) reproduces the structure of the gauge group as given in (12.2.4).

Now let us study what happens when we make the coupling of the last gauge group in (12.2.4) very weak. When we completely turn oﬀ the coupling, we lose the last gauge group $SU\left({N}_{k}\right)$. The new last gauge group is $SU\left({N}_{k-1}\right)$, which is now coupled to ${N}_{k}+{n}_{k-1}$ hypermultiplets in the fundamental representation. Note that ${N}_{k}$ of them originally came from the bifundamental hypermultiplet for $SU\left({N}_{k-1}\right)\times SU\left({N}_{k}\right)$.

This process for the quiver tail characterized by the Young diagram $\left(3,1\right)$ is shown on the right hand side of Fig. 12.10. In terms of the ultraviolet curve, turning oﬀ the coupling of the last gauge group corresponds to splitting oﬀ the last two punctures. When we completely decouple the gauge group, we ﬁnd a new puncture emerging. The type of this new puncture can be determined by the rule explained above, from the resulting gauge theory with one less gauge group. In this case, the newly appearing puncture on the left has the Young diagram $\left(2,1,1\right)$. The decoupled three-punctured sphere on the right hand side represents one hypermultiplet in the doublet representation of $SU\left(2\right)$. We intentionally do not discuss the new puncture arising on this decoupled three-punctured sphere on the right; for more details, see [72, 73].

We can continue the process. Decoupling the next gauge group, the Young diagram becomes $\left(1,1,1,1\right)$, i.e. the full puncture. The situation is shown in Fig. 12.11. The decoupled three-punctured sphere on the right hand side represents two hypermultiplets in the triplet representation of $SU\left(3\right)$.

Note that $SU\left(3\right)$ gauge group before the complete decoupling can be thought of as gauging the $SU\left(3\right)$ subgroup of the $SU\left(4\right)$ ﬂavor symmetry of the full puncture, as shown in the second row of the ﬁgure. This splits four fundamental ﬂavors coupled to $SU\left(4\right)$ into a set of three ﬂavors and an additional one ﬂavor. The $SU\left(3\right)$ gauge group makes the ﬁrst three into the bifundamental hypermultiplet of $SU\left(4\right)\times SU\left(3\right)$, and one ﬂavor remains to couple just to $SU\left(4\right)$ on the upper row.

Another example of decoupling process for the puncture of type $\left(2,2\right)$ is shown in Fig. 12.12. The decoupled three-punctured sphere on the right hand side represents an empty theory.