In Sec. 12.2, we introduced punctures on the ultraviolet curve labeled by Young diagrams in a rather ad hoc manner. Examples for $SU\left(4\right)$ case were shown in Fig. 12.7. In this last subsection of the note, we would like to study the meaning of the Young diagram in slightly more detail. For example, how should we understand the process of changing the full puncture to the simple puncture, i.e. the puncture of type $\left(3,1\right)$, shown in Fig 12.26? We will use this particular example of changing the full puncture $\left(1,1,1,1\right)$ to the simple puncture $\left(3,1\right)$ as a concrete example throughout this section. The extension to the general punctures should be left as an exercise to the reader. The content of this section is based on an unpublished work with Francesco Benini, done sometime between 2009 and 2010.

The Seiberg-Witten curves are both given by

$${\lambda}^{4}+{\varphi}_{2}\left(z\right){\lambda}^{2}+{\varphi}_{3}\left(z\right)\lambda +{\varphi}_{4}\left(z\right)=0.$$ | (12.5.1) |

In both cases, ${\varphi}_{k}\left(z\right)$ has one full puncture at $z=0$ and ﬁve simples punctures at $z={z}_{i}$. For the ﬁrst, the puncture at $z=\infty $ was full and for the second, it is a simple puncture, of type $\left(3,1,1\right)$.

For the ﬁrst, the ﬁelds ${\varphi}_{k}\left(z\right)$ are given by

For the second, they are given by

Here, ${u}_{k}^{\left(i\right)}$ is the dimension-$k$ Coulomb branch operator of the $i$-th gauge group, and the way to determine them from the pole structure was described around (12.2.17).

It is clear that ${\varphi}_{k}\left(z\right)$ in (12.5.3) is obtained by setting ${u}_{4}^{\left(3,4\right)}={u}_{3}^{\left(4\right)}=0$ in (12.5.2). We will explain below that we can start from the ﬁrst theory, set the Coulomb branch parameters to this subspace, and then move to the Higgs branch, realizing the second theory.

To facilitate the analysis of the Higgs branch, we introduce new names to the bifundamentals, see Fig 12.27. We name the rightmost $SU\left(N\right)$ ﬂavor symmetry $SU{\left(N\right)}_{0}$, and the gauge groups $SU{\left(N\right)}_{i=1,2,3,\dots}$ from the right to the left. Introduce an auxiliary $N$-dimensional complex space ${V}_{i}$ for each of them. For each consecutive pair $SU{\left(N\right)}_{i+1}\times SU{\left(N\right)}_{i}$, we have a bifundamental hypermultiplet $\left({Q}_{b}^{a},{\stackrel{\u0303}{Q}}_{a}^{b}\right)$ where $a=1,\dots ,N$ and $b=1,\dots ,N$ are the indices for $SU{\left(N\right)}_{i+1}$, $SU{\left(N\right)}_{i}$ respectively. We regard ${Q}_{b}^{a}$ as a linear map ${A}_{i}:{V}_{i}\to {V}_{i+1}$ and ${\stackrel{\u0303}{Q}}_{a}^{b}$ as a map in the reverse direction ${B}_{i}:{V}_{i+1}\to {V}_{i}$. Note that each pair $\left({A}_{i},{B}_{i}\right)$ comes from one of the several three-punctured spheres comprising the ultraviolet curve, as shown in the ﬁgure. Let us say that there are $k$ three-punctured spheres in total.

Let us introduce the notation

$${M}_{i}^{\prime}:={B}_{i}{A}_{i},\phantom{\rule{2em}{0ex}}{}_{}^{\prime}{M}_{i}:={A}_{i}{B}_{i}.$$ | (12.5.4) |

We will use the trivial identity

repeatedly below.

Note that $tr\phantom{\rule{0.3em}{0ex}}{M}_{i}:=tr\phantom{\rule{0.3em}{0ex}}{M}_{i}^{\prime}=tr\phantom{\rule{0.3em}{0ex}}{}_{}^{\prime}{M}_{i}$ is the mass term for the $i$-th $U\left(1\right)$ ﬂavor symmetry, which can be naturally associated to the simple puncture of the $i$-th three-punctured sphere. We also have two other gauge invariant combinations, namely

${M}_{0}^{\prime}{|}_{\text{traceless}}$ is an adjoint of $SU\left(N\right)$ ﬂavor symmetry associated to the full puncture of the rightmost sphere, at $z=\infty $. Similarly, ${}_{}^{\prime}{M}_{k}{|}_{\text{traceless}}$ is an adjoint of the $SU\left(N\right)$ ﬂavor symmetry at the puncture $z=0$.

Now, we would like to make a local modiﬁcation at the puncture $z=\infty $, by giving a non-zero vev to the adjoint ﬁeld ${M}_{0}^{\prime}{|}_{\text{traceless}}$. Other gauge-invariant combinations $tr\phantom{\rule{0.3em}{0ex}}{M}_{i}$ for $i=1,\dots ,k$ and ${}_{}^{\prime}{M}_{k}{|}_{\text{traceless}}$ are ‘localized’ at other punctures. So we choose to keep them zero.

The F-term relation from the adjoint scalar in the gauge multiplet for $SU{\left(N\right)}_{i}$ is

$${M}_{i+1}^{\prime}{|}_{\text{traceless}}={}_{}^{\prime}{M}_{i}{|}_{\text{traceless}}.$$ | (12.5.7) |

As we are imposing the condition $tr\phantom{\rule{0.3em}{0ex}}{M}_{i}=0$, we can drop the tracelessness condition and just say

$${M}_{i+1}^{\prime}={}_{}^{\prime}{M}_{i}.$$ | (12.5.8) |

Then we have the following relations:

for arbitrary $n$.

This means that the gauge-invariant combination ${M}_{0}^{\prime}$, transforming in the adjoint of the $SU\left(N\right)$ ﬂavor symmetry, is a nilpotent matrix. They can be put in the Jordan normal form by a complexiﬁed $SU\left(N\right)$ rotation:

$${M}_{0}^{\prime}={J}_{{t}_{1}}\oplus {J}_{{t}_{2}}\oplus \cdots \phantom{\rule{0.3em}{0ex}},\phantom{\rule{2em}{0ex}}\sum _{i}{t}_{i}=N$$ | (12.5.10) |

where ${J}_{t}$ is the Jordan cell of size $t$,

We again found a partition $\left({t}_{i}\right)$ of $N$. We argue below that this partition $\left({t}_{i}\right)$ is exactly the Young diagram labeling the punctures introduced in Sec. 12.2. To study the eﬀect of the vev (12.5.10), we need to ﬁnd a choice of hypermultiplet ﬁelds $\left({A}_{i},{B}_{i}\right)$ solving the F-term and the D-term relations.

To write down such a choice, it is useful to introduce a further diagrammatic notation, see Fig 12.28. An $N$-dimensional vector space $V$ has $N$ basis vectors. Let us denote them by a column of $N$ dots. A matrix whose entries are 0 or 1, from $V$ to ${V}^{\prime}$ can be represented by a set of arrows connecting the $a$-th dot for $V$ to the $b$-th dot for ${V}^{\prime}$ if and only if the $\left(a,b\right)$-th entry of the matrix is 1. In the center of Fig 12.28 we denoted a Jordan block ${J}_{4}$ of size $4$. The rightmost diagram of the same ﬁgure is for a projector to the last two basis vectors.

For concreteness, let $N=4$, and give a nilpotent vev to ${M}_{0}^{\prime}$ of type $\left(3,1\right)$, namely it is given by ${J}_{3}\oplus {J}_{1}$. A solution to the F-term relations are given in Fig. 12.29. There, we see that the unbroken gauge group is now $SU\left(4\right)\times SU\left(4\right)\times SU\left(3\right)\times SU\left(2\right)$.

In general, a solution to the F-term relations can be constructed as follows. Let us say we would like to set ${M}_{0}^{\prime}=X$, where $X$ is in a Jordan normal form. We identify the vector spaces ${V}_{0}={V}_{1}={V}_{2}=\cdots \phantom{\rule{0.3em}{0ex}}$. Let us introduce the notation ${W}_{i}=Im{X}^{i}$ and denote the projector to ${W}_{i}$ by ${P}_{{W}_{i}}$. We then set

$${A}_{0}=X,\phantom{\rule{1em}{0ex}}{A}_{1}=X{P}_{{W}_{1}},\phantom{\rule{1em}{0ex}}{A}_{2}=X{P}_{{W}_{2}},\dots $$ | (12.5.12) |

and take

$${B}_{0}={P}_{{W}_{1}},\phantom{\rule{1em}{0ex}}{B}_{1}={P}_{{W}_{2}},\phantom{\rule{1em}{0ex}}{B}_{2}={P}_{{W}_{3}},\dots .$$ | (12.5.13) |

Clearly, the remaining gauge group is of the form

$$\cdots \times SU\left({N}_{3}\right)\times SU\left({N}_{2}\right)\times SU\left({N}_{1}\right)$$ | (12.5.14) |

where

$${N}_{i}=N-dim{W}_{i}=N-rank{X}^{i}.$$ | (12.5.15) |

Deﬁne ${s}_{i}={N}_{i}-{N}_{i-1}$. A short combinatorial computation shows that when $X$ has the type described by a Young diagram whose $i$-th column from the left has height ${t}_{i}$, the sequence $\left({s}_{1},{s}_{2},\dots \right)$ is such that ${s}_{i}$ is the width of the $i$-th row from the bottom. This is exactly the rule we already introduced in Sec. 12.2 for the gauge group. Now let us determine the massless matter content of the resulting theory.

An indirect but fast way to determine the matter content is as follows. We started from a superconformal theory without any parameters. After the Higgsing, the only parameter with mass dimensions is the vev of the hypermultiplet ﬁelds. By the general decoupling of the hypermultiplet and the vector multiplet side of the Lagrangian, which we discussed in Sec. 7.1, we see that there cannot be any mass terms or dynamical scales in the low-energy theory after the Higgsing. Therefore, the resulting theory is also superconformal. We already determined ${N}_{i}$, and we can only have bifundamental ﬁelds or fundamental ﬁelds. This shows that $SU\left({N}_{i}\right)$ should have exactly

$${n}_{i}=2{N}_{i}-{N}_{i+1}-{N}_{i-1}$$ | (12.5.16) |

fundamental hypermultiplets in addition.

Of course this result can also be obtained by a direct computation of the mass terms of the various ﬁelds in the system. Note that originally, there is an $\mathcal{\mathcal{N}}=1$ superpotential $tr\phantom{\rule{0.3em}{0ex}}{A}_{i}{\Phi}_{i}{B}_{i}$ and $tr\phantom{\rule{0.3em}{0ex}}{B}_{i}{\Phi}_{i+1}{A}_{i}$ where ${\Phi}_{i}$ is the adjoint scalar of the $SU{\left(N\right)}_{i}$ vector multiplet. As we gave vevs to some components to ${A}_{i}$ and ${B}_{i}$, we see that certain components of hypermultiplets scalars and vector multiplet scalars pair up, due to the three-point couplings. One example is shown in Fig 12.30. There, the vev of ${A}_{1}$ represented by a red down-left arrow gives a mass term of a component of the vector multiplet scalar of the gauge group for ${V}_{2}$ and a component of ${B}_{1}$.

We see that always a bifundamental in $SU\left({N}_{i+1}\right)\times SU\left({N}_{i}\right)$ remains massless. But from a careful analysis of the mass terms, we see that sometimes more charged hypermultiplets remain massless. For example, as shown in Fig 12.31, the whole bifundamental between ${V}_{3}$ and ${V}_{2}$ remains massless. At ${V}_{2}$, $SU\left(4\right)$ is broken to $SU\left(3\right)$. Therefore, from the point of view of the unbroken $SU\left(4\right)$ at ${V}_{3}$, we see there are an $SU\left(4\right)\times SU\left(3\right)$ bifundamental together with a fundamental of $SU\left(4\right)$. This can be generalized to see that the number of additional fundamental hypermultiplets of $SU\left({N}_{i}\right)$ is given by (12.5.16).

In Sec. 12.2, we said that the puncture at $z=\infty $ carries all the ﬂavor symmetry associated to the additional ${n}_{i}$ fundamental hypermultiplets attached to $SU\left({N}_{i}\right)$. This sounded somewhat counter-intuitive, since the ﬂavor symmetry $SU\left({n}_{i}\right)$ looks more associated to the $i$-th node. Now we understand the physical mechanism operating here. Let us take the puncture of type $\left(3,1\right)$ again for concreteness, see Fig 12.32. The vev $X={M}_{0}^{\prime}$, which is from our rule is given by $X={J}_{3}\oplus {J}_{1}$, is invariant under the $U\left(1\right)$ rotation acting on the three basis vectors, as denoted by black dots in the ﬁgure. This symmetry, if unaccompanied by the gauge rotation, does not ﬁx the Higgs vevs $\u27e8{A}_{i}\u27e9$ and $\u27e8{B}_{i}\u27e9$. To make the symmetry compatible with the Higgs vev, we need to rotate at the same time all the other basis vectors connected from the original black dots by the arrows representing ${A}_{i}$ and ${B}_{i}$.

We see that the Higgs vevs identify the $U\left(1\right)$ ﬂavor symmetry rotating three basis vectors of ${V}_{0}$ and the $U\left(1\right)$ ﬂavor symmetry rotating the last basis vector of ${V}_{3}$. After the Higgsing, this latter $U\left(1\right)$ symmetry is exactly the ﬂavor symmetry carried by the additional one fundamental hypermultiplet of $SU\left(4\right)$ at ${V}_{3}$, denoted by green in the ﬁgure. This analysis can be generalized to arbitrary types of punctures.

Summarizing, we found a new interpretation of the punctures introduced in Sec. 12.2. Such a puncture can always be obtained from the full puncture, by ﬁrst choosing the Coulomb branch vevs to the right subspace, and then giving a nilpotent vev to the hypermultiplet combination ${M}_{0}^{\prime}$ which transforms in the adjoint of the ﬂavor $SU\left(N\right)$ associated to the full puncture. The vev given to ${M}_{0}^{\prime}$ causes some of the other hypermultiplet ﬁelds ${A}_{i}$, ${B}_{i}$ for $i>0$ to have non-zero vevs, breaking the original gauge group $\cdots \times SU\left(N\right)\times SU\left(N\right)\times SU\left(N\right)$ to $\cdots \times SU\left({N}_{3}\right)\times SU\left({N}_{2}\right)\times SU\left({N}_{1}\right)$.