So far we studied the Argyres-Douglas CFTs which were obtained by special limits of $SU\left(2\right)$ gauge theories with ${N}_{f}=1,2,3$ ﬂavors. The data of these and other related CFTs are summarized in Table 10.1. The Argyres-Douglas CFTs are the ﬁrst three rows of the table. The fourth row is for the $SU\left(2\right)$ theory with ${N}_{f}=4$ massless ﬂavors. The next two rows are for slightly diﬀerent classes of theories. Namely, if we consider $SU\left(2\right)$ theory with more than $4$ ﬂavors or $U\left(1\right)$ theory with nonzero charged hypermultiplets, they are infrared free, see (8.1.5) and (8.3.13). They appeared repeatedly as a local behavior close to a singularity on the $u$-plane.

In Table 10.1 we also tabulated the dimension of the Higgs branch. Let us quickly recall how they are obtained. We know $A{D}_{{N}_{f}=1}\left(SU\left(2\right)\right)$ does not have one, since its parent theory $SU\left(2\right)$ with ${N}_{f}=1$ does not have one either. For $A{D}_{{N}_{f}=2}\left(SU\left(2\right)\right)$, we consider $SU\left(2\right)$ with ${N}_{f}=2$ with a $U\left(1\right)$ mass term. Then the Higgs branch is ${\u2102}^{2}\u2215{\mathbb{Z}}_{2}$, whose quaternionic dimension is 1. For $A{D}_{{N}_{f}=3}\left(SU\left(2\right)\right)$, we consider $SU\left(2\right)$ with ${N}_{f}=3$. With a $U\left(1\right)$ mass term, its Higgs branch can be found by studying a point $u={\mu}_{1}^{2}={\mu}_{2}^{2}={\mu}_{3}^{2}$ in a weakly-coupled theory. The physics there is just $U\left(1\right)$ with three charged hypermultiplets, with the Higgs branch of quaternionic dimension 2. For free $SU\left(2\right)$ theory with ${N}_{f}\ge 4$ ﬂavors, the quaternionic dimension is just $2{N}_{f}-dimSU\left(2\right)$, and similarly for $U\left(1\right)$ theory with $N$ ﬂavors it is given just by $N-dimU\left(1\right)$. One funny feature is that we see

$${dim}_{\mathbb{H}}\left(\text{Higgsbranch}\right)={h}^{\vee}\left(\text{\ufb02avorsymmetry}\right)-1$$ | (10.4.1) |

for the ﬁrst six rows, where ${h}^{\vee}\left(G\right)$ is the dual Coxeter number, which is also a contribution to the one-loop running $C\left(\text{adj}\right)$ from the adjoint representation of $G$. These theories have just one Coulomb branch modulus, and the low-energy theory on a generic point on the Coulomb branch is just a free $U\left(1\right)$ theory. Such theories are called rank-1.

$$\begin{array}{cccccc}\hfill \text{name}\hfill & \hfill \text{monodromy}\hfill & \hfill \text{\ufb02avor}\hfill & \hfill \left[u\right]\hfill & \hfill \#\hfill & \hfill {dim}_{\mathbb{H}}\left(\text{Higgs}\right)\hfill \\ \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332\\ \hfill A{D}_{{N}_{f}=1}\left(SU\left(2\right)\right)\hfill & \hfill \left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{array}\right)\hfill & \hfill \hfill & \hfill 6\u22155\hfill & \hfill 2\hfill & \hfill \hfill \\ \hfill A{D}_{{N}_{f}=2}\left(SU\left(2\right)\right)\hfill & \hfill \left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{array}\right)\hfill & \hfill SU\left(2\right)\hfill & \hfill 4\u22153\hfill & \hfill 3\hfill & \hfill 1\hfill \\ \hfill A{D}_{{N}_{f}=3}\left(SU\left(2\right)\right)\hfill & \hfill \left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill -1\hfill \end{array}\right)\hfill & \hfill SU\left(3\right)\hfill & \hfill 3\u22152\hfill & \hfill 4\hfill & \hfill 2\hfill \\ \hfill SU\left(2\right)\phantom{\rule{1em}{0ex}}{N}_{f}=4\hfill & \hfill \left(\begin{array}{cc}\hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right)\hfill & \hfill SO\left(8\right)\hfill & \hfill 2\hfill & \hfill 6\hfill & \hfill 5\hfill \\ \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332\\ \hfill SU\left(2\right)\phantom{\rule{1em}{0ex}}{N}_{f}>4\hfill & \hfill \left(\begin{array}{cc}\hfill -1\hfill & \hfill 4-{N}_{f}\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right)\hfill & \hfill SO\left(2{N}_{f}\right)\hfill & \hfill \hfill & \hfill {N}_{f}+2\hfill & \hfill 2{N}_{f}-3\hfill \\ \hfill U\left(1\right)\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}N\phantom{\rule{1em}{0ex}}\text{\ufb02avors}\hfill & \hfill \left(\begin{array}{cc}\hfill 1\hfill & \hfill N\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\hfill & \hfill SU\left(N\right)\hfill & \hfill \hfill & \hfill N\hfill & \hfill N-1\hfill \\ \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332& \u0332\\ \hfill MN\left({E}_{6}\right)\hfill & \hfill \left(\begin{array}{cc}\hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)\hfill & \hfill {E}_{6}\hfill & \hfill 3\hfill & \hfill 8\hfill & \hfill 11\hfill \\ \hfill MN\left({E}_{7}\right)\hfill & \hfill \left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)\hfill & \hfill {E}_{7}\hfill & \hfill 4\hfill & \hfill 9\hfill & \hfill 17\hfill \\ \hfill MN\left({E}_{8}\right)\hfill & \hfill \left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)\hfill & \hfill {E}_{8}\hfill & \hfill 6\hfill & \hfill 10\hfill & \hfill 29\hfill \end{array}$$

$$\begin{array}{cccccc}\hfill \Gamma \hfill & \hfill {\mathbb{Z}}_{n}\hfill & \hfill {\hat{\mathcal{\mathcal{D}}}}_{n-2}\hfill & \hfill \hat{\mathcal{\mathcal{T}}}\hfill & \hfill \hat{\mathcal{\mathcal{O}}}\hfill & \hfill \hat{\mathcal{\mathcal{I}}}\hfill \\ \u0332& \u0332& \u0332& \u0332& \u0332& \u0332\\ \hfill {G}_{\Gamma}\hfill & \hfill SU\left(n\right)\hfill & \hfill SO\left(2n\right)\hfill & \hfill {E}_{6}\hfill & \hfill {E}_{7}\hfill & \hfill {E}_{8}\hfill \end{array}$$

We have not discussed the theories listed in the remaining three rows. One way to motivate them is to refer to a classical mathematical result of Kodaira. At a given point on the $u$-plane, we have the ultraviolet curve $C$ and the Seiberg-Witten curve $\Sigma $. The curve $\Sigma $ is a torus, whose shape is parameterized by its complex structure $\tau $, which depend holomophically on $u$. Therefore we have a ﬁbration of torus over the complex plane with the coordinate $u$. The $u$-plane together with the ﬁber $\Sigma $ forms a complex two-dimensional space $X$.

Kodaira classiﬁed the possible types of singularities of such ﬁbrations, and the ﬁrst six rows of Table 10.1 is an exact copy of part of that classiﬁcation. The terminologies are of course diﬀerent, since he was a mathematician and we are studying $\mathcal{\mathcal{N}}=2$ gauge theories. Kodaira’s classiﬁcation had three more rows in addition to the ﬁrst six rows, which motivated people that there should be three additional theories corresponding to them. The Seiberg-Witten curves for these were constructed ﬁrst by Minahan and Nemeschansky in [59, 60].

In the mathematical language, a singularity in the torus ﬁbration creates a singularity in the total space $X$ of complex dimension two. It is locally of the form ${\u2102}^{2}\u2215\Gamma $ where $\Gamma $ is a ﬁnite subgroup of $SU\left(2\right)$. They have a natural ADE classiﬁcation, and we can associate a Lie group ${G}_{\Gamma}$, see Table 10.2.

Mathematicians associate this group ${G}_{\Gamma}$ purely mathematically to a torus ﬁbration, and we see that they are exactly the ﬂavor symmetries of the gauge theories, at least to the ﬁrst six. Mathematicians have associated exceptional groups ${E}_{6,7,8}$ to the last three cases. It was thus quite tempting that the putative theories which correspond to the last three rows have these exceptional groups as the ﬂavor symmetries. From the feature (10.4.1) relating the ﬂavor symmetry and the dimension of the Higgs branch, it is also tempting to guess the dimension of the Higgs branch of these theories. We call these CFTs $MN\left({E}_{6}\right)$, $MN\left({E}_{7}\right)$ and $MN\left({E}_{8}\right)$, respectively.

Note that it is rather hard to have an exceptional ﬂavor symmetry in a classical Lagrangian $\mathcal{\mathcal{N}}=2$ theory. We already know a general form of the Lagrangian: the superpotential as an $\mathcal{\mathcal{N}}=1$ theory is forced to be

$$\sum _{i}\int {Q}_{i}\Phi {\stackrel{\u0303}{Q}}_{i},$$ | (10.4.2) |

and it is possible to check explicitly that the ﬂavor symmetry visible in the ultraviolet is a product of $SU$, $SO$ and $Sp$ groups. Therefore, if the exceptional symmetries are to appear, they need to arise via strong-coupling eﬀects. Once the reader comes to Sec. 12.4 of this note, s/he will ﬁnd exactly how this happens in the ﬁeld theory setting.

Another way to construct the theories listed in the table uniformly is to use Type IIB string theory and its non-perturbative version F-theory. This approach originates in [5] for $SU\left(2\right)$ with four ﬂavors. For the general case, see e.g. [61]. The Seiberg-Witten curves of these rank-1 theories can be constructed most uniformly in this approach, see e.g. [62].

The type IIB string theory is ten-dimensional, and it has objects called 7-branes and 3-branes, where a $p$-brane extends along $p$ spatial direction and one time direction. Let us say the spacetime is of the form

$${\mathbb{R}}^{1,3}\times {\mathbb{R}}^{2}\times {\mathbb{R}}^{4}.$$ | (10.4.3) |

Put a 7-brane in the subspace

$${\mathbb{R}}^{1,3}\times \left\{0\right\}\times {\mathbb{R}}^{4}$$ | (10.4.4) |

and a D3-brane in the subspace

$${\mathbb{R}}^{1,3}\times \left\{u\right\}\times \left\{0\right\}.$$ | (10.4.5) |

There are various types of 7-branes in F-theory, corresponding to Kodaira’s classiﬁcation. They can all be obtained by taking a number of the simplest of the 7-branes, called $\left(p,q\right)$ 7-branes, separated along the ${\mathbb{R}}^{2}$ direction and collapsing them at one point. Then the low-energy theory on the D3-brane gives the corresponding $\mathcal{\mathcal{N}}=2$ theories.

Due to its tension, one $\left(p,q\right)$ 7-brane creates deﬁcit angles $\pi \u22156$. With $n$ $\left(p,q\right)$ 7-branes collapsed to a point, the remaining angle is $1-n\u221512$ of $2\pi $. From this the scaling dimension of $u$ can be computed to be

$$u=\frac{12}{12-n},$$ | (10.4.6) |

which explains an interesting pattern in Table 10.1. These 7-branes obtained by collapsing a number of $\left(p,q\right)$ 7-branes has a gauge symmetry $F$ living on its eight-dimensional worldvolume. From the point of view, this gauge symmetry $F$ on the 7-brane becomes a ﬂavor symmetry. The D3-brane can be absorbed into this 7-brane as an instanton in the internal ${\mathbb{R}}^{4}$ direction of (10.4.4). Then, the Higgs branch should be given by the one instanton moduli space of the group $F$. The $k$-instanton moduli space of a group $F$ has quaternionic dimension $k{h}^{\vee}\left(F\right)-1$, explaining the relation (10.4.1).

So far we saw that the structure of rank-1 theories closely follows that of the Kodaira classiﬁcation, listed in Table 10.1. Before going further, it should be mentioned that there are even more rank-1 theories, ﬁrst found through the analysis of S-dualities of various gauge theories in [63]. Their properties are reviewed from the point of view of the 6d construction in Sec. 7 of [64].