In this lecture note, we ﬁrst discussed the Lagrangian of $\mathcal{\mathcal{N}}=2$ supersymmetric gauge theory, and then studied the Coulomb and Higgs branches of $SU\left(2\right)$ gauge theories with various number of ﬂavors. Two related concepts, the Seiberg-Witten curve and the ultraviolet curve played very important roles along the way. We then analyzed what happens when Coulomb branch vevs or exactly-marginal coupling parameters are ﬁnely tuned. Sometimes the limit was described by a dual weakly-coupled gauge theory, as was the case with $SU\left(2\right)$ theory with four ﬂavors. Most often, however, we saw that we end up with new superconformal ﬁeld theories, of Argyres-Douglas-type or of Gaiotto-type.

For example, we saw the theories $A{D}_{{N}_{f}=1,2,3}\left(SU\left(2\right)\right)$ and $MN\left({E}_{6,7,8}\right)$ in Sec. 10.4, the theories ${X}_{N}$ and ${Y}_{N}$ in Sec. 10.5, ${R}_{N}$ in Sec. 12.3 and ${T}_{N}$ in Sec. 12.4. More and more $\mathcal{\mathcal{N}}=2$ superconformal theories are being discovered, see e.g. [65]. This means that, to fully understand the interrelations of $\mathcal{\mathcal{N}}=2$ supersymmetric systems, we cannot restrict our attention to just $\mathcal{\mathcal{N}}=2$ theories composed of vector multiplets and hypermultiplets.

The topics we covered in this lecture note are only a tip of a huge iceberg that is the study of
$\mathcal{\mathcal{N}}=2$
dynamics, and there are many other further directions of research. Let us list some of
them.^{17} First,
we can put an $\mathcal{\mathcal{N}}=2$
theory on a nontrival manifold:

- Using the topological twisting, it can be put on an arbitrary manifold [78]. When the manifold is compact, the partition function is equivalent to what is known as the Donaldson invariant to mathematicians. Applying the Seiberg-Witten solution in the case of pure $SU\left(2\right)$ theory, Witten introduced a new mathematical invariant, now called the Seiberg-Witten invariant [79], which revolutionized four-dimensional diﬀerential geometry twenty years ago.
- We can put it on ${S}^{1}$. Then the theory is eﬀectively three-dimensional. As was ﬁrst analyzed in [80], the Coulomb branch as a three-dimensional theory is naturally a ﬁbration over the Coulomb branch as a four-dimensional theory. The 3d Coulomb branch is hyperkähler, and has the structure of a classical integrable system with ﬁnite degrees of freedom. This integrable system was originally introduced in [70]. For modern developments, see e.g. [81].
- On the so-called $\Omega $ background. Very roughly speaking, it involves a forced rotation of the entire Euclidean system on ${\mathbb{R}}^{4}$ around the origin. The spacetime is eﬀectively compact and we can deﬁne the partition function, which is usually called Nekrasov’s partition function. For a recent comprehensive discussion, see e.g. [71]. In a certain limiting case, it is found in [82] that it gives rise to a quantized integrable system which is a quantized version of the Donagi-Witten integrable system.
- On a round or deformed ${S}^{4}$. The spacetime is compact and the partition function can be computed exactly, see e.g. [83, 84, 85]. The partition function is also known to be related to 2d conformal ﬁeld theories on the ultraviolet curve, see e.g. [86, 87].
- On ${S}^{1}\times {S}^{3}$. The partition function is called the superconformal index, and gives rise to 2d topological ﬁeld theories on the ultraviolet curve. It also has a deep relation to various important orthogonal polynomials, see e.g. [88, 89].
- Other backgrounds can also be considered. See [90] for ${S}^{2}\times {S}^{1}\times \mathbb{R}$. A study of $\mathcal{\mathcal{N}}=1$ theories on ${T}^{2}\times {S}^{2}$ can be found in [91], and surely $\mathcal{\mathcal{N}}=2$ systems can be similarly considered there.

Second, we can study dynamical excitations and externally-introduced operators of these theories:

- We have seen how we can read oﬀ the number of BPS-saturated particle types from the 6d construction. The number is an integer and therefore it cannot usually change, but it does jump at certain loci in the Coulomb branch. This is called the wall-crossing and is an intensively-studied area, see e.g. [51]. The resulting spectrum can often be summarized using a diagram, called the BPS quiver. This point of view was originally introduced in the context of $\mathcal{\mathcal{N}}=2$ supergravity in [92]. For more recent developments, see e.g. [93, 94, 95].
- Instead of dynamical particles, we can introduce worldlines of external objects. These are called line operators. See e.g. [96, 97].
- Once we allow the introduction of external line operators, there is no reason not to introduce higher-dimensional external objects. When they have two spacetime dimensions, they are called surface operators. A Seiberg-Witten curve can be deﬁned intrinsically as the infra-red moduli space of a surface operator [98]. Another interesting recent paper worth studying is [99].
- We can then consider objects with three spacetime dimensions. This is an external domain-wall. A recent study can be found e.g. in [100].

On these topics, the review [101] is a great source of information, although the review itself is meant for mathematicians.

Third, the method described in this lecture note is not yet powerful enough to solve arbitrary $\mathcal{\mathcal{N}}=2$ gauge theories. Many 4d $\mathcal{\mathcal{N}}=2$ theories do come from the 6d $\mathcal{\mathcal{N}}=\left(2,0\right)$ theory, but there are also many which presently do not. Therefore we should also study alternative approaches.

- The 6d construction itself needs to be developed further. For tame punctures, further discussions can be found in e.g. [72, 64, 102] and for wild punctures, more can be found in [103, 104].
- A 6d construction of 4d $\mathcal{\mathcal{N}}=2$ theory can always be uplifted to Type IIB string theory on a non-compact Calabi-Yau manifold, which is a ﬁbration over the ultraviolet curve. Even when the non-compact Calabi-Yau is not a ﬁbration over a curve, Type IIB string theory on it often realizes a 4d $\mathcal{\mathcal{N}}=2$ ﬁeld theory, and this gives an alternative to ﬁnd the solution to the $\mathcal{\mathcal{N}}=2$ systems, see e.g. [69, 105].
- The $\mathcal{\mathcal{N}}={2}^{\ast}$ theories, i.e. $\mathcal{\mathcal{N}}=4$ super Yang-Mills deformed by a mass term for the adjoint hypermultiplet, have been long solved for general gauge group $G$ [14]. Somewhat surprisingly, when $G\ne SU\left(N\right)$, there is no known explicit string theory or M-theory construction of these solutions. This clearly shows how primitive our current understanding is.

Fourth, there are many properties of $\mathcal{\mathcal{N}}=2$ theories which are satisﬁed by all known examples, but we do not currently have any way to derive them. It would be fruitful to devise new methods to study these properties. Let us list a few questions in this direction.

- The chiral operators on the Coulomb branch of the $\mathcal{\mathcal{N}}=2$ gauge theories are clearly always freely generated. For example, in an $SU\left(N\right)$ gauge theory, it is generated by $tr\phantom{\rule{0.3em}{0ex}}{\varphi}^{k}$, ($k=2,\dots ,N$), which have no nontrivial relations. Experimentally, all the non-Lagrangian theories obtained from the 6d construction still satisfy this property: the Coulomb branch operators are freely generated. The author conjectures this is in fact a theorem applicable to every $\mathcal{\mathcal{N}}=2$ supersymmetric systems.
- In [106], it was argued that there is a non-zero lower bound in the change in the central charge $a$ along the RG ﬂow between two $\mathcal{\mathcal{N}}=2$ superconformal ﬁeld theories. Is there are more rigorous derivation of this fact?
- Is it possible to characterize the whole zoo of $\mathcal{\mathcal{N}}=2$
theories itself? As an analogy, consider all the representation of $SU\left(2\right)$.
If we allow only the direct sum, we need all irreducible representations to construct
all possible representations. If we also allow the tensor product and the extraction of
an irreducible summand, we only need the two-dimensional irreducible representation
to generate all others.
We can pose a similar question for $\mathcal{\mathcal{N}}=2$ theories. If we allow only weak gauging, what kind of generalized matter contents, i.e. hypermultiplets and other ‘irreducible’ strongly-coupled theories, are needed to generate all the $\mathcal{\mathcal{N}}=2$ theories? If we also allow the strongly-coupled limit, S-duality, and decomposition into the constituent parts, how much do we need? What ‘percentage’ of the theories can be obtained via 6d, string or M-theory constructions?

$\mathcal{\mathcal{N}}=2$ theories that are complete (in a certain technical sense) were classiﬁed in [94], and $\mathcal{\mathcal{N}}=2$ weakly-coupled gauge theories were classiﬁed in [107]. These are however but two tiny steps into the vast space of all possible $\mathcal{\mathcal{N}}=2$ theories.

Finally, the author would like to emphasize that even such innocent looking gauge theories as

- $\mathcal{\mathcal{N}}=2$ supersymmetric $SU\left(7\right)$ gauge theory with a hypermultiplet in the three-index anti-symmetric tensor representation, or
- $\mathcal{\mathcal{N}}=2$ supersymmetric $SU{\left(2\right)}^{3}$ gauge theory with a massive full hypermultiplet in the trifundamental, $\left({Q}_{aiu},{\stackrel{\u0303}{Q}}^{aiu}\right)$

have not been solved yet. He would be happy to oﬀer a dinner at the Sushi restaurant in the Kashiwa campus to the ﬁrst person who ﬁnds the solution to either of the two theories. There are many other $\mathcal{\mathcal{N}}=2$ gauge theories without known solutions, as listed in [107]. So this ﬁeld should be considered still wide-open.

Hopefully, those readers who came to this point should be at least moderately equipped to tackle these and other recent articles on $\mathcal{\mathcal{N}}=2$ supersymmetric theories. It would be a pleasure for the author if they would continue the study and contribute to extend the frontier of the research.