0 Introduction

The study of $\mathcal{𝒩}=2$ supersymmetric quantum field theories in four-dimensions has been a fertile field for theoretical physicists for quite some time. These theories always have non-chiral matter representations, and therefore can never be directly relevant for describing the real world. That said, the existence of two sets of supersymmetries allows us to study their properties in much greater detail than both non-supersymmetric theories and $\mathcal{𝒩}=1$ supersymmetric theories. Being able to do so is quite fun in itself, and hopefully the general lessons thus learned concerning $\mathcal{𝒩}=2$ supersymmetric theories might be useful when we study the dynamics of theories with lower supersymmetry. At least, the physical properties of $\mathcal{𝒩}=2$ theories have been successfully used to point mathematicians to a number of new mathematical phenomena unknown to them.

These words would not probably be persuasive enough for non-motivated people to start studying $\mathcal{𝒩}=2$ dynamics. It is not, however, the author’s plan to present here a convincing argument why you should want to study it anyway; the fact that you are reading this sentence should mean that you are already somewhat interested in this subject and are looking for a place to start.

There have been many important contributions to the study of $\mathcal{𝒩}=2$ theories since its introduction [1]. The four most significant ones in the author’s very personal opinion are the following:

• In 1994, Seiberg and Witten found in [2, 3] exact low-energy solutions to $\mathcal{𝒩}=2$ supersymmetric $SU\left(2\right)$ gauge theories by using holomorphy and by introducing the concept of the Seiberg-Witten curves.
• In 1996-7, the Seiberg-Witten curves, which were so far mathematical auxiliary objects, were identified as physical objects appearing in various string theory constructions of $\mathcal{𝒩}=2$ supersymmetric theories [4, 5, 6].
• In 2002, Nekrasov found in [7] a concise method to obtain the solutions of Seiberg and Witten via the instanton counting.
• In 2009, Gaiotto found in [8] a huge web of S-dualities acting upon $\mathcal{𝒩}=2$ supersymmetric systems.

The developments before 2002 have been described in many nice introductory reviews and lecture notes, e.g. [9, 10, 11, 12, 13, 14]. Newer textbooks also have sections on them, see e.g. Chap. 29.5 of [15] and Chap. 13 of [16]. A short review on the instanton counting will be forthcoming [17]. A comprehensive review on the newer developments since 2009 would then surely be useful to have, but this lecture note is not exactly that. Rather, the main aim of this lecture note is to present the same old results covered in the lectures and reviews listed above under a new light introduced in 2009 and developed in the last few years, so that readers would be naturally prepared to the study of recent works once they go through this note. A good review with an emphasis on more recent developments can be found in [18, 19].

The rest of the lecture note is organized as follows. First three sections are there to prepare ourselves to the study of $\mathcal{𝒩}=2$ dynamics.

• We start in Sec. 1 by introducing the electromagnetic dualities of $U\left(1\right)$ gauge theories and recalling the basic semiclassical features of monopoles.
• In Sec. 2, we construct the $\mathcal{𝒩}=2$ supersymmetric Lagrangians and studying their classical features. We introduce the concepts of the Coulomb branch and the Higgs branch.
• In Sec. 3, we will first see that the renormalization of $\mathcal{𝒩}=2$ gauge theories are one-loop exact perturbatively. We also study the anomalous R-symmetry of supersymmetric theories. As an application, we will quickly study the behavior of pure $\mathcal{𝒩}=1$ gauge theories.

The next two sections are devoted to the solutions of the two most basic cases.

• In Sec. 4, we discuss the solution to the pure $\mathcal{𝒩}=2$ supersymmetric $SU\left(2\right)$ gauge theory in great detail. Two important concepts, the Seiberg-Witten curve and the ultraviolet curve¹ , will be introduced.
• In Sec. 5, we solve the $\mathcal{𝒩}=2$ supersymmetric $SU\left(2\right)$ gauge theory with one hypermultiplet in the doublet representation. We will see again that the solution can be given in terms of the curves.

The sections 6 and 7 are again preparatory.

• In Sec. 6, we give a physical meaning to the Seiberg-Witten curves and the ultraviolet curves, in terms of six-dimensional theory. With this we will be able to guess the solutions to $SU\left(2\right)$ gauge theory with arbitrary number of hypermultiplets in the doublet representations. This section will not be self-contained at all, but it should give the reader the minimum with which to work from this point on.
• Up to the previous section, we will be mainly concerned with the Coulomb branch. As the analysis of the Higgs branch will become also useful and instructive later, we will study the features of the Higgs branch in slightly more detail in Sec. 7.

We resume the study of $SU\left(2\right)$ gauge theories in the next two sections.

• In Sec. 8, we will see that the solutions of $SU\left(2\right)$ gauge theories with two or three hypermultiplets in the doublet representation, which we will have guessed in Sec. 6, indeed pass all the checks to be the correct ones.
• In Sec. 9, we first study the $SU\left(2\right)$ gauge theory with four hypermultiplets in the doublet representation. We will see that it has an S-duality acting on the $SO\left(8\right)$ flavor symmetry via its outer-automorphism. Then the analysis will be generalized, following Gaiotto, to arbitrary theories with gauge group of the form $SU{\left(2\right)}^n$.

We will consider more diverse examples in the final three sections of the main part.

• In Sec. 10, we will study various superconformal field theories of the type first found by Argyres and Douglas, which arises when electrically and magnetically charged particles become simultaneously very light.
• In Sec. 11, the solutions to $SU\left(N\right)$ and $SO\left(2N\right)$ gauge theories with and without hypermultiplets in the fundamental or vector representation will be quickly described.
• In Sec. 12, we will analyze the S-duality of the $SU\left(N\right)$ gauge theory with $2N$ flavors and its generalization. Important roles will be played by punctures on the ultraviolet curve labeled by Young diagrams with $N$ boxes, whose relation to the Higgs branch will also be explained. As an application, we will construct superconformal field theories with exceptional flavor symmetries $E_{6,7,8}$.

We conclude the lecture note by a discussion of further directions of study in Sec. 13.