0 Introduction

The study of 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 𝒩=1 supersymmetric theories. Being able to do so is quite fun in itself, and hopefully the general lessons thus learned concerning 𝒩=2 supersymmetric theories might be useful when we study the dynamics of theories with lower supersymmetry. At least, the physical properties of 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 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 𝒩=2 theories since its introduction [1]. The four most significant ones in the author’s very personal opinion are the following:

The developments before 2002 have been described in many nice introductory reviews and lecture notes, e.g. [91011121314]. 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 [1819].

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

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

The sections 6 and 7 are again preparatory.

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

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

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