We have seen that the low energy dynamics of the $SU\left(2\right)$ pure gauge theory and the $SU\left(2\right)$ gauge theory with one ﬂavor can both be expressed in terms of the complex curves (4.3.1), (5.2.1). The aim of this section is to explain that these two-dimensional spaces can be given a physical interpretation.

The ideas which will be presented in this section were originally obtained by exploiting various deep properties of string theory and M-theory, namely Calabi-Yau compactiﬁcations, brane constructions, and string dualities. The approach using Calabi-Yau compactiﬁcations goes back to [49, 50, 4] and the brane construction approach was introduced in [6]. Learning these constructions deﬁnitely helps in understanding $\mathcal{\mathcal{N}}=2$ supersymmetric dynamics, and vice versa. This lecture note is not, however, the place where you can learn them.

The presentation here is analogical rather than being logical, and the author intentionally tried to phrase it in such a way that the knowledge of string theory and M-theory required to read it is kept to the minimum. Anyone interested in more details should refer to the original articles, or the reviews such as [9, 13] and Sec. 3 of [51].

6.1 Strings with variable tension

6.2 Strings with variable tension from membranes

6.2.1 General idea

6.2.2 Example: pure $SU\left(2\right)$ theory

6.3 Self-duality of the 6d theory

6.4 Intermediate 5d Yang-Mills theory and its boundary conditions

6.4.1 Five-dimensional maximally-supersymmetric Yang-Mills

6.4.2 $\mathcal{\mathcal{N}}=4$ super Yang-Mills

6.4.3 $\mathcal{\mathcal{N}}=2$ pure $SU\left(2\right)$ theory and the ${N}_{f}=1$ theory

6.4.4 The $SU\left(2\right)$ theories with ${N}_{f}=2,3,4$

6.2 Strings with variable tension from membranes

6.2.1 General idea

6.2.2 Example: pure $SU\left(2\right)$ theory

6.3 Self-duality of the 6d theory

6.4 Intermediate 5d Yang-Mills theory and its boundary conditions

6.4.1 Five-dimensional maximally-supersymmetric Yang-Mills

6.4.2 $\mathcal{\mathcal{N}}=4$ super Yang-Mills

6.4.3 $\mathcal{\mathcal{N}}=2$ pure $SU\left(2\right)$ theory and the ${N}_{f}=1$ theory

6.4.4 The $SU\left(2\right)$ theories with ${N}_{f}=2,3,4$