# Contents

## Idea

The special case of super Yang-Mills theory over a spacetime of dimension 4 and with $N=2$ supersymmetry.

## Properties

### Moduli space of vacua

A speciality of $N=2$, $D=4$ SYM is that its moduli space of vacua has two “branches” called the Coulomb branch and the Higgs branch. This is the content of what is now called Seiberg-Witten theory (Seiberg-Witten 94)

### Reduction to $D=3$ super Yang-Mills

By dimensional reduction on ${ℝ}^{3}×{S}^{1}$ families of $N=2,D=4$ SYM theories interpolate to N=4 D=3 super Yang-Mills theory. (Seiberg-Witten 96).

## References

### General

The terminology “Coulomb branch” and “Higgs branch” first appears in

The dimensional reduction to $D=3$ was first considered in

For references on wall crossing of BPS states see the references given there.

### Introductions and surveys

• Gaiotto, Recent progress in $N=2$ $4d$ field theory (2009) (pdf)

• Greg Moore, Surface Defects and the BPS Spectrum of $4d$ $N=2$ Theories (pdf)

• Gregory Moore, Four-dimensional $N=2$ Field Theory and Physical Mathematics (arXiv:1211.2331)

### Lifts to M-theory

Seiberg-Witten theory (Seiberg-Witten) for $N=2$, $D=4$ SYM has a nice geometrical description in terms of M5-brane compactifications, discussed in

Revised on January 10, 2013 17:28:43 by Urs Schreiber (89.204.153.52)