nLab
N=2 D=4 super Yang-Mills theory

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

Partition function

• AGT correspondence

Related concepts

• N=4 D=4 super Yang-Mills theory

• N=4 D=3 super Yang-Mills theory

• Seiberg-Witten theory, electric-magnetic duality, symplectic duality

• AGT correspondence

References

General

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

• Nathan Seiberg, Edward Witten, Monopoles, Duality and Chiral Symmetry Breaking in $N=2$ Supersymmetric QCD (arXiv:hep-th/9408099)

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

• Nathan Seiberg, Edward Witten, Gauge Dynamics And Compactification To Three Dimensions (arXiv:hep-th/9607163)

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

• A. Klemm, W. Lerche, P. Mayr, Cumrun Vafa, N. Warner, Self-Dual Strings and N=2 Supersymmetric Field Theory (arXiv:hep-th/9604034)

• Edward Witten, Solutions Of Four-Dimensional Field Theories Via M Theory (arXiv:hep-th/9703166)

• Davide Gaiotto, N=2 dualities (arXiv:0904.2715)