nLab Higgs branch

Idea

For N=2 D=4 super Yang-Mills theory the moduli space of vacuum expectation values (VEVs) of the theory is locally a Cartesian product between the space of moduli of the vector multiplet (the gauge field sector) and those of the hypermultiplet (the matter field sector). The former is called the Coulomb branch, and the latter the Higgs branch. These are dual to each other to it under a version of mirror symmetry .

This is the topic of Seiberg-Witten theory.

Definitions of the Coulomb and Higgs branches have been extended to N=4 D=3 super Yang-Mills theory.

Definition

D=4

D=3

Related concepts

• symplectic duality

• fractional D-brane, McKay correspondence

References

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)

Review and exposition:

• Cecilia Albertsson, around p. 31 of Superconformal D-branes and moduli spaces (arXiv:hep-th/0305188)

• Yuji Tachikawa, Section 7 of: $\mathcal{N}=2$ supersymmetric dynamics for pedestrians, Lecture Notes in Physics, vol. 890, 2014 (arXiv:1312.2684, doi:10.1007/978-3-319-08822-8, web version)

On mirror symmetry between Higgs branches/Coulomb branches of D=3 N=4 super Yang-Mills theory (with emphasis of Hilbert schemes of points):

• Jan de Boer, Kentaro Hori, Hirosi Ooguri, Yaron Oz, Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes, Nucl. Phys. B493:101-147, 1997 (arXiv:hep-th/9611063)