nLab
Coulomb branch

Contents

Context

Quantum field theory

Super-Geometry

Contents

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.

References

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

Quick exposition of the basic idea includes

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

For mathematical discussion see

  • Hiraku Nakajima, Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional 𝒩=4\mathcal{N} = 4 gauge theories (arXiv:1706.05154)

  • Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional 𝒩=4\mathcal{N} = 4 gauge theories, I (arXiv:1503.03676)

  • Alexander Braverman, Michael Finkelberg, Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional 𝒩=4\mathcal{N} = 4 gauge theories, II (arXiv:1601.03586)

  • Constantin Teleman, The role of Coulomb branches in 2D gauge theory, (arXiv:1801.10124)

  • Ben Webster, Koszul duality between Higgs and Coulomb categories 𝒪\mathcal{O}, (arXiv:1611.06541)

Last revised on October 26, 2018 at 08:06:39. See the history of this page for a list of all contributions to it.