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 scalar matter field sector). The former is called the Coulomb branch, and the latter the Higgs branch. These are dual to each other 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

Given GG a complex reductive group and MM a complex symplectic representation of the form M=NN *M=N \bigoplus N^*, the Coulomb branch is defined as the spectrum of the ring given by equivariant Borel-Moore homology H G 𝒪()H_\bullet^{G_\mathcal{O}} ( \mathcal{R} ) with convolution product. \mathcal{R} is a space of triples of a GG principle bundle over the formal disk 𝒫\mathcal{P}, a trivialization over the punctured formal disk ϕ\phi and a section ss of 𝒫× GN\mathcal{P} \times_G N with the restriction that ϕ(s)\phi (s) extends over the puncture.

By including *\mathbb{C}^* equivariance as well, this gives a non-commutative deformation. This gives a Poisson structure at the classical level.

Examples

  • If N=0N=0, this gives the Bezrukavnikov-Finkelberg-Mirkovic (BFM) space for the Langlands dual LG^L G.

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 August 24, 2019 at 08:18:35. See the history of this page for a list of all contributions to it.