nLab Coulomb branch



Quantum field theory




For N=2 D=4 super Yang-Mills theory the moduli space of vacuum expectation values (VEVs) of the theory is meant to locally be a Cartesian product of spaces of

  1. moduli of the vector multiplet (the gauge field sector)

    this is called the Coulomb branch

  2. moduli of the hypermultiplet (the scalar matter field sector)

    this is called the Higgs branch.

These are thought to be dual to each other under a version of mirror symmetry. This is largely 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.




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.


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



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

The definition is summarized (specifically for super QCD) in Assel-Cremoni 17, Section 2.1.

Quick exposition of the basic idea includes

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

Mathematical discussion in the case of D=3 N=4 super Yang-Mills theory:

See also:

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


Discussion of Coulomb branch singularities:

Last revised on November 2, 2023 at 10:16:51. See the history of this page for a list of all contributions to it.