nLab Coulomb branch

Idea

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.

Definition

D=4

D=3

Given $G$ a complex reductive group and $M$ a complex symplectic representation of the form $M=N \bigoplus N^*$, the Coulomb branch is defined as the spectrum of the ring given by equivariant Borel-Moore homology $H_\bullet^{G_\mathcal{O}} ( \mathcal{R} )$ with convolution product. $\mathcal{R}$ is a space of triples of a $G$ principle bundle over the formal disk $\mathcal{P}$, a trivialization over the punctured formal disk $\phi$ and a section $s$ of $\mathcal{P} \times_G N$ with the restriction that $\phi (s)$ 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 $N=0$, this gives the Bezrukavnikov-Finkelberg-Mirkovic (BFM) space for the Langlands dual $^L G$.

Related concepts

• symplectic duality

• fractional D-brane, McKay 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, Nucl.Phys. B431 (1994) 484–550 (doi arXiv:hep-th/9408099)

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:

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

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

• Alexander Braverman, Michael Finkelberg, Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, II, Adv. Theor. Math. Phys. 22 (2018) 1071-1147 (arXiv:1601.03586)

• Alexander Braverman, Michael Finkelberg, Hiraku Nakajima, Coulomb branches of 3d $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes), Advances in Theoretical and Mathematical Physics 23:1 (2019) (arXiv:1604.03625)

  (relation to the moduli space of Yang-Mills monopoles)

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

See also:

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

• Justin Hilburn, Joel Kamnitzer, Alex Weekes, BFN Springer Theory, Commun. Math. Phys. 402 (2023) 765–832 (2023) (doi arXiv:2004.14998)

• Zijun Zhou, Virtual Coulomb branch and vertex function (arXiv:2107.06135)

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)

• Alexander Braverman, Michael Finkelberg, Hiraku Nakajima, Line bundles over Coulomb branches (arXiv:1805.11826)

Singularities

Discussion of Coulomb branch singularities:

• Mathew Bullimore, Tudor Dimofte, Davide Gaiotto, The Coulomb branch of 3d $\mathcal{N}=4$ theories, Commun. Math. Phys. (2017) 354: 671 (arXiv:1503.04817)

  (relation to Nahm's equations)

• Benjamin Assel, Stefano Cremonesi, The infrared physics of bad theories, SciPost Phys. 3, 024 (2017) (arXiv:1707.03403)