superalgebra and (synthetic ) supergeometry
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.
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.
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
For 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)
Constantin Teleman, The role of Coulomb branches in 2D gauge theory, (arXiv:1801.10124)
Ben Webster, Koszul duality between Higgs and Coulomb categories $\mathcal{O}$, (arXiv:1611.06541)
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):
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)
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) (arXiv1707.03403)
Last revised on April 30, 2020 at 23:54:35. See the history of this page for a list of all contributions to it.