The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories
Abstract
We propose a construction for the quantumcorrected Coulomb branch of a general 3d gauge theory with N=4 supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkähler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations.
 Authors:
 Inst. for Advanced Study, Princeton, NJ (United States); Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
 Inst. for Advanced Study, Princeton, NJ (United States)
 Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
 Publication Date:
 Research Org.:
 Inst. for Advanced Study, Princeton, NJ (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), High Energy Physics (HEP) (SC25); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 OSTI Identifier:
 1423961
 Grant/Contract Number:
 sc0009988
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Communications in Mathematical Physics
 Additional Journal Information:
 Journal Volume: 354; Journal Issue: 2; Journal ID: ISSN 00103616
 Publisher:
 Springer
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Citation Formats
Bullimore, Mathew, Dimofte, Tudor, and Gaiotto, Davide. The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories. United States: N. p., 2017.
Web. doi:10.1007/s0022001729030.
Bullimore, Mathew, Dimofte, Tudor, & Gaiotto, Davide. The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories. United States. doi:10.1007/s0022001729030.
Bullimore, Mathew, Dimofte, Tudor, and Gaiotto, Davide. Sat .
"The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories". United States.
doi:10.1007/s0022001729030. https://www.osti.gov/servlets/purl/1423961.
@article{osti_1423961,
title = {The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories},
author = {Bullimore, Mathew and Dimofte, Tudor and Gaiotto, Davide},
abstractNote = {We propose a construction for the quantumcorrected Coulomb branch of a general 3d gauge theory with N=4 supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkähler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations.},
doi = {10.1007/s0022001729030},
journal = {Communications in Mathematical Physics},
number = 2,
volume = 354,
place = {United States},
year = {Sat Jun 03 00:00:00 EDT 2017},
month = {Sat Jun 03 00:00:00 EDT 2017}
}
Web of Science

We analyze the low energy behavior of N=1 supersymmetric gauge theories with an SU(N{sub c}){times}SU(N{sub c}) gauge group and a LandauGinzburg type superpotential. These theories contain fundamentals transforming under one of the gauge groups as well as bifundamental matter which transforms as a fundamental under each. We obtain the parametrization of the gauge coupling on the Coulomb branch in terms of a hyperelliptic curve. The derivation of this curve involves making use of Seiberg`s duality for supersymmetric QCD as well as the classical constraints for N{sub f}=N{sub c}+1 and the quantum modified constraints for N{sub f}=N{sub c}. {copyright} {ital 1998}more »

Coulomb branch of N=2 supersymmetric product group theories from branes
We determine the lowenergy description of N=2 supersymmetric {Pi}{sub i} SU(k{sub i}) gauge theories with bifundamental and fundamental matter based on Mtheory fivebrane configurations. The dependence on moduli and scales of the coefficients in the nonhyperelliptic SeibergWitten curves for these theories are determined by considering various field theory and brane limits. A peculiarity in the interpretation of these curves for the vanishing {beta}function case is noted. {copyright} {ital 1998} {ital The American Physical Society} 
6d $$ \mathcal{N}=\left(1,\;0\right) $$ theories on S ^{1}/T ^{2} and class S theories: part II
Here, we study the T ^{2} compactification of a class of 6dmore » 
4d $$ \mathcal{N} $$=2 theories with disconnected gauge groups
In this paper we present a beautifully consistent web of evidence for the existence of interacting 4d rank1more »Cited by 6