The Coulomb Branch of 3d $\mathcal{N}=4$ Theories

Abstract

We propose a construction of the quantum-corrected Coulomb branch of a general 3d gauge theory with $\mathcal{N}=4$ 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 the Bogomolnyi and/or Nahm equations.

arXiv comment98 pages, 6 figures, v2: small typos corrected and references updated

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Cited by in corpus (31)

• Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II
• Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory
• Deformation quantization and superconformal symmetry in three dimensions
• Argyres-Douglas Theories, S^1 Reductions, and Topological Symmetries
• 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)
• Holography and Koszul duality: the example of the $M2$ brane
• M-theory in the Omega-background and 5-dimensional non-commutative gauge theory
• Connecting 5d Higgs Branches via Fayet-Iliopoulos Deformations
• Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety
• Semiclassical framed BPS states
• Twisted M2 brane holography and sphere correlation functions
• Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams
• 3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve
• Refined 3d-3d Correspondence
• Algebraic properties of the monopole formula
• Symmetries of categorical representations and the quantum Ngô action
• Hilbert series and mixed branches of $T[SU(N)]$ theory
• Ring Relations and Mirror Map from Branes
• Questions on provisional Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories
• Little String Defects and Bala-Carter Theory
• 4d/2d $\to$ 3d/1d: A song of protected operator algebras
• Resolved gravity duals of $\mathcal{N}=4$ quiver field theories in 2+1 dimensions
• Masses, Sheets and Rigid SCFTs
• Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type $A$
• Quantum Mechanics With Noncommutative Amplitudes
• Note on some properties of generalized affine Grassmannian slices
• The Hilbert series of 3d N=2 Yang-Mills theories with vectorlike matter
• Dynamics of $\mathcal{N}=4$ supersymmetric field theories in 2+1 dimensions and their gravity dual
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• Fermi-gas correlators of ADHM theory and triality symmetry
• 't Hooft surface operators in five dimensions and elliptic Ruijsenaars operators