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.

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

References in corpus (4)

• Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory
• Ω-deformation and quantization
• Quiver varieties and Beilinson-Drinfeld Grassmannians of type A
• Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence

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