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The Coulomb branch of 3d N=4 theories.

Bullimore, Mathew and Dimofte, Tudor and Gaiotto, Davide (2017) 'The Coulomb branch of 3d N=4 theories.', Communications in mathematical physics., 354 (2). pp. 671-751.


We propose a construction for the quantum-corrected 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.

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Publisher statement:© The Author(s) 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Date accepted:20 March 2017
Date deposited:08 June 2018
Date of first online publication:03 June 2017
Date first made open access:08 June 2018

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