The Coulomb Branch of 3d N=4 Theories

Mathew Bullimore, Tudor Dimofte, Davide Gaiotto

Research output: Contribution to journalArticlepeer-review


We propose a construction of 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 the Bogomolnyi and/or Nahm equations.
Original languageEnglish
Pages (from-to)671–751
Number of pages81
JournalCommunications in Mathematical Physics
Issue number2
Early online date3 Jun 2017
Publication statusPublished - 30 Sep 2017


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