Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 5

Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N} = 4$ gauge theories, II

Pages: 1071 – 1147



Alexander Braverman (Department of Mathematics, University of Toronto, and Perimeter Institute of Theoretical Physics, Waterloo, Ontario, Canada)

Michael Finkelberg (Department of Mathematics, Higher School of Economics, Moscow, Russia; and Institute for Information Transmission Problems, Moscow, Russia)

Hiraku Nakajima (Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan; and the Kavli Institute for the Physics and Mathematics of the Universe, (WPI), University of Tokyo, Kashiwa, Chiba, Japan)


Consider the $3$-dimensional $\mathcal{N} = 4$ supersymmetric gauge theory associated with a compact Lie group $G_c$ and its quaternionic representation $\mathbf{M}$. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold with an $\mathrm{SU}(2)$-action, possibly with singularities. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with $\mathbb{C}^{\times}$-action when $\mathbf{M}$ is of a form $\mathbf{N} \oplus \mathbf{N}^{\ast}$, as the second step of the proposal given in [Nak16].

Published 2 May 2019