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# 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

DOI: https://dx.doi.org/10.4310/ATMP.2018.v22.n5.a1

#### Authors

#### Abstract

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