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 Higgs and Coulomb branches, which are noncompact hyper-K\"ahler manifolds with $SU(2)$-action, possibly with singularities.
Higgs branches are just hyper-K\"ahler quotients of $\mathbf M$ by $G_c$, and the definition is clear. On the other hand, a physical definition of Coulomb branch involves a quantum correction, hence is hard to understand for mathematicians.
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 $\mathbf N\oplus\mathbf N^*$. The definition is motivated by hypothetical $(2+1)D$ topological quantum field theories associated with monopole-like equations.
The talk is based on my joint works with various others, e.g., A.Braverman and M.Finkelberg.
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