  Braverman, Finkelberg and Nakajima have recently given a mathematical
construction of the Coulomb branches of a large class of $3d$ $\mathcal{N} =4$
gauge theories, as algebraic varieties with Poisson structure. They conjecture
that these varieties have symplectic singularities. We confirm this conjecture
for all quiver gauge theories without loops or multiple edges, which in
particular implies that the corresponding Coulomb branches have finitely many
symplectic leaves and rational Gorenstein singularities. We also give a
criterion for proving that any particular Coulomb branch has symplectic
singularities, and discuss the possible extension of our results to quivers
with loops and/or multiple edges.
