  We settle a long-standing question about the hypermultiplet moduli spaces of
the heterotic strings on ALE singularities. These heterotic backgrounds are
specified by the singularity type, an instanton number, and a (nontrivial) flat
connection at infinity. Building on their interpretation as six-dimensional
theories, we determine a class of three-dimensional $\mathcal{N}=4$ quiver
gauge theories whose quantum corrected Coulomb branch coincides with the exact
heterotic hypermultiplet moduli space.
