We settle a longstanding question about the hypermultiplet moduli spaces of the heterotic strings on asymptotically locally Euclidean 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 N = 4 quiver gauge theories whose quantum corrected Coulomb branch coincides with the exact heterotic hypermultiplet moduli space.

We begin a study of the Higgs branch of six-dimensional (1, 0) little string theories governing the worldvolumes of heterotic ALE instantons. We give a description of this space by constructing the corresponding magnetic quiver. The latter is a three-dimensional N = 4 quiver gauge theory that flows in the infrared to a fixed point whose quantum corrected Coulomb branches is the Higgs branch of the six-dimensional theory of interest. We present results for both types of heterotic strings, and mostly for ℂ²/ℤ_k ALE spaces. Our analysis is valid both in the absence and in the presence of small instantons. Along the way, we also describe small SO(32) instanton transitions in terms of the corresponding magnetic quiver, which parallels a similar treatment of the small E8 instanton transitions in the context of the E₈ × E₈ heterotic string.