Conformal field theories on K3 and three-dimensional gauge theories

According to a recent conjecture, the moduli space of the heterotic conformal field theory on a G ⊂ ADE singularity of an ALE space is equivalent to the moduli space of a pure = 4 supersymmetric three-dimensional gauge theory with gauge group G. We establish this relation using geometric engineering of heterotic strings and generalize it to theories with non-trivial matter content. A similar equivalence is found between the moduli of heterotic CFT on isolated Calabi-Yau 3-fold singularities and two-dimensional Kazama-Suzuki coset theories.