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According to a recent conjecture, the moduli space of the heterotic conformalfield theory on a $G\subset$ ADE singularity of an ALE space is equivalent tothe moduli space of a pure $\cx N=4$ supersymmetric three-dimensional gaugetheory with gauge group G. We establish this relation using geometricengineering of heterotic strings and generalize it to theories with non-trivialmatter content. A similar equivalence is found between the moduli of heteroticCFT on isolated Calabi--Yau 3-fold singularities and two-dimensionalKazama-Suzuki coset theories.Comment: 11pp, harvmac, v2: a relation between heterotic CFT on Calabi-Yau  3-fold singularities and Kazama-Suzuki models has been adde

Conformal field theories on K3 and three-dimensional gauge theories