Hyper-Kahler singularities in superstring compactification and two-dimensional N=4 conformal field theory

Using a geometric realization of the SU(2)(R) symmetry and a factorization of the gauge and SU (2)(R) charges, we study the small instanton singularit ies of the Higgs branch of supersymmetric U(1)(r) gauge theories with eight supercharges. We derive new solutions for the moduli space of vacua preser ving manifestly the eight supercharges, In particular, we obtain an extensi on of the ordinary ADE singularities for hyper-Kahler manifolds and show th at the classical moduli space of vacua is, in general, given by cotangent b undles of compact weighted projective spaces describing new models which fl ow in the infrared to two-dimensional (2D) N = (4, 4) scale-invariant model s. We also study the N = 4 conformal Liouville description near an A(n) sin gularity of the metric of the 2D N = 4 Higgs branch using a field-theoretic al approach.