Hyperkahler Singularities in Superstrings Compactification and 2 d N = 4 Conformal Field Theory

We study the singularities of the Higgs branch of supersymmetric U (1) r gauge theories with eight supercharges. We derive new solutions for the mod-uli space of vacua preserving manifestly the eight supercharges by using a geometric realization of the SU (2) R symmetry and a separation procedure of the gauge and SU (2) R charges, which allow us to put the hypermultiplet vacua in a form depending on a parameter γ. For γ = 1, we obtain new models which flow in the infrared to 2d N = (4, 4) conformal models and we show that the classical moduli spaces are given by intersecting cotangent weighted complex projective spaces containing the small instanton singularity, discussed in [17], as a leading special case. We also make comments regarding 1 the 2d N = 4 conformal Liouville description of the Higgs branch throat by following the analysis of [18]. Other features are also discussed.