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Localized tachyons in C/ZN

We study the condensation of localized closed string tachyons in C/ZN nonsupersymmetric noncompact orbifold singularities via renormalization group flows that preserve supersymmetry in the worldsheet conformal field theory and their interrelations with the toric geometry of these orbifolds. We show that for worldsheet supersymmetric tachyons, the endpoint of tachyon condensation generically includes “geometric” terminal singularities (orbifolds that do not have any marginal or relevant Kähler blowup modes) as well as singularities in codimension two. Some of the various possible distinct geometric resolutions are related by flip transitions. For Type II theories, we show that the residual singularities that arise under tachyon condensation in various classes of Type II theories also admit a Type II GSO projection. We further show that Type II orbifolds entirely devoid of marginal or relevant blowup modes (Kähler or otherwise) cannot exist, which thus implies that the endpoints of tachyon condensation in Type II theories are always smooth spaces.