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On nonsupersymmetric $\BC^4/\BZ_N$, tachyons, terminal singularities and flips

We investigate nonsupersymmetric $\BC^4/\BZ_N$ orbifold singularities using their description in terms of the string worldsheet conformal field theory and its close relation with the toric geometry description of these singularities and their possible resolutions. Analytic and numerical study strongly suggest the absence of nonsupersymmetric Type II terminal singularities (i.e. with no marginal or relevant blowup modes) so that there are always moduli or closed string tachyons that give rise to resolutions of these singularities, although supersymmetric and Type 0 terminal singularities do exist. Using gauged linear sigma models, we analyze the phase structure of these singularities, which often involves 4-dimensional flip transitions, occurring between resolution endpoints of distinct topology. We then discuss 4-dim analogs of unstable conifold-like singularities that exhibit flips, in particular their Type II GSO projection and the phase structure. We also briefly discuss aspects of M2-branes stacked at such singularities and nonsupersymmetric $AdS_4\times S^7/\BZ_N$ backgrounds.