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Using geometric engineering method of 4D $\mathcal{N}=2$ quiver gaugetheories and results on the classification of Kac-Moody (KM) algebras, we showon explicit examples that there exist three sectors of $\mathcal{N}=2$ infraredCFT$_{4}$s. Since the geometric engineering of these CFT$_{4}$s involve type IIstrings on K3 fibered CY3 singularities, we conjecture the existence of threekinds of singular complex surfaces containing, in addition to the two standardclasses, a third indefinite set. To illustrate this hypothesis, we giveexplicit examples of K3 surfaces with H$_{3}^{4}$ and E$_{10}$ hyperbolicsingularities. We also derive a hierarchy of indefinite complex algebraicgeometries based on affine $A_{r}$ and T$%_{(p,q,r)}$ algebras going beyond thehyperbolic subset. Such hierarchical surfaces have a remarkable signature thatis manifested by the presence of poles.Comment: 12 pages, 2 figure

Classification of supersymmetric CFT4s: indefinite series

Classification of supersymmetric CFT₄s: indefinite series