Classification of N = 2 supersymmetric CFT 4 s : Indefinite Series

Using geometric engineering method of 4D N = 2 quiver gauge theories and results on the classification of Kac-Moody (KM) algebras, we show on explicit examples that there exist three sectors of N = 2 infrared CFT4s. Since the geometric engineering of these CFT4s involve type II strings on K3 fibered CY3 singularities, we conjecture the existence of three kinds of singular complex surfaces containing, in addition to the two standard classes, a third indefinite set. To illustrate this hypothesis, we give explicit examples of K3 surfaces with H3 and E10 hyperbolic singularities. We also derive a hierarchy of indefinite complex algebraic geometries based on affine Ar and T(p,q,r) algebras going beyond the hyperbolic subset. Such hierarchical surfaces have a remarkable signature that is manifested by the presence of poles.