Classification of N=2 supersymmetric CFT_{4}s: Indefinite Series

Using geometric engineering method of 4D \(\mathcal{N}=2\) quiver gauge theories and results on the classification of Kac-Moody (KM) algebras, we show on explicit examples that there exist three sectors of \(\mathcal{N}=2\) infrared CFT\(_{4}\)s. Since the geometric engineering of these CFT\(_{4}\)s 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 H\(_{3}^{4}\) and E\(_{10}\) hyperbolic singularities. We also derive a hierarchy of indefinite complex algebraic geometries based on affine \(A_{r}\) 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.