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

Abstract

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 algebras going beyond the hyperbolic subset. Such hierarchical surfaces have a remarkable signature that is manifested by the presence of poles.

12 pages, 2 figures

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