  Using Katz, Klemm and Vafa geometric engineering method of $\mathcal{N}=2$
supersymmetric QFT$_{4}$s and results on the classification of generalized
Cartan matrices of Kac-Moody (KM) algebras, we study the un-explored class of
$\mathcal{N}=2$ CFT$_{4}$s based on \textit{indefinite} singularities. We show
that the vanishing condition for the general expression of holomorphic beta
function of $\mathcal{N}=2$ quiver gauge QFT$_{4}$s coincides exactly with the
fundamental classification theorem of KM algebras. Explicit solutions are
derived for mirror geometries of CY threefolds with \textit{% hyperbolic}
singularities.
