Hyperbolic Invariance in Type II Superstrings

We first review aspects of Kac Moody indefinite algebras with particular focus on their hyperbolic subset. Then we present two field theoretical systems where these structures appear as symmetries. The first deals with complete classification of $\mathcal{N}=2$ supersymmetric CFT$_{4}$s and the second concerns the building of hyperbolic quiver gauge theories embedded in type IIB superstring compactification of Calabi-Yau threefolds. We show, amongst others, that $\mathcal{N}=2$ CFT$_{4}$s are classified by Vinberg theorem and hyperbolic structure is carried by the axion modulus. Keywords: Classification of KM algebras, Indefinite KM sector and Hyperbolic subset, Quiver gauge theories embedded in type II superstrings.