  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.
