Hyperbolic Invariance

Motivated by the study of duality cascades in supersymmetric quiver gauge theories beyond affine models, we develop in this paper the analysis of a class of simply laced hyperbolic Lie algebras. These are specific generalizations of affine ADE symmetries which form a particular subclass of the so-called Indefinite Lie algebras. Because of indefinite signature of their bilinear form, we show that these infinite dimensional invariances have very special features and admit a remarkable link type IIB background with non zero axion. We also show that hyperbolic root system ${\mathrm{\Delta }}_{hyp}$ has a ${\mathbb{Z}}_2{\times \mathbb{Z}}_3$ gradation containing two specific and isomorphic proper subsets of affine Kac-Moody root systems baptized as $\Delta {affine}^{\delta}$and$\Delta{affine}^{\gamma}$. We give an explicit form of the commutation relations for hyperbolic ADE algebras and analyze their Weyl groups W${}_{hyp}$. Comments regarding links with Seiberg like dualities and RG cascades are made.