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Hyperbolic Invariance

Malika Ait Ben Haddou·El Hassan Saidi
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Abstract

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 $Δ_{hyp}$ has a $\mathbb{Z}_{2}\mathbb{\times Z}_{3}$ gradation containing two specific and isomorphic proper subsets of affine Kac-Moody root systems baptized as $Δ_{affine}^δ$ and $Δ_{affine}^γ$. 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.

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Malika Ait Ben Haddou, El Hassan Saidi. 2004-05-27. Hyperbolic Invariance. https://arxiv.org/abs/hep-th/0405251

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