Using geometric engineering method of 4D N = 2 quiver gauge theories and results on the classification of Kac-Moody (KM) algebras, we show on explicit examples that there exist three sectors of N = 2 infrared CFT4s. Since the geometric engineering of these CFT4s involve type II strings on K3 fibered CY3 singularities, we conjecture the existence of three kinds of singular complex surfaces containing, in addition to the two standard classes, a third indefinite set. To illustrate this hypothesis, we give explicit examples of K3 surfaces with H 4 3 and E10 hyperbolic singularities. We also derive a hierarchy of indefinite complex algebraic geometries based on affine Ar and T (p,q,r) algebras going beyond the hyperbolic subset. Such hierarchical surfaces have a remarkable signature that is manifested by the presence of poles.Keywords: Geometric engineering of N = 2 QFT 4s, Indefinite and Hyperbolic Lie algebras, K3 fibered CY threefolds with indefinite singularities, N = 2 CFT 4s embedded in type II strings. * aitbenha@fsmek.ac.ma

In this paper, we provide a general classification of supersymmeric QFT4s into three basic sets: ordinary, affine and indefinite classes. The last class, which has not been enough explored in literature, is shown to share most of properties of ordinary and affine super QFT4s. This includes, amongst others, its embedding in type II string on local Calabi-Yau threefolds. We give realizations of these supersymmetric QFT4s as D-brane world volume gauge theories. A special interest is devoted to hyperbolic subset for its peculiar features and for the role it plays in type IIB background with non zero axion. We also study RG flows and duality cascades in case of hyperbolic quiver theories. Comments regarding the full indefinite sector are made.

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Malcev–Poisson–Jordan algebra (MPJ-algebra) is defined to be a vector space endowed with a Malcev bracket and a Jordan structure which are satisfying the Leibniz rule. We describe such algebras in terms of a single bilinear operation, this class strictly contains alternative algebras. For a given Malcev algebra [Formula: see text], it is interesting to classify the Jordan structure ∘ on the underlying vector space of [Formula: see text] such that [Formula: see text] is an MPJ-algebra (∘ is called an MPJ-structure on Malcev algebra [Formula: see text]. In this paper we explicitly give all MPJ-structures on some interesting classes of Malcev algebras. Further, we introduce the concept of pseudo-Euclidean MPJ-algebras (PEMPJ-algebras) and we show how one can construct new interesting quadratic Lie algebras and pseudo-Euclidean Malcev (non-Lie) algebras from PEMPJ-algebras. Finally, we give inductive descriptions of nilpotent PEMPJ-algebras.