Classification of supersymmetric CFT₄s: indefinite series

Abstract

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

Partial breaking in the rigid limit of $\mathcal{N}=2$ gauged supergravity

“…Though the apparent picture looks different from the Toda theory as here one works with algebraic geometry techniques, the key idea is formally similar in the sense it uses as well particular properties of the ADE Lie algebras. In this method, a class of N = 2 supersymmetric QFT 4 are engineered as low energy effective field theories resulting from type IIB string compactified on local Calabi-Yau threefolds preserving eight supersymmetric charges [60][61][62]. The local CY3s are generally speaking realised as ALE surfaces with an ADE singularity fibered on complex projective line.…”

Brane Realizations of Quantum Hall Solitons and Lie Algebras

“…A nice way to introduce the Kac-Moody Lie algebras is in terms of the generalized Cartan matrix [19,20,21,22] and the Dynkin diagrams. For later use, let us give a brief review of some basic facts about this method.…”

RG cascades in hyperbolic quiver gauge theories

“…For precise definitions on these two varieties of indefinite Lie algebras see [25], [26], [15]. In this paper we will be interested in standard hyperbolic algebras, they can be thought of as particular extension of affine simply laced Kac-Moody invariance.…”

“…In fact the C ψ chain should be thought of as just a part of the following closed two-chain 15) which is associated with the imaginary root δ of affine ADE Kac-Moody algebras. In the above eqs, K + ij and K 0 ij are as before, they stand for the usual ordinary and affine ADE Cartan matrices respectively.…”

“…Following [15], the holomorphic beta function factors β i = β (g i ) of N = 2 supersymmetric Π r i=−1 U N i quiver gauge theories depend linearly on the N i ranks. They are proportional to the Cartan matrix K − hyp of the underlying hyperbolic quiver symmetry;…”