Adil Belhaj

Geometric engineering of CFT4 based on indefinite singularities: hyperbolic case

Abstract Using Katz, Klemm and Vafa geometric engineering method of N =2 supersymmetric QFT4 and results on the classification of generalized Cartan matrices of Kac–Moody (KM) algebras, we study the unexplored class of N =2 CFT4 based on indefinite singularities. We show that the vanishing condition for the general expression of holomorphic beta function of N =2 quiver gauge QFT4 coincides exactly with the fundamental classification theorem of KM algebras. Explicit solutions are derived for mirror geometries of CY threefolds with hyperbolic singularities.

Toric Calabi-Yau supermanifolds and mirror symmetry

We study mirror symmetry of supermanifolds constructed as fermionic extensions of compact toric varieties. We mainly discuss the case where the linear sigma A-model contains as many fermionic fields as there are U(1) factors in the gauge group. In the mirror super-Landau-Ginzburg B-model, focus is on the bosonic structure obtained after integrating out all the fermions. Our key observation is that there is a relation between the super-Calabi-Yau conditions of the A-model and quasi-homogeneity of the B-model, and that the degree of the associated superpotential in the B-model is given in terms of the determinant of the fermion charge matrix of the A-model.

Classification of N = 2 supersymmetric CFT 4 s: Indefinite Series

Using the geometric engineering method of 4D quiver gauge theories and results on the classification of Kac?Moody (KM) algebras, we show by explicit examples that there exist three sectors of infrared CFT4s. Since the geometric engineering of these CFT4s involves type II strings on K3 fibred 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 H43 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.

RG Cascades in Hyperbolic Quiver Gauge Theories

Abstract In this paper, we provide a general classification of supersymmatric QFT 4 s 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- QFT 4 s. This includes, amongst others, its embedding in type II string on local Calabi–Yau threefolds. We give realizations of these supersymmetric QFT 4 s 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.

F-theory duals of M-theory on G2 manifolds from mirror symmetry

Using mirror pairs (M3, W3) in type II superstring compactifications on Calabi–Yau threefolds, we study, geometrically, F-theory duals of M-theory on seven manifolds with G2 holonomy. We first develop a way of obtaining Landau–Ginzburg (LG) Calabi–Yau threefolds W3, embedded in four complex-dimensional toric varieties, mirror to the sigma model on toric Calabi–Yau threefolds M3. This method gives directly the right dimension without introducing non-dynamical variables. Then, using toric geometry tools, we discuss the duality between M-theory on S1 × M3/Z2 with G2 holonomy and F-theory on elliptically fibred Calabi–Yau fourfolds with SU(4) holonomy, containing W3 mirror manifolds. Illustrative examples are presented.

On fractional quantum Hall solitons in ABJM-like theory

Abstract Using D-brane physics, we study fractional quantum Hall solitons (FQHS) in ABJM-like theory in terms of type IIA dual geometries. In particular, we discuss a class of Chern–Simons (CS) quivers describing FQHS systems at low energy. These CS quivers come from R–R gauge fields interacting with D6-branes wrapped on 4-cycles, which reside within a blown up CP 3 projective space. Based on the CS quiver method and mimicking the construction of del Pezzo surfaces in terms of CP 2 , we first give a model which corresponds to a single layer model of FQHS system, then we propose a multi-layer system generalizing the doubled CS field theory, which is used in the study of topological defect in graphene.

Manifolds of G 2 Holonomy from N = 4 Sigma Model

Using two dimensional (2D) N = 4 sigma model, with U(1) r gauge symmetry, and introducing the ADE Cartan matrices as gauge matrix charges, we build ” toric” hyperKahler eight real dimensional manifolds X8. Dividing by one toric geometry circle action

Mirror symmetry and Landau-Ginzburg-Calabi-Yau superpotentials in F theory compactifications

We study the Landau–Ginzburg (LG) theory mirror to two-dimensional (2D) N = 2 gauged linear sigma models on toric Calabi–Yau manifolds. We derive and solve new constraint equations for LG elliptic Calabi–Yau superpotentials depending on the physical data of dual linear sigma models. In the Calabi–Yau threefold case, we consider two examples. First, we give the mirror symmetry of the canonical line bundle over the Hirzebruch surfaces Fn. Second, we find a special geometry with the affine so(8) Lie algebra toric data extending the geometry of elliptically fibred K3. This geometry leads to a pure N = 1 six-dimensional SO(8) gauge model from the F-theory compactification. For Calabi–Yau fourfolds, we give a new algebraic realization for ADE hypersurfaces.

ON F-THEORY QUIVER MODELS AND KAC–MOODY ALGEBRAS

We discuss quiver gauge models with bi-fundamental and fundamental matter obtained from F-theory compactified on ALE spaces over a four-dimensional base space. We focus on the base geometry which consists of intersecting F0 = CP1 × CP1 Hirzebruch complex surfaces arranged as Dynkin graphs classified by three kinds of Kac–Moody (KM) algebras: ordinary, i.e. finite-dimensional, affine and indefinite, in particular hyperbolic. We interpret the equations defining these three classes of generalized Lie algebras as the anomaly cancelation condition of the corresponding N = 1 F-theory quivers in four dimensions. We analyze in some detail hyperbolic geometries obtained from the affine base geometry by adding a node, and we find that it can be used to incorporate fundamental fields to a product of SU-type gauge groups and fields.

Engineering of Quantum Hall Effect from Type IIA String Theory on The K3 Surface

Using D-brane configurations on the K3 surface, we give six-dimensional type IIA stringy realizations of the Quantum Hall Effect (QHE) in 1+2 dimensions. Based on the vertical and horizontal lines of the K3 Hodge diamond, we engineer two different stringy realizations. The vertical line presents a realization in terms of D2 and D6-branes wrapping the K3 surface. The horizontal one is associated with hierarchical stringy descriptions obtained from a quiver gauge theory living on a stack of D4-branes wrapping intersecting 2-spheres embedded in the K3 surface with deformed singularities. These geometries are classified by three kinds of the Kac–Moody algebras: ordinary, i.e. finite dimensional, affine and indefinite. We find that no stringy QHE in 1+2 dimensions can occur in the quiver gauge theory living on intersecting 2-spheres arranged as affine Dynkin diagrams. Stringy realizations of QHE can be done only for the finite and indefinite geometries. In particular, the finite Lie algebras give models with fractional filling fractions, while the indefinite ones classify models with negative filling fractions which can be associated with the physics of holes in the graphene.