arXiv subjects

A. Belhaj

Publications and source records attributed to A. Belhaj.

69 records · Page 4Linked to original sources

arXiv2006

Black Holes in Type IIA String on Calabi-Yau Threefolds with Affine ADE Geometries and q-Deformed 2d Quiver Gauge Theories

Motivated by studies on 4d black holes and q-deformed 2d Yang Mills theory, and borrowing ideas from compact geometry of the blowing up of affine ADE singularities, we build a class of local Calabi-Yau threefolds (CY^{3}) extending the local 2-torus model \mathcal{O}(m)\oplus \mathcal{O}(-m)\to T^{2\text{}} considered in hep-th/0406058 to test OSV conjecture. We first study toric realizations of T^{2} and then build a toric representation of X_{3} using intersections of local Calabi-Yau threefolds \mathcal{O}(m)\oplus \mathcal{O}(-m-2)\to \mathbb{P}^{1}. We develop the 2d \mathcal{N}=2 linear σ-model for this class of toric CY^{3}s. Then we use these local backgrounds to study partition function of 4d black holes in type IIA string theory and the underlying q-deformed 2d quiver gauge theories. We also make comments on 4d black holes obtained from D-branes wrapping cycles in \mathcal{O}(\mathbf{m}) \oplus \mathcal{O}(\mathbf{-m-2}%) \to \mathcal{B}_{k} with \mathbf{m=}(m_{1},...,m_{k}) a k-dim integer vector and \mathcal{B}_{k} a compact complex one dimension base consisting of the intersection of k 2-spheres S_{i}^{2} with generic intersection matrix I_{ij}. We give as well the explicit expression of the q-deformed path integral measure of the partition function of the 2d quiver gauge theory in terms of I_{ij}.

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arXiv2006

On Local Calabi-Yau Supermanifolds and Their Mirrors

We use local mirror symmetry to study a class of local Calabi-Yau super-manifolds with bosonic sub-variety V_b having a vanishing first Chern class. Solving the usual super- CY condition, requiring the equality of the total U(1) gauge charges of bosons Φ_{b} and the ghost like fields Ψ_{f} one \sum_{b}q_{b}=\sum_{f}Q_{f}, as \sum_{b}q_{b}=0 and \sum_{f}Q_{f}=0, several examples are studied and explicit results are given for local A_{r} super-geometries. A comment on purely fermionic super-CY manifolds corresponding to the special case where q_{b}=0, \forall b and \sum_{f}Q_{f}=0 is also made.\bigskip

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arXiv2006

On ADE Quiver Models and F-Theory Compactification

Based on mirror symmetry, we discuss geometric engineering of N=1 ADE quiver models from F-theory compactifications on elliptic K3 surfaces fibered over certain four-dimensional base spaces. The latter are constructed as intersecting 4-cycles according to ADE Dynkin diagrams, thereby mimicking the construction of Calabi-Yau threefolds used in geometric engineering in type II superstring theory. Matter is incorporated by considering D7-branes wrapping these 4-cycles. Using a geometric procedure referred to as folding, we discuss how the corresponding physics can be converted into a scenario with D5-branes wrapping 2-cycles of ALE spaces.

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arXiv2005

On Non Commutative G2 structure

Using an algebraic orbifold method, we present non-commutative aspects of $G_2$ structure of seven dimensional real manifolds. We first develop and solve the non commutativity parameter constraint equations defining $G_2$ manifold algebras. We show that there are eight possible solutions for this extended structure, one of which corresponds to the commutative case. Then we obtain a matrix representation solving such algebras using combinatorial arguments. An application to matrix model of M-theory is discussed.

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arXiv2004

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.

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arXiv2003

Geometric Engineering of N=2 CFT_{4}s based on Indefinite Singularities: Hyperbolic Case

Using Katz, Klemm and Vafa geometric engineering method of $\mathcal{N}=2$ supersymmetric QFT$_{4}$s and results on the classification of generalized Cartan matrices of Kac-Moody (KM) algebras, we study the un-explored class of $\mathcal{N}=2$ CFT$_{4}$s based on \textit{indefinite} singularities. We show that the vanishing condition for the general expression of holomorphic beta function of $\mathcal{N}=2$ quiver gauge QFT$_{4}$s coincides exactly with the fundamental classification theorem of KM algebras. Explicit solutions are derived for mirror geometries of CY threefolds with \textit{% hyperbolic} singularities.

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arXiv2003

Classification of N=2 supersymmetric CFT_{4}s: Indefinite Series

Using geometric engineering method of 4D $\mathcal{N}=2$ quiver gauge theories and results on the classification of Kac-Moody (KM) algebras, we show on explicit examples that there exist three sectors of $\mathcal{N}=2$ infrared CFT$_{4}$s. Since the geometric engineering of these CFT$_{4}$s 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$_{3}^{4}$ and E$_{10}$ hyperbolic singularities. We also derive a hierarchy of indefinite complex algebraic geometries based on affine $A_{r}$ 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.

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arXiv2003

On N=1 gauge models from geometric engineering in M-theory

We study geometric engineering of four-dimensional N=1 gauge models from M-theory on a seven-dimensional manifold with G_2 holonomy. The manifold is constructed as a K3 fibration over a three-dimensional base space with ADE geometry. The resulting gauge theory is discussed in the realm of (p,q) webs. We discuss how the anomaly cancellation condition translates into a condition on the associated affine ADE Lie algebras.

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arXiv2002

NC Calabi-Yau Orbifolds in Toric Varieties with Discrete Torsion

Using the algebraic geometric approach of Berenstein et {\it al} (hep-th/005087 and hep-th/009209) and methods of toric geometry, we study non commutative (NC) orbifolds of Calabi-Yau hypersurfaces in toric varieties with discrete torsion. We first develop a new way of getting complex $d$ mirror Calabi-Yau hypersurfaces $H_Δ^{\ast d}$ in toric manifolds $M_{Δ}^{\ast (d+1)}$ with a $C^{\ast r}$ action and analyze the general group of the discrete isometries of $H_Δ^{\ast d}$. Then we build a general class of $d$ complex dimension NC mirror Calabi-Yau orbifolds where the non commutativity parameters $θ_{μν}$ are solved in terms of discrete torsion and toric geometry data of $M_Δ^{(d+1)}$ in which the original Calabi-Yau hypersurfaces is embedded. Next we work out a generalization of the NC algebra for generic $d$ dimensions NC Calabi-Yau manifolds and give various representations depending on different choices of the Calabi-Yau toric geometry data. We also study fractional D-branes at orbifold points. We refine and extend the result for NC $% (T^{2}\times T^{2}\times T^{2})/(\mathbf{Z_{2}\times {Z_{2})}}$ to higher dimensional torii orbifolds in terms of Clifford algebra.

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arXiv2002

On Non-Commutative Orbifolds of K3 Surfaces

Using the algebraic geometry method of Berenstein and Leigh for the construction of the toroidal orbifold (T^2 x T^2 x T^2) / (Z_2 x Z_2) with discrete torsion and considering local K3 surfaces, we present non-commutative aspects of the orbifolds of product of K3 surfaces. In this way, the ordinary complex deformation of K3 can be identified with the resolution of stringy singularities by non-commutative algebras using crossed products. We give representations and make some comments regarding the fractionation of branes. Illustrating examples are presented.

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arXiv2001

On Non Commutative Calabi-Yau Hypersurfaces

Using the algebraic geometry method of Berenstein et al (hep-th/0005087), we reconsider the derivation of the non commutative quintic algebra ${\mathcal{A}}_{nc}(5)$ and derive new representations by choosing different sets of Calabi-Yau charges ${C_{i}^{a}}$. Next we extend these results to higher $d$ complex dimension non commutative Calabi-Yau hypersurface algebras ${\mathcal{A}}_{nc}(d+2)$. We derive and solve the set of constraint eqs carrying the non commutative structure in terms of Calabi-Yau charges and discrete torsion. Finally we construct the representations of ${\mathcal{A}}_{nc}(d+2) $ preserving manifestly the Calabi-Yau condition $ \sum_{i}C_{i}^{a}=0$ and give comments on the non commutative subalgebras.

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arXiv2000

Toric Geometry, Enhanced non Simply laced Gauge Symmetries in Superstrings and F-theory Compactifications

We study the geometric engineering of supersymmetric quantum field theories (QFT), with non simply laced gauge groups, obtained from superstring and F-theory compactifications on local Calabi-Yau manifolds. First we review the main lines of the toric method for ALE spaces with ADE singularities which we extend to non simply laced ordinary and affine singularities. Then, we develop two classes of solutions depending on the two possible realisations of the outer-automorphism group of the toric graph $Δ$(ADE). In F-theory on elliptic Calabi-Yau manifolds, we give explicit results for the affine non simply laced toric data and the corresponding BCFG mirror geometries. The latters extend known results obtained in litterature for the affine ADE cases. We also study the geometric engineering of $ N=1$ supersymmetric gauge theory in eight dimensions. In type II superstring compactifications on local Calabi-Yau threefolds, we complete the analysis for ordinary ADE singularities by giving the explict derivation of the lacking non simply ones. Finally we develop the basis of polyvalent toric geometry. The latter extends bivalent and trivalent geometries, considered in the geometric engineering method, and use it to derive a new solution for the affine $\hat D_4$ singularity. Other features are also discussed.

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arXiv2000

On HyperKahler Singularities

Using a geometric realization of the $SU(2)_R$ symmetry and a procedure of factorisation of the gauge and $SU(2)_R$ charges, we study the small instanton singularities of the Higgs branch of supersymmetric $U(1)^r$ gauge theories with eight supercharges. We derive new solutions for the moduli space of vacua preserving manifestly the eight supercharges. In particular, we obtain an extension of the ordinary ADE singularities for hyperKahler manifolds and show that the classical moduli space of vacua is in general given by cotangent bundles of compact weighted projective spaces.

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arXiv2000

Explicit Derivation of Yang-Mills Self-Dual Solutions on non-Commutative Harmonic Space

We develop the noncommutative harmonic space (NHS) analysis to study the problem of solving the non-linear constraint eqs of noncommutative Yang-Mills self-duality in four-dimensions. We show that this space, denoted also as NHS($η,θ$), has two SU(2) isovector deformations $η^{(ij)}$ and $θ^{(ij)}$ parametrising respectively two noncommutative harmonic subspaces NHS($η,0$) and NHS($0,θ$) used to study the self-dual and anti self-dual noncommutative Yang-Mills solutions. We formulate the Yang-Mills self-dual constraint eqs on NHS($η,0$) by extending the idea of harmonic analyticity to linearize them. Then we give a perturbative self-dual solution recovering the ordinary one. Finally we present the explicit computation of an exact self-dual solution.

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arXiv2000

Hyperkahler Singularities in Superstrings Compactification and 2d N=4 Conformal Field Theory

We study the singularities of the Higgs branch of supersymmetric U(1)^r gauge theories with eight supercharges. We derive new solutions for the moduli space of vacua preserving manifestly the eight supercharges by using a geometric realization of the SU(2)_R symmetry and a separation procedure of the gauge and SU(2)_R charges, which allow us to put the hypermultiplet vacua in a form depending on a parameter $γ$. For $γ=1$, we obtain new models which flow in the infrared to 2d N=(4,4) conformal models and we show that the classical moduli spaces are given by intersecting cotangent weighted complex projective spaces containing the small instanton singularity, discussed in [17], as a leading special case. We also make comments regarding the d$2d N=4 conformal Liouville description of the Higgs branch throat by following the analysis of [18]. Other features are also discussed.

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