arXiv subjects

M. B. Sedra

Publications and source records attributed to M. B. Sedra.

49 records · Page 3Linked to original sources

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.

hep-th↗

arXiv2005

HyperKhaler Metrics Building and Integrable Models

Methods developed for the analysis of integrable systems are used to study the problem of hyperKähler metrics building as formulated in D=2 N=4 supersymmetric harmonic superspace. We show, in particular, that the constraint equation $β\partial^{++2}ω-ξ^{++2}\exp 2βω=0$ and its Toda like generalizations are integrable. Explicit solutions together with the conserved currents generating the symmetry responsible of the integrability of these equations are given. Other features are also discussed

hep-th↗

arXiv2005

Some Aspects of Moyal Deformed Integrable Systems

Besides its various applications in string and D-brane physics, the $θ$-deformation of space (-time) coordinates (naively called the noncommutativity of coordinates), based on the $\star$-product, behaves as a more general framework providing more mathematical and physical informations about the associated system. Similarly to the Gelfand-Dickey framework of pseudo differential operators, the Moyal $θ$-deformation applied to physical problems makes the study more systematic. Using these facts as well as the backgrounds of Moyal momentum algebra introduced in previous works [21, 25, 26], we look for the important task of studying integrability in the $θ$-deformation framework. The main focus is on the $θ$-deformation version of the Lax representation of two principal examples: the $sl_2$ KdV$_θ$ equation and the Moyal $θ$-version of the Burgers systems. Important properties are presented.

hep-th↗

arXiv2005

Moyal Noncommutative Integrability and the Burgers-KdV Mapping

The Moyal momentum algebra is, once again, used to discuss some important aspects of NC integrable models and 2d conformal field theories. Among the results presented, we setup algebraic structures and makes useful convention notations leading to extract non trivial properties of the Moyal momentum algebra. We study also the Lax pair building mechanism for particular examples namely, the noncommutative KdV and Burgers systems. We show in a crucial step that these two systems are mapped to each others through a crucial mapping. This makes a strong constraint on the NC Burgers system which corresponds to linearizing its associated differential equation. From the conformal field theory point of view, this constraint equation is nothing but the analogue of the conservation law of the conformal current. We believe that this mapping might help to bring new insights towards understanding the integrability of noncommutative 2d-systems.

hep-th↗

arXiv2004

Explicit derivation of a Central extended Hyper-Kahler Metric

This work consists in applying the analysis of integrable models to study the problem of Hyper-Kahler metrics building. In this context, we use the harmonic superspace language applied to D=2 N=4 SU(2) Liouville self interacting model and derive an explicit central extended Hyper-Kahler metric as well as the induced scalar potential.

hep-th↗

arXiv2003

Type IIB String Backgrounds on Parallelizable PP-Waves and Conformal Liouville Theory

The scope of this work concerns the adaptation of the parallelizability pp-wave (Ppp-wave) process to D=10 type IIB string backgrounds in the presence of the non-trivial anti-self dual R-R 5-form $\QTR{cal}{F}$. This is important in the sense that it gives rise to some unsuspected properties. In fact, exact solutions of type IIB string backgrounds on Ppp-waves are discussed. For the $u$-dependence of the dilaton field $Φ$, we establish explicitly a correspondence between type IIB supergravity equations of motion and 2d-conformal Liouville field theory. We show also that the corresponding conserved conformal current $T(Φ)$ coincides exactly with the trace of the symmetric matrix $μ_{ij}$ appearing in the quadratic front factor $F=μ_{ij}x^{i}x^{j}$ of the Ppp-wave. Furthermore, we consider the transverse space dependence of the dilaton $Φ$ and show that the supergravity equations are easily solved for the linear realization of the dilaton field. Other remarkable properties related to this case are also discussed.

hep-th↗

arXiv2003

Noncommutative Geometry Framework and The Feynman's Proof of Maxwell Equations

The main focus of the present work is to study the Feynman's proof of the Maxwell equations using the NC geometry framework. To accomplish this task, we consider two kinds of noncommutativity formulations going along the same lines as Feynman's approach. This allows us to go beyond the standard case and discover non-trivial results. In fact, while the first formulation gives rise to the static Maxwell equations, the second formulation is based on the following assumption $m[x_{j},\dot{x_{k}}]=i\hbar δ_{jk}+imθ_{jk}f.$ The results extracted from the second formulation are more significant since they are associated to a non trivial $θ$-extension of the Bianchi-set of Maxwell equations. We find $div_θB=η_θ$ and $\frac{\partial B_{s}}{\partial t}+ε_{kjs}\frac{\partial E_{j}}{\partial x_{k}}=A_{1}\frac{d^{2}f}{dt^{2}}+A_{2}\frac{df}{dt}+A_{3},$ where $η_θ$, $A_{1}$, $A_{2}$ and $A_{3}$ are local functions depending on the NC $θ$-parameter. The novelty of this proof in the NC space is revealed notably at the level of the corrections brought to the previous Maxwell equations. These corrections correspond essentially to the possibility of existence of magnetic charges sources that we can associate to the magnetic monopole since $div_θB=η_θ$ is not vanishing in general.

hep-th↗

arXiv2003

Algebra of Deformed Differential Operators and Induced Integrable Toda Field Theory

We build in this paper the algebra of q-deformed pseudo-differential operators shown to be an essential step towards setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the q-analogues of the generalized KdV hierarchy. We focus in particular on the first leading orders of this q-deformed hierarchy namely the q-KdV and q-Boussinesq integrable systems. We present also the q-generalization of the conformal transformations of the currents and discuss the primarity condition of the fields by using the Volterra gauge group transformations for the q-covariant Lax operators. An induced su(n)-Toda(su(2)-Liouville) field theory construction is discussed and other important features are presented.

hep-th↗

arXiv2003

Non Standard Extended Noncommutativity of Coordinates

We present in this short note an idea about a possible extension of the standard noncommutative algebra to the formal differential operators framework. In this sense, we develop an analysis and derive an extended noncommutative structure given by $[x_{a}, x_{b}]_{\star} = i(θ+ χ)_{ab}$ where $θ_{ab}$ is the standard noncommutative parameter and $χ_{ab}(x)\equiv χ^μ_{ab}(x)\partial_μ ={1/2}(x_a θ^μ_{b} - x_b θ^μ_{a})\partial_μ$ is an antisymmetric non-constant vector-field shown to play the role of the extended deformation parameter. This idea was motivated by the importance of noncommutative geometry framework, with nonconstant deformation parameter, in the current subject of string theory and D-brane physics.

hep-th↗

arXiv2002

The Moyal Momentum algebra applied to (theta)-deformed 2d conformal models and KdV-hierarchies

The properties of the Das-Popowicz Moyal momentum algebra that we introduce in hep-th/0207242 are reexamined in details and used to discuss some aspects of integrable models and 2d conformal field theories. Among the results presented, we setup some useful convention notations which lead to extract some non trivial properties of the Moyal momentum algebra. We use the particular sub-algebra sl(n)-{Sigma}_{n}^{(0,n)} to construct the sl(2)-Liouville conformal model and its sl(3)-Toda extension. We show also that the central charge, a la Feigin-Fuchs, associated to the spin-2 conformal current of the (theta)-Liouville model is given by c(theta)=1+24.theta^{2}. Moreover, the results obtained for the Das-Popowicz Mm algebra are applied to study systematically some properties of the Moyal KdV and Boussinesq hierarchies generalizing some known results. We discuss also the primarity condition of conformal $w_θ$-currents and interpret this condition as being a dressing gauge symmetry in the Moyal momentum space. Some computations related to the dressing gauge group are explicitly presented.

hep-th↗

arXiv2000

On the harmonic superspace language adapted to the Gelfand-Dickey algebra of differential operators

Methods developed for the analysis of non-linear integrable models are used in the harmonic superspace (HS) framework. These methods, when applied to the HS, can lead to extract more information about the meaning of integrability in non-linear physical problems. Among the results obtained, we give the basic ingredients towards building in the HS language the analogue of the G.D. algebra of pseudo-differential operators. Some useful convention notations and algebraic structures are also introduced to make the use of the harmonic superspace techniques more accessible.

hep-th↗

arXiv2000

On the Fairlie's Moyal formulation of M(atrix)- theory

Starting from the Moyal formulation of M-theory in the large N-limit, we propose to reexamine the associated membrane equations of motion in 10 dimensions formulated in terms of Poisson bracket. Among the results obtained, we rewrite the coupled first order Nahm's equations into a simple form leading in turn to their systematic relation with $SU(\infty)$ Yang Mills equations of motion. The former are interpreted as the vanishing condition of some conserved currents which we propose. We develop also an algebraic analysis in which an ansatz is considered and find an explicit form for the membrane solution of our problem. Typical solutions known in literature can also emerge as special cases of the proposed solution

hep-th↗

arXiv1997

Three Graded Modified Classical Yang-Baxter Equations and Integrable Systems

The $6 = 3\times 2$ huge Lie algebra $Ξ$ of all local and non local differential operators on a circle is applied to the standard Adler-Kostant-Symes (AKS) R-bracket sckeme. It is shown in particular that there exist three additional Lie structures, associated to three graded modified classical Yang-Baxter(GMCYB) equations. As we know from the standard case, these structures can be used to classify in a more consitent way a wide class of integrable systems. Other algebraic properties are also presented.

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