arXiv subjects
Publications and source records attributed to A. Rezaei-Aghdam.
We show that the Wess-Zumino-Novikov-Witten (WZW) model on the Heisenberg Lie group $H_{4}$ has Jacobi-Lie symmetry with four dual Lie groups. We construct Jacobi-Lie T-dual sigma models with one of their Jacobi-Lie bialgebra and show that the original model is equivalent to the $H_{4}$ WZW model. The conformality of the dual sigma model up to one-loop order is also investigated.
Constructing M2-brane and its boundary conditions from D2-brane and the related boundary conditions and vice versa has been possible in our recent work by using 3-Lie bialgebra for BLG model with N = 8 supersymmetry. This could be generalized for BL model with N = 6 by the concept of the 3-Leibniz bialgebra. The 3-Lie bialgebra is an especial case of 3-Leibniz bialgebra, then more comprehensive information will be obtained in this work. Consequently, according to the correspondence of these 3-Leibniz bialgebras with Lie bialgebras, we reduce to D2-brane such that with some restrictions on the gauge field this D2-brane is related to the bosonic sector of an N = (4,4) WZW model equipped with one 2-cocycle in its Lie bialgebra structure. Moreover, the Basu-Harvey equation which is found by considering boundary conditions for BL model containing Leibniz bialgebra structure is reduced to Nahm equation and vice versa using this correspondence.
Using the concept of 3-Lie bialgebra; we construct the Bagger- Lambert- Gustavson (BLG) model on the Manin triple $\cal D$ of the especial 3-Lie bialgebra $({\cal D},{\cal A}_{\cal G},{\cal A}_{{\cal G}^*}^*)$ which is in correspondence with Manin triple of Lie bialgebra $({\cal D},{\cal G},{\cal G}^*)$. We have shown that the Nahm equation (with Lie bialgebra ${\cal G}$) can be obtained from the Basu-Harvey equation as a boundary condition of BLG model (with 3-Lie bialgebra ${\cal D}$) and vice versa.
Using the concept of Jacobi-Lie group and Jacobi-Lie bialgebra, we generalize the definition of Poisson-Lie symmetry to Jacobi-Lie symmetry. In this regard, we generalize the concept of Poisson-Lie T-duality to Jacobi-Lie T-duality and present Jacobi-Lie T-dual sigma models on Lie groups, which have Jacobi-Lie symmetry. Using this symmetry, new cases of duality appear and some examples are given. This generalization may provide insights to understand the quantum features of Poisson-Lie T-duality, in a more satisfactory way.
The two-loop (order $α'$) $β$-function equations, which are equivalent to the equations of motion of $α'$-corrected string effective action, are considered for anisotropic homogeneous space-times. These equations are solved for all Bianchi-type models in two schemes of effective action, namely $R^2$ and Gauss-Bonnet schemes with zero cosmological constant and then the metric, dilaton and $B$-field are found at $α'$ perturbative corrections.
We introduce a four-dimensional extension of the Poincaré algebra $(\mathcal{N})$ in (1+1)-dimensional space-time and obtain a (1+1)-dimensional gauge symmetric gravity model using the algebra $\mathcal{N}$. We show that the obtained gravity model is dual (canonically transformed) to the (1+1)-dimensional anti de Sitter ($AdS$) gravity. We also obtain some black hole and Friedmann-Robertson-Walker (FRW) solutions by solving its classical equations of motion. Then, we study $\frac{\mathbf{A}_{\mathbf{4,8}}}{\mathbf{A}_{\mathbf{1}}\otimes \mathbf{A}_{\mathbf{1}}}$ gauged Wess-Zumino-Witten (WZW) model and obtain some exact black hole and cosmological solutions in string theory. We show that some obtained black hole and cosmological metrics in string theory are same as the metrics obtained in solutions of our gauge symmetric gravity model.
The aim of this paper is to extend the notion of bialgebra for Leibniz algebras (and Lie algebras) to $3$-Leibniz algebras (and $3$-Lie algebras) by use of the cohomology complex of $3$-Leibniz algebras. Also, some theorems about Leibniz bialgebras are extended and proved in the case of $3$-Leibniz bialgebras ($3$-Lie bialgebras). Moreover, a new theorem on the correspondence between $3$-Leibniz bialgebra and its associated Leibniz bialgebra is proved. Finally, some examples are discussed in detail.
The non-Abelian T-dualization of the BTZ black hole is discussed in detail by using the Poisson-Lie T-duality in the presence of spectators. We explicitly construct a dual pair of sigma models related by Poisson-Lie symmetry. The original model is built on a $2+1$-dimensional manifold ${\cal M} \approx O \times \bf G$, where $\bf G$ as a two-dimensional real non-Abelian Lie group acts freely on ${\cal M}$, while $O$ is the orbit of $\bf G$ in ${\cal M}$. The findings of our study show that the original model indeed is canonically equivalent to the $SL(2,\mathbb{R})$ Wess-Zumino-Witten (WZW) model for a given value of the background parameters. Moreover, by a convenient coordinate transformation we show that this model describes a string propagating in a spacetime with the BTZ black hole metric in such a way that a new family of the solutions to low energy string theory with the BTZ black hole vacuum metric, constant dilaton field and a new torsion potential is found. The dual model is built on a $2+1$-dimensional target manifold $\tilde {\cal M}$ with two-dimensional real Abelian Lie group ${\tilde {\bf G}}$ acting freely on it. We further show that the dual model yields a three-dimensional charged black string for which the mass $M$ and axion charge $Q$ per unit length are calculated. After that, the structure and asymptotic nature of the dual space-time including the horizon and singularity are determined.
Admissible structure constants related to the dual Lie superalgebras of particular Lie superalgebra $({\cal C}^3 + {\cal A})$ are found by straightforward calculations from the matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation. Then, by making use of the automorphism supergroup of the Lie superalgebra $({\cal C}^3 + {\cal A})$, the Lie superbialgebra structures on the Lie superalgebra $({\cal C}^3 + {\cal A})$ are obtained and classified into inequivalent 31 families. We also determine all corresponding coboundary and bi-r-matrix Lie superbialgebras. The quantum deformations associated with some Lie superbialgebras $({\cal C}^3 + {\cal A})$ are obtained, together with the corresponding deformed Casimir elements. As an application of these quantum deformations, we construct a deformed integrable Hamiltonian system from the representation of the Hopf superalgebra ${{U}_{_λ}}^{\hspace{-1mm}({\cal C}_{p=1}^{2,ε} \oplus {\cal A}_{1,1})}\big(({\cal C}^3+{\cal A})\big)$.
We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.
In this paper we classify all four dimensional real Lie bialgebras of symplectic type. The classical r- matrices for these Lie bialgebras and Poisson structures on all of the related four dimensional Poisson-Lie groups are also obtained. Some new integrable models for which the Poisson-Lie group plays the role as a phase space and its dual Lie group plays the role of a symmetry group of the system, are obtained.
We describe the definition of Jacobi (generalized)-Lie bialgebras $(({\bf{g}},ϕ_{0}),({\bf{g}}^{*},X_{0}))$ in terms of structure constants of the Lie algebras ${\bf{g}}$ and ${\bf{g}}^{*}$ and components of their 1-cocycles $X_{0}\in {\bf{g}}$ and $ϕ_{0}\in {\bf{g}}^{*}$ in the basis of the Lie algebras. Then, using adjoint representations and automorphism Lie groups of Lie algebras, we give a method for classification of real low dimensional Jacobi-Lie bialgebras. In this way, we obtain and classify real two and three dimensional Jacobi-Lie bialgebras.
Using the concept of 3-Lie bialgebra, which has recently been defined in arXiv:1604.04475, we construct Bagger-Lambert-Gustavson (BLG) model for M2-brane on Manin triple of a special 3-Lie bialgebra. Then by using the correspondence and relation between those 3-Lie bialgebra with Lie bialgebra, we reduce this model to an $N=(4,4)$ WZW model (D2-brane), such that, its algebraic structure is a Lie bialgebra with one 2-cocycle. In this manner by using correspondence of 3-Lie bialgebra and Lie bialgebra (for this special 3-Lie algebra) one can construct M2-brane from a D2-brane and vice versa.
In this research we obtain the classical r-matrices of real two and three dimensional Jacobi-Lie bialgebras. In this way, we classify all non-isomorphic real two and three dimensional coboundary Jacobi-Lie bialgebras and their types (triangular and quasitriangular). Also, we obtain the generalized Sklyanin bracket formula by use of which, we calculate the Jacobi structures on the related Jacobi-Lie groups. Finally, we present a new method for constructing classical integrable systems using coboundary Jacobi-Lie bialgebras.
We consider a string model at one-loop related to a $σ$-model whose antisymmetric tensor field is constructed as complex structure on the background manifold, specially on a manifold $R\times N$ where $N$ is a complex manifold. As an example, we consider a homogeneous anisotropic $(1+4)$-dimensional $σ$-model where space part of the background is a $4$-dimensional complex manifold. By solving the related one-loop $β$-functions we obtain a static solution so that by reduction of this solution to $(1+3)$-dimension we obtain a static solution of Einstein equation where the matter sector is effectively interpreted as an inhomogeneous, anisotropic and barotropic matter satisfying all the energy conditions. Finally, the $T$-dual background of the solution is investigated and it is shown that the duality transformation and reduction processes commute with each other.
We show that the WZW model on the Heisenberg Lie group $H_4$ has Poisson-Lie symmetry only when the dual Lie group is ${ A}_2 \oplus 2{ A}_1$. In this way, we construct the mutual T-dual sigma models on Drinfel'd double generated by the Heisenberg Lie group $H_4$ and its dual pair, ${ A}_2 \oplus 2{ A}_1$, as the target space in such a way that the original model is the same as the $H_4$ WZW model. Furthermore, we show that the dual model is conformal up to two loops order. Finally, we discuss $D$-branes and the worldsheet boundary conditions defined by a gluing matrix on the $H_4$ WZW model. Using the duality map obtained from the canonical transformation description of the Poisson-Lie T-duality transformations for the gluing matrix which locally defines the properties of the $D$-brane, we find two different cases of the gluing matrices for the WZW model based on the Heisenberg Lie group $H_4$ and its dual model.
We have determined all Nambu tensors (Nambu structures) of order four and three on four dimensional real Lie groups. Furthermore, we have obtained superintegrable systems by use of the Nambu structures of order four on some of these Lie groups as phase spaces with symmetry groups A4;8 and A4;10.
We have obtained exact three dimensional BTZ type solutions with gauge fields, for string theory on a gauge symmetric gravitational background constructed from semi-simple extension of the Poincare algebra (and the Maxwell algebra) in 2 + 1 dimensions. We have studied the models for two non-Abelian and Abelian gauge fields solutions and shown that the related sigma models for each of these backgrounds is a SL(2;R) WZW (Wess-Zumino-Witten) model and that they are classically canonically equivalent. We have also obtained the dual solution for the Abelian case and by interpreting the new field strength tensors of the Abelian solution as electromagnetic field strength tensors shown that dual models coincide with the charged black string solution.