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L. Snobl

Publications and source records attributed to L. Snobl.

13 recordsLinked to original sources
arXiv2009

Invariant solutions of the supersymmetric sinh-Gordon equation

A systematic group-theoretical analysis of the supersymmetric sinh-Gordon equation is performed. A generalization of the method of prolongations is used to determine the Lie superalgebra of symmetries, and the method of symmetry reduction is applied in order to obtain invariant solutions of the supersymmetric sinh-Gordon equation. The results are compared with those previously found for the supersymmetric sine-Gordon equation. The presence of nonstandard invariants is discussed for the supersymmetric sinh-Gordon equation, as well as for the supersymmetric Korteweg-de Vries equation.

math-ph↗
arXiv2009

Invariant solutions of the supersymmetric sine-Gordon equation

A comprehensive symmetry analysis of the N=1 supersymmetric sine-Gordon equation is performed. Two different forms of the supersymmetric system are considered. We begin by studying a system of partial differential equations corresponding to the coefficients of the various powers of the anticommuting independent variables. Next, we consider the super-sine-Gordon equation expressed in terms of a bosonic superfield involving anticommuting independent variables. In each case, a Lie (super)algebra of symmetries is determined and a classification of all subgroups having generic orbits of codimension 1 in the space of independent variables is performed. The method of symmetry reduction is systematically applied in order to derive invariant solutions of the supersymmetric model. Several types of algebraic, hyperbolic and doubly periodic solutions are obtained in explicit form.

math-ph↗
arXiv2006

Surfaces associated with sigma models and the sine-Gordon equation

We present a unified method of construction of surfaces associated with Grassmannian sigma models, expressed in terms of an orthogonal projector. This description leads to compact formulae for structural equations of two-dimensional surfaces immersed in the su(N) algebra. In the special case of the CP^1 sigma model we obtain constant negative Gaussian curvature surfaces. As a consequence, this leads us to an explicit relation between the CP^1 sigma model and the sine-Gordon equation.

math.DG↗
arXiv2005

Description of surfaces associated with Grassmannian sigma models on Minkowski space

We construct and investigate smooth orientable surfaces in su(N) algebras. The structural equations of surfaces associated with Grassmannian sigma models on Minkowski space are studied using moving frames adapted to the surfaces. The first and second fundamental forms of these surfaces as well as the relations between them as expressed in the Gauss-Weingarten and Gauss-Codazzi-Ricci equations are found. The scalar curvature and the mean curvature vector expressed in terms of a solution of Grassmanian sigma model are obtained.

math.DG↗
arXiv2004

Description of surfaces associated with $CP^{N-1}$ sigma models on Minkowski space

The objective of this paper is to construct and investigate smooth orientable surfaces in $R^{N^2-1}$ by analytical methods. The structural equations of surfaces in connection with $CP^{N-1}$ sigma models on Minkowski space are studied in detail. This is carried out using moving frames adapted to surfaces immersed in the $su(N)$ algebra. The first and second fundamental forms of this surface as well as the relations between them as expressed in the Gauss-Weingarten and Gauss-Codazzi-Ricci equations are found. The Gaussian curvature, the mean curvature vector and the Willmore functional expressed in terms of a solution of $CP^{N-1}$ sigma model are obtained. An example of a surface associated with the $CP^1$ model is included as an illustration of the theoretical results.

math.DG↗
arXiv2004

A class of solvable Lie algebras and their Casimir Invariants

A nilpotent Lie algebra n_{n,1} with an (n-1) dimensional Abelian ideal is studied. All indecomposable solvable Lie algebras with n_{n,1} as their nilradical are obtained. Their dimension is at most n+2. The generalized Casimir invariants of n_{n,1} and of its solvable extensions are calculated. For n=4 these algebras figure in the Petrov classification of Einstein spaces. For larger values of n they can be used in a more general classification of Riemannian manifolds.

math-ph↗
arXiv2004

Poisson-Lie T-plurality of three-dimensional conformally invariant sigma models II: Nondiagonal metrics and dilaton puzzle

We look for 3-dimensional Poisson-Lie dualizable sigma models that satisfy the vanishing beta-function equations with constant dilaton field. Using the Poisson-Lie T-plurality we then construct 3-dimensional sigma models that correspond to various decompositions of Drinfeld double. Models with nontrivial dilaton field may appear. It turns out that for ``traceless'' dual algebras they satisfy the vanishing beta-function equations as well. In certain cases the dilaton cannot be defined in some of the dual models. We provide an explanation why this happens and give criteria predicting when it happens.

hep-th↗
arXiv2004

Poisson-Lie T-plurality of three-dimensional conformally invariant sigma models

Starting from the classification of 6-dimensional Drinfeld doubles and their decomposition into Manin triples we construct 3-dimensional Poisson-Lie T-dual or more precisely T-plural sigma models. Of special interest are those that are conformally invariant. Examples of models that satisfy vanishing beta-function equations with zero dilaton are presented and their duals are calculated. It turns out that for "traceless" dual algebras they satisfy the beta-function equations as well but usually with rather nontrivial dilaton. We also present explicit examples of several kinds of obstacles and difficulties present in construction of quantum dual models. Such concrete examples might be helpful in further development and improvement of quantum version of Poisson-Lie T-duality.

hep-th↗
arXiv2002

On modular spaces of semisimple Drinfeld doubles

We construct modular spaces of all 6-dimensional real semisimple Drinfeld doubles, i.e. the sets of all possible decompositions of the Lie algebra of the Drinfeld double into Manin triples. These modular spaces are significantly different from the known one for Abelian Drinfeld double, since some of these Drinfeld doubles allow decomposition into several non-isomorphic Manin triples and their modular spaces are therefore written as unions of homogeneous spaces of different dimension. Implications for Poisson-Lie T-duality and especially Poisson-Lie T-plurality are mentioned.

hep-th↗
arXiv2002

Classification of 6-dimensional real Manin triples

We present a complete list of 6-dimensional Manin triples or, equivalently, of 3-dimensional Lie bialgebras. We start from the well known classification of 3-dimensional real Lie algebras and assume the canonical bilinear form on the 6-dimensional Drinfeld double. Then we solve the Jacobi identities for the dual algebras. Finally we find mutually non-isomorphic Manin triples. The complete list consists of 78 classes of Manin triples, or 44 Lie bialgebras if one considers dual bialgebras equivalent.

math.QA↗
arXiv2002

Classification of 6-dimensional real Drinfeld doubles

Starting from the classification of real Manin triples done in a previous paper we look for those that are isomorphic as 6-dimensional Lie algebras with the ad-invariant form used for construction of the Manin triples. We use several invariants of the Lie algebras to distinguish the non-isomorphic structures and give explicit form of maps between Manin triples that are decompositions of isomorphic Drinfeld doubles. The result is a complete list of 6-dimensional real Drinfeld doubles. It consists of 22 classes of non-isomorphic Drinfeld doubles.

math.QA↗
arXiv1998

Solution of a Yang-Baxter system

Yang-Baxter system related to quantum doubles is introduced and large class of both continuous and discrete symmetries of the solution manifold are determined. Strategy for solution of the system based on the symmetries is suggested and accomplished in the dimension two.The complete list of invertible solutions of the system is presented.

math.QA↗