Free motion on the Poisson SU(n) group

SL(N,C) is the phase space of the Poisson SU(N). We calculate explicitly the symplectic structure of SL(N,C), define an analogue of the Hamiltonian of the free motion on SU(N) and solve the corresponding equations of motion. Velocity is related to the momentum by a non-linear Legendre transformation.Comment: LaTeX, 10 page

Phase spaces related to standard classical rr-matrices

Fundamental representations of real simple Poisson Lie groups are Poisson actions with a suitable choice of the Poisson structure on the underlying (real) vector space. We study these (mostly quadratic) Poisson structures and corresponding phase spaces (symplectic groupoids).Comment: 20 pages, LaTeX, no figure

Chern-Simons Particles with Nonstandard Gravitational Interaction

The model of nonrelativistic particles coupled to nonstandard (2+1)-gravity [1] is extended to include Abelian or non-Abelian charges coupled to Chern-Simons gauge fields. Equivalently, the model may be viewed as describing the (Abelian or non-Abelian) anyonic dynamics of Chern-Simons particles coupled, in a reparametrization invariant way, to a translational Chern-Simons action. The quantum two-body problem is described by a nonstandard Schr\"{o}dinger equation with a noninteger angular momentum depending on energy as well as particle charges. Some numerical results describing the modification of the energy levels by these charges in the confined regime are presented. The modification involves a shift as well as splitting of the levels.Comment: LaTeX, 1 figure (included), 14 page

On the complexity of the ideal of absolute null sets

Answering a question posed by Banakh and Lyaskovska, we prove that, for an arbitrary countable infinite amenable group G, the ideal of sets having μ-measure zero for every Banach measure μ on G is an Fσδ subset of {0; 1}G.У вiдповiдь на питання, поставлене Банахом i Ляскiвською, доведено, що для будь-якої злiченної аменабельної групи G iдеал множин, що мають нульову μ-мiру для будь-якої мiри Банаха μ на G, є Fσδ-пiдмножиною {0,1}G

Some Comments on BPS systems

We look at simple BPS systems involving more than one field. We discuss the conditions that have to be imposed on various terms in Lagrangians involving many fields to produce BPS systems and then look in more detail at the simplest of such cases. We analyse in detail BPS systems involving 2 interacting Sine-Gordon like fields, both when one of them has a kink solution and the second one either a kink or an antikink solution. We take their solitonic static solutions and use them as initial conditions for their evolution in Lorentz covariant versions of such models. We send these structures towards themselves and find that when they interact weakly they can pass through each other with a phase shift which is related to the strength of their interaction. When they interact strongly they repel and reflect on each other. We use the method of a modified gradient flow in order to visualize the solutions in the space of fields.Comment: 27 pages, 17 figure

Fuzzy Nambu-Goldstone Physics

In spacetime dimensions larger than 2, whenever a global symmetry G is spontaneously broken to a subgroup H, and G and H are Lie groups, there are Nambu-Goldstone modes described by fields with values in G/H. In two-dimensional spacetimes as well, models where fields take values in G/H are of considerable interest even though in that case there is no spontaneous breaking of continuous symmetries. We consider such models when the world sheet is a two-sphere and describe their fuzzy analogues for G=SU(N+1), H=S(U(N-1)xU(1)) ~ U(N) and G/H=CP^N. More generally our methods give fuzzy versions of continuum models on S^2 when the target spaces are Grassmannians and flag manifolds described by (N+1)x(N+1) projectors of rank =< (N+1)/2. These fuzzy models are finite-dimensional matrix models which nevertheless retain all the essential continuum topological features like solitonic sectors. They seem well-suited for numerical work.Comment: Latex, 18 pages; references added, typos correcte

Noncommutative planar particle dynamics with gauge interactions

We consider two ways of introducing minimal Abelian gauge interactions into the model presented in [Ann. Phys. 260 (1997) 224]. These two approaches are different only if the second central charge of the planar Galilei group is nonzero. One way leads to the standard gauge transformations and the other one to a generalised gauge theory with gauge transformations accompanied by time-dependent area-preserving coordinate transformations. Both approaches, however, are related to each other by a classical Seiberg–Witten map supplemented by a noncanonical transformation of the phase space variables for planar particles. We also formulate the two-body problem in the model with our generalised gauge symmetry and consider the case with both CS and background electromagnetic fields, as it is used in the description of fractional quantum Hall effect

Links between different analytic descriptions of constant mean curvature surfaces

Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the system $$\begin{split} &\partial \psi_{1} = (|\psi_{1}|^{2} + |\psi_{2}|^{2}) \psi_{2} \\ &\bar{\partial} \psi_{2} =- (|\psi_{1}|^{2} + |\psi_{2}|^{2}) \psi_{1} \end{split}$$ descriptive of CMC surfaces within the framework of the generalized Weierstrass representation, decouples into a direct sum of the elliptic Sh-Gordon and Laplace equations. Connections of this system with the sigma model equations are established. It is pointed out, that the instanton solutions correspond to different Weierstrass parametrizations of the standard sphere $S^{2} \subset E^{3}$

Stirring Bose-Einstein condensate

By shining a tightly focused laser light on the condensate and moving the center of the beam along the spiral line one may stir the condensate and create vortices. It is shown that one can induce rotation of the condensate in the direction opposite to the direction of the stirring.Comment: 4 pages, 5 figures, published versio

Nonuniversality in level dynamics

Statistical properties of parametric motion in ensembles of Hermitian banded random matrices are studied. We analyze the distribution of level velocities and level curvatures as well as their correlation functions in the crossover regime between three universality classes. It is shown that the statistical properties of level dynamics are in general non-universal and strongly depend on the way in which the parametric dynamics is introduced.Comment: 24 pages + 10 figures (not included, avaliable from the author), submitted to Phys. Rev.