Showing papers on "Winding number published in 1992"

Spectrum of Dirac operator and role of winding number in QCD

Abstract: We show that very general considerations based on the properties of the partition function of QCD allow one to extract information about the eigenvalues of the Dirac operator in vacuum gauge fields. In particular, we demonstrate that the familiar suppression of field configurations with a nontrivial topology occurring for small quark masses is a finite size effect which disappears if the four-dimensional volume [ital V] is large enough. The formation of a quark condensate is connected with the occurrence of small eigenvalues of order [lambda][sub [ital n]][proportional to]1/[ital V].

Studying links via closed braids. V. The unlink

Abstract: The main result is a version of Markov's Theorem which does not involve stabilization, in the special case of the r-component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most 1. To state the result, we need the concept of an "exchange move", which modifies a closed braid without changing its link type or its braid index. For generic closed braids exchange moves change conjugacy class. Theo- rem 1 shows that exchange moves are the only obstruction to reducing a closed n-braid representative of the r-component unlink to the standard closed r-braid representative, through a sequence of braids of nonincreasing braid index. Let K be an oriented link type in oriented S3, K a representative of K, and A an unknotted curve in S3. K is a closed n-braid with braid axis A if K winds monotonically about A with total winding number n. "Mono- tonically" means that if A is the z-axis in R3, parametrized by cylindrical coordinates (r(t), 0(t), z(t)), then 0(t) is a strictly increasing function of t. This is equivalent to the assertion that if {Ht: t E (0, 27X)} is a fibration of S3 - A by meridian discs then K meets each Ht transversally in exactly n coherently oriented intersections. Alexander proved in (A) that every link could be so-represented (in many ways). The theorem which is known as Markov's theorem describes how the various closed braid representatives of a given link type are related. Markov's Theorem was announced in (Ma), with a working outline for a proof. It has been important in recent years because of its central

Loop-erased self-avoiding walks in two dimensions: exact critical exponents and winding numbers

Abstract: Loop-erased self-avoiding walks (LESAWs) are defined as walks resulting from sequential erasing of the loops of random walks. The critical properties of LESAWs are those of a c = −2 conformal theory in two dimensions, and, geometrically, those of subgraphs of spanning trees. In this paper, we derive the exact asymptotic behavior of the set of probabilities P k ( N ) that k ( k ⩾2) LESAWs of length N starting at neighbouring points do not intersect, as well as the exact asymptotic winding angle distribution of a single two-dimensional LESAW.

On the number of determining nodes for the Ginzburg-Landau equation

Abstract: In the case of the complex Ginzburg-Landau equation in one space dimension it is proven that solutions are completely determined by their values at two sufficiently close points. As a consequence, an upper bound for the winding number of stationary solutions is established in terms of the bifurcation parameters. It is also proven that the fractal dimension of the set of stationary solutions is less than or equal to 4.

Topological Interpretation of Electric Charge, Duality and Confinement in 2+1 Dimensions

Abstract: Electric charge in QED is topological in the sense that the electric current is a curl of a local gauge-invariant field — the dual electromagnetic field strength. In 2+1 dimensions it can be explicitly represented as the winding number in terms of a local field V describing the Nielsen-Olesen magnetic vortices: . Electrically charged particles are then visualized as topological solitons of V corresponding to elements of the homotopy group π1(S1)=Z. The quantization of electric charge and the universality of the electromagnetic coupling are thus given a topological interpretation. The low energy physics is described by a “dual” Lagrangian written in terms of the field V only. This dual Lagrangian furnishes the Landau-Ginzburg description of the Coulomb-Higgs phase transition with V as a pertinent local order parameter. The symmetry which is associated with the phase transition is the magnetic flux symmetry. The nonzero VEV of V in the Coulomb phase breaks the magnetic flux spontaneously, leading to the appearance of a massless Goldstone boson—the photon. The vortex field V is also constructed in non-Abelian gauge theories. Here the solitons of V are identified with the “constituent” quarks. The dual Lagrangian contains explicit flux-symmetry-breaking terms. As a result the topological solitons (quarks) are linearly confined. We describe in detail the derivation of the dual Lagrangian and the topological mechanism of confinement in non-Abelian theories with adjoint and fundamental matter. The extension of the above description to the (3+1)-dimensional gauge theories is briefly discussed. Several notions are easily generalizable but the picture is far from being complete.

Winding number formula for Maslov indices.

Abstract: A formula for the Maslov index of closed curves on Lagrangian manifolds is derived. The index is expressed as the number of times the plane tangent to the curve winds around it. Applications include unstable periodic orbits, in which case the Lagrangian manifolds are the stable or unstable manifolds of the orbits, and cycles on the invariant tori of integrable systems, in which case the manifolds are the tori themselves.

Chiral cosmic strings.

Abstract: We consider all global cosmic-string solutions of U(1) scalar field theory with a cylindrically symmetric energy density. These can be characterized as extrema of the field's energy for given angular and linear momenta. This string class comprises, apart from the well-known ordinary and rotating strings, twisted string configurations whose isophase surfaces twist as one moves along the axis, and lightlike-phase strings in which the twisted phase propagates at the speed of light parallel to the string axis. We rule out global strings whose isophase contours are spirals in a plane. We prove, in a unified way and for a broad class of scalar potentials, the stability of ordinary, rotating, and lightlike-phase strings with a unit winding number against small perturbations. Twisted strings are apparently unstable unless they are made into loops. Lightlike-phase strings are stable for causal rather than for topological reasons.

Anyon winding numbers from the Chern-Simons interaction

Abstract: Integrating out the gauge field in the Chern-Simons interaction of anyons, the authors recover in the non-relativistic limit the winding number in the action for particles with fractional statistics.

Higher baryon and winding numbers in the Nambu-Jona-Lasinio model

Abstract: A bosonized version of the SU(2) Nambu-Jona-Lasinio model is solved self-consistently for hedgehog fields on the chiral circle in the solitonic sector with baryon number B=2. The authors consider explicitly the case in which the baryon number is raised by occupying B-bound valence quark states for different winding numbers omega of the chiral angle. It is found that only the cases B>or= omega allow the existence of self-consistent solutions. Fractional windings are also excluded in practice by the self-consistency condition. For the description of nucleon properties the constituent mass has to be chosen around 400 MeV. There no solitons with higher winding numbers are found in contrast to the results of the Skyrme model. On the other hand, in this region the binding energy of the deuteron and the alpha -particle can be roughly reproduced with winding number one.

Topologically entangled polymers

Abstract: The statistical mechanics of a ring polymer confined to a plane and entangled with many randomly placed thin rods perpendicular to the plane are considered. The entanglements are characterized by the Gauss linking number. If the statistics of the random distribution of the rods is given by only the second cumulant then it is shown that the resulting entanglement problem can be solved formally exactly. For this special case the exact solution becomes possible because the problem can be reduced to one involving the winding of the polymer around one infinitely thin rod. The exact solution can be obtained for both the annealed and the quenched random distribution of obstacles. The entanglement of the ring polymer around the obstacles leads to a repulsive topological potential which is an effective interaction between the polymer and the rods. The origin of this potential is solely due to the constraint that the winding number be conserved. It is shown that forR2/Ll≪l (R is the location of the polymer segment,L is the total length of the polymer, andl is the length of the monomer) the topological potential for the annealed random case goes asN ln ln(Ll/R2) whereN is the number of obstacles whereas for the quenched random case the potential is given byC lnLl/R2, whereC is a numerical constant that depends onN.

Quarks as topological defects or what is confined inside a hadron

Abstract: We present a picture of confinement based on representation of quarks as pointlike topological defects. The topological charge carried by quarks and confined in hadrons is explicitly constructed in terms of Yang - Mills variables. In 2+1 dimensions we are able to construct a local complex scalar field $V(x)$, in terms of whichthe topological charge is $Q=-\frac{i}{4\pi}\int d^2x \epsilon_{ij}\partial_i(V^*\partial_i V -c.c.)$. The VEV of the field $V$ in the confining phase is nonzero and the charge is the winding number corresponding to homotopy group $\pi_1(S^1)$. Qurks carry the charge $Q$ and therefore are topological solitons. The effective Lagrangian for $V$ is derived in models with adjoint and fundamental quarks. In 3+1 dimensions the explicit expression for $V$ and therefore a detailed picture is not available. However, assuming the validity of the same mechanism wepoint out several interesting qualitative consequences. We argue that in the Georgi - Glashow model or any grand unified model the photon (in the Higgs phase) should have a small nonperturbative mass and $W^\pm$ should be confined although with small string tension.

Analysis of Singularities of the Standard Map Conjugation Function

Abstract: There are many different approaches to the determination of the breakdown of the invariant curves and the resulting stochastic transition for the area-preserving maps but essentially we can divide them in two classes. Given an invariant curve it is possible to detect its breakdown either analyzing the behaviour of the nearby resonance orbits, using the overlap criteria 1 or the Greene’s method 2, or studying directly the properties of this curve3,4. The Kam theorem5,6 tells us that the diophantine rotation number orbits of a slightly perturbed Hamiltonian system are analytically conjugated to the curves with the same winding number of the unperturbed system, until a certain value of the perturbation parameter is reached. The analytical function Φ that transforms the perturbed tori to the unperturbed ones is called conjugation function; it depends on the winding number and it is analytical in e, strength of the perturbation, until e c , critical value. So a method to find the critical value at which the invariant curve with rotation number ω breakdown is to study the loss of analyticity of Φ.

Generalized Winding Number of a Dynamical System

Abstract: The generalized winding number of the BVP model system stimulated by a periodic current is studied in this letter. As long as the attractors lie on the invariant torus, the winding number diagram shows the well-known devil's staircases scenario. When the system transits to chaotic state, the winding number becomes irregular.

Winding number and the number of real zeros of a function

Abstract: The theorem in this paper shows that the number of real simple zeros of a function of the form f(x) = q(x) + ax + b, x e R, for not too wild q(x) can be obtained counting the winding number of a closed plane curve about the point (a, b) .