Recent mathematical developments in the Skyrme model

In this review we present a pedagogical introduction to recent, more mathematical developments in the Skyrme model. Our aim is to render these advances accessible to mainstream nuclear and particle physicists. We start with the static sector and elaborate on geometrical aspects of the definition of the model. Then we review the instanton method which yields an analytical approximation to the minimum energy configuration in any sector of fixed baryon number, as well as an approximation to the surfaces which join together all the low energy critical points. We present some explicit results for B=2. We then describe the work done on the multibaryon minima using rational maps, on the topology of the configuration space and the possible implications of Morse theory. Next we turn to recent work on the dynamics of Skyrmions. We focus exclusively on the low energy interaction, specifically the gradient flow method put forward by Manton. We illustrate the method with some expository toy models. We end this review with a presentation of our own work on the semi-classical quantization of nucleon states and low energy nucleon-nucleon scattering.Comment: 129 pages, about 30 figures, original manuscript of published Physics Report

Remarks on gauge vortex scattering

In the abelian Higgs model, among other situations, it has recently been realized that the head-on scattering of $n$ solitons distributed symmetrically around the point of scattering is by an angle $\pi/n$, independant of various details of the scattering. In this note, it is first observed that this result is in fact not entirely surprising: the above is one of only two possible outcomes. Then, a generalization of an argument given by Ruback for the case of two gauge theory vortices in the Bogomol'nyi limit is used to show that in the geodesic approximation the above result follows from purely geometric considerations.Comment: 6 pages, revtex, missing authors added to one referenc

Solitons in a Baby-Skyrme model with invariance under area preserving diffeomorphisms

We study the properties of soliton solutions in an analog of the Skyrme model in 2+1 dimensions whose Lagrangian contains the Skyrme term and the mass term, but no usual kinetic term. The model admits a symmetry under area preserving diffeomorphisms. We solve the dynamical equations of motion analytically for the case of spinning isolated baryon type solitons. We take fully into account the induced deformation of the spinning Skyrmions and the consequent modification of its moment of inertia to give an analytical example of related numerical behaviour found by Piette et al.. We solve the equations of motion also for the case of an infinite, open string, and a closed annular string. In each case, the solitons are of finite extent, so called "compactons", being exactly the vacuum outside a compact region. We end with indications on the scattering of baby-Skyrmions, as well as some considerations as the properties of solitons on a curved space.Comment: 30 pages, 5 figures, revtex, major modifications, conclusions modifie

Low Energy Skyrmion-Skyrmion Scattering

We study the scattering of Skyrmions at low energy and large separation using the method proposed by Manton of truncation to a finite number of degrees freedom. We calculate the induced metric on the manifold of the union of gradient flow curves, which for large separation, to first non-trivial order is parametrized by the variables of the product ansatz. (presented at the Lake Louise Winter Institute, 1994)Comment: 6 page

On the Strong Coupling Limit of the Faddeev-Hopf Model

The variational calculus for the Faddeev-Hopf model on a general Riemannian domain, with general Kaehler target space, is studied in the strong coupling limit. In this limit, the model has key similarities with pure Yang-Mills theory, namely conformal invariance in dimension 4 and an infinite dimensional symmetry group. The first and second variation formulae are calculated and several examples of stable solutions are obtained. In particular, it is proved that all immersive solutions are stable. Topological lower energy bounds are found in dimensions 2 and 4. An explicit description of the spectral behaviour of the Hopf map S^3 -> S^2 is given, and a conjecture of Ward concerning the stability of this map in the full Faddeev-Hopf model is proved.Comment: 21 pages, 0 figure

Time perception and its neuropsychological correlates in patients with schizophrenia and in healthy volunteers

Disordered time perception has been reported in schizophrenia. We investigated time perception dysfunction and its neuropsychological correlates in patients with schizophrenia. Participants comprised 38 patients and 38 age- and sex-matched healthy volunteers who were compared in an auditory temporal bisection paradigm using two interval ranges (a 400/800 ins condition and a 1000/2000 ms condition). In the temporal bisection, subjects were required to categorise a probe duration as short or long, based upon the similarity with two reference durations. All subjects also completed a battery of neuropsychological tests measuring sustained attention, short- and long-term memory and executive function. In the 400/800 ins condition, patients judged durations significantly shorter than did control subjects. Patients also exhibited decreased temporal sensitivity in both conditions. We found in both groups a negative association between temporal sensitivity and sustained attention for the 400/800 ms condition, and between temporal sensitivity and long-term memory for the 1000/200 ms condition. In patients, short-term memory performance was negatively associated with duration judgement in both conditions, while executive dysfunction was correlated to a general performance deficit in the 400/800 ms condition. These findings suggest the possibility that time perception abnormalities in schizophrenia are part of neuropsychological dysfunction and are likely to adversely impact upon activity of daily living. (c) 2008 Elsevier Ireland Ltd. All rights reserved

Mechanisms Gating the Flow of Information in the Cortex: What They Might Look Like and What Their Uses may be

The notion of gating as a mechanism capable of controlling the flow of information from one set of neurons to another, has been studied in many regions of the central nervous system. In the nucleus accumbens, where evidence is especially clear, gating seems to rely on the action of bistable neurons, i.e., of neurons that oscillate between a quiescent “down” state and a firing “up” state, and that act as AND-gates relative to their entries. Independently from these observations, a growing body of evidence now indicates that bistable neurons are also quite abundant in the cortex, although their exact functions in the dynamics of the brain remain to be determined. Here, we propose that at least some of these bistable cortical neurons are part of circuits devoted to gating information flow within the cortex. We also suggest that currently available structural, electrophysiological, and imaging data support the existence of at least three different types of gating architectures. The first architecture involves gating directly by the cortex itself. The second architecture features circuits spanning the cortex and the thalamus. The third architecture extends itself through the cortex, the basal ganglia, and the thalamus. These propositions highlight the variety of mechanisms that could regulate the passage of action potentials between cortical neurons sets. They also suggest that gating mechanisms require larger-scale neural circuitry to control the state of the gates themselves, in order to fit in the overall wiring of the brain and complement its dynamics

Baby Skyrmion Strings

We provide analytical and numerical evidence of the existence of classically stable, string-like configurations in a 2+1 dimensional analog of the Skyrme model. The model contains a conserved topological charge usually called the baryon number. Our strings are non-topological solitons which have a constant baryon number per unit length. The energy per length containing one baryon is, however, less than the energy of an isolated baryon (radially symmetric ``baby Skyrmion") in a region of the parameter space, which suggests a degree of stability for our configurations. In a limiting case, our configuration saturates a Bogomolnyi-type bound and is degenerate in energy per baryon with the baby Skyrmion. In another limiting case, the energies are still degenerate but do not saturate the corresponding Bogomolnyi-type bound. Nonetheless, we expect the string to be stable here. Both limiting cases are solvable analytically.Comment: Latex, (revtex), with one figure in a separate postscript file, 12 page