Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 308 (Classical and Quantum Solitons)

Differentiating the potential givesThus $\varphi =\pm 1$ are stationary points, andbecause $|ϵ|<1$. Both are local minima. Their energies are

For $ϵ=0$, the static energy can be completed to a square:The increasing scalar-field kink therefore obeys the first-order Bogomolny equationWith center $X$, its solution and energy areThe antikink uses the opposite sign.

For small positive $ϵ$, the true vacuum $\varphi =-1$ lies below the false vacuum $\varphi =1$ byThis pressure exerts force $\Delta U$ on a kink with $-1$ on its left and $+1$ on its right. Dividing by its leading mass $4/3$ gives acceleration toward the false-vacuum side:

An antikink followed by a kink encloses a region of the lower vacuum while approaching $\varphi =1$ at both infinities. Vacuum pressure pushes the pair apart, whereas their attraction pulls them together. At a static separation $s$,soThis estimate is self-consistent for $ϵ\ll 1$, when the two soliton cores are well separated.

At critical coupling, an $N$-vortex solution of the Abelian Higgs model satisfies the Bogomolny vortex equations. The Abelian Higgs vortex moduli space ${\mathcal{M}}_N$ consists of these solutions modulo gauge transformations. The unordered zeros of the Higgs field specify the $N$ vortex positions, soas a complex manifold. A tangent vector is represented by a linearized solution orthogonal to infinitesimal gauge transformations. Substitution into the field-theory kinetic energy giveswhich defines the natural $L^2$ Riemannian metric on the moduli space. Its geodesics give the slow-motion approximation.

For $N=1$, translation invariance makes ${\mathcal{M}}_1$ a flat plane. For $N=2$, the center-of-mass plane factors from a rotationally symmetric relative moduli space. Far from coincidence, exchanging the vortices identifies the relative separation $z$ with $-z$, so the relative space is asymptotic to a cone of angle $\pi$. The exact metric smoothly rounds its apparent tip at coincident vortices; the regular local coordinate is proportional to $z^2$.

A head-on relative geodesic passes smoothly through this coincidence point. In terms of the asymptotic separation coordinate, it emerges on the perpendicular axis, producing Right-angle scattering of Abelian Higgs vortices. For signed impact parameter $b$, the deflection is odd in $b$, approaches zero as $|b|\to \infty$, and approaches the two signed values $\pm \pi /2$ as $b\to 0^{\pm }$. Within the geodesic approximation the trajectory, and hence this graph, is independent of incoming speed; speed only changes its parametrization in time.

For identical point particles in a bounded repulsive central potential, the relative coordinate moves in an ordinary plane rather than the rounded exchange cone. At energy below the finite central barrier, a head-on encounter is reflected and gives backscattering through $\pi$, so the available deflections range from zero to nearly $\pi$. At energy above the barrier, a head-on trajectory passes through the center and has zero deflection; the magnitude reaches a maximum at nonzero $|b|$ and then returns to zero. Increasing speed reduces this maximum. Thus point particles can exhibit backscattering at low speed and near-transparent head-on passage at high speed, whereas slow identical vortices have the speed-independent right-angle limit forced by the geometry of ${\mathcal{M}}_2$.

In the Rational map approximation for Skyrmions, stereographic coordinate $z$ describes the direction $\hat{\mathbf{x}}\in S^2$, and a degree-$B$ rational map $R(z)=p(z)/q(z)$ defines a unit vector ${\mathbf{n}}_R$. The Skyrme model field is approximated byIts baryon number is the degree $B$ of $R$. Angular integration reduces the energy to a radial variational problem,up to the conventional overall normalization, where the angular functional $\mathcal{I}$ depends only on $R$. One first minimizes $\mathcal{I}$ among degree-$B$ maps and then minimizes over the profile $f$. This efficiently captures the topology, energy, and polyhedral symmetries of many Skyrmions.

Let $\omega =e^{2\pi i/5}$. Since ${\omega }^5=1$,which is a fivefold spatial rotation accompanied by a target-space rotation. The real coefficients also give $R(\bar{z})=\overline{R(z)}$, while direct substitution givesTogether these transformations extend the cyclic symmetry to the stated $D_{5d}$ symmetry.

For $p=z^7-7z^2$ and $q=7z^5+1$, the Wronskian isBesides $z=0$, put $y=z^5$. ThenThus five zeros lie on the circleat arguments $2\pi k/5$, and five lie on the reciprocal circle at arguments $(2k+1)\pi /5$. The polynomial has degree eleven, so the twelfth zero lies at $z=\infty$. On the Riemann sphere, the zeros therefore form two opposite poles and two staggered pentagonal rings: the twelve vertices of an icosahedron.

The angular baryon-density factor is proportional to $|dR/dz{|}^2$ and vanishes at these critical directions. The Wronskian zeros therefore point toward twelve holes in the baryon-density surface. They are the face centers of the dodecahedral $B=7$ Skyrmion, equivalently the vertices of its dual icosahedron, and make its icosahedral symmetry visible directly in the rational map.