Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 308 (Classical and Quantum Solitons)

WriteThe four vacua are $-q,-p,p,q$. Hence there are three adjacent scalar-field kink sectors, $(-q,-p)$, $(-p,p)$ and $(p,q)$, together with their three reversed antikinks. Withthe Bogomolny bound gives $E=|W({\varphi }_{+})-W({\varphi }_{-})|$. The central kink obeys ${\varphi }^{\prime }=W^{\prime }(\varphi )$ and hasThe two outer kinks obey ${\varphi }^{\prime }=-W^{\prime }(\varphi )$ and have equal energyFor the kink passing through zero, choose $\varphi (0)=0$. Its profile is determined implicitly byIf $q=p$, the central equation becomes ${\varphi }^{\prime }=(p^2-{\varphi }^2{)}^2$, soFor $q=p+\epsilon$,so the leading ratio is

Away from a zero of $\varphi$, the first Bogomolny vortex equation givesSubstitution into the second equation, with ${\nabla }^2=4{\partial }_z{\partial }_{\bar{z}}$, givesNow set $\tilde{\Omega }=\Omega |\varphi {|}^2$. The Gaussian curvature formula givesTherefore $(\tilde{K}+1/2)\tilde{\Omega }=(K+1/2)\Omega$ implieswhich is exactly the vortex equation.

The metricis the pullback of the $n=1$ Poincare disc model metric under $w=z^n$. It consequently has $K=-1/2$ away from the origin. Comparing it with $d{s}_1^2$ gives the Witten hyperbolic vortexPutting $n=N+1$ gives winding number $N$. A gauge choice with positive radial factor isand then

For coprime polynomials $p,q$, the algebraic degree of $R=p/q:S^2\to S^2$ isand its Wronskian is $W=p^{\prime }q-p{q}^{\prime }$. Thuswhile forone obtains

For $R=z^2$, a domain rotation $z\mapsto e^{i\alpha }z$ is accompanied by a target rotation $R\mapsto e^{2i\alpha }R$. There is also a half-turn about every axis in the equatorial plane, with the corresponding target half-turn. These generate the axial dihedral rotational symmetry: the distinguished spatial axis is the $z$-axis, and the continuous rotations about it are accompanied by twice the angle in target space.

The Rational map approximation for Skyrmions usesThe rational-map degree is the Skyrmion baryon number, while zeros of the Wronskian identify directions where the angular baryon density vanishes and therefore influence the shape and energy. For $R=z^2$ the result is the $B=2$ toroidal Skyrmion.

Quantization treats spatial and isospin rotations as collective coordinates and imposes the Finkelstein-Rubinstein constraints. For even baryon number, spin and isospin are integral. The axial constraint is $2J_3+I_3=0$, and the equatorial half-turn imposes $(-1{)}^{J+I}=-1$. Restricting to $J,I\le 1$ leavesThe $(0,0)$ and $(1,1)$ representations violate the discrete constraint.