Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 308 / 3 / Solution

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.