Wess–Zumino Phase of 2\pi Isorotations in the Skyrme Model: A Toy-Model Study

This work presents a numerical study of the Wess–Zumino phase associated with a 2\pi isorotation of Skyrmion configurations. The motivation is to examine how the fermionic sign (a phase of -1 under a 360^\circ rotation) can emerge from the topology of an extended object, without inserting a spinor by hand. We consider the SU(2) Skyrme model and embed the field into SU(3) to make the Wess–Zumino five-form non-trivial. The phase is evaluated on an explicit two-dimensional disc filling of the isorotation loop. For baryon numbers B=0, 1, 2, and 3, we find \Gamma \simeq \pi |B| with a numerical uncertainty of order 10^{-4}\pi, yielding the physical phase (-1)^{N_c B} for integer N_c, consistent with known results. The B=1 configuration uses an exact hedgehog profile, while B=2 and B=3 employ smooth rational-map trial configurations. We also document convergence studies, consistency checks, and the limitations of the calculation. The code and numerical implementation are not included in this deposit; the focus is on the manuscript and the reported results.