Quantization of a periodic soliton coordinate (${\alpha }_0\sim {\alpha }_0+2\pi ⟹p_{{\alpha }_0}\in \hslash \mathbb{Z}$)

The conjugate momentum to a periodic internal soliton angle generates shifts of that angle. Single-valued wavefunctions, or Bohr-Sommerfeld quantization with zero Maslov index, quantize this canonical momentum. A theta term can shift the relation between canonical momentum and mechanical charge without changing integer canonical labels.