Dynamics of $\mathrm{CP}^{N-1}$ skyrmions

We derive several exact results for the dynamics of CP$^{N-1}$ skyrmions with arbitrary $N$. Fractonic continuity equation is shown to hold for arbitrary CP$^{N-1}$ fluid implying the conservation of the topological charge and the dipole moment. Inclusion of the Gilbert damping modifies the continuity equation, resulting in the violation of the dipole moment conservation but not of the topological charge. Thiele's equation for the CP$^{N-1}$ skyrmion follows from the modified continuity equation. The Girvin-MacDonald-Platzman (GMP) algebra in the long-wavelength limit is derived for arbitrary CP$^{N-1}$ fluid. In the case of CP$^2$ skyrmions, we identify two kinds of energetically stable skyrmions in which the quadrupolar moments or the ferromagnetic moments are dominant. In the latter case, one can associate a nonzero CP$^1$ charge equal to half the CP$^2$ skyrmion charge and argue that the topological Hall effect of electrons should exist due to their coupling to the ferromagnetic part of the CP$^2$ texture.