We tacitly assume that no obstructions to gauging are present, such as quantum anomalies or central charges in the Lie algebra of the symmetry group.

Whereas in Sec. II the symbol $\mathrm{\Delta }$ represented the form change of fields under coordinate diffeomorphisms, here we abuse the notation somewhat by using it for the combined action of diffeomorphisms and internal symmetries. We believe that no confusion can arise though.

Apart from the terms present in the Lagrangian (19), a nonlocal operator proportional to $g^{\mu \nu }{\partial }_{\mu }\rho {\partial }_{\nu }\rho /\rho$ with $\rho \equiv {\psi }^{†}\psi$ appears upon integrating $v^a$ out. As long as the particle number U(1) symmetry is spontaneously broken by a nonzero vacuum expectation value of $\psi$, this operator will be irrelevant at energies well below the symmetry breaking scale. Being separately invariant under all the symmetries of the problem, we can simply discard it without affecting the symmetry properties of the action.

The $g$ in the volume measure here of course refers to the spatial metric $g_{ij}$, defined by Eq. (13). The opposite sign in front of the $\frac{g^{\mu \nu }}{2m}{D}_{\mu }\pi D_{\nu }\pi$ term as compared to Ref. [4] is just a matter of a sign convention for the field $\pi$.

Since the relation $\mathfrak{g}\sim {\mathfrak{g}}^{\prime }$ indicating the existence of $\mathfrak{h}\in \mathcal{H}$ such that ${\mathfrak{g}}^{\prime }=\mathfrak{g}\mathfrak{h}$ is an equivalence, any two cosets ${\chi }_{\mathfrak{g}}$ and ${\chi }_{{\mathfrak{g}}^{\prime }}$ are either disjoint or identical. The coset space $\mathcal{G}/\mathcal{H}$ provides a decomposition of the group $\mathcal{G}$ into equivalence classes of $\sim$.

This can always be achieved for compact Lie algebras by a suitable choice of basis of generators.