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For a massive positive-energy orbit, choose $p_{\ast }=(m,0,0)$ with $m>0$. Its little group is $SO(2)$, whose unitary irreducible representations are the characters $e^{is\theta }$. Induction gives one-particle states $|p,s⟩$ on the mass shell $p^2=m^2$, $p^0>0$, labeled by mass and spin. For the Poincare group itself, single-valuedness gives $s\in \mathbb{Z}$; its double cover permits half-integers, and its universal cover permits any real $s$, producing anyonic spin in $(2+1)$ dimensions.

Substituting $a=W^T{a}^{\prime }$ into the Chern--Simons term givesThis matrix is integral and symmetric. It is invertible becauseFor $p^{\prime }=Wp$ and $q^{\prime }=Wq$,Thus every anyon braiding phase is unchanged, and bijectivity of $W$ on the charge lattice shows that the full sets coincide. The torus ground-state degeneracy of an Abelian Chern--Simons theory is also invariant: