Fermion-boson correspondence on the circle (${\mathcal{F}}_P\cong {⨁}_{q\in \mathbb{Z}}{\mathcal{F}}_{\mathrm{boson}}$)

In charge $q$, let ${\Omega }_q$ fill all integer modes below $q$. Its rotation energy is $q(q-1)/2$. The negative Heisenberg current algebra of a fermion on the circle modes create mutually commuting oscillators $b_n^{\ast }=J_{-n}/\sqrt{n}$ with $[b_n,b_m^{\ast }]={\delta }_{nm}$. Their normalized occupation monomials applied to ${\Omega }_q$ are orthonormal. The Jacobi triple product identifies the fermionic charge-energy generating function with ${\sum }_q{u}^q{t}^{q(q-1)/2}{\prod }_{n>0}(1-t^n{)}^{-1}$, proving completeness in each finite-dimensional energy space. Thus the rotation Hamiltonian is $Q(Q-1)/2+{\sum }_{n>0}n{b}_n^{\ast }{b}_n$.