= Fermion-boson correspondence on the circle
{title2=$\mathcal F_P\cong\bigoplus_{q\in\mathbb Z}\mathcal F_{\mathrm{boson}}$}
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 $b_n^*=J_{-n}/\sqrt n$ with $[b_n,b_m^*]=\delta_{nm}$. 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 $Q(Q-1)/2+\sum_{n>0}n b_n^*b_n$.
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