= Heisenberg current algebra of a fermion on the circle
{c}
{title2=$[J_n,J_m]=n\delta_{n+m,0}I$}
For integer modes filled at indices below zero, set $E_{rs}=c_r^\dagger c_s-\delta_{rs}\mathbf1_{r<0}$ and $J_n=\sum_r E_{r,r+n}$ on the finite-energy domain. The <canonical anticommutation relations> give a central term $\delta_{st}\delta_{ur}(\mathbf1_{r<0}-\mathbf1_{s<0})$ in $[E_{rs},E_{tu}]$. Summing across the filled-empty boundary gives $[J_n,J_m]=n\delta_{n+m,0}I$, with $J_n^*=J_{-n}$. The current's level is the coefficient of this central term.
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