= Free-boson Virasoro central term
{title2=$[L_m,L_n]_{\rm central}=\tfrac d{12}m(m^2-1)\delta_{m+n,0}$}
For $d$ free target coordinates, <normal ordering> of $L_m=\sum_n:\alpha_{m-n}\cdot\alpha_n:/2$ gives a matter <central charge> $d$. On the formal zero-momentum vacuum, the double contractions yield $\langle L_mL_{-m}\rangle=(d/2)\sum_{r=1}^{m-1}r(m-r)=d(m^3-m)/12$. Two target-metric contractions give $\eta_{ab}\eta^{ab}=d$, so timelike target coordinates do not subtract from this <central charge>. Transverse-only and covariant generators must not be confused.
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