Virasoro central extension

The Virasoro central extension is the term $c(m^3-m){\delta }_{m+n,0}/12$ in $[L_m,L_n]$. It records the conformal anomaly and commutes with every Virasoro generator.

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 $⟨L_m{L}_{-m}⟩=(d/2){\sum }_{r=1}^{m-1}r(m-r)=d(m^3-m)/12$. 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.