= Virasoro zero-mode shift
{c}
{title2=$\mathcal L_m=L_m-a\delta_{m,0}$}
Shifting the zero mode changes the algebra's central representative:
$$
[\mathcal L_m,\mathcal L_n]=(m-n)\mathcal L_{m+n}
+\left(\frac c{12}m(m^2-1)+2am\right)\delta_{m+n,0}.
$$
The linear term follows from replacing $L_{m+n}$ by $\mathcal L_{m+n}+a\delta_{m+n,0}$. It changes the zero-mode convention, not the cubic coefficient or <central charge>. This distinction separates the intercept from the <Virasoro central extension>.
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