Bosonic Virasoro Central Extension

For fixed $m,n$, the oscillator commutator and $[AB,C]=A[B,C]+[A,C]B$ give $[L_m,{\alpha }_n^{\rho }]=\frac{1}{2}{\sum }_k({\alpha }_{m-k}^{\mu }[{\alpha }_{k\mu },{\alpha }_n^{\rho }]+[{\alpha }_{m-k}^{\mu },{\alpha }_n^{\rho }]{\alpha }_{k\mu })=-n{\alpha }_{m+n}^{\rho }$. More explicitly, the first nonzero Kronecker term has $k=-n$ and coefficient $k=-n$; the second has $k=m+n$ and coefficient $m-k=-n$. Normal ordering changes only scalar commutators and does not affect this commutator with a single oscillator.

Set $C_{m,n}=[L_m,L_n]-(m-n)L_{m+n}$. The Jacobi identity and step 1.1 show $[C_{m,n},{\alpha }_r^{\rho }]=0$ for every $r,\rho$. An operator commuting with every creation and annihilation mode is scalar on the fixed-momentum irreducible Fock module: it sends the vacuum to a scalar vacuum because all positive modes annihilate it, and commuting with creation modes gives the same scalar on every finite-particle descendant. Mode degree then forces that scalar to vanish unless $m+n=0$.

Let $m>0$. In $[L_m,L_{-m}]$ the noncentral terms combine to $2m{L}_0$; the remaining double contractions pair the finitely many creation/annihilation modes with indices $r$ and $m-r$, $1\le r\le m-1$. There are $D$ target components and the quadratic-generator factor contributes $1\mathrm{/}2$, so the scalar remainder is $\frac{D}{2}{\sum }_{r=1}^{m-1}r(m-r)$. Since ${\sum }_{r=1}^{m-1}r=m(m-1)\mathrm{/}2$ and ${\sum }_{r=1}^{m-1}{r}^2=m(m-1)(2m-1)\mathrm{/}6$, this scalar equals $D(m^3-m)\mathrm{/}12$. The cases $m=0$ and $m<0$ follow from antisymmetry of the commutator.

Step 2.1 restricts the scalar $C_{m,n}$ to $m+n=0$, and step 3.1 computes that scalar for all integer $m$. Hence $C_{m,n}=\frac{D}{12}(m^3-m){\delta }_{m+n,0}I$, which is the asserted Virasoro relation.