Charged rank-one Fock spaces carry a central-charge-one Virasoro action

Multiplication and differentiation give $[a_k,a_{-\mathrm{\ell }}]=k{\delta }_{k\mathrm{\ell }}I$ for $k,\mathrm{\ell }\ge 1$, while all two multiplication operators commute, all two derivative operators commute, and $a_0=pI$ commutes with every mode. Therefore $[a_m,a_n]=m{\delta }_{m+n,0}I$ for all integers $m,n$.

For fixed $n$ and a fixed polynomial, terms in ${\sum }_j:a_j{a}_{n-j}:$ with a sufficiently large positive index kill that polynomial; terms with two negative indices have $j+(n-j)=n$ and hence occur only finitely many times. Thus $L_n^{(p)}$ is well defined on $P$.

In the commutator $[L_m^{(p)},a_n]$, the identity $[AB,C]=A[B,C]+[A,C]B$ leaves only the summands whose one index is $-n$. Their equal contributions and the prefactor $1\mathrm{/}2$ give $[L_m^{(p)},a_n]=-n{a}_{m+n}$.

Set $C_{m,n}=[L_m^{(p)},L_n^{(p)}]-(m-n)L_{m+n}^{(p)}$. The Jacobi identity and step 3.1 show $[C_{m,n},a_k]=0$ for every $k$. An operator on $P$ commuting with every multiplication $x_j$ and every derivative $j{\mathrm{\partial }}_{x_j}$ is scalar: its value on $1$ is killed by each derivative, so is constant, and commutation with multiplication determines its value on every monomial. Hence each $C_{m,n}$ is scalar. Since $L_j^{(p)}$ has oscillator degree $-j$, this scalar is zero unless $m+n=0$.

For $m>0$, normal ordering gives $L_{-m}^{(p)}1=p{x}_m+\frac{1}{2}{\sum }_{k=1}^{m-1}{x}_k{x}_{m-k}$, $L_m^{(p)}1=0$, and $L_0^{(p)}1=(p^2\mathrm{/}2)1$. Using step 3.1, $[L_m^{(p)},L_{-m}^{(p)}]1=(m{p}^2+\frac{1}{2}{\sum }_{k=1}^{m-1}k(m-k))1$. Subtracting $2m{L}_0^{(p)}1=m{p}^{2}1$ leaves $\frac{1}{2}{\sum }_{k=1}^{m-1}k(m-k)1=(m^3-m)1\mathrm{/}12$. Antisymmetry gives the same polynomial for negative $m$, and $m=0$ gives zero. This proves the Virasoro bracket with central charge $1$ in [F2]'s convention.

Direct evaluation of the normally ordered sums for $n=0,-1$ gives the two displayed formulas. For monomials, the inner product in [F1] gives $⟨a_{k}f,g⟩=⟨f,a_{-k}g⟩$ and $⟨a_{-k}f,g⟩=⟨f,a_{k}g⟩$ for every $k\ge 1$. Since each normally ordered sum acts finitely on a polynomial, termwise adjunction gives $⟨L_n^{(p)}f,g⟩=⟨f,L_{-n}^{(p)}g⟩$ for $f,g\in P$. These are algebraic-core identities for the quadratic modes. The weight formula follows by applying the Euler operator in $L_0^{(p)}$ to each monomial.

Let $e_u$ be the normalized monomial basis in the Statement. On this orthonormal basis, the polynomial restrictions act by the displayed weighted shifts. If a sequence and its weighted-shift images converge in $F_p$, coordinatewise convergence gives the same shift formulas for the limits, so the maximal weighted shifts are closed. Truncation to finitely many basis vectors converges in each graph norm, hence these are the closures of the polynomial restrictions. For $v={\sum }_u{c}_u{e}_u$, the adjoint coefficient identity against each $e_u$ forces $A_k^{\ast }v$ to have the coefficients of $A_{-k}v$. Because finite-support vectors are graph-dense, this identity extends to the full domain exactly when the forced coefficient sequence is square-summable, which is equivalent to ${\sum }_{u}k(u_k+1)\mathrm{\mid }{c}_u{\mathrm{\mid }}^2<\mathrm{\infty }$. Reversing the roles gives $A_{-k}^{\ast }=A_k$. The zero mode is the bounded self-adjoint scalar $pI$ because $p\in \mathbb{R}$.