Rank-one Heisenberg oscillator fields and vacuum space

Let $P=\mathbb{C}[x_1,x_2,\dots ]$ be the polynomial algebra whose elements involve finitely many indeterminates, as in The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials. Give $x_r$ degree $r$ and write $P_d$ for its degree-$d$ subspace. On $P$, put

$a_{-r}=x_r\cdot ,a_r=r\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_r}(r\ge 1),a_0=0,\mathbf{1}=1,T={\sum }_{r\ge 1}r{x}_{r+1}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_r}\mathrm{.}$

Every sum of operators here is applied to a polynomial and is then finite. Define $a(z)={\sum }_{n\in \mathbb{Z}}{a}_n{z}^{-n-1}$, in the conventions of Formal distributions and fields on a vector space, and the divided derivative $b_r(z)={\mathrm{\partial }}_z^{r-1}a(z)\mathrm{/}(r-1)!$ for $r\ge 1$. In normal ordering, all modes $a_n$ with $n<0$ are moved to the left of all modes with $n\ge 0$, preserving their respective internal orders. For a polynomial monomial $v=x_{r_1}\cdots x_{r_s}$ define the candidate field

$Y(v,z)=:b_{r_1}(z)\cdots b_{r_s}(z):,Y(1,z)=I\mathrm{.}$

The empty normal product is the identity. The same-sign modes commute, so the formula is symmetric in the listed factors; it is extended linearly from the polynomial monomial basis. Set $\omega =\frac{1}{2}{x}_1^2$. The well-defined state-field and conformal-VOA structure on this vacuum space are proved in The rank-one Heisenberg vacuum space is a conformal VOA of central charge one ↗. The symbol $a_n$ is a current mode; in particular $Y(x_1,z)=a(z)$ and $Y(\omega ,z)={\sum }_{n}L(n)z^{-n-2}$ use different mode shifts.