Rank-one Feigin–Fuchs oscillator spaces and long-screening coefficients

Fix $p\in \mathbb{C}\setminus \{0\}$. Put $P=\mathbb{C}[x_1,x_2,\dots ]$, where each polynomial involves only finitely many variables, and assign weighted degree $\mathrm{deg}{x}_m=m$. For $t\in \mathbb{C}$, let ${\mathcal{F}}_t=P{e}^{t\alpha }$ be a labelled copy of $P$; the symbol $e^{t\alpha }$ is a label for its constant polynomial, not an analytic exponential. For $m\ge 1$, define ${\alpha }_{-m}$ to be multiplication by $x_m$, ${\alpha }_m=2pm{\mathrm{\partial }}_{x_m}$, and ${\alpha }_0=2ptI$ on ${\mathcal{F}}_t$. Write $P[N]$ for the subspace of weighted degree $N$, with $P[N]=0$ for $N<0$, and set $D_t={\sum }_{m\ge 1}m{x}_m{\mathrm{\partial }}_{x_m}+p{t}^2-(p-1)tI\mathrm{.}$

For an integer $r$ and a charge $t$ satisfying $2prt\in \mathbb{Z}$, define the coefficientwise formal map ${\mathcal{E}}_{r,t}(z):{\mathcal{F}}_t\to {\mathcal{F}}_{t+r}$ by ${\mathcal{E}}_{r,t}(z)(f{e}^{t\alpha })=z^{2prt}\exp \text{\hspace{0.17em}⁣}\left({\sum }_{m\ge 1}\frac{r{x}_m}{m}{z}^m\right)f(x_1-2pr{z}^{-1},x_2-2pr{z}^{-2},\dots )e^{(t+r)\alpha }\mathrm{.}$ Here $\exp$ means its formal power-series expansion, and ${\mathrm{Res}}_z$ means the coefficient of $z^{-1}$. The annihilation substitution is finite on each polynomial. For every fixed power of $z$, only finitely many creation monomials contribute, so every coefficient map is an algebraic linear map; no analytic convergence is asserted.

When $2pt\in \mathbb{Z}$, define the long-screening coefficient $Q_t:{\mathcal{F}}_t\to {\mathcal{F}}_{t+1}$ by $Q_t={\mathrm{Res}}_z{\mathcal{E}}_{1,t}(z)$. Write $Q^j$ for a composition along successive charges whenever all of its coefficient maps are defined, and put $Q^0=I$. In particular, if $p$ is an integer and $2pt\in \mathbb{Z}$, every $Q^j$ for $j\ge 0$ is defined.

For each $n\in \mathbb{Z}$, define $L_t(n)=\frac{1}{4p}{\sum }_{m\in \mathbb{Z}}:{\alpha }_m{\alpha }_{n-m}:-\frac{p-1}{2p}(n+1){\alpha }_n\mathrm{.}$ Normal ordering places every negative-index oscillator to the left of every nonnegative-index oscillator. For fixed $n$ and a fixed polynomial input, the sum is finite: terms with two negative indices have only finitely many possible indices, and a positive-index derivative vanishes once its index exceeds the polynomial's finite support and degree.

Adamović–Milas, Section 1

Ridout–Wood, Section 2

The explicit operators satisfy $[{\alpha }_m,{\alpha }_n]=2pm{\delta }_{m+n,0}I$: for $m>0$, $[{\alpha }_m,{\alpha }_{-m}]=2pm[{\mathrm{\partial }}_{x_m},x_m]=2pmI$, while same-sign modes and distinct opposite modes commute. Directly normal ordering at $n=0$ gives $L_t(0)=D_t$. The Virasoro relations and the commutation of $Q_t$ with the Virasoro modes are not included in this definition; they are proved separately.