Charged rank-one Heisenberg Fock space

Let $P=\mathbb{C}[x_1,x_2,\dots ]$, where each polynomial involves only finitely many variables. For a real charge $p$, the rank-one charged Heisenberg Fock space has algebraic core $F_p^{\mathrm{a}\mathrm{l}\mathrm{g}}=P$. Its modes act by

$a_{-k}f=x_{k}f,a_{k}f=k\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_k},a_{0}f=pf(k\ge 1,f\in P)\mathrm{.}$

Give the monomial basis the Hermitian inner product

$⟨{\prod }_{k\ge 1}{x}_k^{u_k},{\prod }_{k\ge 1}{x}_k^{v_k}⟩={\mathbf{1}}_{u=v}{\prod }_{k\ge 1}{k}^{u_k}{u}_k!\mathrm{.}$

and let $F_p$ be its Hilbert completion. The current is $a(z)={\sum }_{j\in \mathbb{Z}}{a}_j{z}^{-j-1}$. Normal ordering moves every negative-index mode to the left of every nonnegative-index mode. The conformal modes are

$L_n^{(p)}=\frac{1}{2}{\sum }_{j\in \mathbb{Z}}:a_j{a}_{n-j}:\mathrm{.}$

These sums are interpreted on $P$; their finite action is established by the cited well-definedness result. The vacuum vector $1\in P$ has charge $p$, oscillator number $0$, and declared conformal weight $p^2\mathrm{/}2$.

Rank-one Heisenberg oscillator fields and vacuum space

Ginsparg, Section 8.1

Mnev–Reshetikhin, Section 6.1.6

For $p=0$, the algebraic core and current modes agree with the vacuum oscillator space in Rank-one Heisenberg oscillator fields and vacuum space. For $p\mathrm{\ne }0$, the zero mode acts by the scalar $p$ and the lowest conformal weight is $p^2\mathrm{/}2$. This is one chiral sector; it is not a full left/right theory.