Rank-one lattice and square-zero screening data for the triplet construction

Fix an integer $p\ge 2$. Let $L=\mathbb{Z}\alpha$ with bilinear form $⟨r\alpha ,s\alpha ⟩=2prs$, and put $P=\mathbb{C}[x_1,x_2,\dots ]$, where every polynomial uses finitely many variables. Write ${\mathcal{F}}_t=P{e}^{t\alpha }$ for a copy of $P$ labelled by $t\in \mathbb{Q}$. On it define ${\alpha }_{-n}=x_n$, ${\alpha }_n=2pn{\mathrm{\partial }}_{x_n}$ for $n\ge 1$, and ${\alpha }_0=2ptI$. Write $\alpha (z)={\sum }_{n\in \mathbb{Z}}{\alpha }_n{z}^{-n-1}$. The vector $e^{t\alpha }$ means $1\in {\mathcal{F}}_t$; it is not an analytic exponential.

Use two distinguished sectors indexed by $ϵ=0,1$: $\mathcal{A}={⨁}_{ϵ=0}^1{⨁}_{n\in \mathbb{Z}}{\mathcal{F}}_{n-ϵ\mathrm{/}p},\mathcal{M}={⨁}_{ϵ=0}^1{⨁}_{n\in \mathbb{Z}}{\mathcal{F}}_{n+1\mathrm{/}2-ϵ\mathrm{/}p}\mathrm{.}$ The sector-one subspace of $\mathcal{A}$ is denoted $I$ and its sector-zero subspace $V_L$. A state charge $r=m-\delta \mathrm{/}p$ carries sector $\delta \in \{0,1\}$. The charge shift $e_r$ sends $f{e}^{t\alpha }$ in sector $ϵ$ to $f{e}^{(t+r)\alpha }$ in sector $ϵ+\delta$ when $ϵ+\delta \le 1$, and to zero otherwise. This rule applies on both $\mathcal{A}$ and $\mathcal{M}$. In particular two sector-one actions have zero composite. The scalar cocycle for allowed shifts is $\epsilon (r,s)=1$; there is no extra phase. For two state charges with sector sum at most one, $⟨r\alpha ,s\alpha ⟩=2prs$ is an even integer, so this choice has commutator $(-1{)}^{2prs}=1$. The multiplication involving two sector-one states is defined to be zero, rather than a fractional-power generalized lattice product.

Define exponential fields on either vector space by ${\mathcal{E}}_r(z)=\exp \text{\hspace{0.17em}⁣}\left({\sum }_{n\ge 1}\frac{r{\alpha }_{-n}}{n}{z}^n\right)\exp \text{\hspace{0.17em}⁣}(-{\sum }_{n\ge 1}\frac{r{\alpha }_n}{n}{z}^{-n})e_r{z}^{r{\alpha }_0},$ where $z^{r{\alpha }_0}f{e}^{t\alpha }=z^{2prt}f{e}^{t\alpha }$. For every allowed action the exponent $2prt$ is an integer. On a disallowed action the field is zero. Each coefficient is defined algebraically: the annihilation exponential is finite on a polynomial, and the creation exponential has only finitely many contributions to a fixed power of $z$.

Put $b_n(z)={\mathrm{\partial }}_z^{n-1}\alpha (z)\mathrm{/}(n-1)!$. For a monomial state $x_{n_1}\cdots x_{n_k}{e}^{r\alpha }$ define its field by $Y(x_{n_1}\cdots x_{n_k}{e}^{r\alpha },z)=:b_{n_1}(z)\cdots b_{n_k}(z){\mathcal{E}}_r(z):$. Here normal ordering means the following explicit finite distribution over the $k$ oscillator factors: split each $b_n$ into its part with oscillator index less than zero and its part with index at least zero; sum over all choices, placing every chosen negative part to the left of ${\mathcal{E}}_r$ and every chosen nonnegative part to its right. Factors within either group commute. Extend linearly, and distinguish $Y_{\mathcal{A}}$ and $Y_{\mathcal{M}}$ by the space acted upon. This definition gives candidate state fields, not a waiver of their vertex/module Jacobi obligations.

Set $\mathbf{1}=e^0$, $\omega =(x_1^2\mathrm{/}(4p)+(p-1)x_2\mathrm{/}(2p))e^0$, $v=e^{-\alpha \mathrm{/}p}$, $Q=(e^{\alpha }{)}_{(0)}$ and $F=e^{-\alpha }$, $H=QF$, $E=Q^{2}F$, using coefficients $Y(a,z)={\sum }_{k\in \mathbb{Z}}{a}_{(k)}{z}^{-k-1}$. Let $W$ be the smallest linear subspace of $V_L$ containing $\mathbf{1},\omega ,F,H,E$ and closed under every product $a_{(k)}b$ of its elements, for $k\in \mathbb{Z}$; it exists as the intersection of all such subspaces. Once the vertex and conformal identities are established, this is the vertex subalgebra generated by the four displayed states, denoted $W(p)$ here. Equality with the full kernel ${\ker }_{V_L}{v}_{(0)}$ is a further structural assertion, not part of the definition and not required for the four-generator construction.

For $a\in W$ use the explicit screening-deformation candidate $\tilde{Y}(a,z)=Y_{\mathcal{M}}(a,z)+{\sum }_{n\ge 1}\frac{(-1{)}^{n+1}}{n}{z}^{-n}{Y}_{\mathcal{M}}(v_{(n)}a,z)\mathrm{.}$ Define $P_{+}^{p-1}$ as the span of all finite words of its modes applied to $e^{\alpha \mathrm{/}2}$, and $M_1$ as the corresponding cyclic span generated by $e^{-\alpha \mathrm{/}2}$. These spans are defined within $\mathcal{M}$; their module, containment and grading properties are separate assertions. The endpoint convention is $\mathrm{\Lambda }(p-1)=W{e}^{(p-2)\alpha \mathrm{/}(2p)}\subset {⨁}_n{\mathcal{F}}_{n+(p-2)\mathrm{/}(2p)}$ and $\mathrm{\Pi }(1)=W{e}^{-\alpha \mathrm{/}2}\subset {⨁}_n{\mathcal{F}}_{n+1\mathrm{/}2}$ for the undeformed lattice fields. These definitions specify particular cyclic spaces; their irreducibility and their identification with the extension endpoints also require proof.

Formal distributions and fields on a vector space

Adamović–Milas, lattice construction, §§3–5

Adamović–Milas, triplet algebra, §§1 and 3