Self-dual and logarithmic representations of the twisted Heisenberg--Virasoro algebra at level zero

TL;DR

This paper advances the understanding of the twisted Heisenberg-Virasoro algebra at level zero by providing explicit formulas for singular vectors, constructing self-dual and logarithmic modules, and exploring their extensions and relations.

Contribution

It introduces new free field realizations of self-dual modules and constructs a broad class of logarithmic modules with potential links to projective covers.

Findings

Explicit formulas for singular vectors in Verma modules

Construction of self-dual modules via free field realization

Existence of non-split self-extensions of irreducible modules

Abstract

This paper is a continuation of arXiv:1405.1707. We present certain new applications and generalizations of the free field realization of the twisted Heisenberg-Virasoro algebra ${\mathcal H}$ at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for ${\mathcal H}$ is presented by combining a bosonic construction of Whittaker modules from arXiv:1409.5354 with a construction of logarithmic modules for vertex algebras. As an application, we prove that there exists a non-split self-extension of irreducible self-dual module which is a logarithmic module of rank two. We construct a large family of logarithmic modules containing different types of highest weight modules as subquotients. We believe that these logarithmic modules are related with projective covers of irreducible modules in a suitable category of ${\mathcal H}$-modules.

Equations

$$h_I/c_{L,I}-1=p\in {\mathbb{Z}}_{>0}$$

$$S_p(c)=S_p(c(-1),\dots ,c(-p))=S_p\left(\frac{-1}{c_{L,I}}I(-1),\dots ,\frac{-1}{c_{L,I}}I(-p)\right)$$

$$h_I/c_{L,I}-1=-p(p\in {\mathbb{Z}}_{>0}).$$

$$\sum _{i=1}^p\left(L(-i)S_{p-i}(-c)\right)v_{h,h_I}+\left(h+\frac{c_L-2}{24}(p-1)\right)S_p(-c)v_{h,h_I}$$

$$-\frac{c_L-26}{24}\left(\sum _{i=1}^p(i-1)c(-i)S_{p-i}(-c)\right)v_{h,h_I}.$$

$$\widetilde{\Pi (p,r)}\cong \bigcup _{m\ge 0}{Z}^{(m)},$$

$$\{L(n),I(n):n\in \mathbb{Z}\}\cup \{C_L,C_{LI},C_I\}$$

$$\left[L(n),L(m)\right]=(n-m)L(n+m)+{\delta }_{n,-m}\frac{n^3-n}{12}{C}_L,$$

$$\left[L(n),I(m)\right]=-mI(n+m)-{\delta }_{n,-m}(n^2+n)C_{LI},$$

$$\left[I(n),I(m)\right]=n{\delta }_{n,-m}{C}_I,$$

$$\left[\mathcal{H},C_L\right]=\left[\mathcal{H},C_{LI}\right]=\left[\mathcal{H},C_I\right]=0.$$

$$⟨x,x⟩=⟨y,y⟩=0,⟨x,y⟩=1.$$

$$M(1,\gamma ):=U(\hat{\mathfrak{h}}){\otimes }_{U(\mathbb{C}[t]\otimes \mathfrak{h}\oplus \mathbb{C}K)}\mathbb{C}$$

$$I$$

$$\omega$$

$$I(z)=Y(I,z)=\sum _{n\in \mathbb{Z}}I(n)z^{-n-1},L(z)=Y(\omega ,z)=\sum _{n\in \mathbb{Z}}L(n)z^{-n-2}$$

$$\left[L(n),c(m)\right]$$

$$\left[L(n),d(m)\right]$$

$$Q=\text{mbox}{Res}_{z}Y(u,z)=u_0.$$

$$[Q,I(n)]=[Q,L(n)]=0\forall n\in \mathbb{Z},$$

$$\Pi (0)=M(1)\otimes \mathbb{C}[\mathbb{Z}c]$$

$$d^1=d+\frac{c_L-26}{12}c,d^2=d-\frac{c_L-26}{12}c.$$

$$\Pi (p,r):=\Pi (0)\cdot e^{\frac{p-1}{2}{d}^2+\frac{1-r}{2}c}\text{mbox}wherep\in \mathbb{Z},r\in \mathbb{C}.$$

$$h_{p,r}=(1-p^2)\frac{c_L-26}{24}+1-p+(1-r)\frac{p}{2}.$$

$$W_{p,r}=U(\mathcal{H})\cdot e^{\frac{p-1}{2}{d}^2+\frac{1-r}{2}c}\subset \Pi (p,r).$$

$$v_{p,r-2\ell }^{(m)}=\frac{(-1{)}^m}{2^{m}m!}d(-p{)}^m{v}_{p,r-2\ell }$$

$$Q^m{w}_{p,r-2\ell }^{(m)}=v_{p,r-2(\ell +m)}.$$

$$[d(-k),e_j^c]=2e_{j-k}^c,[e_j^c,c(-k)]=0,[e_0^c,e_j^c]=0.$$

$$Q^j{v}_{p,r-2\ell }^{(m)}=\{\begin{array}{ll}{v}_{p,r-2(\ell +j)}^{(m-j)}\mathrm{mod}{\mathrm{Ker}}_{\Pi (p,r)}{Q}^m,& j<m,\\ v_{p,r-2(\ell +m)},& j=m.\end{array}$$

$$\overline{\Pi (0)}={\mathrm{Ker}}_{\Pi (0)}Q.$$

$$\overline{\Pi (0)}=\underset{n\in \mathbb{Z}}{⨁}{W}_{1,1-2n}\cong \underset{n\in \mathbb{Z}}{⨁}{L}^{\mathcal{H}}(n,0).$$

$$\Pi (0)=\bigcup _{n\ge 0}\Pi (0{)}^{(n)},\Pi (0{)}^{(n)}\cdot \Pi (0{)}^{(m)}\subset \Pi (0{)}^{(n+m)}.$$

$$\overline{\Pi (0)}$$

$$\underset{n\in \mathbb{Z}}{⨁}{W}_{p,r-2n}\cong \underset{n\in \mathbb{Z}}{⨁}{L}^{\mathcal{H}}(h_{p,r}+np,(1-p)c_{L,I}).$$

Full text

1. Introduction

The twisted Heisenberg–Virasoro Lie algebra $\mathcal{H}$ is an important example of a Lie algebra whose associated vertex algebra has many applications in the representation theory and conformal field theory. If the level of the corresponding Heisenberg vertex subalgebra is non-zero, the Heisenberg–Virasoro vertex algebra is isomorphic to the tensor product of Heisenberg vertex algebra and the (universal or simple) Virasoro vertex algebra (cf. [1],[14]). The study of the twisted Heisenberg–Virasoro algebra at level zero was initiated by Y. Billig [12] motivated by applications to the toroidal Lie algebras. New results on the representation theory were obtained in recent papers [2], [19], [24].

In this paper we continue our study of free field realization of the twisted Heisenberg–Virasoro algebra from [2]. Our main motivation is to present free field realization of self-dual modules and certain Verma modules which we were unable to construct using methods from [2].

Let $\mathcal{H}$ be the twisted Heisenberg–Virasoro algebra at level zero. Let $V^{\mathcal{H}}(h,h_{I})$ (resp. $L^{\mathcal{H}}(h,h_{I})$) denote the Verma module (resp. the irreducible highest weight module) with highest weight $(c_{L},c_{I},c_{L,I},h,h_{I})$ such that $c_{I}=0$, $c_{L,I}\neq 0$ is fixed but arbitrary, and with highest weight vector $v_{h,h_{I}}$. Then $V^{\mathcal{H}}(h,h_{I})$ is reducible if and only if $h_{I}/c_{L,I}-1\in{\mathbb{Z}}\setminus\{0\}$ (cf. [12]).

A free field realization of irreducible highest weight modules for the twisted Heisenberg–Virasoro algebra at level zero was presented by the authors in [2]. Using free field realization we calculated the fusion rules for non-generic irreducible modules i.e., for those irreducible modules which are not isomorphic to a Verma $\mathcal{H}$–module. In our approach, the screening operator $Q$ introduced in [2, Section 2] has played an important role. In particular, the singular vector in $V^{\mathcal{H}}(h,h_{I})$ in the case

[TABLE]

is expressed as $Qv_{h+p,h_{I}}=S_{p}(c)v_{h,h_{I}}$ where $S_{p}(c)$ is $p^{th}$–Schur polynomial in variables $(c(-1),c(-2),\cdots)$ defined by (4.16) where $c=-I/c_{L,I}$:

[TABLE]

(see (2.7) and [2, Theorem 4.3]).

In the present paper we found a similar approach for the singular vector in the case

[TABLE]

The formula is

[TABLE]

Verma modules $V^{\mathcal{H}}(h,h_{I})$ in the case (1.1) are explicitly constructed in Section 5 by using certain deformation studied previously in [4]. Singular vector (1.2) admits a nice interpretation as an element $e^{\frac{p-1}{2}d^{2}+\frac{1-r}{2}c+c}$ of the group algebra in the explicit lattice realization (cf. Theorem 5.3).

It turned out that the methods from [2] do not provide a free field realization of self-dual modules $L^{\mathcal{H}}(h,h_{I})$ such that $h\neq\frac{c_{L}-2}{24}$, $h_{I}=c_{L,I}$. On the other hand, the same paper contains certain results on vanishing of the fusion rules in the category of modules which contains self-dual module $L^{\mathcal{H}}(h,h_{I})$ as above (see [2, Theorem 5.4(3), Remark 6.5]). In order to understand these fusion rules properties from the vertex–algebraic point of view, one needs to find a bosonic realization of self-dual modules. We shall see that the realization includes both the concepts of Whittaker and logarithmic modules for vertex algebras. We shall prove that the Whittaker module $\Pi_{\lambda}$ for the vertex algebra $\Pi(0)$ introduced in [7], after certain logarithmic deformation (cf. Theorem 5.2) becomes a highly reducible $\mathcal{H}$–module $\widetilde{\Pi}_{\lambda}$ which contains $L^{\mathcal{H}}(h,h_{I})$ as a submodule. Since $\widetilde{\Pi}_{\lambda}$ is not a module for the Heisenberg vertex algebra $M(1)$, it is clear that such module could not have appeared in the fusion rules analysis made in [2].

In Section 2 we recall from [2] a free field realization of $\mathcal{H}$, the definition of vertex algebra $\Pi(0)$ and its modules $\Pi(p,r)$. We study an extension of the Heisenberg–Virasoro vertex algebra $\overline{\Pi(0)}\subset\Pi(0)$ and present a structure of $\overline{\Pi(0)}$–modules $\Pi(p,r)$ in Section 3. By using certain relation in $\Pi(0)$–modules we recover formula 1.2 in Section 4. Then we consider a deformed action of $\mathcal{H}$ on these modules and obtain a family of modules $\widetilde{\Pi(p,r)}$ [cf. Theorem 5.2, Theorem 5.5] with the following properties:

• •

$\widetilde{\Pi(p,r)}$ is a logarithmic $\mathcal{H}$–module with the following action of the element $\widetilde{L(0)}$ of the Virasoro algebra:

• –

${\mathbb{C}}[\widetilde{L(0)}]v$ is finite–dimensional for every $v\in\widetilde{\Pi(p,r)}$, $p\geq 1$,

• –

${\mathbb{C}}[\widetilde{L(0)}]v$ is infinite–dimensional for every $v\in\widetilde{\Pi(p,r)}$, $p\leq 0$.

• •

For $p\geq 1$, $\widetilde{\Pi(p,r)}$ admits a filtration

[TABLE]

such that $Z^{(m)}$ is a logarithmic $\mathcal{H}$–module of rank $m+1$ and $Z^{(m)}/Z^{(m-1)}$ is a weight $\mathcal{H}$–module. [cf. Theorem 5.4]

In Section 6 we consider a deformed action on Whittaker $\Pi(0)$–modules and obtain a realization of self-dual $\mathcal{H}$–modules $L^{\mathcal{H}}(h,c_{L,I})$ , $h\neq\frac{c_{L}-2}{24}$ [cf. Theorem 6.3]

Finally, we find new applications of our results on the vertex algebra associated to the $W(2,2)$–algebra. We present a non-local bosonic formula (7.26) for the screening operator introduced by the authors in [3]. We hope that our new expression for screening operator can be applied to construction of a quantum group which would play the role of Kazhdan–Lusztig dual of the vertex algebra $W(2,2)$.

We also discuss a connection of our approach with a realization of the BMS3–algebra obtained in [11] in the case $c_{L}=26$.

Acknowledgements. The authors are partially supported by the Croatian Science Foundation under the project 2634 and by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004). We thank the referee for his/her valuable comments.

2. Results from [2]

In this section we recall from [2] the free field realization of the twisted Heisenberg–Virasoro algebra, the construction of vertex algebra $\Pi(0)$ and their modules $\Pi(p,r)$. The definition of the lattice is slightly changed, but the action of the generators of the Heisenberg–Virasoro algebra is the same as in [2]. Main results of Section 2 of [2] are stated in Proposition 2.1.

We also present new, explicit formulas for sub–singular vectors introduced in [2, Proposition 2.7].

Recall that the twisted Heisenberg–Virasoro algebra is an infinite–dimensional complex Lie algebra $\mathcal{H}$ with basis

[TABLE]

and commutation relation:

[TABLE]

Let $V^{\mathcal{H}}(c_{L},c_{I},c_{L,I},h,h_{I})$ denote the Verma module with highest weight $(c_{L},c_{I},c_{L,I},h,h_{I})$, and $L^{\mathcal{H}}(c_{L},c_{I},c_{L,I},h,h_{I})$ its irreducible quotient (cf. [12]). In this paper we consider the case $c_{I}=0$. For simplicity we shall denote the Verma module $V^{\mathcal{H}}(c_{L},0,c_{L,I},h,h_{I})$ with $V^{\mathcal{H}}(h,h_{I})$ and its irreducible quotient with $L^{\mathcal{H}}(h,h_{I})$.

Define the following hyperbolic lattice $Hyp={\mathbb{Z}}x+{\mathbb{Z}}y$ such that

[TABLE]

Let ${\mathfrak{h}}={\mathbb{C}}\otimes_{\mathbb{Z}}Hyp$ and extend the form $\langle\cdot,\cdot\rangle$ on ${\mathfrak{h}}$. We can consider ${\mathfrak{h}}$ as an abelian Lie algebra. Let $\widehat{\mathfrak{h}}={\mathbb{C}}[t,t^{-1}]\otimes{\mathfrak{h}}\oplus{\mathbb{C}}K$ be the affinization of ${\mathfrak{h}}$. Let $\gamma\in{\mathfrak{h}}$ and consider $\widehat{\mathfrak{h}}$–module

[TABLE]

where $t{\mathbb{C}}[t]\otimes{\mathfrak{h}}$ acts trivially on ${\mathbb{C}}$, ${\mathfrak{h}}$ acts as $\langle\delta,\gamma\rangle$ for $\delta\in{\mathfrak{h}}$ and $K$ acts as $1$. We shall denote the highest weight vector in $M(1,\gamma)$ by $e^{\gamma}$.

We shall write $M(1)$ for $M(1,0)$. For $h\in{\mathfrak{h}}$ and $n\in{\mathbb{Z}}$ we write $h(n)$ for $t^{n}\otimes h$. Set $h(z)=\sum_{n\in{\mathbb{Z}}}h(n)z^{-n-1}$. Then $M(1)$ is a vertex algebra which is generated by the fields $h(z)$, $h\in{\mathfrak{h}}$ (cf. [20]. Moreover, $M(1,\gamma)$ for $\gamma\in{\mathfrak{h}}$, are irreducible $M(1)$–modules.

Let $V_{Hyp}=M(1)\otimes{\mathbb{C}}[Hyp]$ be the vertex algebra associated to the lattice $Hyp$, where ${\mathbb{C}}[Hyp]$ is the group algebra of $Hyp$.

In this paper we shall consider the lattice $L={\mathbb{Z}}c+{\mathbb{Z}}d$ such that $c=x$ and $d=2y$. Then $V_{L}$ is a vertex subalgebra of $V_{Hyp}$.

Define the Heisenberg and the Virasoro vector:

[TABLE]

Then the components of the fields

[TABLE]

satisfy the commutation relations for the twisted Heisenberg–Virasoro Lie algebra $\mathcal{H}$ and $I$ and $\omega$ generate the simple Heisenberg–Virasoro vertex algebra $L^{\mathcal{H}}(c_{L},0,c_{L,I},0,0)$ which we shall denote by $L^{\mathcal{H}}(c_{L},c_{L,I})$.

We have:

[TABLE]

Let $u=e^{c}$. Then by [2, Lemma 2.4]

[TABLE]

is a screening operator. This means that

[TABLE]

i.e., $Q$ commutes with the action of the Heisenberg-Virasoro algebra.

Recall that screening operators provide an important tool for construction of singular vectors in free-field realizations (cf. [26]).

One can show that

[TABLE]

is a simple vertex algebra (cf. [13]). Let $Y(\cdot,z)$ be the associated vertex operator. The vertex operator $Y(u,z)$ and the screening operator $Q$ are well-defined on every $\Pi(0)$–module.

Let

[TABLE]

Consider the following irreducible $\Pi(0)$–modules

[TABLE]

(The irreducibility of $\Pi(p,r)$ was also proved in [13]).

Let

[TABLE]

Set $v_{p,r}=e^{\frac{p-1}{2}d^{2}+\frac{1-r}{2}c}$. Note that $h_{p,r+2}=h_{p,r}-p$.

The following result is proved in Propositions 2.5 and 2.7 of [2].

Proposition 2.1**.**

We have:

$W_{p,r}\cong V^{\mathcal{H}}(h_{p,r},(1-p)c_{L,I})$* if and only if $p\in{\mathbb{C}}\setminus{\mathbb{Z}}_{\geq 1}$.*

$W_{p,r}\cong L^{\mathcal{H}}(h_{p,r},(1-p)c_{L,I})$* if $p\in{\mathbb{Z}}_{\geq 1}$.*

For every $r\in{\mathbb{C}}$, $W_{0,r}\cong L^{\mathcal{H}}(\frac{c_{L}-2}{24},c_{L,I})$.

The following modules were not constructed in [2]:

• •

Reducible Verma modules $V^{\mathcal{H}}(h_{p,r},(1-p)c_{L,I})$ in the case $p\in{\mathbb{Z}}_{\geq 1}$.

• •

Self-dual modules $L^{\mathcal{H}}(h,c_{L,I})$ for $h\neq\frac{c_{L}-2}{24}$.

We shall present the construction of these modules in Sections 5 and 6.

We shall also need the following result.

Lemma 2.2**.**

Assume that $p\in{\mathbb{Z}}_{>0}$. As an ${\mathcal{H}}$–module, $\Pi(p,r)$ is generated by a family of ${\mathcal{H}}$–singular vectors $\{v_{p,r-2\ell}\ |\ \ell\in{\mathbb{Z}}\}$ and a family of $\mathcal{H}$–cosingular vectors $\{v_{p,r-2\ell}^{(m)}\ |\ \ell,m\in{\mathbb{Z}},\ m\geq 1\},$ where

[TABLE]

Proof.

By [2, Proposition 2.7] we have that $\Pi(p,r)$ is generated as ${\mathcal{H}}$–module by a family of singular vectors $\{v_{p,r-2\ell}\ |\ell\in{\mathbb{Z}}\}$ and by a family of cosingular vectors $\{w^{(m)}_{p,r-2\ell}\ |\ \ell,m\in{\mathbb{Z}},\ m\geq 1\}$ satisfying

[TABLE]

Let us prove that for cosingular vectors $w^{(m)}_{p,r-2\ell}$ we may choose $v_{p,r-2\ell}^{(m)}$.

Recall the commutator relations in $\Pi(0)$:

[TABLE]

By induction on $m$ we see that $e^{c}_{-kp}v_{p,r-2\ell}^{(m)}$ is spanned by $S_{(k+i-1)p}(c)v_{p,r-2(\ell+1)}^{(m-i)}$, $i=k-1,\dots m$ for each $k\in\mathbb{Z}_{\geq 0}$. Therefore we have

[TABLE]

The proof follows. ∎

See Figure 1 in the Appendix A for structure of $\Pi(0)$–module $\Pi(p,r)$.

Remark 2.3**.**

Note that $\Pi(p,r)=\Pi(p,s)$ if and only if $r-s\in 2\mathbb{Z}$. Moreover, $\Pi(1,1)=\Pi(0)$ and $U(\mathcal{H})v_{1,1}\cong L^{\mathcal{H}}(c_{L},c_{L,I})$.

3. An extension of the vertex algebra $L^{\mathcal{H}}(c_{L},c_{L,I})$

An extension of the Heisenberg–Virasoro vertex algebra Now we study modules $\Pi(p,r)$ in more details. There is a vertex subalgebra of $\Pi(0)$ which can be treated as an extension of the vertex algebra $L^{\mathcal{H}}(c_{L},c_{L,I})$:

[TABLE]

In this section we obtain filtrations of $\Pi(p,r)$ by $\overline{\Pi(0)}$–modules such that the subquotients are irreducible over $\overline{\Pi(0)}$.

Proposition 3.1**.**

Let $\Pi(0)^{(n)}=\operatorname{Ker}_{\Pi(0)}Q^{n+1}$. Then we have

• (1)

$\overline{\Pi(0)}$* is a vertex subalgebra of $\Pi(0)$ which is as $L^{\mathcal{H}}(c_{L},c_{L,I})$–module isomorphic to*

[TABLE]

• (2)

For every $n\in{\mathbb{Z}}_{\geq 0}$, $\Pi(0)^{(n)}$ and $\Pi(0)^{(n+1)}/\Pi(0)^{(n)}$ are $\overline{\Pi(0)}$–modules. Moreover we have

[TABLE]

Proof.

Since $Q$ is a screening operator, $\overline{\Pi(0)}=\operatorname{Ker}_{\Pi(0)}Q$ is a vertex subalgebra of $\Pi(0)$. By using [2, Proposition 2.7] we get

[TABLE]

The proof of assertion (2) is clear. ∎

Condition (2) from Proposition 3.1 shows that $\mathcal{H}$–modules (and $\overline{\Pi(0)}$–modules) $\Pi(0)^{(n)}$ give a $\mathbb{Z}_{\geq 0}$ filtration on the vertex algebra $\Pi(0)$. In the same way we can construct a filtration on certain $\Pi(0)$–modules.

Theorem 3.2**.**

Assume that $p\in{\mathbb{Z}}_{>0}$. Let $\Pi(p,r)^{(m)}=\operatorname{Ker}_{\Pi(p,r)}Q^{m+1}$. Then we have

• (1)

$\Pi(p,r)\cong\bigcup_{m\geq 0}\Pi(p,r)^{(m)},\quad\Pi(0)^{(n)}\cdot\Pi(p,r)^{(m)}\subset\Pi(p,r)^{(n+m)}$.

• (2)

For every $m\in{\mathbb{Z}}_{\geq 0}$ $\Pi(p,r)^{(m)}$ is $\overline{\Pi(0)}$–module and $\Pi(p,r)^{(m)}\subset\Pi(p,r)^{(m+1)}$.

• (3)

$\Pi(p,r)^{(m+1)}/\Pi(p,r)^{(m)}$* is an irreducible $\overline{\Pi(0)}$–module which is as $\mathcal{H}$–module isomorphic to*

[TABLE]

Proof.

The proof of assertions (1) and (2) is clear. The decomposition (3.14) essentially follows from [2, Proposition 2.7]. Let us prove the irreducibility result in (3). It suffices to prove that

[TABLE]

Recall that

[TABLE]

where $u_{p,r-2\ell}$ is an $\mathcal{H}$–cosingular vector $v_{p,r-2(\ell-m-1)}^{(m+1)}$ such that $Q^{m+1}u_{p,r-2\ell}=v_{p,r-2\ell}$.

Since $e^{nc}_{k_{0}}v_{p,r-2\ell}=v_{p,r-2(\ell+n)}$ for $k_{0}=-n(p-1)-1$ we get that

[TABLE]

So $e^{nc}_{k_{0}}u_{p,r-2\ell}+\Pi(p,r)^{(m)}$ generates $W_{p,1-2(n+\ell)}$ in $\Pi(p,r)^{(m+1)}/\Pi(p,r)^{(m)}$. The proof follows. ∎

Figure 1 in Appendix A represents a portion of module $\Pi(p,r)$ with action of $Q$ and $e^{c}_{-p}$ on (sub)singular generators obtained in Lemma 2.2. Quotient module $\Pi(p,r)^{(m+1)}/\Pi(p,r)^{(m)}$ is a direct sum of ”slices”, each generated by $v_{p,r-2\ell}^{(m+1)}+\Pi(p,r)^{(m)}$ and isomorphic to $W_{p,r-2(\ell+m+1)}$.

Let us now consider modules $\Pi(-p,r)$. Recall that $h_{-p,r+2}=h_{-p,r}+p$.

Theorem 3.3**.**

As an $\mathcal{H}$–module $\Pi(0,r)$ is isomorphic to

[TABLE]

Let $p\in{\mathbb{Z}}_{>0}$. Consider $Q$ as $\overline{\Pi(0)}$–endomorphism of $\Pi(-p,r)$, and let $\Pi(-p,r)^{(m)}=\operatorname{Im}Q^{m}$. Then for $m\in{\mathbb{Z}}_{>0}$ we have

• (a)

$\Pi(-p,r)\cong\Pi(-p,r)^{(m)}$* as $\overline{\Pi(0)}$–modules,*

• (b)

$\Pi(-p,r)^{(m)}/\Pi(-p,r)^{(m+1)}$* is an irreducible $\overline{\Pi(0)}$–module which is as an $\mathcal{H}$–module isomorphic to*

[TABLE]

Proof.

It was shown in [2] that $\Pi(p,r)\cong\bigoplus_{\ell\in\mathbb{Z}}W_{p,r+2\ell}$ as $\mathcal{H}$–module when $p\notin{\mathbb{Z}}_{>0}$. Decomposition in (1) then follows from Proposition 2.1 (3). Since

[TABLE]

we see that $Q(W_{-p,r+2\ell})\subset W_{-p,r+2(\ell-1)}$ and since $Q$ commutes with the action of $\mathcal{H}$ we have $\operatorname{Ker}Q=0$. Therefore $\Pi(-p,r)\cong\operatorname{Im}Q=\Pi(-p,r)^{(1)}$, so claim (a) follows by iteration.

Let us prove assertion (b). It suffices to prove the claim for $m=0$. General statement then follows from (a). Recall that $W_{-p,r+2\ell}\cong V^{\mathcal{H}}(h_{-p,r+2\ell},(1+p)c_{L,I})$ and notice that $Qv_{-p,r+2(\ell+1)}=S_{p}(c)v_{-p,r+2\ell}$ is an $\mathcal{H}$–singular vector in $W_{-p,r+2\ell}$ which generates the maximal submodule. Now $U(\mathcal{H})\cdot(v_{-p,r+2\ell}+\operatorname{Im}Q)\cong L^{\mathcal{H}}(h_{-p,r+2\ell},(1+p)c_{L,I})$. This proves the decomposition in (b). Proof of irreducibility in (b) is similar to the proof of Theorem 3.2. ∎

See Figure 2 in Appendix A for reference.

4. Relations in $\Pi(0)$–modules

In this section we shall apply a relation in the vertex algebra $\Pi(0)$ on its modules and recover an explicit formula for a singular vector in the Verma module $V^{\mathcal{H}}(h,(1-p)c_{L,I})$, for $p\geq 1$. We shall use this formula in Section 5 when we construct a free field realization of these Verma modules.

Recall first that the Schur polynomials $S_{r}(x_{1},x_{2},\dots)$ in variables $x_{1},x_{2},\dots$ are defined by

[TABLE]

Then for any $\gamma\in L$ we set $S_{r}(\gamma):=S_{r}(\gamma(-1),\gamma(-2),\dots)$.

Let

[TABLE]

By direct calculation we get

[TABLE]

Let

[TABLE]

Then we get

Lemma 4.1**.**

On every $\Pi(0)$–module we have

[TABLE]

Proof.

The assertion follows from (4.17) and the fact that

[TABLE]

in every vertex operator algebra. ∎

Now we shall see some consequences of the relation $\mathcal{Q}=0$ for irreducible $\mathcal{H}$–modules $L^{\mathcal{H}}(h,h_{I})$ such that $h_{I}=(1-p)c_{L,I}$, $p\in{\mathbb{Z}}_{>0}$, which are realized as $W_{p,r}$, for $r\in\mathbb{C}$.

We have

[TABLE]

In this way we have proved:

Proposition 4.2**.**

Let $p\in{\mathbb{Z}}_{>0}$. In $W_{p,r+2}$ we have:

[TABLE]

In particular, let $h_{I}=(1-p)c_{L,I}$, $h=h_{p,r+2}$. Then the singular vector of level $p$ in $V^{\mathcal{H}}(h,h_{I})$ is $\Phi_{p}(L,c)\cdot v_{h,h_{I}}$ where

[TABLE]

5. Deformed action of $\mathcal{H}$ on weight $\Pi(0)$–modules and realization of Verma modules

As we noticed in Section 2, the free field realization from [2] does not provide realization of Verma modules $V^{\mathcal{H}}(h,(1-p)c_{L.I})$ and their singular vectors in the case $p\geq 1$. In order to understand these Verma modules, we shall use certain deformation of free field realization from [2]. We shall use the construction from [4] to deform the action of the twisted Heisenberg–Virasoro algebra on $\Pi(0)$–modules (see also [8], [17]). Let

[TABLE]

First we recall a definition of logarithmic modules. More information about structure theory of logarithmic modules can be found in literature on logarithmic vertex operator algebras (see [6], [15], [18], [22], [23] and reference therein).

Definition 5.1**.**

• (1)

A module $(M,Y_{M})$ for the conformal vertex algebra with conformal vector $\omega$ is a logarithmic module of rank $m\in{\mathbb{Z}_{\geq 1}}$ if

[TABLE]

where $L_{ss}(0)$ is the semisimple part of $L(0)$.

• (2)

If for every $m\in{\mathbb{Z}_{\geq 1}}$ $(L(0)-L_{ss}(0))^{m}\neq 0$ on $M$, we say that $(M,Y_{M})$ is a logarithmic module of infinite rank.

Theorem 5.2**.**

For every $\Pi(0)$–module $(M,Y_{M}(\cdot,z))$,

[TABLE]

is a $\overline{\Pi(0)}:=\operatorname{Ker}_{\Pi(0)}Q$–module. The action of Heisenberg–Virasoro algebra is

[TABLE]

In particular,

[TABLE]

and $\widetilde{L(0)}-\widetilde{L_{ss}(0)}=u_{0}=Q$.

Recall the definition of $\Pi(0)$–modules

[TABLE]

Then $\widetilde{\Pi(p,r)}$ are logarithmic ${\mathcal{H}}$–modules which are uniquely determined by the action (5.19).

We shall also consider the cyclic submodules:

[TABLE]

5.1. Case $h_{I}=(1-p)c_{L,I}$

We saw that for the undeformed action of ${\mathcal{H}}$ studied in [2], vector $v_{p,r}=e^{\frac{p-1}{2}d^{2}+\frac{1-r}{2}c}$, for $p\geq 1$, generates the irreducible highest weight module $W_{p,r}$. But we shall see below that $\widetilde{W_{p,r}}$ is isomorphic to a Verma module.

Theorem 5.3**.**

Assume that $p\in{\mathbb{Z}}_{>0}$, $r\in{\mathbb{C}}$. We have

$\widetilde{W_{p,r}}\cong V^{\mathcal{H}}(h_{p,r},(1-p)c_{L,I})$.

Singular vectors in $\widetilde{W_{p,r}}\cong V^{\mathcal{H}}(h_{p,r},(1-p)c_{L,I})$ are

[TABLE]

Proof.

It is clear that $v_{p,r-2n}=e^{\frac{p-1}{2}d^{2}+\frac{1-r}{2}c+nc}$ is a singular vector for any $n\geq 0$. We only need to prove that

[TABLE]

Assume that (5.20) holds for $n\in{\mathbb{Z}}_{\geq 0}$. Since $e^{c}_{k}v_{p,r-2n}=0$ for every $k\geq 1-p$ we have

[TABLE]

But since $e^{c}_{-p}v_{p,r-2n}=v_{p,r-2(n+1)}$ we have

[TABLE]

By using the expression for singular vector in $V^{\mathcal{H}}(h_{p,r},(1-p)c_{L,I})$ from Proposition 4.2 we get for $h=h_{p,r-2n}$

[TABLE]

(Above we used the fact that $v_{p,r-2n}$ generates the irreducible $\mathcal{H}$–module $L^{\mathcal{H}}(h_{p,r},(1-p)c_{L,I})$ for the undeformed action, so $\Phi_{p}(L,c)\cdot v_{p,r-2n}=0$.)

Thus we get that $v_{p,r-2(n+1)}=e^{\frac{p-1}{2}d^{2}+\frac{1-r}{2}c+(n+1)c}$ belongs to $\widetilde{W_{p,r}}$. The claim now follows by induction. ∎

Finally, we obtain a deformed version of Theorem 3.2.

Theorem 5.4**.**

Let $Z^{(m)}=\operatorname{Ker}_{\widetilde{\Pi(p,r)}}Q^{m+1}$. Then we have

• (1)

$\widetilde{\Pi(p,r)}\cong\bigcup_{m\geq 0}Z^{(m)},\quad\Pi(0)^{(n)}\cdot Z^{(m)}\subset{Z}^{(n+m)}$.

• (2)

For every $m\in{\mathbb{Z}}_{\geq 0}$, $Z^{(m)}$ is a logarithmic $\overline{\Pi(0)}$–module of rank $m+1$ with respect to $\widetilde{L(0)}$.

• (3)

$Z^{(m)}/Z^{(m-1)}$* is a weight $\overline{\Pi(0)}$–module which is as $\mathcal{H}$–module isomorphic to*

[TABLE]

Proof.

Assertion (1) is clear. Using relation (2.13) in the proof of Lemma 2.2 we see that $v_{p,r-2\ell}^{(m)}\in Z^{(m)}\setminus Z^{(m-1)}$. Since $\widetilde{L(0)}-\widetilde{L_{ss}(0)}=Q$, we have that $Z^{(m)}$ is a logarithmic module of $\widetilde{L(0)}$–nilpotent rank $m+1$ so (2) holds. Assertion (3) results from following facts:

• (a)

As an $\mathcal{H}$–module $\widetilde{\Pi(p,r)}$ is generated by set of vectors

[TABLE]

• (b)

$Z^{(m)}/Z^{(m-1)}$ is a weight $\mathcal{H}$–module (i.e., non-logarithmic) generated by vectors $\{v_{p,r-2j}^{(m)}+Z^{(m-1)}|\ \ell\in{\mathbb{Z}}\}$.

• (c)

$v_{p,r-2\ell}^{(m)}+Z^{(m-1)}$ generates the Verma module $\widetilde{W_{p,r-2\ell}}$.

Since $Q$ and $e^{c}_{-j}$ commute and by using (2.12) we get

[TABLE]

so $e^{c}_{-j}v_{p,r-2\ell}^{(m)}\in Z^{(m-1)}$ for $j<p$. Therefore

[TABLE]

The proof of claims (a) and (b) easily follows from Theorem 3.2 and (5.22). Let us prove claim (c).

We have proved in (5.22) that $v_{p,r-2\ell}^{(m)}+Z^{(m-1)}$ is a highest weight vector with highest weight $(h_{p,r-2\ell},(1-p)c_{L,I})$. Now, repeating the same arguments as in the proof of Theorem 5.3 we get

[TABLE]

This implies that $v_{p,r-2\ell}^{(m)}+Z^{(m-1)}$ generates the Verma module $\widetilde{W_{p,r-2\ell}}$ which contains all Verma modules $\widetilde{W_{p,r-2(\ell+j)}}$, $j\in{\mathbb{Z}}_{\geq 1}.$ This completes the proof. (See also Figure 3 in Appendix A where one can follow steps in the proof.) ∎

5.2. Case $h_{I}=c_{L,I}$

Note that $\widetilde{\Pi(0,r)}$ is an ${\mathcal{H}}$–module on which $I(0)$ acts as multiplication by $c_{L,I}$.

In particular, $\widetilde{W_{0,r}}$ is a ${\mathbb{Z}_{\geq 0}}$–graded logarithmic $\mathcal{H}$–module whose lowest component is

[TABLE]

Moreover, since

[TABLE]

we conclude that $\widetilde{W_{0,r}}$ is a $\mathbb{Z}_{\geq 0}$–graded logarithmic module of infinite rank. See Figure 4 in Appendix A.

5.3. Case $h_{I}=(1+p)c_{L,I}$

We saw that $\Pi(-p,r)$ contains a descending chain of submodules $\Pi(-p,r)^{(m)}$ isomorphic to $\Pi(-p,r)$ (Theorem 3.3).

Theorem 5.5**.**

Let $p\in{\mathbb{Z}}_{>0}$. Then $\widetilde{\Pi(-p,r)}$ is a logarithmic $\overline{\Pi(0)}$–module such that

[TABLE]

Quotient $\widetilde{\Pi(-p,r)}/\widetilde{\Pi(-p,r)}^{(1)}$ is a weight module such that

[TABLE]

Proof.

Let $S=\{v_{-p,r-2\ell}\ |\ \ell\in\mathbb{Z}\}$. Let $\langle S\rangle$ be the $\mathcal{H}$–submodule generated by the set $S$. We shall first prove that $\widetilde{\Pi(-p,r)}=\langle S\rangle$. Since, as a vector space $\widetilde{\Pi(-p,r)}\cong\Pi(-p,r)\cong\bigoplus_{\ell\in\mathbb{Z}}W_{-p,r-2\ell}$, it suffices to show that $W_{-p,r-2\ell}\subset\langle S\rangle$ for each $\ell$.

Take an arbitrary basis vector

[TABLE]

of $W_{-p,r-2\ell}$, where $\ell\in{\mathbb{Z}}$, $p_{1},\dots,p_{s},q_{1},\dots,q_{m}\geq 1$. Then by applying formula for $\widetilde{L(n)}$ and relation $[e^{c}_{m},L(n)]=me^{c}_{m+n}$ we get

[TABLE]

where $w$ is a linear combination of vectors

[TABLE]

such that $\ell^{\prime}\in{\mathbb{Z}}$, $m^{\prime}<m$ . The assertion now follows by induction on $m$.

Furthermore, we have

[TABLE]

Since $Q$ commutes with action of $\mathcal{H}$, we proved the first claim.

Taking a quotient by $\widetilde{\Pi(-p,r)}^{(1)}=\operatorname{Im}Q$ results in a weight module (i.e., non logarithmic module) on which $\widetilde{L(0)}\equiv L(0)$. ∎

See Figure 5 in Appendix A.

Remark 5.6**.**

As far as we know, modules $\widetilde{\Pi(-p,r)}/\widetilde{\Pi(-p,r)}^{(1)}$, and their cyclic submodules generated by images of $v_{-p,r-2\ell}$ are weight $\mathcal{H}$–modules which haven’t been analysed in the literature.

6. Realization of self-dual modules via Whittaker $\Pi(0)$–modules

In Section 5 we slightly refined the free field realization from [2], but these results still don’t give a realization of all irreducible self-dual modules. In order to construct all self-dual modules we shall apply the deformation from Section 5 on the Whittaker $\Pi(0)$–module $\Pi_{\lambda}$ which was constructed in [7, Section 11] and used for a realization of Whittaker $A_{1}^{(1)}$–modules at the critical level. As a by-product we shall see that self-dual modules for $\mathcal{H}$ have non-trivial self-extensions which are logarithmic modules.

6.1. Whittaker $\Pi(0)$–module $\Pi_{\lambda}$

We shall recall the construction of a Whittaker $\Pi(0)$–module $\Pi_{\lambda}$ from [7, Section 11]. Let $u=e^{c}$, $u^{-1}=e^{-c}$.

Theorem 6.1**.**

[7]** Assume that $\lambda\neq 0$. There is a $\Pi(0)$–module $\Pi_{\lambda}$ generated by the cyclic vector $w_{\lambda}$ such that $c(0)=-\mbox{Id}$ on $\Pi_{\lambda}$ and

[TABLE]

As a vector space

[TABLE]

$\Pi_{\lambda}$* is ${\mathbb{Z}_{\geq 0}}$–graded*

[TABLE]

and lowest component is isomorphic to ${\mathbb{C}}[d(0)]$.

Recall also (cf. [7]) that the lowest component $\Pi_{\lambda}(0)$ is an irreducible Whittaker module for the associative algebra $\mathcal{A}$ defined by generators

[TABLE]

and relations

[TABLE]

6.2. Realization of self-dual modules

Now we can apply Theorem 5.2 on the Whittaker $\Pi(0)$–module $\Pi_{\lambda}$. We get $\mathcal{H}$–module $\widetilde{\Pi}_{\lambda}$ , which is as a vector space isomorphic to $\Pi_{\lambda}$ and the (deformed) action of $\mathcal{H}$ is as follows:

[TABLE]

and on the lowest component $\widetilde{\Pi}_{\lambda}(0)$ we have

[TABLE]

This implies:

[TABLE]

Define also the following (logarithmic) cyclic module:

[TABLE]

Lemma 6.2**.**

We have:

[TABLE]

Proof.

By using (6.24) one can easily see that for $0\leq m\leq n$ there is a polynomial $P(x)$ such that

[TABLE]

This proves that $\widetilde{\Pi}_{\lambda}^{(n+1)}\supset\widetilde{\Pi}_{\lambda}^{(n)}$ for $n\in{\mathbb{Z}}_{\geq 0}$.

Take an arbitrary basis vector

[TABLE]

of $\widetilde{\Pi}_{\lambda}$, where $\ell\in{\mathbb{Z}}_{\geq 0}$, $p_{1},\dots,p_{s},q_{1},\dots,q_{r}\geq 1$. Then

[TABLE]

where $A\neq 0$ and $w$ is a linear combination of vectors

[TABLE]

such that $\ell^{\prime}\in{\mathbb{Z}}_{\geq 0}$, $r^{\prime}<r$ or $r=r^{\prime}$ and $u_{1}+\cdots+u_{r^{\prime}}<q_{1}+\cdots+q_{r}$. The assertion now follows by induction. ∎

Theorem 6.3**.**

For every $\lambda\in{\mathbb{C}}$, $\lambda\neq 0$ we have:

$\widetilde{\Pi}_{\lambda}$* is a logarithmic $\mathcal{H}$–module of infinite rank with respect to $\widetilde{L(0)}$.*

$\widetilde{\Pi}_{\lambda}^{(0)}$* is an irreducible self-dual $\mathcal{H}$–module with highest weight*

[TABLE]

$\mathcal{H}$–module $\widetilde{\Pi}_{\lambda}$ admits the $\mathbb{Z}_{\geq 0}$–filtration:

[TABLE]

such that

[TABLE]

Every $\widetilde{\Pi}_{\lambda}^{(n)}$ is a logarithmic $\mathcal{H}$–module of rank $n$ with respect to $\widetilde{L(0)}$.

Proof.

(1) follows from the fact that on the top component $\widetilde{\Pi}_{\lambda}(0)$ we have $Q=u_{0}=\widetilde{L(0)}-\frac{c_{L}-2}{24}$.

(2) Using (6.25) and the fact that the Verma module with highest weight $(h,h_{I})=(\frac{c_{L}-2}{24}+\lambda,c_{L,I})$ is irreducible we get $L^{\mathcal{H}}(h,h_{I})=U(\mathcal{H})\cdot w_{\lambda}=\widetilde{\Pi}_{\lambda}^{(0)}$.

(3) First we notice that for $m\geq 0$ we have

[TABLE]

Therefore we have isomorphism $L^{\mathcal{H}}(h,h_{I})\rightarrow\widetilde{\Pi}_{\lambda}^{(n+1)}/\widetilde{\Pi}_{\lambda}^{(n)}.$ The proof now follows from Lemma 6.2. ∎

See Figure 6 in Appendix A for reference.

We list two interesting consequences of previous theorem.

Corollary 6.4**.**

Logarithmic $\mathcal{H}$–module $\widetilde{\Pi}_{\lambda}^{(1)}$ is a non-split self-extension of irreducible self-dual module $L^{\mathcal{H}}(h,h_{I})$:

[TABLE]

Note that the vertex algebra $\overline{\Pi(0)}$ is not ${\mathbb{Z}}_{\geq 0}$–graded since for every $n\in{\mathbb{Z}}$, $e^{nc}$ has weight $n$. Irreducible $\overline{\Pi(0)}$–modules from Theorem 5.4 (3) are not $\mathbb{Z}_{\geq 0}$–graded. But, quite surprisingly, the vertex algebra $\overline{\Pi(0)}$ admits a large family of ${\mathbb{Z}}_{\geq 0}$–graded modules which are self-dual. We also construct a family of intertwining operators which haven’t appeared in [2].

Corollary 6.5**.**

We have:

$L^{\mathcal{H}}(h,c_{L,I})$* is an irreducible $\mathbb{Z}_{\geq 0}$–graded $\overline{\Pi(0)}$–module.*

For every $n\in{\mathbb{Z}}$ there is a non-trivial intertwining operator of type

[TABLE]

7. Some applications to the $W(2,2)$–algebra

In [2] we introduced a free field realization of the $W(2,2)$-algebra as a subalgebra of the Heisenberg Virasoro algebra.

Recall that $W(2,2)$ is realized as a subalgebra of $L^{\mathcal{H}}(c_{L},c_{L,I})$ generated by $L(z)$ and

[TABLE]

where

[TABLE]

In the paper [3] we discussed a free field realization of highest weight $W(2,2)$–modules. We constructed in [3, Section 4] $W(2,2)$–homomorphism $S_{1}:L^{\mathcal{H}}(c_{L},c_{L,I})\rightarrow L^{\mathcal{H}}(1,0)$ such that $\mbox{Ker}_{L^{\mathcal{H}}(c_{L},c_{L,I})}S_{1}$ is the simple vertex algebra $L^{W(2,2)}(c_{L},-24c_{L,I}^{2}).$ In this paper shall present a bosonic, non-local expression for the screening operator $S_{1}$.

The vertex algebra $W(2,2)$ has appeared in physics literature as the Galilean Virasoro algebra ([25], [9], [10]) and as $\text{BMS}_{3}$ algebra ([11]). We noticed a free field realization of the $W(2,2)$-algebra in terms of the $\beta\gamma$ systems in [11]. We shall see how this realization relates to our approach.

7.1. A bosonic formula for the second screening operator and $W(2,2)$–algebra

Our approach is motived by the realization of screening operators in LCFT from [5] and [6]. Recall the definition of modules $W_{p,r}$ from (2.11). For $r\in{\mathbb{Z}}$ we define:

[TABLE]

Lemma 7.1**.**

We have:

[TABLE]

Proof.

In the proof we use the following formulas

[TABLE]

We have:

[TABLE]

Relation $[c(m),S]=2(1-\delta_{m,0})e^{c}_{m}$ follows directly from the definition of the operator $S$. Next we have

[TABLE]

The proof follows. ∎

Now, we will see that in the case $r=1$ our operator $S$ is a multiple of the screening operator $S_{1}$ from [3]:

Corollary 7.2**.**

Let $r=1$, and consider $S:W_{p,1}\rightarrow W_{p,-1}.$ Then $S$ commutes with the action of the $W(2,2)$-algebra:

[TABLE]

Moreover, $S$ is a $W(2,2)$–homomorphism which is proportional to $S_{1}$.

Proof.

In the case $r=1$ we have that $d^{1}(0)$ and $Q=e^{c}_{0}$ act trivially on $W_{p,1}$, and therefore Lemma 7.1 implies that

[TABLE]

It is clear that the $W(2,2)$–homomorphism $S_{1}:W_{p,1}\rightarrow W_{p,-1}$ from [3, Section 4] is uniquely determined by the properties

[TABLE]

which gives $[S_{1},c(n)]=\frac{1}{c_{L,I}}e^{c}_{n}$. Now Lemma 7.1 gives that $S=2c_{L,I}S_{1}$. ∎

7.2. On the Banerjee, Jatkar, Mukhi, Neogi’s free field realization of the $\text{BMS}_{3}$–algebra

Recently, Banerjee, Jatkar, Mukhi and Neogi in [11] have discovered a new free field realization of the $W(2,2)$–algebra for central charge $c_{L}=26$. The vertex algebra $L^{W(2,2)}(26,c_{W})$ is realized inside of the $\beta\gamma$ system. Since the $\beta\gamma$–system can be embedded into the vertex algebra $\Pi(0)$, one may try to extend this realization in order to obtain an arbitrary central charge $c_{L}$. Quite surprisingly, even in the case of the larger vertex algebra $\Pi(0)$, one gets the $W(2,2)$–structure only for $c_{L}=26$.

Recall the definition of following Virasoro vector of central charge $c_{L}\in{\mathbb{C}}$.

[TABLE]

We shall now deform this vector in a different way:

Lemma 7.3**.**

For every $\mu\in{\mathbb{C}}$

[TABLE]

is a Virasoro vector of central charge $c_{L}$.

The proof of lemma follows from a more general statement (which is also noticed in [11]):

Claim: Assume that $(V,Y,{\bf 1},\omega)$ is a VOA of central charge $c$, and $v$ is a primary, commutative vector of conformal weight $-1$. Then $\widetilde{\omega}=\omega+\frac{1}{6}D^{3}v$ is a Virasoro vector of central charge $c$.

The following result is obtained in [11, Section 2]. We include a proof of this result from which one can see that such construction works only for $c_{L}=26$.

Proposition 7.4**.**

[11]**. The vertex algebra $L^{W(2,2)}(c_{L},c_{W})$ for $c_{L}=26$ is isomorphic to a vertex subalgebra of $\Pi(0)$ generated by

[TABLE]

where

[TABLE]

Proof.

By direct calculation we get

[TABLE]

By using formulas

[TABLE]

we get that

[TABLE]

which easily implies that

[TABLE]

Since

[TABLE]

we get

[TABLE]

Claim now follows from (7.30)-(7.32), (7.36) and Lemma 7.3. ∎

Remark 7.5**.**

The Weyl vertex algebra (also called the $\beta\gamma$ system in the physical literature) can be realized as a subalgebra of the vertex algebra $\Pi(0)$ as follows:

[TABLE]

Then the components of the fields $Y(\beta,z)=\sum_{n\mathbb{Z}}\beta(n)z^{-n-1}$, $Y(\gamma,z)=\sum_{n\in{\mathbb{Z}}}\gamma(n)z^{-n}$ satisfy the commutation relation for the Weyl algebra

[TABLE]

The vertex algebra $\Pi(0)$ can be treated as a certain localization of the Weyl vertex algebra (for details see [16], [21], [7]).

By using (7.37) we see that for $c_{L}=26$ $L^{W(2,2)}(c_{L},c_{W})$ is realized as a subalgebra of the Weyl vertex algebra, which corresponds to the realization in [11].

Remark 7.6**.**

It would be interesting to investigate the structure of $W(2,2)$–modules $\Pi(p,r)$ with this new action. Since

[TABLE]

we have that $\Pi(p,r)$ and $\Pi_{\lambda}$ are weight $L^{W(2,2)}(c_{L},c_{W})$–modules on which $\widetilde{L(0)}=L(0)$. In our forthcoming papers we plan to investigate the appearance of these modules in the fusion rules analysis at $c_{L}=26$.

Appendix A Figures

Here we present some visualizations of modules $\Pi(p,r)$ and $\widetilde{\Pi(p,r)}$.