Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra

1.   Introduction

Let $\mathcal{L}$ be the twisted Heisenberg-Virasoro algebra. It is the universal central extension of the Lie algebra of differential operators on a circle of order at most one (cf. [3]):

$\mathcal{L}$ contains an infinite-dimensional Heisenberg algebra and the Virasoro algebra as subalgebras (cf. [3], [4]). The induced module $V_{\mathcal{L}}(\ell_{123},0) = U(\mathcal{L})\otimes_{U(\mathcal{L}_{(\leq 1)})} {\mathbb C}_{\ell_{123}}$ is a vertex operator algebra of central charge $\ell_{1}$ with conformal vector $\omega = L_{-2}{\bf 1}$ (cf. [12]). $V_{\mathcal{L}}(\ell_{123},0)$ is a nonrational vertex operator algebra and is not $C_{2}$-cofinite. The structure theory and representation theory of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$ are closely related to the three scalars $\ell_{1},\ell_{2},\ell_{3}$ (cf. [3], [1], [2], [4], [5], [8], [12], etc.).

Determining the automorphism group $\mbox{Aut}(V)$ of a vertex operator algebra $V$ is an important subject in vertex operator algebra theory. It is related to the orbifold theory which studies the fixed point subalgebras of vertex operator algebras and their modules under certain finite subgroups of the full automorphism groups. The orbifold conjecture says that under some conditions on $V$, every simple $V^{G}$-module is contained in some $g$-twisted $V$-module, where $G$ is a finite subgroup of $\mbox{Aut}(V)$, $g\in G$, $V^{G}$ is the fixed point subalgebra of $V$ under the group $G$.

The automorphism group of the twisted Heisenberg-Virasoro algebra $\mathcal{L}$ has been studied in [15]. In this paper, we study the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$. We know that $V_{\mathcal{L}}(\ell_{123},0)$ is generated by $\omega = L_{-2}{\bf 1}$ and $I_{-1}{\bf 1}$. By definition, any homomorphism of a vertex operator algebra $(V,Y,{\bf 1}, \omega)$ takes $\omega$ to $\omega$. Therefore, it suffices to determine the action on the generator $I_{-1}{\bf 1}$. It turns out that automorphisms of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$ depend on the numbers $\ell_{2}$ and $\ell_{3}$, and there exists automorphism of order other than 2, which makes the study of twisted modules for $V_{\mathcal{L}}(\ell_{123},0)$ interesting.

In [12], all irreducible modules for the vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$ are classified: that is, every irreducible module for $V_{\mathcal{L}}(\ell_{123},0)$ is isomorphic to some $L_{\mathcal{L}}(\ell_{123}, h_{1}, h_{2})$, $h_{1},h_{2}\in {\mathbb C}$. In this paper, for any integer $t>1$, we classify $\sigma_{t}$-twisted irreducible modules for $V_{\mathcal{L}}(\ell_{123},0)$, where $\sigma_{t}$ is an order $t$ automorphism of $V_{\mathcal{L}}(\ell_{123},0)$. The way we deal with this problem is similar to the one used for modules of vertex algebra (cf. [10], [11], [12], [13], etc.), but in the context of twisted modules. We first introduce another Lie algebra $\mathcal{L}_{t}$. It is a Lie algebra with basis $\{L_{n},I_{n+\frac{1}{t}},k_{1},k_{3}\ | \ n\in {\mathbb Z} \}$, and Lie brackets

Note that when $t\neq 2$, $\{I_{n+\frac{1}{t}} \ | \ n\in {\mathbb Z} \}$ is an abelian Lie algebra. Then we construct irreducible $\mathcal{L}_{t}$-modules $L_{\mathcal{L}_{t}}(\Bbbk_{1},\Bbbk_{3},h)$ as quotient of the induced modules $M_{\mathcal{L}_{t}}(\Bbbk_{1},\Bbbk_{3},h)$, where $\Bbbk_{1},\Bbbk_{3},h\in\mathbb{C}$. We show that $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-modules ($\ell_{2} = 0$) are in one-to-one correspondence with restricted $\mathcal{L}_{t}$-modules of level $\ell_{13}$. Using this result, we get a complete list of irreducible $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-modules, where $\ell_{2} = 0$, $\ell_{1}$, $\ell_{3}\in {\mathbb C}$. Let $V_{\mathcal{L}}(\ell_{123},0)^{\sigma_{t}}$ be the fixed point subalgebra of $V_{\mathcal{L}}(\ell_{123},0)$ under the automorphism $\sigma_{t}$. $V_{\mathcal{L}}(\ell_{123},0)^{\sigma_{t}}$ is a vertex operator subalgebra of $V_{\mathcal{L}}(\ell_{123},0)$. It is important and meaningful to study the representations of the fixed point subalgebra $V_{\mathcal{L}}(\ell_{123},0)^{\sigma_{t}}$. We remark at the end of the paper that except for the case of order 2, the complete list of irreducible modules for $V_{\mathcal{L}}(\ell_{123},0)^{\sigma_{t}}$ needs to be further investigated.

This paper is organized as follows. In Section 2, we review the notions and some results of vertex operator algebras, automorphisms and twisted modules for vertex operator algebras. In Section 3, we study the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$. In Section 4, we first study the twisted modules of $V_{\mathcal{L}}(\ell_{123},0)$($\ell_{2} = 0$) under an order $t$ automorphism $\sigma_{t}$ for any integer $t>1$. Then we give a complete list of irreducible $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$($\ell_{2} = 0$)-modules.

2.   Preliminaries

For later use, we recall the following result (cf. Proposition 2.3.7 of [13]).

Lemma 2.1.

for $m>n$, $m,n\in {\mathbb N}$, where $\delta\Big(\frac{x_{1}}{x_{2}}\Big) = \sum\limits_{n\in {\mathbb Z}}x_{1}^{n}x_{2}^{-n}$.

Let $(V,Y,\omega,{\bf 1})$ be a vertex operator algebra (cf. Definition 3.1.22 of [13]). Then, a priori, ($V$, $Y$, 1) is a vertex algebra (cf. Definition 3.1.1 of [13]). Let $D$ be the endomorphism of the vertex algebra $V$ defined by $D(v) = v_{-2}{\bf 1}$ for $v\in V$. We have $Y(Dv,x) = \frac{d}{dx}Y(v,x)$. Denote by $L(n) = \omega_{n+1}$ for any $n\in {\mathbb Z}$. It can be proved that ${\bf 1}\in V_{(0)}$ and $D = L(-1)$.

Definition 2.2. An automorphism of a vertex operator algebra $(V,Y,\omega,{\bf 1})$ is a linear isomorphism $\sigma$ of $V$ such that

for any $u,v\in V$, $n\in\mathbb{Z}$.

The group of all automorphisms of a vertex operator algebra $V$ is denoted by $\mbox{Aut}(V)$. Any automorphism of a vertex operator algebra $V$ is grading-preserving, i.e. it preserves each homogeneous subspace $V_{(n)}$ of $V$, $n\in {\mathbb Z}$. Let $\sigma$ be an order $t$ automorphism of a vertex operator algebra $V$, $t$ is a positive integer. Then $\sigma$ acts semisimply on $V$. Therefore

where $V^{k}$ is the eigenspace of $V$ for $\sigma$ with eigenvalues $\eta^{k}$, where $\eta = \mbox{exp}(\frac{2\pi \sqrt{-1}}{t})$, $k = 0,\ldots,t-1$. It is easy to see that the fixed points set $V^{0}: = V^{\sigma} = \{v\in V\ | \ \sigma(v) = v\}$ is a vertex operator subalgebra of $V$

Now we recall some notions regarding to twisted modules from [14].

Definition 2.3. Let $(V, {\bf 1}, Y)$ be a vertex algebra with an automorphism $\sigma$ of order $t$. A $\sigma$-twisted $V$-module is a triple $(W,d,Y_{W})$ consisting of a vector space $W$, an endomorphism $d$ of $W$ and a linear map

satisfying the following conditions:

(TW1) $\mbox{For any } v\in V, w\in W$, $v_{n}w = 0\;\mbox{for}\; n\in\frac{1}{t} {\mathbb Z}\;\mbox{sufficiently large};$

(TW2) $Y_{W}(\textbf{1},z) = Id_{W};$

(TW3) $[d, Y_{W}(v,z)] = Y_{W}(D(v),z) = \frac{d}{dz}Y_{W}(v,z)\;\mbox{for any}\; v\in V;$

(TW4) For any $u,v\in V$, the following $\sigma$-twisted Jacobi identity holds:

If $V$ is a vertex operator algebra, a $\sigma$-twisted $V$-module for $V$ as a vertex algebra is called a $\sigma$-twisted weak module for $V$ as a vertex operator algebra.

Definition 2.4. For $V$ a vertex operator algebra, a $\sigma$-twisted $V$-module $W$ is a $\sigma$-twisted weak module for $V$ as a vertex algebra and $W = \coprod\limits_{h\in {\mathbb C}}W_{(h)}$ such that

(TW5) $L(0)w = h w\;\;\;\;\mbox{for}\; h\in {\mathbb C}, w\in W_{(h)};$

(TW6) $\mbox{For any fixed } h, W_{(h+n)} = 0\;\mbox{for} \;n\in \frac{1}{t} {\mathbb Z}\;\mbox{sufficiently small};$

(TW7) $\mbox{dim}\; W_{(h)}<\infty\;\;\mbox{for any} \;h\in {\mathbb C}.$

Remark 2.5. Let $W$ be a $\sigma$-twisted $V$-module. Then

for $v\in V$. Hence, if $v\in V$ is homogeneous,

It follows that a $\sigma$-twisted $V$-module $W$ decomposes into twisted submodules corresponding to the congruence classes mod $\frac{1}{t} {\mathbb Z}$: For $h\in\mathbb{C}/\frac{1}{t} {\mathbb Z}$, let

Then

In particular, if $W$ is irreducible, then

for some $h$.

Remark 2.6. Let $W$ be a $\sigma$-twisted $V$-module. For $u\in V^{k}$, $v\in V$, there is the following twisted iterate formula (cf. (2.32) of [14])

where

Then it can be easily deduced that for any $n\in {\mathbb Z}$,

where

$\binom{\frac{k}{t}}{j} = \frac{\frac{k}{t}(\frac{k}{t}-1)\cdots(\frac{k}{t}-j+1)}{j!}$. Note that when $k = 0$, $\frac{k}{t} = 0$, we have $j = 0$, and then (2.7) is the usual formula for (untwisted) $V$-modules (cf. (3.8.16) of [13]).

Definition 2.7. A homomorphism between two $\sigma$-twisted weak $V$-modules $M$ and $W$ is a linear map $f:M\longrightarrow W$ such that for any $v\in V$,

If $V$ is a vertex operator algebra, then a $V$-module homomorphism $f$ is compatible with the gradings:

Using the formula (2.7), similarly as Proposition 4.5.1 of [13], there is the following result.

Proposition 2.8. Let $W_{1}$ and $W_{2}$ be $\sigma$-twisted $V$-modules and let $\psi\in Hom_{ {\mathbb C}}(W_{1}, W_{2})$. Suppose that

where $S$ is a given generating set of $V$. Then $\psi$ is a $\sigma$-twisted $V$-module homomorphism.

In the following, we review from Section 3 of [14] the local systems of twisted vertex operators.

Definition 2.9. Let $W$ be a vector space, let $t$ be a fixed positive integer. A ${\mathbb Z}_t$-twisted weak vertex operator on $W$ is a formal series $a(z) = \sum_{n\in\frac{1}{t} {\mathbb Z}}a_{n}z^{-n-1}\in (\mbox{End}\; W)[[z^{\frac{1}{t}}, z^{-\frac{1}{t}}]]$ such that for any $w\in W$, $a_{n}w = 0$ for $n\in\frac{1}{t} {\mathbb Z}$ sufficiently large.

Definition 2.10. Two ${\mathbb Z}_t$-twisted weak vertex operators $a(z)$ and $b(z)$ are said to be mutually local if there is a positive integer $n$ such that

A ${\mathbb Z}_t$-twisted weak vertex operator is called a ${\mathbb Z}_t$-twisted vertex operator if it is local with itself.

Denote by $F(W,t)$ the space of all ${\mathbb Z}_{t}$-twisted weak vertex operators on $W$. Let $\sigma$ be the endomorphism of $(\mbox{End}\; W)[[z^{\frac{1}{t}},z^{-\frac{1}{t}}]]$ defined by: $\sigma f(z^{\frac{1}{t}}) = f(\eta^{-1}z^{\frac{1}{t}})$. Denote by $F(W,t)^{k} = \{f(z)\in F(W,t)\ | \ \sigma f(z) = \eta^{k}f(z)\}$ for $0\leq k\leq t-1$. For any mutually local ${\mathbb Z}_t$-twisted vertex operators $a(z), b(z)$ on $W$, define $a(z)_{n}b(z)$ as follows (cf. Definition 3.7 of [14]).

Definition 2.11. Let $W$ be a vector space and let $a(z)$ and $b(z)$ be mutually local ${\mathbb Z}_t$-twisted vertex operators on $W$ such that $a(z)\in F(W,t)^{k}$. Then for any integer $n$ we define $a(z)_{n}b(z)$ as an element of $F(W,t)$ as follows:

where

For any set $S$ of mutually local ${\mathbb Z}_{t}$-twisted vertex operators on $W$, by Zorn's Lemma, there exists a local system $A$ of ${\mathbb Z}_{t}$-twisted vertex operators on $W$ (cf. Section 3 of [14]). Denote by $\langle S\rangle$ the vertex algebra generated by $S$ inside $A$ via the operations $a(z)_{n}b(z)$, $n\in {\mathbb Z}$.

3.   Automorphism group

In this section, we first recall the definition of the twisted Heisenberg-Virasoro algebra $\mathcal{L}$ and the construction of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$. Then we determine the automorphism group of the vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$.

Recall the definition of the twisted Heisenberg-Virasoro algebra $\mathcal{L}$ (see [3] or [4]).

Definition 3.1. The twisted Heisenberg-Virasoro algebra $\mathcal{L}$ is a Lie algebra with basis $\{L_{n},I_{n},c_{1},c_{2},c_{3}\ | \ n\in {\mathbb Z} \}$, and the following Lie brackets:

Clearly, $\mbox{Span}\{L_{n},\;c_{1}\ | \ n\in {\mathbb Z}\}$ is a Virasoro algebra, $\mbox{Span}\{I_{n},\;c_{3}\ | \ n\in {\mathbb Z} \backslash\{0\}\}$ is an infinite-dimensional Heisenberg algebra, we denote them by $Vir$, $\mathcal{H}$ respectively.

Let

then the defining relations of $\mathcal{L}$ become to be

We recall the construction of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$ from [12]. Let

They are graded subalgebras of $\mathcal{L}$ and

Let $\ell_{i},i = 1,2,3,$ be any complex numbers. Consider ${\mathbb C}$ as an $\mathcal{L}_{(\leq 1)}$-module with $c_{i}$ acting as the scalar $\ell_{i},i = 1,2,3,$ and with $\coprod_{n\leq 1} {\mathbb C} L_{-n}\oplus\coprod_{n\leq 0} {\mathbb C} I_{-n}$ acting trivially. Denote this $\mathcal{L}_{(\leq 1)}$-module by ${\mathbb C}_{\ell_{123}}$. Form the induced module

where $U(\cdot)$ denotes the universal enveloping algebra of a Lie algebra. Set ${\bf 1} = 1\otimes 1\in V_{\mathcal{L}}(\ell_{123},0)$. $V_{\mathcal{L}}(\ell_{123},0)$ is a vertex operator algebra with vacuum vector ${\bf 1}$ and conformal vector $\omega = L_{-2}{\bf 1}$. And $\{\omega = L_{-2}{\bf 1},I: = I_{-1}{\bf 1}\}$ is a generating subset of $V_{\mathcal{L}}(\ell_{123},0)$. Recall the grading on $V_{\mathcal{L}}(\ell_{123},0)$:

where $V_{\mathcal{L}}(\ell_{123},0)_{(0)} = {\mathbb C}$ and $V_{\mathcal{L}}(\ell_{123},0)_{(n)}$, $n\geq 1$, has a basis consisting of the vectors

for $r, s\geq 0$, $m_{1}\geq\cdots\geq m_{r}\geq 2$, $k_{1}\geq\cdots\geq k_{s}\geq 1$ with $\sum\limits_{i = 1}^{r}m_{i}+\sum\limits_{j = 1}^{s}k_{j} = n.$

Now we give our first main result. The automorphism group of $V_{\mathcal{L}}(\ell_{123},0)$ is determined in the following theorem.

Theorem 3.2. (1) If $\ell_{2}\neq 0$, then $Aut(V_{\mathcal{L}}(\ell_{123},0)) = \{id\}.$

(2) If $\ell_{2} = 0$ and $\ell_{3}\neq 0$, then $Aut(V_{\mathcal{L}}(\ell_{123},0))\cong \mathbb{Z}_{2}$.

(3) If both $\ell_{2}$ and $\ell_{3}$ are 0, then $Aut(V_{\mathcal{L}}(\ell_{123},0))\cong\mathbb{C}^{\times} = \mathbb{C}\backslash \{0\}.$

Proof. Let

be an automorphism of the vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$. Then $\varphi({\bf 1}) = {\bf 1}$ and $\varphi(\omega) = \omega.$ Since $V_{\mathcal{L}}(\ell_{123},0)$ is generated by $\omega$ and $I = I_{-1}{\bf 1}$, it suffices to determine $\varphi(I)$. $\varphi$ is grading-preserving, so $\varphi(I) = aI$ for some $a\in\mathbb{C}^{\times}$.

Then, on the one hand, we have

on the other hand, we have

Therefore if $\ell_{2}\neq 0$, we get that $a = 1$, i.e. when $\ell_{2}\neq 0$, $\mbox{Aut}(V_{\mathcal{L}}(\ell_{123},0)) = \{id\}$ only consists of the identity map.

Suppose now $\ell_{2} = 0$. Let's consider $\varphi(I_{1}I)$. On the one hand,

on the other hand,

So if $\ell_{3}\neq 0$, we get $a^{2} = 1$, i.e. $a = 1$ or $a = -1$, then $\mbox{Aut}(V_{\mathcal{L}}(\ell_{123},0))\cong\mathbb{Z}_{2}$.

Now let $\ell_{2} = 0 \;\mbox{and}\; \ell_{3} = 0$, then $a$ can be any nonzero complex number, so $\mbox{Aut}(V_{\mathcal{L}}(\ell_{123},0))\cong\mathbb{C}^{\times}.$

4.   $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-modules

In this section, we always assume $\ell_{2} = 0$. We study twisted modules for the vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$. More precisely, for any integer $t>1$, we introduce an infinite-dimensional Lie algebra $\mathcal{L}_{t}$. We show that there is a one-to-one correspondence between restricted $\mathcal{L}_{t}$-modules of level $\ell_{13}$ and $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-modules, where $\sigma_{t}$ is an order $t$ automorphism of $V_{\mathcal{L}}(\ell_{123},0)$. And we give a complete list of irreducible $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-modules.

Note that if $\ell_{2} = 0$ and $\ell_{3}\neq0$, then $t$ can only be the integer 2 (Theorem 3.2). As we need, we introduce the following Lie algebra.

Definition 4.1. Let $\mathcal{L}_{t}$ be a Lie algebra with basis $\{L_{n},I_{n+\frac{1}{t}},k_{1},k_{3}\ | \ n\in {\mathbb Z} \}$, and the Lie brackets are given by:

Note that if $t\neq 2$, then $[I_{m+\frac{1}{t}}, I_{n+\frac{1}{t}}] = 0$ for any $m,n\in {\mathbb Z}$.

Form the generating function as

Then the defining relations (4.1), (4.2), (4.3) amount to:

Now we construct irreducible $\mathcal{L}_{t}$-modules (cf. [12], [13], etc.). Let

It is a subalgebra of $\mathcal{L}_{t}.$

Let ${\mathbb C}$ be an $(\mathcal{L}_{t})_{\geq 0}$-module, where $L_{m}$, $I_{n+\frac{1}{t}}$ act trivially for all $m\geq 1$, $n\geq 0$, and $L_{0}, k_{1}, k_{3}$ act as scalar multiplications by $h, \Bbbk_{1}, \Bbbk_{3}$ respectively. Denote this $(\mathcal{L}_{t})_{\geq 0}$-module by $\mathbb{C}_{\Bbbk_{13},h}$. Form the induced module

Set

Then $M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)$ is ${\mathbb C}$-graded by $L_{0}$-eigenvalues:

where $M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)_{(h)} = \mathbb{C}_{\Bbbk_{13},h}$ and $M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)_{(n+h)}$ is the $L_{0}$-eigenspace of eigenvalue $n+h$ for $n> 0$. $M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)_{(n+h)}$ has a basis consisting of

for $r, s\geq 0$, $m_{1}\geq\cdots\geq m_{r}\geq 1$, $k_{1}\geq\cdots\geq k_{s}\geq 1$ with $\sum\limits_{i = 1}^{r}m_{i}+\sum\limits_{j = 1}^{s}(k_{j}- \frac{1}{t}) = n$, $n> 0$.

Remark 4.2. As a module for $\mathcal{L}_{t}$, $M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)$ is generated by ${\bf 1}_{\Bbbk_{13},h}$ with the relations

and

$M_{\mathcal{L} _{t}}(\Bbbk_{1}, \Bbbk_{3}, h)$ is universal in the sense that for any $\mathcal{L}_{t}$-module $W$ of level $\Bbbk_{13}$ equipped with a vector $v$ such that $L_{0}v = hv,\; L_{m}v = 0, I_{n+\frac{1}{t}}v = 0\; \mbox{for}\; m\geq 1, n\geq 0$, there exists a unique $\mathcal{L}_{t}$-module map $M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)\longrightarrow W$ sending ${\bf 1}_{\Bbbk_{13},h}$ to $v$.

In general, $M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)$ as an $\mathcal{L}_{t}$-module may be reducible. Since $\mathbb{C}_{\Bbbk_{13},h}$ generate $M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)$ as $\mathcal{L}_{t}$-module, for any proper submodule $U$ of $M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)$, $U_{(h)} = U\bigcap M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)_{(h)} = 0$. Hence there exists a maximal proper $\mathcal{L}_{t}$-submodule $T_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)$. Set

Then $L_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)$ is an irreducible $\mathcal{L}_{t}$-module.

Definition 4.3. An $\mathcal{L}_{t}$-module $W$ is said to be restricted if for any $w\in W, n\in {\mathbb Z}$, $L_{n}w = 0$ and $I_{n+\frac{1}{t}}w = 0$ for $n$ sufficiently large. We say an $\mathcal{L}_{t}$-module $W$ is of level $\Bbbk_{13}$ if the central element $k_{i}$ acts as scalar $\Bbbk_{i}$ for $i = 1,3.$

It is easy to see that $M_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)$, $L_{\mathcal{L}_{t}}(\Bbbk_{1}, \Bbbk_{3}, h)$ are restricted $\mathcal{L}_{t}$-modules of level $\Bbbk_{13}$ for any $h\in {\mathbb C}$. Now we are going to relate $\mathcal{L}_{t}$-modules with twisted $V_{\mathcal{L}}(\ell_{123},0)$-modules. On the one hand, we have

Theorem 4.4. If $W$ is a restricted $\mathcal{L}_{t}$-module of level $\ell_{13}$, then $W$ is a $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-module for $V_{\mathcal{L}}(\ell_{123},0)$ as a vertex algebra with

Proof. Let $U_{W} = \{L(z), I_{\sigma_{t}}(z), {\bf 1}_{W}\}$, where ${\bf 1}_{W}$ is the identity operator on $W$. Clearly, $L(z), I_{\sigma_{t}}(z)$ are ${\mathbb Z}_{t}$-twisted weak vertex operators on $W$. From (4.4), (4.5), (4.6), using (2.1), we see that $L(z), I_{\sigma_{t}}(z)$ are mutually local ${\mathbb Z}_{t}$-twisted vertex operators on $W$. Hence, by Corollary 3.15 of [14], $\langle U_{W}\rangle$ is a vertex algebra with $W$ a faithful $\sigma$-twisted module, where $\sigma$ is an order $t$ automorphism of the vertex algebra $\langle U_{W}\rangle$. To say that $W$ is a $\sigma_{t}$-twisted module for $V_{\mathcal{L}}(\ell_{123},0)$, from Proposition 3.17 of [14], it suffices to show that there exists a vertex algebra homomorphism from $V_{\mathcal{L}}(\ell_{123},0)$ to $\langle U_{W}\rangle$.

By Lemma 2.11 of [14], $Y(L(z),z_{1})$ and $Y(I_{\sigma_{t}}(z),z_{1})$ satisfy the twisted Heisenberg-Virasoro algebra relations (3.4), (3.5), (3.6). Then $\langle U_{W}\rangle$ is an $\mathcal{L}$-module with $L_{n}, I_{n}$ acting as $L(z)_{n+1}$, $I_{\sigma_{t}}(z)_{n}$ for $n\in {\mathbb Z},$ $c_{i}$ acting as $\ell_{i}$ with $\ell_{2} = 0$, $i = 1,2,3.$

By the universal property of $V_{\mathcal{L}}(\ell_{123},0)$ (c.f. Remark 2.7 of [12]), there exists a unique $\mathcal{L}$-module homomorphism

Then

for all $v\in V_{\mathcal{L}}(\ell_{123},0)$, $n\in {\mathbb Z}.$ Hence $\psi$ is a vertex algebra homomorphism. Therefore, $W$ is a weak $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-module with $Y_{\sigma_{t}}(L_{-2}{\bf 1}, z) = L(z)$, $Y_{\sigma_{t}}(I_{-1}{\bf 1}, z) = I_{\sigma_{t}}(z).$

Conversely, we have

Theorem 4.5. If $W$ is a $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$ ($\ell_{2} = 0$)-module, then $W$ is a restricted $\mathcal{L}_{t}$-module of level $\ell_{13}$ with $L(z) = Y_{W}(L_{-2}{\bf 1},z)$, $I_{\sigma_{t}}(z) = Y_{W}(I_{-1}{\bf 1},z).$

Proof. Let $W$ be a $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)(\ell_{2} = 0)$-module. Recall the following formula (c.f. (2.40) of [14])

where $k$ is determined by $a$. In our case, when $a = L_{-2}{\bf 1}$, $k = 0$, when $a = I_{-1}{\bf 1}$, $k = 1.$ For $a = b = L_{-2}{\bf 1}$, we have ((2.21) of [12])

so

For $a = L_{-2}{\bf 1}, b = I_{-1}{\bf 1}$, we have ((2.22) of [12] with $\ell_{2} = 0$)

so

For $a = b = I_{-1}{\bf 1}$, we have ((2.23) of [12])

so

Note that for $t\neq 2$, we have to require $\ell_{3} = 0$ (Theorem 3.2). Therefore, with (4.4), (4.5) and (4.6), $W$ is a $\mathcal{L}_{t}$-module with $L(z) = Y_{W}(L_{-2}{\bf 1},z)$, $I_{\sigma_{t}}(z) = Y_{W}(I_{-1}{\bf 1},z)$, $k_{i} = \ell_{i}$, $i = 1,3$. Then $W$ is restricted of level $\ell_{13}$ is clear.

Let

Then $\mathcal{L} = \coprod\limits_{n\in {\mathbb Z}}\mathcal{L}_{t}^{(\frac{n}{t})}$ is a $\frac{1}{t} {\mathbb Z}$-graded Lie algebra, and the grading is given by $\mbox{ad} L_{0}$-eigenvalues.

By Theorem 4.4 and Theorem 4.5 we have the following result.

Theorem 4.6. The $\sigma_{t}$-twisted modules for $V_{\mathcal{L}}(\ell_{123},0)$ ($\ell_{2} = 0$) viewed as a vertex operator algebra (i.e. ${\mathbb C}$-graded by $L_{0}$-eigenvalues and with the two grading restrictions (TW6), (TW7)) are exactly those restricted modules for the Lie algebra $\mathcal{L}_{t}$ of level $\ell_{13}$ that are ${\mathbb C}$-graded by $L_{0}$-eigenvalues and with the two grading restrictions. Furthermore, for any $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-module $W$, the $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-submodules of $W$ are exactly the submodules of $W$ for $\mathcal{L}_{t}$, and these submodules are in particular graded.

Hence irreducible restricted $\mathcal{L}_{t}$-modules of level $\ell_{13}$ corresponds to irreducible $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-modules. Recall that for any $h\in {\mathbb C}$, $L_{\mathcal{L}_{t}}(\ell_{1}, \ell_{3}, h)$ is an irreducible restricted $\mathcal{L}_{t}$-module of level $\ell_{13}$, so it is an irreducible $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-module with $\ell_{2} = 0$.

Now we give the complete list of irreducible $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-modules.

Theorem 4.7. Let $\ell_{2} = 0$, $\ell_{1},\ell_{3}\in\mathbb{C}$. Then $\{L_{\mathcal{L}_{t}}(\ell_{1}, \ell_{3}, h)\ | \ h\in {\mathbb C}\}$ is a complete list of irreducible $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-modules.

Proof. Let $W = \coprod\limits_{r\in {\mathbb C}}W_{(r)}$ be an irreducible $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-module. By Theorem 4.6, $W$ is an irreducible restricted $\mathcal{L}_{t}$-module of level $\ell_{13}$. So $k_{i}$ acts on $W$ as a scalar $\ell_{i}$ for $i = 1,3$. From Remark 2.5, there exists $h\in {\mathbb C}$ such that $W_{(h)}\neq 0$ and $W_{(h-n)} = 0$ for all $n\in\frac{1}{t} {\mathbb Z}_{\geq 1}$. Let $0\neq w\in W_{(h)}$. Then

for $m\geq 1, n\geq 0$. In view of Remark 4.2, there is a unique $\mathcal{L}_{t}$-module homomorphism

such that $\varphi({\bf 1}_{\Bbbk_{13},h}) = w$. By Proposition 2.8, $\varphi$ is a $\sigma$-twisted $\langle U_{W}\rangle$-module homomorphism (since $\mathcal{L}_{t}$ generates the vertex algebra $\langle U_{W}\rangle$), where $\sigma$ is an order $t$ automorphism of the vertex algebra $\langle U_{W}\rangle$. Recall that $M_{\mathcal{L}_{t}}(\ell_{1}, \ell_{3}, h)$ is a weak $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-module via the vertex algebra homomorphism $\psi$ in Theorem 4.4. So $\varphi$ can be viewed as a $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-module homomorphism. Since $W$ is irreducible and $T_{\mathcal{L}_{t}}(\ell_{1}, \ell_{3}, h)$ is the (unique) largest proper submodule of $M_{\mathcal{L}_{t}}(\ell_{1}, \ell_{3}, h)$, it follows that

and

Thus $\varphi$ reduces to a $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$-module isomorphism from $L_{\mathcal{L}_{t}}(\ell_{1}, \ell_{3}, h)$ to $W$.

It is interesting and important to classify the irreducible modules for the fixed point subalgebra $V_{\mathcal{L}}(\ell_{123},0)^{\sigma_{t}}: = \{v\in V_{\mathcal{L}}(\ell_{123},0) \ | \ \sigma_{t}(v) = v \}$, $t\in {\mathbb Z}_{\geq 1}$. In the case of $\ell_{2} = 0$ and $\ell_{3}\neq 0$, we have $t = 2$. Then $\sigma_{t}$ is the order 2 automorphism of $V_{\mathcal{L}}(\ell_{123},0)$ which is induced from its Heisenberg vertex operator subalgebra. Denote by $\sigma_{2} = \sigma$. Precisely, the automorphism

is defined on the basis elements by

and extended linearly, where $r,s\geq 1$, $m_{1}\geq\cdots\geq m_{r}\geq 2$, $k_{1}\geq\cdots\geq k_{s}\geq 1$. Let

be the fixed point subalgebra under $\sigma$.

For $\ell_{3}\neq 0$, denote by

Let $V_{\mathcal{H}}(\ell_{3},0)$ be the vertex operator algebra constructed from the Heisenberg subalgebra $\mathcal{H}$ which is equipped with the nonstandard conformal vector $\omega_{H} = \frac{1}{2\ell_{3}}I_{-1}I_{-1}{\bf 1} + \frac{\ell_{2}}{\ell_{3}}I_{-2}{\bf 1}$ (of central charge $1-12 \frac{\ell_{2}^{2}}{\ell_{3}}$). Let $\tilde{Vir}$ be the Virasoro algebra constructed by $\tilde{\omega} = \omega-\omega_{H}$. Let $V_{\tilde{Vir}}(c_{\tilde{Vir}},0)$ be the corresponding Virasoro vertex operator algebra. Recall that when $\ell_{3}\neq 0$, we have (cf. Theorem 3.16 of [12])

as vertex operator algebras.

It is well known that there exists an order 2 isomorphism of the Heisenberg vertex operator algebra $V_{\mathcal{H}}(\ell_{3},0)$. Let again

be the order 2 automorphism which is defined on the basis elements by

and extended linearly, where $s\geq 1$, $k_{1}\geq\cdots\geq k_{s}\geq 1$. The fixed point subalgebra $V_{\mathcal{H}}(\ell_{3},0)^{+} = \{v\in V_{\mathcal{H}}(\ell_{3},0) \ | \ \sigma(v) = v\}$ has beed extensively studied (cf. [6] etc.).

Then it is immediate to see that when $\ell_{3}\neq 0$ and $\ell_{2} = 0$, we have an isomorphism of vertex operator algebras

Up to isomorphism, irreducible modules for the vertex operator algebra $V_{\mathcal{H}}(\ell_{3},0)^{+}$ are (cf. [6], etc.) $V_{\mathcal{H}}(\ell_{3},0)^{\pm}, V_{\mathcal{H}}(\ell_{3},0)(\sigma)^{\pm},V_{\mathcal{H}}(\ell_{3},\lambda)\cong V_{\mathcal{H}}(\ell_{3},-\lambda)$ for any $0\neq \lambda\in {\mathbb C}.$ Up to isomorphism, irreducible modules for the Virasoro vertex operator algebra $V_{\tilde{Vir}}(c_{\tilde{Vir}},0)$ are $L_{\tilde{Vir}}(c_{\tilde{Vir}},h)$ for all $h\in {\mathbb C}$ (cf. [9], [16], etc.). Therefore, in the case of $\ell_{2} = 0$ and $\ell_{3}\neq 0$, irreducible modules of $V_{\mathcal{L}}(\ell_{123},0)^{+}$ are one-to-one correspond to the tensor product of irreducible modules of $V_{\mathcal{H}}(\ell_{3},0)^{+}$ and irreducible modules of $V_{\tilde{Vir}}(c_{\tilde{Vir}},0)$ (cf. Proposition 4.7.2 and Theorem 4.7.4 of [7]).

For other automorphism $\sigma_{t}$, the complete list of irreducible $V_{\mathcal{L}}(\ell_{123},0)^{\sigma_{t}}$-modules remains to be investigated.