On ${\mathbb {Z}}_p$-orbifold constructions of the Moonshine vertex operator algebra

Abstract

For $p = 3,5,7,13$, we consider a ${\mathbb {Z}}_p$-orbifold construction of the Moonshine vertex operator algebra $V^\natural$. We show that the vertex operator algebra obtained by the ${\mathbb {Z}}_p$-orbifold construction on the Leech lattice vertex operator algebra $V_\Lambda$ and a lift of a fixed-point-free isometry of order p is isomorphic to the Moonshine vertex operator algebra $V^\natural$. We also describe the relationship between those ${\mathbb {Z}}_p$-orbifold constructions and the ${\mathbb {Z}}_2$-orbifold construction in a uniform manner. In Appendix, we give a characterization of the Moonshine vertex operator algebra $V^\natural$ by the existence of an orthogonal pair of Ising vectors.

Appendix A: A characterization of the Moonshine VOA

In this appendix, we give another characterization of the Moonshine VOA $V^\natural$ using Ising vectors. It also provides an alternative proof that $\widetilde{V}_{\Lambda ,\tau } \cong V^\natural$ for $p=3$ and 5. The main theorem is as follows.

Theorem A.1

Let V be a simple,  rational,  $C_2$-cofinite,  holomorphic VOA of CFT-type with central charge 24 such that the weight 1 subspace $V_1 = 0$. If there is an orthogonal pair of Ising vectors of V,  then V is isomorphic to the Moonshine VOA $V^\natural$.

The idea is essentially the same as in [25] and is also similar to that in Sect. 4. We try to obtain the Leech lattice VOA $V_\Lambda$ by using some ${\mathbb {Z}}_2$-orbifold construction on V. The extra assumption on Ising vectors is used to define an involution on V such that the corresponding twisted module has conformal weight one.

An element $e\in V_2$ is called an Ising vector if the vertex subalgebra $\mathrm {Vir}(e)$ generated by e is isomorphic to the simple Virasoro VOA L(1 / 2, 0) of central charge 1 / 2. Let $V_e(h)$ be the sum of all irreducible $\mathrm {Vir}(e)$-submodules of V isomorphic to L(1 / 2, h) for $h=0,1/2,1/16$. Then one has an isotypical decomposition:

$$\begin{aligned} V=V_e(0)\oplus V_e\left( \frac{1}{2}\right) \oplus V_e\left( \frac{1}{16}\right) . \end{aligned}$$

Recall from [28] that the linear automorphism $\tau _e$ which acts as 1 on $V_e(0)\oplus V_e(\frac{1}{2})$ and $-1$ on $V_e(\frac{1}{16})$ defines an automorphism of the VOA V. On the fixed point subVOA $V^{\langle \tau _e\rangle }=V_e(0)\oplus V_e(\frac{1}{2})$, the linear automorphism $\sigma _e$ which acts as 1 on $V_e(0)$ and $-1$ on $V_e(\frac{1}{2})$ also defines an automorphism on $V^{\langle \tau _e\rangle }$.

Let e and f be two Ising vectors in V. If $e_{(i)}f=0$ for any $i\in {\mathbb {Z}}_{\ge 0}$ then it is said that e and f are orthogonal. We take an orthogonal pair (e, f) of Ising vectors, and let U be the subVOA generated by e and f. Then

$$\begin{aligned} U = \mathrm {Vir}(e) \otimes \mathrm {Vir}(f) \cong L\left( \frac{1}{2},0\right) \otimes L\left( \frac{1}{2},0\right) . \end{aligned}$$

For any $h_1, h_2\in \{0, 1/2, 1/16\}$, we define the space of multiplicities of the irreducible U-module $L(\frac{1}{2}, h_1) \otimes L(\frac{1}{2},h_2)$ in V by

$$\begin{aligned} M(h_1, h_2)=\hom _{U}\left( L\left( \frac{1}{2}, h_1\right) \otimes L\left( \frac{1}{2},h_2\right) ,V\right) . \end{aligned}$$

Then we have the isotypical decomposition

$$\begin{aligned} V= \bigoplus _{h_1, h_2\in \{ 0, \frac{1}{2}, \frac{1}{16}\}} L\left( \frac{1}{2}, h_1\right) \otimes L\left( \frac{1}{2},h_2\right) \otimes M(h_1, h_2). \end{aligned}$$

Notice that $M(0,0) =\mathrm {Com}_V(U) = U^c$ is a subVOA of central charge 23 and $M(h_1, h_2)$, $h_1, h_2\in \{0, 1/2, 1/16\}$, are M(0, 0)-modules. Note also that

$$\begin{aligned} (V^{\langle \tau _e, \tau _f\rangle })^{\langle \sigma _e, \sigma _f\rangle } = L\left( \frac{1}{2}, 0\right) \otimes L\left( \frac{1}{2},0\right) \otimes M(0, 0). \end{aligned}$$

Proposition A.2

The subVOA M(0, 0) is $C_2$-cofinite and rational. Moreover,  $M(h_1, h_2),$ $h_1, h_2\in \{0, 1/2\},$ are simple current modules of M(0, 0).

Proof

Let $E=\langle \tau _e, \tau _f\rangle \subset {{\mathrm{Aut}}}V$ be the subgroup generated by the Miyamoto involutions $\tau _e$ and $\tau _f$. Then E is elementary abelian of order 4 and the fixed point subVOA is

$$\begin{aligned} V^E = \bigoplus _{h_1, h_2\in \{0, \frac{1}{2}\}} L\left( \frac{1}{2}, h_1\right) \otimes L\left( \frac{1}{2},h_2\right) \otimes M(h_1, h_2). \end{aligned}$$

Then $V^E$ is $C_2$-cofinite and rational by a result of Carnahan and Miyamoto [3, 31].

Let $S=\langle \sigma _e, \sigma _f\rangle \subset {{\mathrm{Aut}}}V^E$. Then S is also elementary abelian and hence the fixed point subVOA

$$\begin{aligned} W= (V^E)^S = L\left( \frac{1}{2}, 0\right) \otimes L\left( \frac{1}{2},0\right) \otimes M(0, 0) \end{aligned}$$

is $C_2$-cofinite and rational. Therefore, M(0, 0) is also $C_2$-cofinite and rational.

That $M(h_1, h_2)$, $h_1, h_2\in \{0, 1/2\}$, are simple current modules of M(0, 0) follows from the fact that $L(\frac{1}{2}, h_1) \otimes L(\frac{1}{2},h_2)\otimes M(h_1, h_2)$ are common eigenspaces of S on $V^E$ and [8, Remark 6.4]. $\square$

Notation A.3

For $i,j\in \{0,1\}$, let $V^{(i,j)} = \{v\in V\mid \tau _e v=(-1)^iv, \tau _f v=(-1)^jv\}$. Then

$$\begin{aligned} V^{\left( 0,0\right) }= & {} V^E = \bigoplus _{h_1, h_2\in \{0, \frac{1}{2}\}} L\left( \frac{1}{2}, h_1\right) \otimes L\left( \frac{1}{2},h_2\right) \otimes M\left( h_1, h_2\right) ,\\ V^{\left( 1,0\right) }= & {} L\left( \frac{1}{2}, \frac{1}{16}\right) \otimes L\left( \frac{1}{2},0\right) \otimes M\left( \frac{1}{16}, 0\right) \oplus L\left( \frac{1}{2}, \frac{1}{16}\right) \otimes L\left( \frac{1}{2},\frac{1}{2}\right) \otimes M\left( \frac{1}{16}, \frac{1}{2}\right) , \\ V^{\left( 0,1\right) }= & {} L\left( \frac{1}{2},0\right) \otimes L\left( \frac{1}{2}, \frac{1}{16}\right) \otimes M\left( 0, \frac{1}{16}\right) \oplus L\left( \frac{1}{2},\frac{1}{2}\right) \otimes L\left( \frac{1}{2}, \frac{1}{16}\right) \otimes M\left( \frac{1}{2},\frac{1}{16}\right) , \\ V^{\left( 1,1\right) }= & {} L\left( \frac{1}{2}, \frac{1}{16}\right) \otimes L\left( \frac{1}{2}, \frac{1}{16}\right) \otimes M\left( \frac{1}{16}, \frac{1}{16}\right) . \end{aligned}$$

Notice that $V^{(i,j)}, i,j\in \{0,1\},$ are simple current modules of $V^E$ [8].

Lemma A.4

Let V be a simple,  rational,  $C_2$-cofinite,  holomorphic VOA of CFT type with central charge 24 such that $V_1=0$. Assume that there exists an orthogonal pair of Ising vectors. Then $M(h_1, h_2)\ne 0$ for any $h_1, h_2\in \{0,\frac{1}{2}, \frac{1}{16}\}$.

Proof

Recall $U = \mathrm {Vir}(e) \otimes \mathrm {Vir}(f)\cong L(\frac{1}{2}, 0) \otimes L(\frac{1}{2},0)$. Then the double commutant $(U^c)^c$ is an extension of U. Note that there is only one non-trivial extension of U, which is isomorphic to $L(\frac{1}{2}, 0) \otimes L(\frac{1}{2},0) \oplus L(\frac{1}{2}, \frac{1}{2}) \otimes L(\frac{1}{2},\frac{1}{2})$ and the weight one subspace is non-zero. Hence $(U^c)^c=U$, for $V_1=0$. Therefore, by a result of Krauel and Miyamoto [22], all irreducible modules of U must appear as a submodule of V since V is holomorphic and U and M(0, 0) are $C_2$-cofinite and rational. It implies $M(h_1, h_2)\ne 0$ for any $h_1, h_2\in \{0,\frac{1}{2}, \frac{1}{16}\}$. $\square$

From now on, let V be as in Lemma A.4. The following two results follow immediately from the general arguments on simple current extensions [26, 36].

Lemma A.5

Let $M= L(\frac{1}{2}, \frac{1}{2}) \otimes L(\frac{1}{2}, \frac{1}{2}) \otimes M(0, 0)$ and set

$$\begin{aligned} \widetilde{M} = \bigoplus _{h_1, h_2\in \{0, \frac{1}{2}\}} L\left( \frac{1}{2}, \frac{1}{2}-h_1\right) \otimes L\left( \frac{1}{2}, \frac{1}{2}-h_2\right) \otimes M(h_1, h_2). \end{aligned}$$

Then $\widetilde{M}$ is an irreducible module of $V^E$.

Theorem A.6

Let $t=\tau _e\tau _f$ and set

$$\begin{aligned} X=\bigoplus _{i,j \in \{0,1\}} V^{(i,j)} \boxtimes _{V^E} \widetilde{M}. \end{aligned}$$

Then X is an irreducible t-twisted module of V.

Define

$$\begin{aligned} \widetilde{V}= V^{\langle t\rangle }\oplus X^{\langle t\rangle }, \end{aligned}$$

where $X^{\langle t\rangle }$ is the irreducible $V^{\langle t\rangle }$-submodule of X which has integral weights. Then $\widetilde{V}$ is a simple, rational, $C_2$-cofinite, holomorphic VOA of CFT-type. Notice that the conformal weight of $M= L(\frac{1}{2}, \frac{1}{2}) \otimes L(\frac{1}{2}, \frac{1}{2}) \otimes M(0, 0)$ is 1 and M is an $L(\frac{1}{2}, 0) \otimes L(\frac{1}{2}, 0) \otimes M(0, 0)$-submodule of $X^{\langle t \rangle }$. Hence, $(X^{\langle t\rangle })_1\ne 0$ and $(\widetilde{V})_1\ne 0$. Since $V_1=0$, we have $\tilde{V}_1= (X^{\langle t\rangle })_1$ and hence the Lie algebra on $\tilde{V}_1$ is abelian. Thus we have the following theorem.

Theorem A.7

The VOA $\widetilde{V}$ is isomorphic to the Leech lattice VOA $V_\Lambda$.

Now we are ready to prove our main theorem.

Proof of Theorem A.1

By Theorem A.7, we know that the VOA $\widetilde{V}$ is isomorphic to the Leech lattice VOA $V_\Lambda$. Let g be the automorphism of $\widetilde{V}$ which acts as 1 on $V^{\langle t\rangle }$ and $-1$ on $X^{\langle t\rangle }$. Then g is conjugate to the lift $\theta$ of the $-1$ map on $\Lambda$ since g acts on $\widetilde{V}_1$ as $-1$ (cf. Theorem 4.3). Therefore, we have $\widetilde{V}^{\langle g\rangle }=V^{\langle t\rangle }\cong V_\Lambda ^+$. Then by the same argument as in Theorem 4.5, we have

$$\begin{aligned} V\cong V_\Lambda ^+ \oplus V_\Lambda ^{T,+}\cong V^\natural \end{aligned}$$

as $V_\Lambda ^+$-modules. Then by the uniqueness of simple current extensions, we can establish the desired isomorphism between V and $V^\natural$. $\square$

Remark A.8

Recall that the Leech lattice $\Lambda$ contains a sublattice isometric to $\sqrt{2}E_8^{\oplus 3}$. For $p=3, 5$, we can choose a fixed-point-free isometry $\tau$ of order p such that each direct summand of $\sqrt{2}E_8^{\oplus 3}$ is stabilized; indeed, $\sqrt{2}E_8^{\oplus 3}$ contains $\sqrt{2}A_2^{\oplus 12}$ and $\sqrt{2}A_4^{\oplus 6}$ as sublattices and the fixed-point-free isometry of $\Lambda$ of order 3 (resp. 5) can be induced by the Coxeter element of $A_2$ (resp. $A_4$). Thus, we have $(V_{\sqrt{2}E_8}^{\langle \tau \rangle } )^{\otimes 3} \subset V_{\Lambda }^{\langle \tau \rangle }$. Let $\theta \in {{\mathrm{Aut}}}V_{\sqrt{2}E_8}$ be a lift of the $-1$-isometry of $\sqrt{2}E_8$. Then $\theta$ and $\tau$ commutes. Since $V_{\sqrt{2}E_8}^{\langle \theta \rangle }$ has exactly 496 Ising vectors [23, Proposition 4.3] and 496 is relatively prime to p, there exists an Ising vector in $V_{\sqrt{2}E_8}^{\langle \theta \rangle }$ fixed by $\tau$. Hence $V_{\Lambda }^{\langle \tau \rangle }$ contains a pair of (in fact, three) orthogonal Ising vectors. By Theorem A.1, we have $\widetilde{V}_{\Lambda , \tau } \cong V^\natural$, also.