数理解析研究所講究録904 巻 1995 年 78-86

78

MODULAR FORMS ASSOCIATED WITH THE MONSTERMODULE
KOICHIRO HARADA AND MONG LUNG LANG

1. INTRODUCTION

In Harada-Lang [4], we associated to each irreducible charactermonster simple group
Thompson series forseries$t_{g}(z)$$\chi$$\mathrm{M}$.

a modular function$t_{\chi}(z)$for the element$g$$t_{\chi}(z)$$g$ranges over$t_{\chi}(z)= \frac{1}{|\mathrm{M}\mathrm{I}|}\sum_{g\in \mathrm{M}}\chi(g)t(gz)$

If$\Gamma_{\chi}$is the invariance subgroup of$t_{\chi}(z)$of the, called in [4], the McKay-

is a weighted average of allof MI as$\chi$$\mathrm{M}$$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{a}\mathrm{y}$-Thompson:.

, then we showed$\Gamma_{\chi}=^{\mathrm{r}}0(N)x=\cap g\in \mathrm{M}\mathrm{r}_{g}$

where

$g$

ranges over all the elements of$N_{\chi}=1\mathrm{c}\mathrm{m}${$n_{g}h_{g}$: for all$\mathrm{M}$$g\in$

such that $\chi(g)\neq 0$ andNI with $\chi(g)\neq 0$ }.

As shown in Conway-Norton [1], the invariance groupsubgroup of index$h$$\Gamma_{g}$of$t_{g}(z)$is a certainof the conjugate byof$\Gamma_{0}(\frac{n}{h})+e,$$f,$ $\cdots$

This is a preliminary version. A full version with a table will be published elsewhere.

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KOICHIRO HARADA AND MONG LUNG LANG

where$e,$$f$, etc. denote the Atkin-Lehner involutions. In [1], such a conjugate

is denoted by$n|h+e,$ $f,$

The numbers $n,$$t_{\chi}(z)$$h$$\cdots$.

depend on , hence our notation$g$$n_{g},$$h_{g}$. Obviously every

is invariant by$\bigcap_{g\in \mathrm{M}}\Gamma_{g}=^{\mathrm{r}_{0}}(N\mathrm{o})$where $N_{0}=2^{6}3^{3}5^{2}7,11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71\sim 10^{21}$ . The
level$N_{\chi}$can be very large or relatively small. For example,$N_{x1}=N_{0},$ $N_{x16}6=2^{6}3^{3}7=4032$

where $\chi_{1}=1$ is the trivial character and the character numbering such as$\chi_{166}$is taken from the Atlas. In this paper, we will investigate the relation between$t_{\chi}(z)$and the generating functions of the highest weight vectors (also called

singular vectors, primary fields or lowest weight vectors.)

2. THE MONSTER MODULE AS A $Vir$ MODULE
The monster module V is constructed in Frenkel-Lepowsky-Meurman [3] as avertex operator algebra and is denoted byalgebra. Then$V$$\mathrm{V}^{\mathfrak{h}}$there. Let$V$

be a vertex operator

possesses two distinguished elements 1 andvacuum and the conformal vector (or the Virasoro element) of$V$$\omega$, called the, respectively.

If $Y( \omega, z)=\sum\omega_{n}z^{-n-1}$ is the vertex operator corresponding to the conformal vector$\omega$and if we set $L(n)=\omega_{n+1}$ for$n\in \mathbb{Z}$, then $L(n)$ satisfies thecommutation relation:$[L(n), L(m)]=(n-m \mathrm{I}^{L_{n}}+m+\frac{1}{12}(n^{3}-n)c\delta n+m,0$80

MCKAY-THOMPSON SERIES

where$c$

is a constant called the central charge of

V, $c=24$ .

Let$\mathcal{L}$

$c$

$V$

. For the monster moduleis also called the rank of the vertex operator algebrabe the Lie algebra generated by $L(n),$ $n\in$ Z.

else where. The subalgebras$L(n),$ $n\in \mathbb{Z}^{-}$ ,$L^{+}$

and$\mathcal{L}$.

is denoted by $Vir$

are generated by $L(n),$$\mathcal{L}^{-}$$V$$n\in \mathbb{Z}^{+}$

andrespectively. It is known that V possesses a positive definite

invariant bilinear form and so V is completely reducible as an$\mathcal{L}$module andis a sum of highest weight modules.
Let $M(h, c)$ be the Verma module of the Virasoro algebra of central charge$c$generated by the highest weight vector$M(h, c)=\mathcal{L}v,$$v$

of height$\mathcal{L}^{+}v=0,$$h$: i.e.

$L(\mathrm{O})v=hv$.

The module structure of $M(h, c)$ has been determined by Feigin-Fuchs [2].

We will use their results to determine the module structure of V as an. Feigin-Fuchs showed that$\overline{111}\mathrm{o}\mathrm{d}\mathrm{u}^{1}1\mathrm{e}$$\mathrm{e}\overline{\backslash }\overline{\prime}\mathrm{e}\iota\overline{\mathrm{y}}$submodule of$M_{(}’h,$$C\grave{)}$$\mathcal{L}$is a sum of

submodules that are also Verma modules. Therefore, the knowledge of allembeddings among Verma modules gives all submodules of a given Vermamodule. The main theorem of Feigin-Fuchs states that there are six types of

embeddings of the Verma modules into other Verma modules. Let$p\alpha-q\beta=m$

(1)

$\{$

$c=2 \frac{(3p-2q)(3q-2p)}{q)^{2}pq}h=\frac{m^{2}-(p-4=}{4pq}$

where

$p,$

$q$

and $m$ are complex variables. We next solve for integers

Let$\epsilon=\frac{-11\pm i\sqrt{23}}{2}$,$\overline{\epsilon}=\frac{-11\mp i\sqrt{23}}{2}$$\alpha$and$\beta$.

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KOICHIRO HARADA AND MONG LUNG LANG

We compute$\epsilon\overline{\epsilon}=1,$$\epsilon+\overline{\epsilon}=\frac{-11}{6},$$\epsilon^{2}+\overline{\epsilon}^{2}=\frac{49}{36}$.

Using the second equality of (1), we obtain$(p\alpha-q\beta)22==m4pq+(q-p)2$ ,which may be rewritten as$(\alpha-\epsilon\beta)2h=4\epsilon+(\epsilon-1)2$.

We therefore obtain two equations :$\alpha^{2}-2\epsilon\alpha\beta+\epsilon^{2}\beta 2=4\epsilon h+(\epsilon-1)2$,$\alpha^{2}-2\overline{\epsilon}\alpha\beta+\overline{\epsilon}^{2}\beta 2=4\overline{\epsilon}h+(\overline{\epsilon}-1)2$.

andTaking the sum of them, we get$72\alpha^{2}+132\alpha\beta+49\beta^{2}=-264h+253$.

By subtracting one from the other, we get$-12\alpha\beta-11\beta^{2}=24h-23$.

Therefore$\alpha^{2}-\beta^{2}=0$or$\alpha=\pm\beta$. Setting$\alpha=\delta\beta$

with$\delta=\pm 1$,, we have$\beta^{2}=\frac{24h-1}{11-12\delta}$.

If $h=0$ , then we must have $\delta=1$ and so $\alpha=\beta=\pm 1$ . In particular,$\alpha\beta=1>0$. On the other hand, if$h\in \mathbb{Z}^{+}$, then $\delta=-1$ and so $\alpha=-\beta=\pm 1$ ,82

MCKAY-THOMPSON SERIES

and hence $\alpha\beta=-1<0$ . Using the results of Feigin-Fuchs [2], we conclude(which must be well known to experts) :Theorem.
$M(1,24)$ .

$M(0,24)$has a unique submodule, which is isomorphic toFor all positive integers$M(h, 24)$$h,$

is irreducible.

Let $L(c, h)$ be the unique irreducible highest weightcharge$c$and height$h$$\mathcal{L}$-module of central. ThenCorollary. We have(1). $L(\mathrm{O}, 24)=M(0,24)/M(1,24),$(2). $L(h, 24)=M(h, 24)$ if$h\in \mathbb{Z}^{+}$$and_{f}$.

Let us now express the monster module V as a sum of$\mathrm{V}=\sum_{h=0}S_{h}L(h, 24)\infty$Then$h_{\nu},$$s_{h}$’$L(h, 24)_{\mathrm{S}}$.

is the number of linearly independent singular vectors$\mathrm{R}\mathrm{h}\mathrm{n}\underline{\mathrm{r}\rho}v_{n}|\in \mathrm{V}_{h}$the monster$\mathrm{M}$.

$\underline{\mathrm{S}}\mathrm{i}.\mathfrak{n}\mathrm{A}$the$\mathrm{V}\mathrm{i}\mathrm{r}\wedge \mathrm{a}\mathrm{S}\mathrm{o}\mathrm{r}\mathrm{o}\underline{\mathrm{a}}\mathrm{l}\mathrm{g}\mathrm{e}\underline{\mathrm{b}}\mathrm{r}\mathrm{a}L$, we can actually split$s_{h}$

corresponds to the irreducible character$v_{h}$of height

commutes with the action of

into the sum of$\chi_{k}$as follows$s_{h}^{k}$where the index$k$. More precisely, let$\mathrm{V}_{h}^{k}=C_{h}kxk$

where$c_{hk}$is the multiplicity of$\chi_{k}$in$\mathrm{V}_{h}$and$\mathrm{V}^{k}=\coprod_{=h0}^{\infty}\mathrm{V}^{k}h$Thus$\mathrm{V}^{k}$is anisomorphic to$\mathrm{M}$$\chi_{k}$.

submodule of V consisting entirely of irreducible submodules
and$\mathrm{V}_{h}^{k}$is an[ submodule of$\mathrm{M}$of height . We also define$h$$\mathrm{V}^{k}$$W_{h}^{k}=\mathcal{L}(\coprod_{h0\leq i<}\mathrm{V}^{k}i)\cap \mathrm{V}^{k}h$’83

KOICHIRO HARADA AND MONG LUNG LANG

which is an$\mathrm{M}$submodule of$\mathrm{V}_{h}^{k}$that is generated by elements of lower heights.

Let$s_{h}^{k}=\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{V}_{h}^{k}/W_{h}^{k}$Then$s_{h}^{k}$.

is the number of linearly independent singular vectors in$\mathrm{V}_{h}^{k}$. Obvi-ously$s_{h}= \sum_{k=1}^{94}S^{k}1h$

For a graded module$X=\Sigma_{h\in \mathbb{Z}}X_{h}$.

, the character of $X$ (or the partition

function of $X$ ) is defined to be a formal sum

char$(X)= \sum_{h\in \mathbb{Z}}\dim X_{h}Xh$.

Using this notation, we have, as is well known,
char $M(h, c)=x^{h} \sum_{n\geq 0}p(n)x^{n}$where $p(n)$ is the partition function of . For convenience, set$n$$p(n)=0$ if$n\in \mathbb{Z}^{-}$. Let us consider the$\mathcal{L}$$p(\mathrm{O})=1$, andsubmodule generated by the vacuum

1. We have $V_{1}=0$ but the height 1 component of$M(\mathrm{O}, 24)$

is nonzero, weconclude that$\mathcal{L}\cdot 1\simeq M(0,24)/M(1,24)$

.

Hence
char $( \mathcal{L}\cdot 1)=\sum_{n\geq 0}p(n)x^{n}-xn\sum p(n)X=\sum_{n\geq 0\geq^{0}}n(p(n)-p(n-1))_{X^{n}}$ .

Writing$\mathrm{C}\mathrm{h}_{\mathrm{a}\mathrm{r}}(\mathcal{L}\cdot 1)--\sum_{\geq h0}a_{h}1Xh$,

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MCKAY-THOMPSON SERIES

we get a partial list:
2 3 4 5 6 7 8 9 10 11
1 1 1 2 2 4 4 7 8 12 14$h$$0$$a_{h1}$

In [4], we had a partial list ofcharacter$\chi_{1}$occuring in$\mathrm{V}_{h}$$c_{h1}$where$c_{h1}$is the multiplicity of the trivial.

2 3 4 5 6 7 8 9 10 11
1 1 1 2 2 4 4 7 8 12 14$h$$0$$c_{h1}$

The coincidence $c_{h1}=a_{h1}$ stops there and we have$h$$a_{h1}$$c_{h1}$This means$11$$s_{12}^{1}=1$, namely,, do not. The number$\mathrm{V}_{h}^{1}(0\leq h\leq 30)$$d$$\mathrm{V}_{12}^{1}$

1221
22

contains a singular vector, while$\mathrm{V}_{h}^{1},0<h\leq$of linearly independent singular vectors occuring in

is as follows

$h$

$d$

12 16 18 20 22 24 26 27 28 29 30
1 1 1 1 1 3 2 1 4 2 6

We are now lead to consider its generating function for each$k,$$1\leq k\leq 194$ .

Define$G^{k}(x)= \sum_{h\geq 0}s^{k}x^{h}h$

The character of$\mathrm{V}^{k}$ischarwhere$t_{\chi}(z)$is the.

$( \mathrm{V}k)=\sum c_{h}kh\geq 0(\deg x_{k})X^{h}=x\deg\chi_{k}t_{\chi}(x)$$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{a}\mathrm{y}$-Thompson series for the irreducible characterthe other hand, using the expression$\mathrm{V}^{k}=\sum_{\geq h0}S_{h}L(h, 24)k$,$\chi$. On

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KOICHIRO HARADA AND MONG LUNG LANG

we obtainchar

Suppose $k>0$ . Then$( \mathrm{v}k)=\sum_{h\geq 0}S_{h}\mathrm{C}\mathrm{h}k\mathrm{r}\mathrm{a}L(h, 24)$$s_{0}^{k}=0$

char.

and so$( \mathrm{V}^{k})=\sum_{h\geq 1}s_{h}^{k}x^{h}n\geq 0\sum p(n)x^{n}$

On the other hand if $k=1$ , then$L(\mathrm{O}, 24)$.

occurs only once as a constitient of. Therefore$\mathrm{V}^{1}$char $( \mathrm{V}^{1})=(1-X+\sum Sh\geq 2h1X^{h})\sum_{0n\geq}p(n)x^{n}$.

Using the Dedekind$\mathrm{e}\mathrm{t}\mathrm{a}$-function and replacing$x$

by $q=e^{2\pi iz}$ , we obtain, bysetting $s_{1}^{1}=-1$ for convenience,$\deg\chi kt_{\chi k}(q)=\frac{q^{-1}(\Sigma_{h\geq h}0^{S}q^{h})kq\frac{1}{24}}{\eta(q)}$

.

Hence
$\deg x_{k}t_{\chi k}(q)\eta(q)=q-\frac{23}{24}\sum_{\geq h0}s_{h}q^{h}k$,which implies$q^{-\frac{23}{24}}G^{k}(q)=\deg\chi_{k}t_{\chi}(kq)\eta(q)$where as defined beforein$\mathrm{V}^{k}$. WritingTheorem.
$N_{\chi}$$G^{k}(q)$$G^{k}=G^{\chi}$$q^{-\frac{23}{24}G^{\chi}}(q)$is the generating function of the singular vectors

in general, we obtain :is a meromorphic modular form of weight$\frac{1}{2}$and level.

Corollary.

12 and level$q^{-\frac{23}{24}}G^{x}(q)\eta(q)^{2}3$$N_{\chi}$.

is a holomorphic modular function of weight

86

MCKAY-THOMPSON SERIESREFERENCES

1. Conway, J. H., Norton, S. P., Monstrous Moonshine, Bull. London Math. Soc.11 (1979),

308-338.
2. Feigin, B.L., Fuchs, D.B., Verma Modules over the Virasoro Algebras, Lecture Notes
Math., 1060 (1984).

3. Frenkel, I., Lepowsky, J., Meurman, A., Vertex Operator Algebras and the Monster,
Academic Press, Inc. (1988).

4. Harada, K., Lang, M.L., The$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{a}\mathrm{y}$-Thompson Series associated with the Irreducible

Characters of the Monster, to appear.

Department of Mathematics,
The Ohio State University,
Columbus, Ohio, 43210

U.S.A

Department of Mathematics,
National University of Singapore,

Singapore, 0511
Republic of Singapore