%This is a plain tex paper.
\magnification=\magstep1
\vbadness=10000
\hbadness=10000
\tolerance=10000
\def\R{{\bf R}} % real numbers
\def\Z{{\bf Z}} % integers
\def\Q{{\bf Q}} % rational numbers
\def\C{{\bf C}} % complex numbers
\def\Ad{{\rm Ad}}
\def\mult{{\rm mult}}
\def\Aut{{\rm Aut}}
\def\q{***}
\proclaim Automorphic forms and Lie algebras. 30 Sept 1996
Richard E. Borcherds,\footnote{$^*$}{ Supported by a Royal Society
professorship and by NSF grant
DMS9401186.}
D.P.M.M.S.,
16 Mill Lane,
Cambridge,
CB2 1SB,
England.
email: reb@pmms.cam.ac.uk
home page: http://www.dpmms.cam.ac.uk/\hbox{\~{}}reb
\bigskip
\proclaim 0.~Introduction.
This paper is mainly an advertisement for one particular Lie algebra
called the fake monster Lie algebra. The justification for looking at
just one rather obscure object in what is supposed to be a general
survey is that the fake monster Lie algebra has already led directly
to the definition of vertex algebras, the definition of generalized
KacMoody algebras, the proof of the moonshine conjectures, a new
family of automorphic forms, and it is likely that there is still more
to come from it.
The reader may wonder why we do not start by looking at a simpler
example than the fake monster Lie algebra. The reason is that the fake
monster is the simplest known example in the theory of nonaffine Lie
algebras; the other examples are even worse.
Here is a quick summary of the rest of this paper. The first sign of
the existence of the fake monster came from Conway's discovery that
the Dynkin diagram of the lattice $II_{25,1}$ is essentially the Leech
lattice. We explain this in section 1. For any Dynkin diagram we can
construct a KacMoody algebra, and a first approximation to the fake
monster Lie algebra is the KacMoody algebra with Dynkin diagram the
Leech lattice. It turns out that to get a really good Lie algebra we
have to add a little bit more; more precisely, we have to add some
``imaginary simple roots'', to get a generalized KacMoody
algebra. Next we can look at the ``denominator function'' of this fake
monster Lie algebra. For affine Lie algebras the denominator function
is a Jacobi form which can be written as an infinite product [K,
chapter 13]. For the fake monster Lie algebra the denominator
function turns out to be an automorphic form for an orthogonal group,
which can be written as an infinite product. There is an infinite
family of such automorphic forms, all of which have explicitly known
zeros. (In fact it seems possible that automorphic forms constructed
in a similar way account for all automorphic forms whose zeros have a
``simple'' description.) Finally we briefly mention some connections
with other areas of mathematics, such as reflection groups and moduli
spaces of algebraic surfaces.
I thank S.T. Yau for inviting me to this conference, U.~Gritsch,
I.~Grojnowski, G.~Heckman, and M.~Kleber for many helpful remarks
about earlier drafts, and the NSF for financial support.
\proclaim 1 The Leech lattice and $II_{25,1}$.
In this section we explain Conway's calculation of the reflection
group of the even Lorentzian lattice $II_{25,1}$, so we
first recall the definition of this lattice. A lattice is
called even if the norms $(v,v)$ of all vectors $v$ are even and is
called odd otherwise. The lattice is called unimodular if every
element of the dual of $L$ is given by the inner product with some
element of $L$. The positive or negative definite lattices seem
impossible to classify in dimensions above about 30 as there are too
many of them, but the indefinite unimodular lattices have a very
simple description: there is exactly one odd unimodular lattice
$I_{r,s}$ of any given dimension $r+s$ and signature $rs$ for
positive integers $r$ and $s$, and there is exactly one even
unimodular lattice $II_{r,s}$ if the signature $rs$ is divisible by
$8$ and no even unimodular ones otherwise. (Note that the $II$ in
$II_{r,s}$ is two $I$'s and not a capital $\Pi$!) In particular there
is a unique even unimodular 26 dimensional Lorentzian lattice
$II_{25,1}$.
The odd lattice $I_{r,s}$ can be constructed easily as the set
of vectors $(n_1,\ldots,n_{r+s})\in\R^{r,s}$ with all $n_i\in \Z$.
(The norm of this vector in $\R^{r,s}$ is $n_1^2+\cdots n_r^2n_{r+1}^2
\cdots n_{r+s}^2$.) The even lattice can be constructed
in the same way, except that the conditions on the coordinates are
that their sum is even, and they are either all integers or all
integers $+1/2$. Notice that the vector $(1/2,\ldots,1/2)$ has even
norm $(rs)/4$ because the signature $rs$ is divisible by 8. For
$r=8, s=0$ this is of course just the usual construction of the $E_8$
lattice. There is a second way to construct $II_{25,1}$: if $\Lambda$
is the Leech lattice (the unique 24 dimensional even unimodular
positive definite lattice with no roots) then $\Lambda\oplus II_{1,1}$
is an even 26 dimensional Lorentzian lattice and is therefore
isomorphic to $II_{25,1}$. For the lattice $II_{1,1} $ it is
convenient to use a different coordinate system: we represent its
vectors as pairs $(m,n)\in \Z^2$ with the norm of $(m,n)$ defined to
be $2mn$, so that $(1,0)$ and $(0,1)$ are norm 0 vectors. We use
this to write vectors of $II_{25,1}$ in the form $(\lambda,m,n)$,
where $\lambda \in \Lambda$ and $m,n\in \Z$, where this vector has
norm $\lambda^22mn$.
Conversely Conway showed [CS] that we can run this backwards and give a
very short construction of the Leech lattice. If we let
$\rho=(0,1,2,\ldots,22,23,24,70)\in II_{25,1}$ then $\rho^2=0$ and
$\rho^\perp/\rho$ is isomorphic to the Leech lattice.
In the rest of this section we describe Vinberg's algorithm for
finding simple roots of reflection groups and Conway's application
of it.
If $L$ is any
Lorentzian lattice the norm 0 vectors form two cones and the negative
norm vectors are the ``insides'' of these cones. We select one cone
and call it the positive cone $C$. The set of norm $1$ vectors in $C$
forms a copy of hyperbolic space $H$. (The metric on $H$ is just the
pseudo Riemannian metric on $L\otimes \R$ restricted to $H$, where it
becomes Riemannian.) The group $\Aut(L\otimes\R)^+$ of all rotations
of $L\otimes \R$ mapping $C$ to itself acts on the hyperbolic space
$H$, and is in fact the group of all isometries of $H$. In particular
the group $\Aut(L)^+$ of all automorphisms of $L$ fixing the positive
cone can be thought of as a discrete group of isometries of $H$. We
let $W$ be the subgroup of $\Aut(L)^+$ generated by reflections. These
reflections can be described as follows: if $r$ is a positive norm
vector of $L$ such that $(r,r)2(r,s)$ for all $s\in L$ then the
reflection in $r^\perp$ given by $s\mapsto s2r(r,s)/(r,r)$ acts on
$L\otimes \R$ and by restriction on $H$, where it is reflection in the
hyperspace $r^\perp\cap H$. The reflection hyperspaces divide
hyperbolic space into cells called the Weyl chambers of $W$.
Just as in the case of finite Weyl groups the Weyl group $W$ acts
transitively on the Weyl chambers. We select one Weyl chamber $D$ and
call it the fundamental Weyl chamber. Then the full group $\Aut(L)^+$
is the semidirect product of $W$ and the subgroup of $\Aut(L)^+$
fixing the fundamental Weyl chamber $D$. In particular if we can
describe $D$ this more or less determines the group $\Aut(L)^+$.
Vinberg invented the following algorithm for finding the shape of
$D$. First choose any point $c$ in the fundamental Weyl chamber $D$.
Then find the faces of $D$ in order of their distance from $c$.
Vinberg showed that a reflection hyperplane is a face of $D$ if and
only if it makes an angle of at most $\pi/2$ with each face of $D$
nearer $c$ (where the angle between faces means the angle as seen from
inside $D$). This means we can find all the walls of $D$ recursively
in order of their distances from $c$.
It is convenient to rephrase this algorithm in terms of the lattice
$L$ rather then the hyperbolic space $H$. Instead of a point $c$ we
choose a vector $\rho$ inside the closed positive cone $C$. For
simplicity we assume that all the roots of $W$ have norm $2$ (which is
the only case we will use later). Then we replace the distance from a
hyperplane to the point $c$ by the height $(\rho,r)$ of $r$ where $r$
is any root having inner product at most $0$ with $\rho$. Then as before
we can find all the simple roots (i.e., positive roots $r$ orthogonal
to a face of $D$) in order of their heights, by observing that a
positive root $r$ is simple if and only if it has inner product at
most 0 with all simple roots of smaller height.
We can now describe Conway's calculation [C] of the simple roots of
the reflection group $W$ of $II_{25,1}$ using Vinberg's algorithm. We
choose the vector $\rho$ to be $(0,0,1)\in \Lambda\oplus
II_{1,1}$. There are no roots of height 0 because the Leech lattice
$\Lambda$ has no roots. The vectors $(\lambda,1,\lambda^2/21)$ for
$\lambda\in \Lambda$ are the norm 2 simple roots of height 1 and it is
easy to check that they form a set isometric to the Leech lattice
$\Lambda$.
Conway's amazing discovery was that these are the only simple roots of
$W$. To see this suppose that $(v,m,n)$ with $v^22mn=2$ is any other
simple root of height $m>1$. Then $(v,m,n)$ has inner product at most
0 with all the simple roots $(\lambda,1,\lambda^2/21)$ for all
$\lambda\in \Lambda$ and an easy calculation shows that this implies
that $v/m\lambda>\sqrt 2$ for all $\lambda$. But Conway, Parker,
and Sloane [CS, chapter 23] showed that the Leech lattice has covering
radius exactly $\sqrt 2$, in other words $\Lambda\otimes\R$ is just
covered by closed balls of radius $\sqrt 2$ around each lattice point
of $\Lambda$. In particular there is some vector $\lambda$ of
$\Lambda$ with $v/m\lambda\le\sqrt 2$, and this contradiction
proves that $(v,m,n)$ cannot be a simple root.
To summarize, this gives a complete description of the
group $\Aut (II_{25,1})^+$ as a semidirect product $W.G$ where $W$ is
the reflection group with Dynkin diagram given by the Leech lattice as
above, and where $G$ is the group of automorphisms of this Dynkin
diagram. It is not hard to see that $G$ is just the group of all
isometries of the ``affine'' Leech lattice (which is the Leech lattice
with the origin ``forgotten''), and so is a semidirect
product $\Lambda.\Aut(\Lambda)$ where
$\Aut(\Lambda)$ is the double cover $(\Z/2\Z).Co_1$ of Conway's
largest sporadic simple group $Co_1$. So $\Aut(II_{25,1})$
can be written as $$W.\Lambda.(\Z/2\Z).Co_1.$$ This description of
$Co_1$ as the group sitting at the top of $\Aut(II_{25,1})$
seems to be the simplest and most natural description of any of the
sporadic simple groups.
Given any Dynkin diagram there is an associated KacMoody algebra. As
the Dynkin diagram of the reflection group of $II_{25,1}$ is so nice,
this suggests that the associated KacMoody algebra $M_{KM}$ should
also be very nice. This turns out to be almost but not quite true: we
first have to modify the Dynkin diagram by adding some imaginary
simple roots in order to get a nice Lie algebra (which will be the
fake monster Lie algebra $M$, whose maximal KacMoody subalgebra is
$M_{KM}$.). Before discussing this we recall some facts about
KacMoody algebras.
\proclaim 2 KacMoody algebras.
Suppose that $G$ is a finite dimensional simple complex Lie algebra
with Cartan subalgebra $H$. Then $G$ has a symmetric invariant
bilinear form $(,)$ which induces a form on $H$, which we use to
identify $H$ with its dual $H^*$. The roots $\alpha\in H^*$ are the
nonzero eigenvalues of the adjoint action of $H$ on $G$ and are the
roots of a finite reflection group $W$ acting on $H$ called the Weyl
group of $G$. The simple roots $\alpha$ of some fundamental Weyl
chamber $D$ of $W$ can be identified with the points of the Dynkin
diagram of $G$.
We define the numbers $a_{ij}$ to be the inner products
$(\alpha_i,\alpha_j)$ of the simple roots of $G$ and are the entries
of the ``symmetrized Cartan matrix'' $A$ of $G$. We can and will
normalize the inner product so that all the diagonal entries are
positive reals. These numbers have the following properties:
\item{} $a_{ii}>0 $
\item{} $a_{ij}=a_{ji}$
\item{} $a_{ij}\le 0$ if $i\ne j$
\item{} $2a_{ij}/a_{ii}\in \Z$.
The Cartan matrix is normalized so that it has integral entries
and diagonal entries all equal to 2
but is not symmetric; for our purposes it is better to
use matrices that are symmetric but do not in general have integral entries
or diagonal coefficients equal to 2. It is easy to get
from a symmetrized Cartan matrix to the Cartan matrix just
by multiplying all the rows by suitable constants.
We can recover the Lie algebra $G$ from the matrix $A$ as the Lie algebra
generated by an $sl_2=\langle e_i,f_i,h_i\rangle$ for each simple root
$\alpha_i$, subject to the
following relations (due to
Serre and HarishChandra) depending on the numbers $a_{ij}$:
\item{} $[e_i,f_i]=h_i$
\item{} $[e_i,f_j]=0$ for $i\ne j$
\item{} $[h_i,e_j]=a_{ij}e_j$
\item{} $[h_i,f_j]=a_{ij}f_j$
\item{} $\Ad(e_i)^{12a_{ij}/a_{ii}}e_j=0$
\item{} $\Ad(f_i)^{12a_{ij}/a_{ii}}f_j=0$
Kac and Moody noticed that we can define a Lie algebra in the same way
for any matrix $A$ satisfying the conditions above; these are the
symmetrizable KacMoody algebras. (They also defined Lie algebras for
non symmetrizable Cartan matrices, which we will not use.) KacMoody
algebras have many of the properties of finite dimensional simple Lie
algebras: we can define roots, Weyl chambers, Weyl groups, Cartan
subalgebras, Verma modules, and so on by copying the usual definitions
for finite dimensional Lie algebras. There is a WeylKac character
formula for the characters of some simple quotients of Verma modules,
which for finite dimensional Lie algebras is just the usual Weyl
character formula for finite dimensional representations. The only
case of this we will use is the WeylKac denominator formula, which is
the WeylKac character formula for the trivial one dimensional module
of character 1, which says that $$\sum_{w\in W}\det(w)w(e^\rho)=
e^{\rho}\prod_{\alpha>0}(1e^\alpha)^{\mult(\alpha)}$$ where $\rho$ is
a special vector called the Weyl vector, the product is over all
positive roots $\alpha$, and $\mult$ is the multiplicity of a root, in
other words the dimension of the corresponding root space.
Kac showed that we recover the Macdonald identities if we apply this
denominator formula to certain special KacMoody algebras called
affine KacMoody algebras, which are roughly central extensions of the
Lie algebras $G[t,1/t]$ of Laurent polynomials with coefficients in a
finite dimensional Lie algebra $G$ or twisted versions of these. For
example the denominator formula of the Lie algebra $sl_2[t,1/t]$ is the
Jacobi triple product identity $$\sum_{n\in \Z}(1)^nq^{n^2}z^n =
\prod_{n>0}(1q^{2n})(1q^{2n1}z)(1q^{2n1}/z).$$ We would like to find some
generalizations of the Macdonald identities corresponding to some
KacMoody algebras other than the affine ones. To do this we need to
find some KacMoody algebras for which both the root multiplicities
$\mult(\alpha)$ and the simple roots are known explicitly in some
easy form. Knowing the simple roots is equivalent to knowing the
Weyl group and hence the sum in the WeylKac denominator formula, and
knowing the root multiplicities is equivalent to knowing the product
in the denominator formula. Unfortunately there are no known examples
of KacMoody algebras (other than sums of finite dimensional and
affine ones) for which both the simple roots and root multiplicities
are known explicitly. It is of course always possible to calculate the
multiplicity of any given root of any Lie algebra if we know the
simple roots by using the denominator formula. This can be used to
give either a recursive formula for the root multiplicities (due to
Peterson) or a large and complicated alternating sum for the
multiplicities. Unfortunately neither of these seems to give a
satisfactory simple and explicit formula for the root multiplicities
of any non affine KacMoody algebra. There have been several
numerical calculations of the multiplicities of some of the easier
KacMoody algebras, and the impression one gets from looking at these
tables is that the root multiplicities look rather complicated and
random.
\proclaim 3 Vertex algebras.
As motivation for the definition of vertex algebras we first recall
a short construction for the finite dimensional simple complex Lie algebras
from their root lattice. We will just do the cases of the Lie algebras
$A_n$, $D_n$, and $E_6$, $E_7$, $E_8$ (the others can be obtained
as fixed points of these Lie algebras under diagram automorphisms).
Suppose that $L$ is the root lattice of $G$, so that the roots
of $G$ are exactly the norm 2 vectors of $L$. We construct a
central extension $\hat L$ of $L$ by a group of order 2
generated by an element $\zeta$ with the property
that $e^ae^b=\zeta^{(a,b)}e^be^a$ if $e^a,e^b$ are lifts to $\hat L$ of
$a,b\in L$. This central extension is unique up to (nonunique)
isomorphism. We define $G$ to be the
$Z$module $$L\oplus \sum_{\alpha^2=2}e^\alpha$$
where the sum is over a set of lifts of the norm 2 vectors
of $L$, and where we identify $\zeta e^a$ with $e^a$.
We define the Lie bracket on $G$ by
\item{} $[a,b]=0$ for $a,b\in L$
\item{} $[a,e^b]=[e^b,a]=(a,b)e^b$ if $a,b\in L$, $(b,b)=2$
\item{} $[e^a,e^b]=a$ if $a=b$, $e^ae^b$ if $(a,b)=1$, and 0 otherwise.
Then it is not hard to check that this bracket is antisymmetric and
satisfies the Jacobi identity, so it defines an integral form of the
complex Lie algebras $G$. (Notice that if we did not first take a
central extension of $L$ the product would not be antisymmetric. In
fact there is no canonical way to construct $G$ from $L$ because the
automorphism group of $L$, which is more or less the Weyl group of
$G$, is not usually a subgroup of the automorphism group of $G$ but is
only a subquotient.)
This construction gives a completely explicit basis for the Lie algebra
$G$. We would like to do something similar for all KacMoody algebras.
Unfortunately the construction above breaks down as soon as the lattice
$L$ is indefinite and there are norm 2 vectors $a,b$ with $(a,b)\le 2$
but $a\ne b$. The vertex algebra of a lattice will provide a
generalization of the construction above for all lattices.
For simplicity we will only construct the vertex algebra of a lattice
$L$ in which all inner products are even; in general if some inner
products are odd it is necessary to first replace $L$ by a central
extension $\hat L$ as above. We will also only do the construction
over the complex numbers, although with slightly more effort it can be
done over the integers. We define the underlying space of the vertex
algebra $V$ of $L$ to be the universal commutative ring with
derivation $D$ generated by the complex group ring $\C(L)$ of $L$. It
is not hard to work out the structure of the ring $V$: it is just the
tensor product for the group ring of $L$ with the symmetric algebra of
the sum of a countable infinite number of copies $L_i$ of $L$. It is
best to think of $V$ as a commutative ring together with an action of
the additive formal group $\C$ of the complex numbers, the action of
the formal group being given by the derivation $D$ (which spans the
Lie algebra of this formal group).
A vertex operator is just a formal series $v(z)=\sum_{n\in
\Z}v_{n1}z^n$ where the $v_n$'s are $\C$linear operators from $V$
to $V$ such that for any $w\in V$ the elements $v_nw$ vanish for $n$
sufficiently large. We can think of $v(z)$ as being a sort of formal
operator valued meromorphic function of $z$, with $v_n$ given formally
as the residue of $v(z)z^ndz$ at $z=0$. We will now define some
vertex operators on $V$. For each $a\in L$ we define the vertex
operator $a^+$ by $a^+(z)=\sum_{n\ge 0}D^n(e^a)z^n/n!$ (where the
operator $D^n(e^a)$ is the operator of multiplication by the element
$D^n(e^a)$). We define another vertex operator $a^$ for $a\in L$ by
saying that $a^(z)$ is the derivation from $V$ to $V[z,1/z]$ taking
$e^b$ to $z^{(a,b)}e^b$. Then we can check that all the vertex
operators of the form $a^+$ commute with each other, all the operators
$a^(z)$ commute with each other. Finally we define the vertex
operators $a(z)$ by $a(z)=a^+(z)a^(z)$ for $a\in L$, so that the
operators $a(y)$ and $b(z)$ formally commute with each other for any
$a,b\in L$. in the sense that
$$(yz)^N\big(a(y)b(z)b(z)a(y)\big)=0$$ for $N$ sufficiently
large depending on $a$ and $b$.
Notice that the coefficient of $y^mz^n$
in $a(y)b(z)$ is not usually the same as the coefficient of $y^mz^n$
in $b(z)a(y)$; it is only when both sides are multiplied by a
power of $yz$ that the coefficients become equal.
We can now define a vertex algebra (roughly) as
a vector space $V$ such that for each element $v$ of
$V$ we are given a vertex operator $v(z)$, and these vertex operators
all formally commute with each other. We also require that $V$ should
have an identity element 1 such that $v(0)1=v$ and $1(z)v=v$.
The best way to think of a vertex algebra is as a sort of commutative
ring with a formal action of the group $\C$, where the
action of $z\in \C$ on $v$ is $v(z)$, and $v(z)w$ is
the ring product of $v(z)$ and $w$. If the vertex operator $v(z)$
is ``holomorphic'' for all $v$, which means that $v_n=0$ for $n\ge 0$,
then a vertex algebra is just a commutative ring with derivation
and the ring product is defined by $vw=v_{1}w$. In general
we can think of $V$ as behaving like a commutative ring whose
multiplication is not defined everywhere because it has poles;
if we perturb one of the elements $v,w$ by an element $z$ of
the group $\C$ then we get a meromorphic function $v(z)w$ of $z$
(behaving like the product of $v$ acted on by $z$ and of $w$)
which will in general have a pole at $z=0$.
Suppose we have a vector space $V$ acted on by a set of commuting
operators, such that $V$ is generated as a vector space by the images
of an element $1\in V$ under the ring $R$ generated by these
operators. Then it is easy to see that the map from $R$ to $V$ taking
$r$ to $r(1)$ is an isomorphism of vector spaces and hence gives $V$
the structure of a commutative ring with unit element 1. There is a
similar theorem for vertex operators: if $V$ is a vector space acted
on by an operator $D$ with a compatible set of commuting vertex
operators acting on it, such that $V$ is generated from an element
$1\in V$ with $D(1)=0$ by the components of these vertex operators,
then we can make $V$ into a vertex algebra. If we apply this to the
space $V$ above constructed from a lattice and to the commuting set of
operators $a(z)$ for $a\in L$ we see that $V$ can be given a vertex
algebra structure. If the inner product on $L$ is identically zero
then this vertex algebra structure on $V$ is essentially the same as
the commutative ring structure defined above; in general the vertex
algebra structure defined by some other inner product on $V$ can be
thought of as a sort of ``meromorphic deformation'' of this ring
structure.
If $u$, $v$, and $w$ are elements of a vertex algebra then
$u(x)v(y)w=v(y)u(x)w$ is a meromorphic function of $x$ and $y$ with
poles only at $x=0$, $y=0$, and $x=y$. This function is also equal to
$(u(xy)v)(y)w$, which is easy to understand if one interprets $u(x)$
as the action of the group element $x$ on the element $u$. (This is
easier to see if we denote the action of a group element $x$ on a ring
element $u$ by $u^x$, when it becomes $u^xv^yw=(u^{x/y}v)^yw$.) If
$f(x)$ is any function of $x$ with poles only at $x=0$ or $x=y$ then
Cauchy's formula shows that the residue at $0$ plus the residue at $y$
is the integral around a large circle of $f(x)dx/2\pi i$. If we apply
this to $f(x)=u(x)v(y)w$ and then take residues at $y=0$, we find that
$$\int_y\int_x (u(xy)v)(y)wdxdy
=
\int_y\int_x u(x)v(y)wdxdy

\int_y\int_x v(y)u(x)wdxdy
$$
where the paths of integration of $x$ are a small circle around $y$,
a large circle around both $y$ and $0$, and a small circle around $0$
in the 3 integrals,
or in other words
$$(u_0v)_0w = u_0(v_0w)v_0(u_0w).$$
If we first multiply the integrands by $(xy)^qx^my^n$ we
get more complicated identities involving the operators
$u_iv$ and some binomial coefficients, which are (non trivially) equivalent
to the identities first used to define vertex algebras in [B86].
\proclaim 4 The noghost theorem and I.~Frenkel's upper bound.
Recall that the Virasoro algebra is a central extension of the Lie
algebra of polynomial complex vector fields on the circle, which is
spanned by elements $L_i$, $i\in \Z$, with $[L_i,L_j]=(ij)L_{i+j}$.
The vertex algebra of a nonsingular lattice $L$ has a natural action
of the Virasoro algebra on it and we define the physical subspace $M$ of
$V(L)$ to be the quotient of $P^1/DP^0$ by the kernel of a certain bilinear
form, where $P^i$ is the space of (lowest weight)
vectors with $L_0(v)=iv$, $L_i(v)=0$, $i>0$. (The motivation for this
comes from string theory, where this space is roughly the space of
physical states of a chiral string moving on the torus $L\otimes
\R/L$.) The vertex algebra $V(L)$ also has a natural $L$grading
induced by the obvious $L$ grading on the group ring of $L$. If $L$
is 26 dimensional and Lorentzian then the noghost theorem states
(among other things) that the piece of degree $\alpha\in
L,\alpha\ne 0$ has dimension $p_{24}(1\alpha^2/2)$, where
$p_{24}(n)$ is the number of partitions of $n$ into parts of 24
colors. (In spite of several statements to the contrary in the
mathematical literature, the original proof of the noghost theorem by
Goddard and Thorn is mathematically rigorous.)
I. Frenkel showed [F] that if $L$ is the lattice $II_{25,1}$ then
the KacMoody algebra $M_{KM}$ (with Dynkin diagram given by
the Dynkin diagram of $II_{25,1}$) can be embedded as a subspace of
the space of physical states of $V(II_{25,1})$, and in particular
this gives $p_{24}(1\alpha^2/2)$ as an upper bound on
the multiplicities of the root spaces of this Lie algebra.
(This work of Frenkel's was the main motivation for the definition
of vertex algebras.)
Frenkel's method gives the same upper bound for the multiplicities of
the roots of any KacMoody algebra of rank 26 all of whose roots have
norm 2. For KacMoody algebras of rank $k$ not equal to 26 it gives
the weaker bound $p_{k1}(1\alpha^2/2)p_{k1}(\alpha^2/2)$ which is
slightly larger than $p_{k2}(1\alpha^2/2)$. There are examples of
Lie algebras of ranks not equal to 26, (such as $E_{10}$, due to Kac
and Wakimoto), some of whose root multiplicities are strictly larger
than $p_{k2}(1\alpha^2/2)$, so the upper bound given by the no ghost
theorem in rank 26 cannot be generalized in the obvious way to all
ranks.
We can next ask how good Frenkel's upper bound $p_{24}(1\alpha^2/2)$
for the root multiplicities of $M_{KM}$ is. If we calculate the
multiplicities of the roots of $M_{KM}$ (for example by using the
Peterson recursion formula) we find the following results. All norm 2
vectors have multiplicity 1, equal to Frenkel's upper bound
$p_{24}(12/2)$. For norm 0 vectors there are 24 orbits of primitive
norm 0 vectors corresponding to the 24 Niemeier lattices (24
dimensional even unimodular positive definite lattices). The
correspondence is given as follows: if $w$ is a nonzero norm 0 vector
in $II_{25,1}$ then the quotient $w^\perp/w$ of the orthogonal
complement $w^\perp$ of $w$ by the space generated by $w$ is a
Niemeier lattice, and conversely if $N$ is a Niemeier lattice then
$N\oplus II_{1,1}$ is a 26 dimensional even Lorentzian lattice and is
therefore isomorphic to $II_{25,1}$. A norm 0 vector in the
$II_{1,1}$ gives a norm 0 vector $w$ in $II_{25,1}$ with
$w^\perp/w=N$. If $z$ is any nonzero norm 0 vector in $II_{25,1}$
then it is not hard to check using the theory of affine Lie algebras
that the multiplicity of the root $nz$ is equal to the rank of the
corresponding Niemeier lattice, which is 0 for the Leech lattice and
24 for any other Niemeier lattice. So the multiplicity of the vector
$nz$ is equal to Frenkel's upper bound $p_{24}(10^2/2)=24$ except
when $z$ is a norm 0 vector corresponding to the Leech lattice. For
norm $2$ vectors the calculations take much more effort. There are
121 orbits of norm $2$ vectors. One of the orbits has multiplicity 0,
one has multiplicity 276, and the other 119 orbits all have
multiplicity $324= p_{24}(1(2)/2)$ equal to Frenkel's upper
bound. Similarly there are 665 orbits of norm $4$ vectors, all but 3
of which turn out to have multiplicity equal to
$3200=p_{24}(1(4)/2)$. Furthermore when calculating these
multiplicities using (say) the Peterson recursion formula, it is
apparent that the ``deficiencies'' in the multiplicities of vectors of
negative norm are ``caused'' by the fact that the multiplicities of
the norm 0 vectors corresponding to the Leech lattice are 0 rather
than 24. This suggests that if we somehow added a 24 dimensional root
space for each multiple of the norm 0 vector $\rho$ then we would get
a Lie algebra whose root multiplicities were exactly equal to
Frenkel's upper bound. We already have a space $M$ containing
$M_{KM}$ given by the space of all physical states of a chiral string
in 26 dimensions. The remarks and calculations above strongly suggest
that $M$ itself should be a Lie algebra with simple roots given by the
simple roots of $M_{KM}$ together with multiples of $\rho$.
It is easy to construct a Lie algebra product on $M$ using the theory
of vertex algebras (which is not surprising given that vertex algebras
were invented partly to construct such a Lie bracket). At the end of section 3
we saw that the vertex
algebra product $u_0v=u_0(v)$ satisfies the identity $$(u_0v)_0w =
u_0(v_0w)v_0(u_0w)$$ which is one version of the Jacobi identity, but
it is not antisymmetric. However the sum $u_0v+v_0u$ does at least lie
in the image of the operator $D=L_{1}$, and using this it is easy to
check that defining $[u,v]$ to be $u_0v$ defines a Lie bracket on the
quotient $V/DV$ for any vertex algebra $V$ (where $DV$
is the image of $V$ under the derivation $D$). The space $M$ is a
subquotient of the Lie algebra $V/DV$ where $V$ is the vertex algebra
of the lattice $II_{25,1}$, so this defines a Lie algebra structure on
$M$. (In fact $M$ is even a subalgebra of $V/DV$, but this is harder
to prove and depends heavily on the structure of $M$ described later.)
There is another way to construct the Lie algebra $M$ using
semiinfinite cohomology. It follows from [FGZ] that the vector space
$M$ can be identified with a certain semiinfinite cohomology group,
which therefore has a natural Lie algebra structure. Lian and
Zuckerman study this in [LZ] and reconstruct this Lie algebra from an
algebraic structure called a Gerstenhaber algebra on the full
semiinfinite cohomology. In the case of the fake monster Lie algebra
$M$ we do not really get anything new because the semiinfinite
cohomology with its algebraic structure can be reconstructed from the
Lie algebra $M$ with its bilinear form, but in more general cases this
is no longer true, and semiinfinite cohomology is probably the correct
way to construct Lie algebras.
We would now like to work out the structure of $M$. As $M$ is close to
being a KacMoody algebra this suggests that we should try to
``force'' $M$ to be a KacMoody algebra and find out what the
obstruction to this is. We summarize the facts we know about $M$: it
is graded by $II_{25,1}$ with root multiplicities given by
$p_{24}(1\alpha^2/2)$, it has an invariant bilinear form $(,)$
induced by the form on the vertex algebra $V$, it has an involution
$\omega$ induced by the automorphism $1$ of $II_{25,1}$ (or more
precisely by a lift of this to a double cover of $II_{25,1}$), and the
contravariant form $(a,b)_0=(a,\omega(b))$ is positive definite on the
root spaces of all nonzero roots. (The form $(,)_0$ is not positive
definite on the zero weight space $M_0$; this weight space is 26
dimensional and the form $(,)_0$ on it is Lorentzian. So $M$ is an
infinite dimensional Lorentzian space under $(,)_0$, in other words,
it has a negative norm vector whose orthogonal complement is positive
definite.) We try to make $M$ into a KacMoody algebra by finding the
elements $e_i$ and then defining the remaining generators $f_i$ and
$h_i$ by $f_i=\omega(e_i), h_i=[e_i,f_i]$. Any KacMoody algebra $G$
can be written as a direct sum of subspaces $E\oplus H\oplus F$ where
$H$ is the Cartan subalgebra and $E$ and $F$ are the sums of the root
spaces of the positive and negative roots. (Of course $G$ is not the
Lie algebra sum of the subalgebras $E$, $F$, and $H$, but only the
vector space sum.) If we choose a positive integer $n_i$ for each
simple root $\alpha_i$ then we can $\Z$grade $G$ by giving $e_i$
degree $n_i$, $f_i$ degree $n_i$, and $h_i$ degree 0. Then
$E=\oplus_{n>0}E_n$ is the sum of the positive degree subspaces $E_n$
of $G$. The elements $e_i$ are then a minimal set of generators of the
space $E$, in the sense that the elements $e_i$ of any fixed degree
$n$ are a basis for the quotient of $E_n$ by the elements in the
subalgebra generated by the $E_k$'s for $k0}E_n$ of the
positive root spaces of $M$. We recursively construct the elements
$e_i$ in $E_n$ as a basis for the space of elements of $E_n$ that are
orthogonal to the subalgebra generated by all the $e_i$'s we have
previously constructed in $E_k$ for $k0$
\item{} $[e_i,f_i]=h_i$
\item{} $[e_i,f_j]=0$ for $i\ne j$
\item{} $[h_i,e_j]=a_{ij}e_j$
\item{} $[h_i,f_j]=a_{ij}f_j$
\item{} $\Ad(e_i)^{12a_{ij}/a_{ii}}e_j=0$ if $a_{ii}>0$,
$[e_i,e_j]=0$ if $a_{ij}=0$.
\item{} $\Ad(f_i)^{12a_{ij}/a_{ii}}f_j=0$ if $a_{ii}>0$,
$[f_i,f_j]=0$ if $a_{ij}=0$.
If all the diagonal entries $a_{ii}$ are positive, in other words if
all the simple roots are real (positive norm), then these conditions
are exactly those defining symmetrizable KacMoody algebras. We will
call any Lie algebra generated by relations satisfying the conditions
above a generalized KacMoody algebra. (In fact generalized KacMoody
algebras are slightly more general than this, because we also allow a
few extra operations, such as adding outer derivations, quotienting by
subspaces of $H$ in the center of $G$, and taking central extensions.)
As the name implies, the theory of generalized KacMoody algebras is
similar to that of KacMoody algebras (and finite dimensional simple
Lie algebras), with some minor changes. The main difference is that we
may have imaginary (norm $\le 0$) simple roots. These contribute some
extra terms to the sum in the WeylKac character formula. In particular the
denominator formula now looks like
$$\sum_{w\in W}\det(w)\sum_S (1)^{S}w(e^{\rho+\sum S})=
e^{\rho}\prod_{\alpha>0}(1e^\alpha)^{\mult(\alpha)}.$$
The inner sum is over sets $S$ of pairwise orthogonal
imaginary simple roots, of cardinality $S$ and sum $\sigma S$.
Notice that knowing the sum on the left is essentially equivalent
to knowing all the simple roots, because the real simple roots
determine the Weyl group $W$ in the outer sum, and the imaginary
simple roots determine the inner sum.
There is an explicit example of such a denominator formula
at the end of this section.
Roughly speaking the imaginary simple
roots do not really add anything very interesting to $G$; the
complexity of $G$ is closely related to the complexity of the Weyl
group $W$ of $G$, which only depends on the real simple roots of
$W$. The imaginary simple roots only contribute some extra flab, which
usually looks a bit like free or free abelian Lie algebras [J]. The main
advantage of generalized KacMoody algebras is that, as we have seen
above for $M$, it is sometimes possible to prove that some naturally
occurring Lie algebra is a generalized KacMoody algebra using only
general properties of $M$ (in particular the existence of an ``almost
positive definite'' contravariant bilinear form $(,)_0$). Most
``natural'' examples of KacMoody algebras other than affine ones
(such as the KacMoody algebra $M_{KM}$ with Dynkin diagram the Leech
lattice, or the FrenkelFeingold algebra [FF]) seem to be the
subalgebra of some more natural generalized KacMoody algebra (such as
the fake monster Lie algebra $M$) generated by some of the real simple root
spaces. See the end of this section for an example.
So far we have seen that $M$ is a generalized KacMoody algebra with
known root multiplicities, so the next obvious thing to do is to try
to work out the simple roots of $M$. The fact that we will be able to
do this for $M$ may be misleading: there are very few known examples
of generalized KacMoody algebras for which the simple roots and root
multiplicities are both known. For example it is possible to
construct a generalized KacMoody algebra similar to $M$ for any
Lorentzian lattice $L$, but if $L$ is any Lorentzian lattice other
than $II_{25,1}$ then the simple roots of the generalized KacMoody
algebra of $L$ seem to be too messy to describe explicitly. Most of
the several hundred known examples of generalized KacMoody
superalgebras whose simple roots and root multiplicities are both
known explicitly can be obtained by ``twisting'' the denominator
formula for the fake monster Lie algebra in some way.
There are some obvious simple roots of $M$: the simple roots
$(\lambda,1,\lambda^2/21)$ of $M_{KM}$ are the real simple roots of
$M$, and the positive multiples of the norm 0 Weyl vector
$\rho=(0,0,1)$ of $M$ are simple roots of multiplicity 24. The
calculations of root multiplicities above turn out to be equivalent to saying
that $M$ has no simple roots of norm $2$ or $4$, and suggest that
$M$ has no simple roots other than the ones of norm 2 or 0 above. This
can be proved roughly as follows. We can get information about the
imaginary simple roots using the fact that they appear in the sum defining
the denominator function. The denominator function is not easy to
describe completely, so the the key step is to look at the denominator
function restricted to the 2 dimensional space of vectors of the form
$(0,\sigma,\tau)\in II_{25,1}\otimes \C$ with $\Im(\sigma)>0$,
$\Im(\tau)>0$. The restriction to this space of the infinite product
defining the denominator function is $$p^{1}\prod_{m>0,n\in
\Z}(1p^mq^n)^{c'(mn)}$$ where $p=e^{2\pi i
\sigma}$, $q=e^{2\pi i \tau}$, and the numbers $c'(mn) $ are defined by
$$\sum_{n\in \Z}c'(n)q^n = \Theta_\Lambda(\tau)/\Delta(\tau) =
j(\tau)720 = q^{1}+24+196884q+21493760q^2+\cdots$$
The function $1/\Delta(\tau)$ has the numbers $p_{24}(n+1)$ as the
coefficient of $q^n$, and the function $\Theta_\Lambda$ is the theta function
of the Leech lattice, which appears because the Leech lattice is the
kernel of the projection of $II_{25,1}$ to the 2 dimensional lattice
$II_{1,1}$. Their product is, up to a constant, the elliptic modular function
$j(\tau)$. This infinite product can be evaluated explicitly
by rewriting it as
$$p^{1}\exp\big(\sum_{m>0}p^mT_m(j(\tau)720)\big)$$ where the
$T_m$'s are Hecke operators. The point is that if $T_m$ is applied to
a modular function such as $j$ then the result is still a modular
function (and in fact a polynomial in $j$). In particular with respect
to the variable $\tau$ the infinite product behaves like a modular
form of weight 12. It is also antisymmetric in $\sigma$ and $\tau$,
and these two conditions (plus some technical conditions about growth
at infinity) are sufficient to characterize it up to multiplication by
a constant. The result is that
$$p^{1}\prod_{m>0,n\in \Z}(1p^mq^n)^{c'(mn)}=
\Delta(\sigma)\Delta(\tau)(j(\sigma)j(\tau)).$$
This identity turns out to mean that in some sense the multiplicities
of the root spaces of imaginary simple roots of $II_{25,1}$ with the
same image in $II_{1,1}$ have an ``average'' value of 0 (in some
rather weird sense of the word average). As the multiplicity of a
simple root is the dimension of the simple root space
and hence nonnegative, the only way in which the
simple root multiplicities can have an average value of 0 is if they
are all 0. In other words there are no simple roots of negative
norm. For more precise details of this argument see [B90].
The denominator formula for $M$ can now be written out explicitly as
we know both the simple roots and the root multiplicities, and it is ([B90])
$$e^\rho{\prod_{r>0}(1e^r)^{p_{24}(1r^2/2)}} = \sum_{w\in W\atop
n\in Z}\det(w)\tau(n)e^{w(n\rho)}$$ where $\tau(n)$ is the Ramanujan
tau function defined by $\sum_{n\in \Z}\tau(n)q^n=
q\prod_{n>0}(1q^n)^{24} =q24q^2+\cdots$. The terms with $n=1$ on the
right hand side are exactly what one would get for the denominator
formula of the KacMoody algebra $M_{KM}$; the terms with $n>1$ are
the extra correction terms coming from the imaginary simple roots (=
positive multiples of $\rho$) of $M$.
We have seen that the KacMoody algebra with Dynkin diagram given by
the Leech lattice is best thought of as a subalgebra of
a larger generalized KacMoody algebra whose root multiplicities
are known explicitly. A similar phenomenon also happens for
some other non affine KacMoody algebras; they are best thought of
as large subalgebras of generalized KacMoody algebras.
We will illustrate this by discussing the case of Frenkel
and Feingold's algebra [FF] with Cartan matrix
$$
\pmatrix{
2&1&0\cr
1&2&2\cr
0&2&2\cr
}
$$
which can be thought of as the affine $A_1$ Dynkin diagram with
an extra node attached. Frenkel and Feingold calculated many of the
root multiplicities of roots $\lambda$, and observed that
many of them were given by values
$1,1,2,3,5,7,11,\ldots$ of the partition
function $p(1\lambda^2/2)$. A suitable generalized KacMoody algebra
containing it was described by Niemann in his thesis [N].
The root lattice of this Lie algebra is $K\oplus II_{1,1}$ where
$K$ is generated by $x$ and $y$ with $(x,x)=4$, $(y,y)=6$, $(x,y)=1$.
This lattice has determinant 23 and no roots, and corresponds to a non
principal ideal in the imaginary quadratic field of discriminant $23$.
We define $p_\sigma(n)$ by
$$\sum_n p_\sigma(n)q^n = {1\over \eta(\tau)\eta(23\tau)}
= q^{1}+1+2q+3q^2+5q^3+7q^4+11q^5+\cdots
$$
Notice that the first 23 values of $p_\sigma(n)$
are the same as the values of the partition function $p(n+1)$.
Niemann showed that there is a generalized KacMoody algebra
with root lattice $K$, whose root multiplicities are
given by
$$\mult(\lambda)=p_\sigma(\lambda^2/2)$$
if
$\lambda\in K, \lambda\notin 23K'$ and by
$$\mult(\lambda)=p_\sigma(\lambda^2/2)+p_\sigma(\lambda^2/46)$$ if
$\lambda\in 23K'$. Moreover this Lie algebra has a norm 0 Weyl vector
and its simple roots can be described explicitly in a way similar to
those of the fake monster Lie algebra. In particular the norm 2
simple roots correspond to points of the lattice $K$, and there are
also some norm 46 simple roots corresponding to two cosets of $K$, and
some norm 0 simple roots of multiplicity 1 or 2 corresponding to
positive multiplicities of the Weyl vector. If we take the norm 2
simple roots corresponding to 0 and the 2 basis vectors of $K$ we find
they have the same Dynkin diagram as the FeingoldFrenkel Lie algebra,
so in particular the FeingoldFrenkel Lie algebra is a subalgebra and
its root multiplicities are bounded by the multiplicities given above.
It is also easy to check that in some sense the FeingoldFrenkel Lie
algebra accounts for all the ``small'' roots, so the first few root
multiplicities are given exactly by $p_\sigma(\lambda^2/2)$, which
are values of the partition function, and this explains the
observation that the FeingoldFrenkel Lie algebra's root
multiplicities are often given by values of the partition function.
So this answers Frenkel and Feingold's question about whether it is
possible to write down an explicit denominator formula for their Lie
algebra, as long as we are willing to modify their question a bit and
look at a slightly bigger Lie algebra.
Niemann also proved similar results for some other KacMoody algebras
obtained by adding an extra node to an affine Dynkin diagram.
\proclaim 5 Relations with moonshine.
The identities above are used in the proof of the moonshine
conjectures, which state that the monster sporadic group has an
infinite dimensional graded representation $V=\oplus V_n$ such that
the traces of elements of the monster on this representation are given
by certain Hauptmoduls. In particular the dimensions of the graded
pieces $V_n$ are given by the coefficients of $q^{n1}$ of the
elliptic modular function $j(\tau)744$. As there are already several
survey articles ([B94], [LZ], [J], [G], [Y]) about the proof of the
moonshine conjectures we will only briefly discuss its relation with
the fake monster Lie algebra.
A candidate for the representation $V$ was constructed by
Frenkel,Lepowsky and Meurman [FLM] and has the structure of a vertex
algebra [B86]. This vertex algebra is a twisted version of the vertex
algebra of the Leech lattice. If we look at the restriction of the
denominator function of the fake monster Lie algebra to the 2
dimensional subspace (see the end of section 4) we see that it says an
infinite product is equal to a simple expression, so we can ask if
this restriction is itself the denominator function of some
generalized KacMoody algebra. This is not quite right, but if we
divide both sides by $\Delta(\sigma)\Delta(\tau)$ we find the identity
$$p^{1}\prod_{m>0,n\in Z} (1p^mq^n)^{c(mn)}=j(p)j(q).$$ This is the
denominator formula of a $II_{1,1}$graded generalized KacMoody
algebra whose piece of degree $(m,n)\ne (0,0)$ is equal to the
coefficient $c(mn)$ of $q^{mn}$ in
$j(\tau)744=\sum_nc(n)q^n=q^{1}+196884q+\cdots$. These
numbers are exactly the same as the dimensions of the graded pieces of
$V$, and using this as a hint it is easy to guess how to construct a
generalized KacMoody algebra from $V$ called the monster Lie algebra:
we first tensor the vertex algebra $V$ with the vertex algebra of
$II_{1,1}$ to get a vertex algebra similar to that of the lattice
$II_{25,1}$, and then we apply the same construction to $V\otimes
V(II_{1,1})$ that we used to construct the fake monster Lie algebra
from $V(II_{25,1})$. In other words the monster Lie algebra is just
the space of physical states in $V\otimes V(II_{25,1} )$. As $V$ is a
twisted version of $V(\Lambda)$, we see that the monster Lie algebra
is a twisted version of the fake monster Lie algebra; more precisely,
both Lie algebras have automorphisms of order 2 such that the fixed
point subalgebras are isomorphic. The monster Lie algebra
can also be constructed via semi infinite cohomology as in [FGZ, LZ].
The monster Lie algebra by construction has an action of the monster on it
with the property that the pieces of degrees $(a,b)$ and $(c,d)$
are isomorphic as representations of the monster whenever
$ab=cd\ne 0$. This can be used to prove Conway and Norton's
moonshine conjectures; see [B92], [J], [G], [B94].
\proclaim 6 The denominator function.
The sum in the denominator formula for the fake monster Lie algebra
defines a function $\Phi$ on a subset of $II_{25,1}\otimes \C$. It is
not hard to check that this sum converges whenever the imaginary part
of the argument lies in the interior of the cone $C$. The fact that
$\Phi$ can also be written as an infinite product suggests that we can
find the zeros of $\Phi$ by looking at the zeros of the factors in the
infinite product. This turns out to be incorrect because the infinite
product does not converge everywhere and the function $\Phi$ has zeros
outside the region of convergence. (This is a quite common
phenomenon; for example, it also happens for the Riemann zeta
function!) We can find one zero of $\Phi$ which cannot be seen in any
of its factors as follows.
First suppose that we consider $\Phi$ just for purely imaginary values
of its argument $v$. Then we can work out the exact region of
convergence of the infinite product, because we know the asymptotic
behavior of the coefficients $p_{24}(1+n)$ by the HardyRamanujan
theorem, and this region turns out to be exactly the region with
$v^2>2$. (Notice the region is $v^2>2$ rather than $v^2<2$ because
$v$ is purely imaginary.) Next we can see that $\Phi$ vanishes
whenever $v$ is purely imaginary and $v^2=2$. To see this recall the
well known lemma from complex analysis which says that if $f(z)=\sum
a_nz^n$ is a power series with radius of convergence $r$ and all the
$a_n$'s are non negative, then $f$ has a singularity at $z=r$. All the
coefficients of the series for $\log(\Phi)$ are non negative (as can
be seen by taking the log of the infinite product and using the fact
that $p_{24}(1+n)$ is nonnegative), so by using a higher dimensional
version of the lemma mentioned above we see that $\log(\Phi(v))$ has a
singularity whenever $v$ is purely imaginary and $v^2=2$. However
$\Phi$ itself is holomorphic at these points as this is inside the
region of convergence of the infinite sum defining $\Phi$, so the only
way that $\log(\Phi(v))$ can be singular is if $\Phi$ vanishes at
these points. Finally as $\Phi$ is holomorphic it must vanish at all
points of the divisor $v^2=2$ inside the region where it is
defined. Notice that this divisor is not a zero of any factor of the
infinite product for $\Phi$ and (hence) lies entirely outside the
region of convergence of this infinite product.
The fact that $\Phi$ vanishes for $v^2=2$ suggests that perhaps $\Phi$
satisfies some sort functional equation forcing it to vanish at these
points, perhaps something like $\Phi(2v/(v,v))=f(v)\Phi(v)$ where $f$
is some function not equal to 1 when $(v,v)=2$. We can guess the exact
form of $f$ by looking at $\Phi$ restricted to the 2 dimensional space
of vectors $(0,\sigma,\tau)$, where we evaluated $\Phi$ explicitly as
$\Delta(\sigma)\Delta(\tau)(j(\sigma)j(\tau))$. Using the functional
equations $\Delta(1/\tau)=\tau^{12}\Delta(\tau)$,
$j(1/\tau)=j(\tau)$, and the fact that if $v=(0,\sigma,\tau)$ then
$v^2=2\sigma\tau$ we see that on this 2 dimensional subspace $\Phi$
satisfies the functional equation
$$\Phi(2v/(v,v))=((v,v)/2)^{12}\Phi(v).$$
Once we have guessed the correct form of the functional equation above
it is surprisingly easy to prove. As before we restrict to purely
imaginary values of $v$ because if we prove it for these $v$ it will
follow for all $v$ by analytic continuation. We first observe that
both $\Phi(v)$ and $(v,v)^{12}\Phi(2v/(v,v))$ are solutions to the
wave equation. For $\Phi$ this follows from the fact that it is a sum
of terms of the form $e^{2\pi i(z,v)}$ with $z^2=0$, which are all
solutions of the wave equation. If $\Phi$ is any solution of the wave
equation in $\R^{25,1}$ then
$(v,v)^{1\dim(\R^{25,1})/2}\Phi(2v/(v,v))$ is automatically also a
solution by the standard behavior of the wave operator under the
conformal transformation $v\mapsto 2v/(v,v)$. (More generally we get
an action of not just isometries but also the whole conformal group on
the space of solutions of the wave equation.) Secondly, the fact that
$\Phi$ vanishes for $v^2=2$ easily implies that both $\Phi $ and
$((v,v)/2)^{12}\Phi(2v/(v,v))$ have the same partial derivatives of
order at most 1 on this surface. (It is obvious they both vanish
there and have vanishing first partial derivatives tangent to this
surface, so we only need to check the first derivatives normal to the
surface which is not hard.) Hence $\Phi$ and
$((v,v)/2)^{12}\Phi(2v/(v,v))$ are both analytic solutions to a
second order differential equation and both have the same derivatives
of order at most 1 along some (non characteristic) Cauchy surface, so
by the Cauchy Kovalevsky theorem they are equal.
\proclaim 7 The automorphic form $\Phi$.
We have seen that $\Phi$ satisfies the functional equation
$$\Phi(2v/(v,v))=((v,v)/2)^{12}\Phi(v),$$ and it is easy to check that
it also satisfies the functional equations $\Phi(v+\lambda)=\Phi(v)$
for $\lambda\in II_{25,1}$ and $\Phi(w(v))=\det(w)\Phi(v)$ for $w\in
\Aut(II_{25,1} )^+$. We can work out the group generated by these three
sorts of transformations and it turns out to be isomorphic to the
group $\Aut(II_{26,2})^+$, which is a discrete subgroup of
the group $O_{26,2}(\R)$ of conformal transformations of (a conformal
completion of) $\R^{25,1}$. The action of $O_{26,2}(\R)^+$ on the
domain of definition of $\Phi$ is given as follows. We can identify
this domain of vectors $v\in II_{25,1}\otimes \C$ with $\Im(v)\in C$
with a subset of norm 0 vectors in the projective space
$P((II_{25,1}\oplus II_{1,1})\otimes\C)$ by mapping $v$ to the norm 0
vector $(v,1,v^2/2)\in II_{25,1}\oplus II_{1,1}=II_{26,2}$. The image is
the set of vectors in projective space represented by norm 0 vectors
whose real and imaginary parts form a positively oriented orthonormal
base for a negative definite subspace of $\R^{26,2}$. As
$O_{26,2}(\R)^+$ obviously acts naturally on this space we get an
action on the domain of definition of $\Phi$.
We can think of the domain of $\Phi$ as a generalization of the upper
half plane, and $O_{26,2}(\R)^+$ as a generalization of the group
$SL_2(\Z)$ acting on the upper half plane. The function $\Phi$ should
then be thought of as a generalization of a modular form on the upper
half plane; these generalizations are called automorphic forms. We can
summarize much of what we have discussed so far by saying that the
denominator function of the fake monster Lie algebra is an automorphic
form of weight 12 for the discrete subgroup $\Aut(II_{26,2})^+$ of
$O_{26,2}(\R)^+$.
\proclaim 8 The zeros of $\Phi$.
We saw above that the naive guess for the zeros of $\Phi$ was wrong:
there are zeros which cannot be seen in the factors of $\Phi$. However
we can now find all the zeros using the fact that $\Phi$ is an
automorphic form. We have seen that $\Phi$ has a zero along the
divisor $v^2=2$, and obviously also has zeros along all conjugates of
this divisor under the group $\Aut(II_{26,2})^+$ because of the
transformations of $\Phi$ under this group. If we identify the domain
of $\Phi$ with a subset of complex projective space as above these
conjugates are easy to visualize: the divisor $v^2=2$ is just the set
of points $(v,1,v^2/2)$ orthogonal to the norm 2 vector $(0,1,1)\in
II_{26,2}$, and $\Aut(II_{26,2})^+$ acts transitively on these norm 2
vectors, so the zeros of $\Phi$ conjugate to $v^2=2$ just correspond
to pairs $\{r,r\}$ of norm 2 vectors in $II_{26,2}$. These are
exactly the zeros of $\Phi$ that are ``forced'' by its functional
equations. Now we want to see that $\Phi$ has no other zeros. Recall
that the zero $v^2=2$ of $\Phi$ was somehow very closely related to
the asymptotic behavior of the function $p_{24}(1+n)$. The
HardyRamanujanRademacher circle method gives a much finer
description of this asymptotic behavior as
$$p_{24}(1+n) = 2\pi n^{13/2}
\sum_{k>0}{I_{13}(4\pi \sqrt{n}/k)\over k}\sum_{0\leq h,h'0}T_m(f))$ where $f$ is a
certain Jacobi form with poles at cusps and the $T_m$'s are Hecke operators
acting on this Jacobi form. (The idea of considering
$\sum_{m>0}T_m(f)$ for holomorphic Jacobi forms
$f$ was used by Maass in his work on the
SaitoKurokawa conjecture described in [EZ] and then extended
by Gritsenko [Gr] to Jacobi forms on higher dimensional lattices.)
Notice that this is a sort of blown up
version of the formula we found for the restriction of $\Phi$ to a 2
dimensional subspace in section 4. This second expression for $\Phi$
easily implies that it transforms nicely under a certain subgroup
$SL_2(\Z)$ of $\Aut(II_{26,2})^+$. By looking at the Fourier
coefficients of $\Phi$ it is easy to check that it also transforms
nicely under $\Aut(II_{25,1})^+$. As this group together
with $SL_2(\Z)$ generate $\Aut(II_{26,2})^+$ we see that
$\Phi$ transforms correctly under all elements of $\Aut(II_{26,2})^+$
and hence shows that $\Phi$ is an automorphic form.
The advantage of the proof sketched above is that it does not really
depend on all the special properties of the Leech lattice, and can be
extended to more general lattices than $II_{25,1}$ and more general
exponents that $p_{24}(1+n)$. If we carry out this extension for even
unimodular lattices we find:
\proclaim Theorem. [B95]
Suppose that $L$ is the even unimodular lattice $II_{s+2,2}$. Suppose
that $f(\tau)=\sum_nc(n)q^n$ is a meromorphic modular form of weight
$s/2$ for $SL_2(\Z)$ with integer coefficients, with poles only at
cusps, and with $24c(0)$ if $s=0$. There is a unique vector $\rho\in
L$ such that
$$\Phi(v)=e^{2\pi i (\rho, v)}\prod_{r>0}(1e^{2\pi i
(r,v)})^{c((r,r)/2)}$$ is a meromorphic automorphic form of weight
$c(0)/2$ for $\Aut(II_{s+2,2})^+$.
It is also possible to describe the zeros and poles of $\Phi$ explicitly.
The denominator function for the fake monster Lie algebra is the
special case of the theorem above with $s=24$, $f(\tau)=1/\Delta(\tau)$.
The proof sketched above is not very enlightening, as it is
essentially a brute force check that $\Phi$ transforms correctly under
a set of generators of $\Aut(II_{s+2,2})^+$.
It is possible to extend this a bit to non unimodular lattices,
but it becomes increasingly complicated as the level and determinant
increase. Gritsenko and Nikulin [GN] have worked out a some
higher level examples explicitly using the method above.
In a recent
preprint [HM], Harvey and Moore have sketched a conceptual proof of
the fact that $\Phi$ is an automorphic form, and also given a better
explanation of why its only zeros are as described above,
using Siegel theta functions of indefinite lattices
(which seem to have been independently rediscovered
by physicists working in string theory). Their method
generalizes much better to higher levels; see [B].
The Siegel theta function $\Theta_L(\tau_1,\tau_2,V)$ of a lattice
$L=II_{r,s}$ (which we will assume is even and unimodular for
simplicity) is defined as $$\Theta_L(\tau_1,\tau_2,V) = \sum_{\l\in
L}\exp(2\pi i \tau_1\l_1^2/2+2\pi i \tau_2\l_2^2/2)$$ where
$\Im(\tau_1)>0$, $\Im(\tau_2)<0$, $V$ is a negative definite subspace
of $L\otimes \R$ of maximal dimension $n$, and $\l_1$ and $\l_2$ are
the projections of $\l$ into $V^\perp$ and $V$. If $L$ is positive
definite this is just the usual theta function of $L$, as it does not
depend on $\tau_2$ or $V$, and for indefinite lattice the series still
converges absolutely because of the condition that $\Im(\tau)_2<0$.
It satisfies the functional equations $$\Theta_L\left({a\tau_1+b\over
c\tau_1+d},{a\tau_2+b\over c\tau_2+d},V\right)
=((c\tau_1+d)/i)^{r/2}((c\tau_2+d)i)^{s/2}\Theta_L(\tau_1,\tau_2,V),$$
and is obviously invariant under the natural action of
$\Aut(II_{r,s})$ on $V$.
Now suppose that $f$ is any linear functional from functions of
$\tau_1$ and $\tau_2$ to $\C$. Then $f(\Theta_L)(V)$
is a function of $V$ which is automatically invariant under
$\Aut(II_{r,s})$.
Harvey and Moore make the following choices for $L$ and $f$. They
choose $L$ to be a lattice $II_{s+2,2}$. The set of subspaces $V$ can
be identified with the hermitian symmetric space of $L$ as follows:
the norm 0 vector $x+iy \in L\otimes \C$ representing a point of the
projective space $P(L\otimes\C)$ corresponds to the negative
definite space spanned by $x$ and $y$. The linear functional they
use is (more or less) given by
$$f(\Theta_L(\tau_1,\tau_2,V)) = \int_D
\Theta_L(\tau,\bar\tau,v)g(\tau)d\tau d\bar\tau/\Im(\tau)$$
where $D$
is a fundamental domain for the action of $SL_2(\Z)$ on the upper half
plane and $f$ is a meromorphic modular form of weight $s/2$ with
poles only at cusps. The function $\Im(\tau)\Theta_L$ transforms like
a modular form of weight $s/2$ and the differential $d\tau
d\bar\tau/\Im(\tau)^2$ is invariant under $SL_2(\Z)$ so the integrand
is invariant under $SL_2(\Z)$ and the integral does not depend on the
choice of fundamental domain $D$. The integral needs to be interpreted
rather carefully as it is wildly divergent because of terms of the
form $e^{2\pi i n\tau}$ with $n<0$. Harvey and Moore get round this by first
integrating with respect to $\Re(\tau)$ and only then integrating
with respect to $\Im(\tau)$ (although this still leaves a problem with
integrating $1/\Im(\tau)$ from 1 to infinity, which they deal with
by first subtracting a suitable function from the integrand).
The integral above is similar to the integral used by Niwa [Ni] in his
work on the Shimura correspondence. This can be used to explain the
formal similarities observed in [B95] between the Shimura
correspondence and the infinite products above; see section 11 below. In
particular Harvey and Moore's formula can be thought of as a sort of
version of the Howe correspondence for the dual reductive pair
$O_{s+2,2}(\R)$ and $SL_2(\R)$ for functions with singularities.
Harvey and Moore formally evaluate this integral and find that it is
essentially given by the logarithm of the absolute value of the
function $\Phi$ of the theorem above, plus a few elementary factors
which account for the fact that the integral is invariant under
$\Aut(II_{r,s})^+$ while $\Phi$ is not quite invariant.
Jorgenson and Todorov found a quite different way of constructing
similar automorphic functions on moduli spaces as analytic discriminants
[JT]. The relation to the work above is that the moduli spaces of Enriques
surfaces and polarized K3 surfaces are (roughly) quotients
of the hermitian symmetric spaces of lattices of signature $(n,2)$
for $n=10$ or $18$. It seems likely that some of the functions
constructed by Jorgensen and Todorov can also be constructed
using some variation of Harvey and Moore's method.
Automorphic forms which are infinite products have recently turned up
in several papers by physicists on string theory, but I do not
understand this well enough to report on it. See [HM] and [D] for
example.
\proclaim 10 Some superalgebras of rank 10.
Harvey and Moore's formula for $\Phi$ has the big advantage
that it generalizes easily to arbitrary lattices $L$ of any dimension,
signature, and determinant; see [B] for details. We will
give a few examples in this section.
One extra feature that appears in the higher level case is that
an automorphic form can have several apparently quite different
infinite product expansions, converging in different regions. A classical
example of this is the different product expansions for
the theta function of a one dimensional lattice.
If we put
$$\theta(\tau)=\sum_{n\in \Z}(1)^nq^{n^2/2}$$
then it
is a modular form for the group $\Gamma(2)$, and
has the following infinite product
expansions at cusps of $\Gamma(2)$.
$$\eqalign{
\theta(\tau)&=12q^{1/2}+2q^{4/2}2q^{9/2}+\cdots\cr
& =
(1q^{1/2})^2(1q)(1q^{3/2})^2(1q^2)\cdots\cr
(\tau/i)^{1/2}\theta(1/\tau)&=
2q^{1^2/8}+2q^{3^2/8}+2q^{5^2/8}+\cdots\cr
&=
2q^{1/8}(1q)^{1}(1q^2)(1q^{3})^{1}(1q^4)\cdots
}$$
So two apparently quite different infinite products are really the
same function expanded around different cusps.
We will take $L$ to be the lattice $II_{9,1}\oplus II_{1,1}(2)$, where
$II_{1,1}(2)$ is the lattice $II_{1,1}$ with all norms multiplied by
2. We define a vector valued modular form $F$ with components
$f_{ij}$ for $i,j\in \Z/2\Z$ by
$$\eqalign{
f_{00}(\tau)&= 8\eta(2\tau)^8/\eta(\tau)^{16} = 8+128q+1152q^2+\cdots\cr
f_{10}(\tau)=f_{01}(\tau) &=8\eta(2\tau)^8/\eta(\tau)^{16}
= 8128q1152q^2\cdots\cr
f_{11}(\tau) =&8\eta(2\tau)^8/\eta(\tau)^{16} +\eta(\tau/2)^8/\eta(\tau)^{16}
= q^{1/2} +36q^{1/2}+402q^{3/2}+\cdots\cr
}
$$
Then there is an automorphic form such that the log of
its absolute value is (more or less) given by
$$\int_D
\Theta_L(\tau,\bar\tau,v)F(\tau)d\tau d\bar\tau/\Im(\tau)$$
(where the Siegel theta function is now a certain vector valued
function taking values in the group ring of $L'/L$).
By theorem 13.3 of [B] we can find an infinite product expansion
of this automorphic form for every primitive norm 0 vector of
$L$ (or equivalently to every one dimensional cusp).
Define $c(n)$ by
$$\sum_nc(n)q^n=f_{00}(\tau)+f_{11}(\tau) =
q^{1/2}+8+36q^{1/2}+O(q).$$ Then the infinite product at one cusp
(for $K$ the even sublattice of $I_{9,1}$) is
$$\eqalign{
&e^{2\pi i(\rho,v)}\prod_{\lambda\in K'\atop(\lambda,W)>0}
(1e^{2\pi i(v,\lambda)})^{\pm c(\lambda^2/2)}\cr
=&
\sum_{w\in G}\det(w)e^{2\pi i(w(\rho),v)}
\prod_n(1e^{2\pi in(w(\rho),v)})^{(1)^n8}\cr
}
$$
where the sign in the exponent is 1 if $\lambda\in K$ or if $\lambda$
has odd norm, and $1$ if $\lambda$ has even norm but is not in $K$.
The group $G$ is the reflection group generated by reflections
of the norm $1$ vectors of $K$.
The infinite product at the other cusp (with $K=II_{9,1}$) is
$$\prod_{\lambda\in K\atop (\lambda,W)>0} (1e^{(v,\lambda)})^{c(\lambda^2/2)}
(1+e^{(v,\lambda)})^{c(\lambda^2/2)}
=1+ \sum_{\lambda}a(\lambda)e^{2\pi i(v,\lambda)}
$$
where $a(\lambda)$ is 1 if $\lambda=0$, the coefficient
of $q^n$ of
$$\eta(\tau)^{16}/\eta(2\tau)^8$$
if $\lambda$ is $n$ times a primitive norm 0 vector in the closure of the
positive cone $C$, and 0 otherwise.
In both cases we can identify the infinite product as a sum using the
fact that it is an automorphic form of singular weight, so that all
its Fourier coefficients corresponding to values of $\lambda$ of
nonzero norm vanish, and we get the formulas above. Both of these
formulas are denominator formulas for generalized KacMoody
superalgebras of rank 10 [R], and both superalgebras can be constructed as
spaces of states of a superstring moving on a 10 dimensional torus.
The automorphic form can also be considered as a function on the
period space of marked Enriques surfaces as in [B96], which can be
used to show that the moduli space of Enriques surfaces is
quasiaffine.
\proclaim 11.~The Shimura correspondence.
The Shimura correspondence is a map from modular forms of weight
$k+1/2$ to modular forms of weight $2k$. Shimura's original definition
in [S] was rather roundabout and involved taking an eigenform of
weight $k+1/2$ under the Hecke operators, taking the corresponding
Euler product, changing it in a mysterious way to a new Euler product,
and then using Weil's theorem to reconstruct a modular form from this
new Euler product. Niwa [Ni] and Kohnen [Ko] reformulated Shimura's
result and found a more straightforward way of constructing Shimura's
map. Combining their results in the level 1 case (and making an easy
extension to the case $k=0$) we get the following theorem as a special
case.
\proclaim Theorem. Suppose that $f(\tau)=\sum_nc(n)q^n$
is a modular form of weight $k+1/2>0$ for $\Gamma_0(4)$ (with $k$
even) such that the Fourier coefficients $c(n)$ vanish
unless $n\equiv 0,1\bmod 4$.
Then the function $\Phi(\tau)$ defined by
$$\Phi(\tau)=c(0)B_{k}/2k +\sum_{n\ne 0}q^n\sum_{00$. If
$k=0$ and all the coefficients $c(n)$ are integers then for some
rational $h$ the functions $q^h\exp(\Phi(\tau))$ is a modular form of
weight $c(0)$ for some character of $SL_2(\Z)$ (at least if we first
remove the infinite constant term from the expression for $\Phi$).
Example. If $k=0$ then we take
$f(\tau)=\theta(\tau)=\sum_nq^{n^2}$.
Then
$$\Phi(\tau)=\log\left(\prod_{n>0}(1q^n)^2\right)$$
so that $q^h\exp(\Phi(\tau))$ is $\eta(\tau)^2$ and $h=1/12$.
Example. Put
$f_{13/2}=\theta F(\theta^416F)(\theta^42F)$ of weight $6+1/2$ where
$F(\tau) = \sum_{n>0}\sigma_1(2n+1)q^{2n+1}$.
Then $\Phi(\tau)$ must be a form of weight 12, so must be
(a multiple of) $\Delta(\tau)$.
We find
$$\eqalign{ &f_{13/2}(\tau)=q  56q^4 + 120q^5  240q^8 + 9q^9 +
1440q^{12}  1320q^{13} 704q^{16}+O(q^{17})\cr & \Delta(\tau) =
\sum_n\tau(n)q^n = q24q^2+252q^31472q^4+O(q^5)\cr }
$$
and can check explicitly in the example above that
$$\tau(n) = \sum_{dn} d^5c(n^2/d^2). $$
Niwa showed that the function $\Phi$ could be obtained by considering
the integral of $\bar\Theta(\tau)f(\tau)$, where $\Theta$ is the
Siegel theta function of a 3 dimensional lattice. This is very similar
to Harvey and Moore's formula in section 9, and in fact both formulas
turn out to be special cases of a sort of Howe correspondence for
modular forms, possibly with poles at cusps [B].
In particular the Shimura correspondence stated above
still works perfectly well if the form $f$ is allowed to have poles at
cusps, except that the form $\Phi$ will now have singularities at
imaginary quadratic irrationals coming from the poles of $f$. The
singularities will be poles of order $k$ if $k>0$ and logarithmic
singularities (corresponding to the poles and zeros of
$q^h\exp(\Phi)$) if $k=0$.
Example. Put
$$\eqalign{
f(\tau)&=f_{13/2}(\tau)E_8(4\tau)/\Delta(4\tau)+6720\sum_nH(2,n)q^n\cr
&=q^{3}+64q32384q^4+131535q^54257024q^8+11535936q^9+O(q^{12})\cr
&=\sum_n c(n)q^n\cr
}$$
(where $H(2,n)=L(1,\chi_n)$ is Cohen's function [Co])
so that $F$ has weight $5/2$.
We see that $\Psi(\tau)$ should is a modular form
of weight $2(5/21/2)=4$, and we can work out the
singularities and find that they are poles of order 2
at the conjugates of a cube root of 1.
We find that $\Psi(\tau)$ is
$$\eqalign{
64\Delta(\tau)/E_4(\tau)^2
&= 64(q  504q^2 + 180252q^3  56364992q^4 + O(q^5))\cr
&=\sum_nq^n \sum_{dn} dc(n^2/d^2)\cr
}$$
which has poles of order 2 at cube roots of 1 and their conjugates
because $E_4$ has zeros of order 1 at these points.
Example. Now we look at an example where $k=0$ and $f$
has poles at the cusp. We take
$$\eqalign{
f(\tau)
&= 2f_{13/2}(\tau)E_6(4\tau)/\Delta(4\tau) 108 \theta(\tau) \cr
&=2q^{3} + 4 + 504q  53496q^4 + 171990q^5 
3414528q^8 + 8192504q^9 + O(q^{12}) \cr
&= \sum_nc(n)q^n.\cr
}
$$
This time $q^h\exp(\Psi)$ should again have weight equal to the
constant term 4, and also turns out to have poles of order 2 at
conjugates of a cube root of 1. These poles come from the term
$2q^{3}$ as follows: the coefficient $2$ is the order of the zero,
and the exponent 3 is the discriminant of the quadratic equations
whose roots are conjugates of a cube root of 1.
We find that
$$\eqalign{
q^h\exp(\Psi)&=\Delta(\tau)/E_4(\tau)^2\cr
&=q\prod_{n>0}(1q^n)^{c(n^2)}\cr
}$$
is the
same as the function in the previous example up to a factor of 64.
So both the coefficients of this function and the exponents
in its infinite product expansion can be given in terms of
modular forms of half integral weight with poles at the cusps.
As we already know an infinite product for $\Delta$
we can use the example above to find an explicit infinite product
for the Eisenstein series $E_4(\tau)$ and hence
also for the elliptic modular function $j(\tau)=E_4(\tau)^3/\Delta(\tau)$.
More generally we can find an infinite product expansion for
any level 1 modular function with integral coefficients all of
whose zeros and poles are at imaginary quadratic irrationals or cusps.
\proclaim 12 Finiteness theorems.
One can ask whether the examples discussed above are
isolated and exceptional objects or whether they are part of an infinite
family. It is possible to generalize most of the constructions above to
produce a few hundred similar examples of things similar
to (say) the fake monster Lie algebra. However there are several theorems
suggesting that the total number of examples like this may be finite.
The nice behavior of the monster Lie algebra depends on the fact that
the reflection group of $II_{25,1}$ is very nice, and in particular on
the fact that it has a (Weyl) vector $\rho$ which has bounded inner
product with all simple roots. If such a vector exists in some
Lorentzian lattice $L$ with negative norm then there are only a finite
number of simple roots of $L$ and the reflection group of $L$ has
finite index in the full automorphism group. V. Nikulin has shown [N]
that there are essentially only a finite number of such lattices, up
to multiplication by constants, and Esselmann has shown that the only
one of dimension greater than $20 $ is the 22 dimensional lattice of
determinant 4 consisting of the even vectors of $I_{21,1}$. Nikulin
recently extended his theorem to cover the case when the vector $\rho$
has zero norm. The largest lattice in this case is presumably the
lattice $II_{25,1}$ though this has not been proved. So Nikulin's
work suggests that there may be only a finite number of interesting
Lie algebras similar to the monster Lie algebra. However there are
some examples of generalized KacMoody algebras with no real roots
which still have known simple roots and root multiplicities and Weyl
vectors equal to 0, so it is still conceivable
(but unlikely) that there are an
infinite number of these. Nikulin and Gritsenko have given some
examples of generalized KacMoody algebras related to some hyperbolic
reflection groups in [GN] and have suggested that maybe most
crystallographic reflection groups in Lorentzian lattices $L$ which
have finite index in the full automorphism group are associated with
some nice generalized KacMoody algebra or superalgebra whose
denominator formula is an automorphic form. In particular they show
that the Siegel modular form $\Delta_5$ (one of the standard
generators of the ring of Siegel modular forms of genus 2) can be
written as such an infinite product.
Many of the Lie algebras similar to the fake monster Lie algebra
(and all the known ones of rank at least 3) are closely related
to certain products of (positive and negative) powers of $\eta$ functions
with multiplicative coefficients. For example the fake monster Lie algebra
itself is related to the function $\Delta(\tau)=\eta(\tau)^{24}$,
whose coefficients $\tau(n)$ were proved to be multiplicative
by Mordell and Hecke. Y. Martin [M] has recently found
all such products of eta functions and in particular has
proved that there are only a finite number of them. This again hints
that there are only finitely many analogues of the fake monster Lie algebra.
Some of the generalized KacMoody algebras of rank 2, such as the
monster Lie algebra itself, are not related to multiplicative products of
eta functions. However most of the known ones that are not are instead
closely related to certain Hauptmoduls of genus 0 congruence
subgroups of $SL_2(\R)$; for example, the monster Lie algebra is
related to the Hauptmodul $j(\tau)$ of the subgroup $SL_2(\Z)$.
It is easy to find an infinite number of non congruence genus 0 subgroups,
but J. G. Thompson showed that there are only a finite number of
conjugacy classes of congruence subgroups of any given genus.
The theorem in section 9 produces an infinite supply of examples of
automorphic forms with infinite product expansions. Unfortunately most
of these cannot possibly be the denominator formulas of generalized
KacMoody algebras. The point is that most of the infinite products
involve vectors of norm at least 4 in the lattice, so such vectors
would have to be roots of any generalized KacMoody algebra. However
if $v$ is a positive root of a generalized KacMoody algebra, then
$(v,v)2(v,w)$ for any other root $w$. This means that if the root
lattice is unimodular then positive norm vectors cannot have
norms greater than 2. In fact the only infinite products given by
the theorem in section 9 which can be the denominator formulas of generalized
KacMoody algebras are the ones associated to the fake monster Lie
algebra and the monster Lie algebra that we have already seen above.
There are of course many automorphic forms of higher level which have
nice infinite product expansions, and it seems conceivable that there
may be infinite families of these with no positive norm vectors
appearing in the infinite product. These could then be interpreted as
the denominator formulas for generalized KacMoody superalgebras with
known differences of the root multiplicities and simple root multiplicities.
\proclaim References.
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