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\proclaim Problems in moonshine.
Richard E. Borcherds, %author
\footnote{$^*$}{ Supported by
NSF grant DMS-9970611.}
Mathematics department,
Evans Hall \#3840,
University of California at Berkeley,
CA 94720-3840
U. S. A.
e-mail: reb@math.berkeley.edu
www home page www.math.berkeley.edu/\hbox{\~{}}reb
\bigskip
The talk at the I.C.C.M. was an introduction to moonshine. As there
are already several survey articles on what is known about moonshine
([B94], [B99], [D-M94], [D-M96], [F-H98], [J]),
this paper differs from the talk and will concentrate
mainly on what we do not know about it.
Moonshine is not a well defined term, but everyone in the area
recognizes it when they see it. Roughly speaking, it means weird
connections between modular forms and sporadic simple groups. It can
also be extended to include related areas such as infinite dimensional
Lie algebras or complex hyperbolic reflection groups. Also, it should
only be applied to things that are weird and special: if there are an
infinite number of examples of something, then it is not moonshine.
We first quickly review the original moonshine conjectures of
McKay, Thompson, Conway and Norton [C-N].
The classification of finite simple groups shows that every finite simple group
either fits into one of about 20 infinite families, or is one
of 26 exceptions, called sporadic simple groups.
The monster simple group is the largest of the sporadic finite simple groups,
and was discovered by Fischer and Griess [G].
Its order is
$$
\eqalign{
&8080,17424,79451,28758,86459,90496,17107,57005,75436,80000,00000 \cr
=&
2^{46}.3^{20}.5^9.7^6.11^2.13^3.17.19.23.29.31.41.47.59.71\cr
}$$
(which is roughly the number of elementary particles in the earth).
The smallest irreducible representations have dimensions
$1, 196883, 21296876, \ldots$.
On the other hand
the elliptic modular function $j(\tau)$, defined by
$$
j(\tau)= {\left(1+240\sum_{n>0}\sigma_3(n)q^n\right)^3
\over
q\prod_{n>0}(1-q^n)^{24} }
$$
has the power
series expansion
$$j(\tau) = q^{-1}+744 + 196884q + 21493760q^2+\ldots$$ where
$q=e^{2\pi i \tau}$.
John McKay noticed some rather weird
relations between coefficients of the elliptic modular function and
the representations of the monster as follows:
$$
\eqalign{
1&=1\cr
196884&=196883+1\cr
21493760&= 21296876+196883+1\cr
}
$$
where the numbers on the left are coefficients of $j(\tau)$ and the
numbers on the right are dimensions of irreducible representations of
the monster. The term ``monstrous moonshine''
(coined by Conway) refers to various extensions of McKay's observation,
and in particular to relations between sporadic simple groups and
modular functions.
McKay and Thompson suggested that there should be a graded representation
$V=\oplus_{n\in \Z} V_n$ of the monster, such that $\dim(V_n)=c(n-1)$,
where $j(\tau)-744 =\sum_nc(n)q^n=q^{-1}+196884q+\cdots$.
To characterize $V$,
Thompson suggested looking at the McKay-Thompson series
$$T_g(\tau)=\sum_nTr(g|V_n)q^{n-1}$$ for each element $g$ of the
monster. For example, $T_1(\tau)$ should be the elliptic modular
function. Conway and Norton [C-N] calculated the first few terms of
each McKay-Thompson series by making a reasonable guess for the
decomposition of the first few $V_n$'s into irreducible
representations of the monster. They discovered the astonishing fact
that all the McKay-Thompson series appeared to be Hauptmoduls for
certain genus 0 subgroups of $SL_2(\R)$. (A Hauptmodul for a subgroup
$\Gamma$ is an isomorphism from $\Gamma\backslash H$ to $\C$,
normalized so that its Fourier series expansion starts off
$q^{-1}+O(1)$.)
The module $V$ was constructed explicitly as a representation of the monster
group in [F-L-M], and it was shown
in [B92] that this module satisfies the moonshine conjectures,
using the fact that it has the structure of a vertex algebra. In the
rest of this paper we will look at various conjectural ways to generalize
this.
{\bf Problem 1.} Find a ``natural'' construction for the monster
vertex algebra $V$. All known constructions for it construct it as the
sum of several (usually two) different pieces, and it takes a lot of
work to show that the vertex algebra structure can be defined on this
sum, and to show that the monster acts on it. It would be much nicer
to have some sort of construction which gives $V$ as ``just one
piece''.
One idea for doing this might be to ``stabilize'' the monster vertex
algebra by tensoring it with copies of the vertex algebra of the
2-dimensional even Lorentzian lattice $II_{1,1}$. The reason for this
is that if we add this lattice to any Niemeier lattice, we get the
same answer $II_{25,1}$, so the Niemeier lattices can be recovered
from norm 0 vectors in $II_{25,1}$. Moreover the proof of the
moonshine conjectures involves taking a tensor product with the vertex
algebra of $II_{1,1}$, showing that this is a natural operation. The
most naive version of this question would be to ask if the monster
vertex algebra tensored with the vertex algebra of $II_{1,1}$ is
isomorphic to the vertex algebra of $II_{25,1}$. This seems rather
unlikely, though as far as I know it has not been disproved.
{\bf Problem 2.} Construct a good integral form on the monster vertex
algebra $V$. This should have the property that each of the
homogeneous pieces of $V$ has a self dual symmetric bilinear form on
it. This integral form might follow from a good answer to problem 1,
but might also be possible to prove independently as follows. The
usual construction for the monster vertex algebra gives a self dual
integral form ``up to 2-torsion''. If we could carry out a similar
construction ``up to $p$-torsion'' for some other prime $p$ then it
might be possible to splice these together to get a good integral
form. There are some constructions [D-M92], [M96] of the monster vertex
algebra which look as if they might extend to give a construction up to
3-torsion. One advantage of a good integral form for the monster
vertex algebra is that it would make the study of modular moonshine
[R], [B-R], [B98] somewhat easier.
{\bf Problem 3.} Find an affine ind group scheme over the integers
for the monster Lie algebra. (Here an affine ind group scheme is
roughly a commutative linearly topologized Hopf algebra, in much the
same way that an affine group scheme is more or less the same as a
commutative Hopf algebra. For infinite dimensional groups, it seems to
be necessary to allow linear topologies on the Hopf algebra because
there is often no natural map from $H$ to $H\otimes H$, but only to
the completion $H\hat\otimes H$.) Over the rational numbers it is not
hard to find an ind group scheme whose Lie algebra is the monster Lie
algebra and whose group of connected components is the monster
group. The question is whether one can find a good integral form for
this ind group scheme. There are also several intermediate questions:
can one find a good integral form for the monster Lie algebra (which
would follow from a good integral form for the monster vertex algebra)
and can one find a formal group over the integers for the monster Lie
algebra? At the moment it is not really clear what one would do with
such an ind group scheme. Perhaps it could be used to study infinite
dimensional automorphic forms corresponding to the monster Lie
algebra, or perhaps its points with values in finite fields might be
useful groups. For the related case of the fake monster Lie algebra
there is some progress on these questions in [B99a], but the monster
Lie algebra seems harder because it is not generated by the root
spaces of roots of zero or positive norm.
{\bf Problem 4.}
Prove Norton's generalized moonshine conjectures [N86].
Roughly speaking, these conjectures assign modular functions to pairs
of commuting elements of the monster, rather than to elements. The
point is that the group $SL_2(\Z)$ acts not only on (genus 0) modular functions
but also on pairs of commuting elements of any group. Some progress
has been made on Norton's conjectures by Dong, Li, and Mason [D-L-M],
who proved the generalized moonshine conjectures in the case when $g$
and $h$ generate a cyclic group by reducing to the case when $g=1$
(the ordinary moonshine conjectures). G. H\"ohn [H] has made some
progress in the harder case when $g$ and $h$ do not generate a cyclic
group by constructing the required modules for the baby monster (when
$g$ is of type $2A$). It seems likely that his methods would also
work for the Fischer group $Fi_{24}$, but it is not clear how to go
further than this. Dong has recently nearly proved the generalized
moonshine conjectures. Very roughly speaking, his proof is an
extension of Zhu's proof that characters of certain vertex algebras
are modular functions to the equivariant case.
{\bf Problem 5.} Explain the connection (if any) between moonshine
and the ``Y presentation'' of the monster [ATLAS]. The latter is a
particularly easy presentation of the monster group conjectured by
Conway and Norton and finally proved by Ivanov (see [I]), which looks
roughly like the presentation of a Coxeter group with one extra
relation. Miyamoto [M95] found a very suggestive relationship between
this presentation and the monster vertex algebra, by finding a set of
involutions of the monster vertex algebra satisfying the necessary
relations. His proof used properties of the $E_6^4$ Niemeier lattice.
{\bf Problem 6.} Classify the generalized Kac-Moody algebras that
have Weyl vectors and whose denominator functions are automorphic
forms (possibly with singularities). The simplest example of such a
Lie algebra is the monster Lie algebra itself, with denominator
function $j(\sigma)-j(\tau)$. There are plenty of other similar rank 2
Lie algebras, some related to moonshine and sporadic groups, and some
which do not seem to be related to sporadic groups. There are also
many examples in various ranks up to 26, which is probably the highest
possible rank.
It is possible to relax the condition on the generalized Kac-Moody
algebra that a Weyl vector should exist, and ask just for those whose
denominator function is an automorphic form. Gritsenko and Nikulin
have classified some of these algebras in low ranks [G-N], [N].
{\bf Problem 7.} Find all ``interesting'' hyperbolic reflection
groups (this includes the arithmetic ones and some others). This is
closely related to the problem of finding interesting generalized
Kac-Moody algebras, because the Weyl group of an interesting
generalized Kac-Moody algebra is often an interesting hyperbolic
reflection group. Nikulin has proved that the number of hyperbolic
reflection groups which are close to being arithmetic (in various
senses) is finite, and known examples suggest that there are probably
a few thousand of them. There seems to be a rather mysterious
connection of these hyperbolic reflection groups to genus 0 modular
groups; slightly more precisely, most of the interesting hyperbolic
reflection groups seem to be associated with certain modular forms
with poles of low order at cusps. For example, Conway's reflection group
of $II_{1,25}$ is associated with the function $1/\Delta(\tau)$.
However this correspondence is not well
understood; see [B99b] for some examples.
{\bf Problem 8.} Find ``natural'' constructions of the generalized
Kac-Moody algebras in the previous problems. All these algebras can
be constructed using generators and relations, but this is not a
satisfactory way of constructing them; for example, it is very hard to
see interesting symmetry groups of these algebras (such as the monster
group) in this approach. What we would prefer is a construction in
which one can explicitly see the interesting symmetry groups. For
example, the monster Lie algebra is constructed from the monster
vertex algebra as the Lie algebra of physical states of strings on a
26 dimensional orbifold, so the action of the monster can be seen
directly because it acts on the monster vertex algebra. Scheithauer [S]
has recently found a similar construction for the ``fake monster Lie
superalgebra'' of rank 10. Harvey and Moore [H-M] have made an
exciting suggestion for realizing these algebras using BPS states, or
by constructing them using the cohomology groups of moduli spaces of
vector bundles of surfaces.
{\bf Problem 9.} Is there anything like moonshine for the 6 sporadic
groups not involved in the monster group? Note that most and probably
all the Chevalley groups act naturally on vertex algebras over finite
fields, and many of the sporadic groups involved in the monster act
naturally on vertex algebras, sometimes over finite fields as in
modular moonshine and sometimes over the rational numbers as for the
monster. This means that most finite simple groups act naturally on
vertex algebras, and an obvious question is whether they all do.
Presumably any simple group acting on a natural vertex algebra would
have some sort of connection with modular functions. Unfortunately (as
far as I know) no one has ever found any serious evidence that the
remaining 6 sporadic groups (the Janko groups $J_1$, $J_3$, and $J_4$,
the O'Nan group, the Lyons group, and the Rudvalis group) have any
connection with vertex algebras or modular functions.
{\bf Problem 10.} Explain McKay's weird observation (described in
[B99] for example) relating the Dynkin diagrams $E_8$, $E_7$, and
$E_6$ with the monster, the baby monster, and the group $Fi_{24}$. The
point is that conjugacy classes of pairs of commuting involutions of
the monster seem to correspond to vertices of the affine $E_8$
diagram, and there is a similar connection between the baby monster
and the $E_7$ Dynkin diagram, and between the group $Fi_{24}.2$ and
$E_6$. This may be related to the (well understood) McKay
correspondence between Dynkin diagrams and finite rotation groups in 3
dimensions.
{\bf Problem 11.} Lian and Yau [L-Y] showed that mirror maps for K3
surfaces are sometimes the inverses of Hauptmoduls; for example one of
their mirror maps is
$$q-744q^2+356652q^3+\cdots$$ which is the inverse of the elliptic
modular function $j(\tau)$. They found that several other Hauptmoduls
of elements turned up. They asked the rather speculative question
about whether there is some direct connection between the monster and
K3 surfaces. This seems rather wild as there is no obvious way in
which the monster could be connected with K3 surfaces, but on the
other hand this is what most people said about McKay's original
observation connecting the monster and elliptic modular functions!
{\bf Problem 12.} Is there a monster manifold? More precisely,
Hirzebruch asked if there was a 24 dimensional manifold acted on by
the monster with Witten genus $j(\tau)-744$. If so, it might be
possible to use this to construct the monster vertex algebra, or at
least its underlying space. Hopkins and Mahowald recently constructed
a manifold with the correct dimension and Witten genus, but so far it
is unclear how to construct an action of the monster on it.
{\bf Problem 13.} Complex hyperbolic reflection groups. Allcock [A]
recently constructed some striking examples of complex hyperbolic
reflection groups from the Leech lattice, or more precisely from the
complex Leech lattice, a 12 dimensional lattice over the Eisenstein
integers. This complex reflection group looks similar in several ways
to Conway's real hyperbolic reflection group of the lattice
$II_{1,25}$. Allcock also showed that there is an automorphic form on
complex hyperbolic space vanishing exactly on the reflection
hyperplanes of this reflection group. Several other complex hyperbolic
reflection groups found by Allcock are closely related to various
moduli spaces, for example the moduli space of cubic surfaces [A-C-T].
Most of these complex hyperbolic reflection groups seem to have
something to do with moonshine, though it is hard to be precise about
what the relationship is. For example many of them are related to
automorphic forms that in turn are related to moonshine. So a general
and rather vague question is: what is going on? For some more specific
questions we could ask for a classification of arithmetic complex
hyperbolic reflection groups, and for each of them we can ask if it is
related to some moduli space and some automorphic form. There is some
speculation that finite complex reflection groups should be related to
some so far unknown algebraic structures provisionally called
``spetses'' [M99] in the same way that the real reflection groups are
related to Lie algebras. As the real hyperbolic reflection groups are
closely related to Kac-Moody algebras, an obvious question is to ask
if there is an extension of spetses to complex hyperbolic reflection
groups.
{\bf Problem 14.} Is there a ``nice'' moduli space related to
$II_{1,25}$? The denominator function of the rank 10 fake monster Lie
superalgebra turns out to be an automorphic form on the period space
of Enriques surfaces vanishing exactly along the points of singular
Enriques surfaces [B96]. A direct construction of this automorphic
form was found by Harvey and Moore [H-M98] and Yoshikawa [Y],
following a suggestion of Jorgenson and Todorov [J-T]. (Note that, as
pointed out in [Y], the main theorem stated in [J-T] is not correct.)
The fake monster Lie superalgebra and the fake monster Lie algebra
seem similar in many ways, so we can ask if there is a similar moduli
space related to the fake monster Lie algebra and its root lattice
$II_{1,25}$. So the period space should be the hermitian symmetric
space corresponding to the lattice $II_{2,26}$, with singular points
along the divisors of norm $-2$ vectors, and the moduli space should
be the quotient by the automorphism group of the lattice
$II_{2,26}$. One can also ask similar
questions about other generalized Kac-Moody algebras. Freitag has
suggested that maybe most ``interesting'' moduli spaces should arise
in a similar way, related to automorphic forms on the hermitian symmetric
space of $\R^{2,n}$ that have an infinite product expansion and all of
whose zeros correspond to roots of some lattice. See [F-H] for some
examples.
{\bf Problem 15.} Modular forms with poles at cusps often turn up in
the questions in this paper. Is there some useful analogue for these
forms of the $L$-functions or Dirichlet series of classical cusp
forms? Note that the obvious Dirichlet series formed by the
coefficients does not converge anywhere, so we have to use a different
way to define the ``$L$-function''. One possibility is to just use the
Mellin transform. This does not converge if there are poles at cusps,
so it would be necessary to regularize the integral in some
way. Another possibility would be to look at functions with poles at
some cusps and zeros at others, and just integrate between two zeros.
The resulting Mellin transforms would satisfy some functional equation
coming from the functional equation of modular forms in the usual
way, but it is hard to see what else one can say about them. Perhaps
one could look at a set of modular forms with singularities invariant
under the Hecke algebra and ask what the corresponding properties of
the Mellin transforms are. (In the case of forms without
singularities, the Hecke algebra has eigenfunctions and the action of
the Hecke algebra on the Eigenfunctions corresponds to an Euler
product decomposition on the $L$-series. However if the modular forms
have cusps then the Hecke algebra tends to act freely because Hecke
operators make singularities worse, so there are no eigenfunctions.)
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\bye