Video summary
This lecture serves as the concluding part of a series exploring the deep, yet somewhat mysterious, connections between hyperbolic reflection groups, Kac-Moody algebras, and automorphic forms. While not every object in these categories corresponds to another, there are significant cases where strong relationships exist. The central theme is that both hyperbolic reflection groups and specific Kac-Moody algebras appear to be intimately linked to automorphic forms. To understand this link, the speaker first reviews the definition of an automorphic form, starting with modular forms on the upper half-plane. These are functions that transform under the action of the group SL(2, Z) up to a specific factor, effectively acting as sections of a line bundle rather than strictly invariant functions. The lecture extends this concept by replacing SL(2, Z) with larger Lie groups and their discrete subgroups, leading to more complex transformation factors.
The primary example discussed is the denominator function associated with the 26-dimensional even unimodular Leech lattice. This function, which can be viewed as an automorphic form on a specific Lorentzian space analogous to the upper half-plane, satisfies a wave equation and exhibits symmetries corresponding to translations and lattice automorphisms. A crucial property of this function is that it vanishes on a specific hypersurface defined by vectors with a norm of two. By comparing this sum representation with a product representation derived from partition theory, the speaker demonstrates that the function must vanish on this surface. This behavior allows for the application of the Schwarz reflection principle (referred to as the Koshi-Govi theorem in the transcript), proving that the function transforms under a larger group than initially apparent, thereby establishing its status as an automorphic form for a specific orthogonal group related to the Leech lattice.
The lecture then investigates how these automorphic forms relate to Vinberg's classification of hyperbolic reflection groups. For many of these groups, particularly those derived from sublattices of the Leech lattice, the corresponding automorphic forms vanish precisely on the reflection hyperplanes of the root system. However, for cases involving infinite Dinkin diagrams, such as the I21 group or extensions related to E10, the situation is more complex. In these instances, the automorphic form vanishes not only on the hyperplanes corresponding to actual roots but also on additional hyperplanes associated with vectors of norm two that are not roots. This suggests that the automorphic form encodes information about the geometry of a larger lattice structure, effectively "causing" extra zeros that reflect the infinite nature of the underlying algebraic system.
Finally, the speaker highlights several open questions and future directions in this field. One major goal is to classify all hyperbolic reflection groups that correspond to automorphic forms, noting recent progress by researchers like Brandon Williams and Yang Song in connecting these structures to conformal field theories. Another significant challenge is finding natural constructions for the associated Kac-Moody algebras beyond mere presentations by generators and relations, similar to how string theory's no-ghost theorem constructs the algebra for the 26-dimensional Leech lattice. The talk concludes by mentioning potential generalizations over other number fields, such as Eisenstein integers, and the existence of "magic functions" related to densest lattice packings, suggesting that analogues of these magical properties might exist for sublattices of the Leech lattice, opening new avenues for research in this intersection of geometry and algebra.
Read the full video transcript
now this video is the fourth and final
part of um a series of videos on my
vinberg lecture on um wiberg's algorithm
and cats Moody algebras if you want to
see the first three parts there should
be a link to them in the description of
the video down below so this part will
be um mostly about automorphic
forms um so in the previous
lecture um we described there was a sort
relation between hyperbolic reflection
groups
and um Lee
algebras um this relation is a little
bit mysterious um so there are plenty of
holic reflection groups that don't
correspond to Lee algebras or even cats
Moody algebras and plenty of cats Moody
algebras that don't correspond to holic
reflection groups but there seem to be a
few cases when there was a sort of
strong relation between them um the one
we looked at in particular was where you
take the even 26 dimensional lenan
latis and this corresponds to a a sort
of generalized cats Moody algebra with
um dinkin diagram the leech
ltis plus some Norm zero vectors so this
was the subm of the Third lecture so I
won't say too much more about it just
now um what we're going to discuss is
that both of
these um objects seem to be related to
automorphic
forms again this correspondence isn't
exact there are plenty of automorphic
forms that don't correspond to Le
algebras or hyperbolic reflection groups
and not all cats moody algebras or
hyperbolic reflection groups seem to be
connected with automorphic forms but
there are several examples suggesting
there is some sort
of um interesting um connection between
these um so I'll start by just recalling
what an automorphic form
is um so the simplest examples of
automorphic forms are just modular
forms um and a modular form is um a
function f such that F of a to B over C
to +
D is equal to C to + D the K times F of
to here and the imaginary part of to is
greater than zero and there are some
other minor conditions that I won't
worry about too
much um and and what's going on is well
we we have this mysterious fudge Factor
here and if we ignore this fudge factor
for a bit um you can see this is almost
as saying say this is almost saying that
f is
invariant
under SL2 Z so here um these matrices um
a b c d are in SL2 of Z which is just 2x
two matrices with integer coefficients
and determinant one and this acts on the
upper half plane by a b c d
epon to is a to + B over C to plus
D um so without this purple fudge Factor
we're just saying the function f is
invariant under the action of SL2 z um
this fudge factor means that F isn't
quite
invariant um but transforms up to an
elementary Factor um what it's really
saying is that f is a section of a line
bundle
under SL2 Z that is invariant and line
bundle is a sort of High um is a sort of
high level way of saying that F
transforms up to this this fudge
factor and an example of an automorph
form we had in the previous lecture was
um Delta of to which is Q * product over
n greater than Z 1 - Q to the N to the
24 where Q is of course e to 2 pi I to
um now this function Delta has uh some
obvious symmetries so so to of Delta
plus the Delta of to + 1 is equal to
Delta of to this is just
obvious because you know Q is invar
under to goes to to + one and there's a
there's a sort of hidden magical
symmetry which says that Delta of minus1
/ to is equal to to 12 time Delta of to
and now you see this says that Delta of
to transforms like this for the
following two Matrix we have the Matrix
1 1
01 um corresponding to Tower goes to to
+ 1 and we have Matrix 0 -1 1 0
corresponding to to goes to minus one/
to now these two
matrices actually
generate
um SL2
Z so um from this we can easily see that
Delta actually satisfies this relation
for all matrices and SL2 Z so Delta is
an example of a modular
form um so what's an automorphic form
well an automorphic
form we just
replace SL2 Z by a
larger
group now SL2 Z is contained in SL2 of
the reals and you what we're going to do
is we're going to replace SL2 of the
reals by some larger Le group and have a
larger discrete subgroup in it and then
um there's going to be some relations
saying a function is invariant onto some
fudge factor and the fudge Factor will
be more complicated and we'll see some
examples of it um fairly soon um
so the the first example of an
automorphic
form is is is going to be the
denominator
function of um the Lee Algebra I
mentioned earlier related to the 26
dimensional even lenan
lce and we remember from last lecture
that this had a denominator formula
which look like this if we take the sum
over all elements of the vile group of
some s
times um Omega
of um
e to the sorry sum of to n time e to the
N
row where to n is the coefficients of
the of Delta function this is equal the
product over Alpha greater than Z of 1 -
e to the
alpha power of the multiplicity of alpha
so let's just recall what the various
bits in it are here sum of to
n q to the N is is just Delta of to
which is Q - 24 Q squared and so on um
and the multiplicity of alpha was given
by P
24 of 1 - Alpha 2 / 2 where p24 are the
coefficients of 1 over Delta which is
q-1 + 24 + 324 q and so on so this is
quite
complicated um and what this is doing is
it's saying that a certain
sum is equal to a certain
product
and what I'm going to do is to show that
um if we think of this as being a
function we can think of this as being a
function on the following um space um we
take
um um I I 25 comma 1 and then we tensor
it with the
reals and then then we have I
times um C where C is the positive
cone um so you remember this is a
lorenzian
space which has a um double cone of Norm
zero vectors and inside this we can see
sort of positive cone here which I'm
going to call C and we're going to say
that
this this is going to be a function on
the set of vectors whose real part is
anything in this space whose imaginary
part is in this positive cone you can
think of this as being an analog of the
upper half plane so the upper half plane
says the real part is in a
one-dimensional vector space and the
imaginary part is in a cone in the reals
which is just the just the positive
reals um so let's see why this thing is
an automorphic form well you remember um
Delta is an automorphic form because
first of all it had some had an obvious
transformation under to go to to + one
and had a mysterious one and the to goes
to minus one over
to um well so our function here has some
obvious
Transformations first of all we can we
can uh
translate by elements of the latice i i
251 which sort of correspond to to goes
to to + one we also have automorphisms
of the ltis II 251 well um we don't
quite get all automorphism but op to a
factor of two um automorphisms of this
lce gives us Elementary
Transformations there's also a
nonobvious
transformation um and we can get this as
follows so here I had
this um sum that I'm not going to write
out again um was equal to some product
that I'm not going to write out again
either and what we notice is the sum is
a
solution of of the wave
equation and this is because if you look
at the the terms of the sum carefully
you see that all these vectors appearing
at h of Norm zero and um X Norm Z Vector
basically gives you um um a solution of
the wave
equation on the other hand this product
if you take its
logarithm the logarithm is singular
um
when um um V is imaginary and v^ 2 um is
equal to um 2 um so
here what's happening is the imaginary
part lies inside this cone C here and
inside this cone there's the
hypersurface of um vectors with v ^2 = 2
it's 2 rather than minus 2 because we're
looking at imaginary vectors and this
product actually vanishes here and we
can ask why does this product
vanish well what we do is we recall that
um if we've got a power series sum of a
n z to the n and if the radius of
convergence is r and all the a ns are
greater than or equal to zero
then um then this has a
singularity
at R so um you know power series with
positive coefficients has to have a
singularity at the um at at the real
point of its radius of
convergence um now if you look at this
product we can work out um ex we can
work out where it converges by using the
Hardy ranagen R ramaka formula for the
ASM totic behavior of of um um the
partition function or rather partitions
into parts of 24
colors furthermore if we take the
logarithm of this all its coefficients
are well they're not all positive
they're all negative but that's good
enough so um using this we can we can
see this product must actually be
singular on uh this surface here on the
other hand this sum here is non singular
so this sum must actually
vanish
when um V is imaginary and V ^2 equals
2 um well now what we notice is that um
if if we let's call this function f so
we've got these two functions got this
function f of V so it's going so this is
a a solution of the wave equation that
vanishes on this hypersurface on the
other hand if we look at the the
function v v over two to the 12 * 5 of
minus vus 2 V over
VV um we can check this is also a
solution of the wave equation I mean
this is just the transformation of the
wave equation under under a
certain um conformal map and um the fact
that this function vanishes on this
hyper plane means that these two
functions actually have the same zero
and first derivatives on this
hypersurface so um we can now apply the
Koshi Govi theorem which says that two
functions that satisfy the wave equation
and have the same zero and first
derivatives on a Koshi hypers surface
must be equal um so these two functions
are actually the same well I guess I
should have put a minus sign in there so
here we're applying the Koshi kvki
theorem so um what we have now is a a
magical extra
transformation of this function so this
kind of
corresponds
to this transformation of the function
Delta and what we saw is that for Delta
these two
Transformations um mean that we're
actually transforming under the group
SL2
z um well if we put together all these
trans formations of this function fi
what we see is that
fi is an automorphic
form for the
group well what we do is we take an
orthogonal group of the ltis 25 comma
1 um over the integers well again it's
up to a factor of two I I should really
take a subgroup of index two but I won't
worry about this and this is contained
in the lead group where you um just take
um it's a 26 dimensional um Lorent group
sorry um should be 26 comma 2 here
things mysteriously go up by
one
um so um what we have is an automorphic
form for
um a group which you should think of you
should think of this as being an analog
of SL2 Z contained in SL2
the um and we can do this with for
several other hyperbolic reflection
groups so we get the same for um
hyperbolic reflection
groups
corresponding to um well if you take the
leech lce and take the fixed Point under
some automorphism that gives you a lce
rather like the leech lce and you can
sort of um go through and get a similar
reflection group and a similar
automorphic form so this is worked out
in a few cases by um nean and um um and
generally by
shiow who showed that if you take any
automorphism of the leech ltis with a
non- of your fix point sublattice then
you can get a similar automorphic form
for it so that shows there are some
hyperbolic reflection groups um
corresponding to automorphic
forms um well now we have the
problem what
about vinberg
groups so taking a a fixed sublattice of
the leech lce only gives you some rather
special
hyperbolic reflection groups and these
automorphic forms are rather special
they turn out to be automorphic forms of
singular
weight um and turns out you can um for
um all the groups vinberg studied and
the answer is we take the form fi on I I
uh 26 comma 2 um which um uh is actually
automor form on I I 25A 1 T C with
imaginary
part um in in the cone and we can just
um pick a dinken diagram
in um
the
um uh dinkin diagram of I I25 comma one
which is just the leech
latice so if we take some dinking
diagram let's call it D and we look at
the orthogonal complement of D which is
contained an i i 25 comma 1 so this will
be some
lce um and then this automorphic
form
restricts to an automorphic
form for D per plus not one
one0 um there only one slight problem
this restriction is identically zero so
although it transforms like an
automorphic form this is completely
uninteresting because it's it's a zero
automorphic form and the problem is that
five
vanishes on the um orthogonal complement
of um any root R so in particular um D
contains roots are so the oral
complement of any Inc diagram the
automorphic form you get will be
identically zero so that seems to be a
little bit of a
problem um well there's a solution to
it we can
differentiate
um by
before
restricting and what we can do is we can
differentiate once for for each
hyperplane um that it vanishes
on um in D so um uh this will be half
the number of
roots of um of the root system of D um
so so you know if D has has n Roots
there'll be n/2 hyperplanes on all of
which five vanishes so what you do is
you sort of make F non Vanishing on all
these hyper planes by first
differentiating
it um and um the effect of
differentiating it is it
increases the
weight of the form five well what's the
weight of five well you remember five
has this property that F
of
um um
2 V over v
v is equal to minus v v over 2 12 * y of
uh
v um well the weight is just this this
bit
here um
and um so what we get is
automorphic forms
for um various luses who whose weight is
a little bit bigger than you might guess
so let's have an
example suppose we take um vinberg
reflection
group um for
i91 so he showed that the reflection
group of this has a finite dinkin
diagram just um which was a little bit
too complicated to do by hand so vinberg
and Kaplan Sky um got a computer to work
it
out and
um um the even
sublattice is given by it's the
orthogonal complement of D6 in in the
leech lattice so so we take a a D6 dink
diagram in the leech lattice and take
its orthogonal complement um by the way
in case you're thinking that leech ltis
has no roots at all we're think thinking
of the leech lce as being the Dink and
diagram of the 26 dimensional even
verenium
Lattis um so um we get an automorphic
form for um I corresponding to
i91 of
weight well it would be
12 um
um plus 60 over 2 well what's 60 well
this is the number of
roots of
D6 um and similarly for all other for
all the other um reflection groups of
even unimodular latis that vinberg
studied we can get um an automorphic
form of some weight for example for I
211 which was the largest just one we
get a form of
weight 12 plus well this time um we
notice this is the orthogonal complement
of D4 so uh D4 is 24 Roots so we take
24/
2 um and these automorphic
forms
vanish on the um
reflection hyperplanes
so um each of these reflection
groups um has reflection hyperplanes and
the automorphic form very neatly
vanishes on these hyperplanes um and um
if you want to just describe all the
places where the automorphic form
vanishes it actually
um um vanishes on reflection hyperplanes
not just of this lattice but you know we
have to make this lattice biger by
adding on a little two-dimensional renan
lce and in fact the automorphic form
vanishes on on hyperplanes of roots of
this bigger lce so the automorphic form
corresponds very nicely to the
hyperbolic reflection group and tells
you what Its Reflection hyp planes
are um well if now if if we look at
i21 um something a little bit strange
happens here this is the one where
vinberg
showed the dinkin diagram is
infinite and we get an automorphic
form corresponding to it and this
automorphic form
vanishes on the
hyperplanes um corresponding to orthal
complements of vectors R where R 2 is
equal to 1 two or three and these ones
are not
Roots um so so something more
complicated seems to be going on the
automorphic form sort of notices the
hyperplanes where the reflection group
vanishes but it also notices some other
hyperplanes that don't correspond to um
um reflections of the reflection group
and this sort of seems to be related to
the fact that the Dink and diagram is
infinite um somehow the the automorphic
form is this kind of causes the
automorphic form to pick up pick up some
extra zeros not corresponding to to
Reflections um we can also look at the
example of um even um unimodular lses so
so vinberg looked at these two
cases so
i91 um this corresponds to the E10 10 1
2 3 4 5 6 7 um 8 9 10 so this
corresponds the E10 dinken diagram and
we recall that this corresponds to um um
bigger dinkin diagrams I'm just going to
a sketch that looks like this we take
two copies of the
E9 diagram and join them like that and
the question is can we find automorphic
forms corresponding to
these
um and the answer is yes um and this is
going to give us a bonus because it is
also going to tell us what is the nice
Le algebra corresponding to this so the
question is um can you extend the
E10 um cat Moody algebra to a bigger Le
algebra with a nice denominator formul
and the automorphic form tells you how
to do
this um so what goes on here is again we
just restrict
um um the automorphic form f after
differentiating of course and for this
we get a form of
weight
12 um plus
um well um we need to figure out what
E10 is well E10 is given by taking E8
plus E8 and then taking its orthogonal
complement um in the 26 dimensional even
ran lce so we need to know how many
roots does this have well it is 248 plus
sorry 240 + 240 Roots
so um which is 480 so it would have
waight we have to add 480 over
2 and this gives us an automorphic
form and this automorphic form um turn
turns out to be the denominator
function of um some sort of generalized
um cats Moody
algebra um so this cats Moody algebra
the real simple roots are just E10 but
it also has quite a lot of imaginary
simple Roots um furthermore this
automorphic form again vanishes exactly
on um hyperplanes corresponding to Norm
two root of this bigger lce I I
um 10 comma 2 um so this seems to be
arguably the correct Lee algebra
corresponds to the E10 root system it's
a it's a Lee algebra corresponding to a
nice automorphic form um acted on by the
the orthogonal group of this
ltis and of course you can do the same
thing for this um reflection group just
by taking just by regarding this as the
orthogonal complement of
E8 um this sort of shows that you can't
really understand the
E10 um Lee algebra without going up to
26 dimensions and and then restricting
the corresponding automorphic form um
there are several other examples
of
um um finite reflection groups corres
automorphic forms these these were
studied by
gritsenko and
nicolin um several years ago who showed
there were some other examples of
hyperbolic finite hyperbolic reflection
groups with finite dinker diagrams that
could actually be extended to correspond
to certain automorphic
forms
um um so I I just finish by um
mentioning a few open
questions um so so the most obvious
question is can we
classify
various um
hyperbolic reflection
groups for instance you can um you know
try calculating the the the ones
corresponding to latices over the
integers whose um reflection group has
finite volume and there's been quite a
lot of work on this by nickeline and
others in fact I sort of heard a rumor
that um this has recently been done but
I haven't yet manag to figure out what
the details of this are um we shouldn't
actually restrict the ones of finite
volume
because in some sense the most
interesting cases don't have finite
volume for instance we have Conway's
reflection group um which doesn't
correspond to finite volume as I
mentioned the analoges of these
um have been partly classified by
shiow and recently Brandon
Williams um Yang so
Wang um and um Sun have some recent
preprints where they get uh quite close
to classifying all the cases that look
like this by by classifying the
corresponding automorphic forms um I
might put a link to their paper in the
in the description of this
video um one of the interesting things
they showed us is that the the laes you
get are um closely related to Shell's
list of conformal field
theories so um so we can ask
which um of these holic reflection
groups
correspond to automorphic
forms um so
these um classify rather special ones
that correspond to automorphic forms of
singular weight but as we've seen in the
examples there are quite a lot of
automorphic forms that aren't of
singular weight that also appear to be
quite interesting corresponding to you
know F finite dinking diagrams or
um um reflection groups whose
fundamental domain is infinite
volume another problem is find Al
constructions
of the Lee
algebras um the problem is that although
we can construct Le algebras from this
they're just given by generation and
relations which is a rather untidy way
of describing a Lee algebra um for some
of the Lee algebras this is known for
example um in the case of the Lee
algebra of the 26 dimensional rents in
ltis there was a
construction using the no ghost
theorem from string
theory and for a few of the
other um similar Le algebras that there
are also similar constructions known but
for most of the Lee algebras we get
corresponding automorphic forms I don't
think anyone has found a really natural
construction that doesn't rely on just
writing down generations and
relations
um next we can ask what about
analoges of II
251 over other number
fields and sometimes you can find
analoges over other number fields for
instance if you take a fix. fre
automorphism of this of order three you
can use that to make this into a lce
over the eisenstein integers and that
seems in some way to be a sort of
eisenstein integer analog of this and
then you can you know you can do things
like look at reflection groups over the
eisenstein integers and Daniel
oock um showed you could get um several
rather striking um complex reflection
groups related to
this
um um and then I got a really crazy idea
um so recently um bovska
um managed to prove that the leech lce
was the densest ltis packing sorry the
densest packing in 24 Dimensions using
various magic
functions um so she had some magic
functions Associated to the leech lce
and the E8 lce and we can ask um do
analogs of these magic functions exist
for um fixed point sublattices of the
leech lce and this so what can you do
with these
functions
um
um so yeah so I think I'll leave it at
that