>> Welcome back to the second
part of the workshop.
It's our great pleasure
to have Liam McAllister
from Cornell here who will start
the second day of
the meeting and talk
about combinatorial
cosmology. Thank you, Liam.
>> Am I audible? Yes. Thanks, Fabian,
and thanks to the organizers
for inviting me.
It's a real privilege to address all
of you at this very timely meeting.
This talk was written
for those of you who
have not worked on string theory.
My goal is to give a high-level
overview that will hopefully set
the stage for some of the later talks
and for our discussions afterwards.
But I'd like to explain the structure
of one of the core
problems in our subject,
which is trying to analyze
the string landscape.
The talk is about realistic
solutions of string theory,
and I'll make it clear during
the remainder what I
mean by realistic.
What I aim to do is to explain
why the solutions allowed by
cosmological measurements
are completely
specified by finite
lists of integers.
We can think of these integers
as quantize fundamental parameters.
If we'd like to understand
what string theory predicts,
we need to connect
those integers to observations.
So what I'll do is formulate
this task of getting from
fundamental integer parameters to
observables as
a computational problem,
and then eventually as
a target for machine learning.
So I'll start with a gentle and
brief invitation to quantum gravity.
I'll explain why parameters are
quantized in quantum gravity.
I'll define what I mean
by the string landscape.
Then after those generalities,
I'll go into one class of examples of
compactification of string theory on
hyper-surfaces in toric spaces.
This is a particularly
useful combinatorial problem
that illustrates some
of the main problems
we face in the landscape.
Then, I'll close with some targets
for machine learning.
So first, what is quantum gravity?
If you take the fundamental
constants in nature,
three of them:
Planck's constant h bar,
the speed of light, and
Newton's gravitational constant,
you can make quantities that have
dimensions of mass, energy, time,
and length: the Planck mass,
Planck energy, Planck
time, Planck length.
Quantum gravity is what happens when
these are the characteristic
scales in a system.
Now, you can note that
the Planck energy
is actually a lot of energy.
This is the energy in a tank of
gas and quantum gravity phenomenon,
or what occur when you put
that much energy in one quantum,
say in one electron or one-photon.
When you do that, the Compton
wavelength of the particle,
so the characteristic scale of
quantum indeterminacy of the particle
and the Schwarzschild radius,
the length-scale setting strong
gravitational effects,
become comparable.
This ruins most of
our abilities to work in
various simplifying
approximations that describe
quantum mechanics in flat space
and classical general relativity.
So this is where the conflict
comes from. When does it matter?
Well, it matters in
very extreme conditions
in black holes singularities,
conceivably at black hole horizons,
in collisions and Planck energies,
in the big bang singularity,
and in the period of inflation
in the early universe.
Given that you might ask,
do we even need to be
mindful of quantum gravity
if it only matters in
the most extreme cases,
the answer to that is simply that
every time we've tried to study
much more extreme circumstances we've
eventually learned
more fundamental laws.
The subject of
a complete talk on its own,
something I won't delve into it all,
is that quantum gravity
turns out to be required to
interpret certain
observations in cosmology,
observations of the cosmic
microwave background.
That's just an aside. It's clearly
key to understanding what gravity is.
In fact, it's also the best hope it's
the current frontier for
understanding what physical laws are,
what kinds of physical
laws are possible,
and part of the landscape problem
I'll be describing consists
of asking in a quantum
gravitational theory,
what are the possible laws of nature?
We'll see that there are discrete.
Okay. So string theory is a finite
theory of quantum gravity.
Finite means that the
predictions it makes
for given scattering processes
are always finite,
there's no regularization and
renormalization of infinities needed,
as we sometimes have to do in
regular quantum field theories.
In string theory, the fundamental
constituents are strings,
either closed strings or loops
or open strings with two end points,
rather than this whole panoply of
particles in the standard model
of particle physics.
Now there are a lot of
different string theories: type I,
type IIA, type IIB,
two different so-called
heterotic string theories.
These are 10-dimensional theories.
There's an 11-dimensional
theory called M theory,
which doesn't have strings but
has the parents of strings,
and then there's a 12-dimensional
construction called F-theory.
But actually, each of these
things is related to the others
by an intricate network of
exact duality relationships.
By dualities we mean
perfect equivalences
for these theories
being dual means that they
describe exactly the same physics,
but manifest in different ways,
described in different ways.
So up to these dualities,
there's really just one kind of,
well, for the experts,
supersymmetric string.
So we're really thinking
about one string theory
or one grand theory that
consists of all of
these string theories and
M-theory and so F-theory.
These are different windows on
the properties of that theory,
different ways of
looking at what it is.
Now despite there being
these different faces of the theory,
they do share some commonalities.
One of them is that
the string theories described here
are naturally 10-dimensional.
So the fundamental solution
of these theories is
10-dimensional Minkowski
space pictured here.
So here the zero is time 1, 2,
3 or xyz, and then these are
six additional spatial coordinates.
What I've drawn is R31,
so Minkowski space times R6.
That's the natural simplest
solution of the theory.
Compactification is
the name we give to
studying solutions that are
a little bit more involved.
They involve, for
example, Minkowski space,
now times a compact space,
rather than a non-compact
Euclidean space.
This is still a space
of signature or plus.
It's a Riemannian manifold
that's compact.
Now, a vacuum solution
of string theory means
a solution in the absence
of sources of
stress energy, so no matter.
Nothing there except
the geometry itself.
The Einstein equations in vacuum,
even in four-dimensions, read
the Ricci tensor vanishes.
So that's still true in
10-dimensions in string theory.
The Einstein equations are
equivalent to the vanishing
of the Ricci tensor.
If I write that out in components,
so here are the 0, 1, 2,
3 directions to the Minkowski space,
and let indices m,
n run over the six
internal directions.
Then, the vanishing of
the Ricci tensor means
that the four-dimensional
Ricci tensor,
and the six-dimensional
Ricci tensor have to vanish.
Minkowski space is flat,
the Ricci tensor
already vanishes here.
So all we've got to do
is ensure this one.
We've got to ensure that
the six-dimensional
compact space is Ricci-flat.
So what this leads us
to is the idea that
a vacuum solution of string
theory corresponds to a
compact Ricci-flat six manifold.
So much of our work consists of
finding manifolds like that.
Now, one very striking thing about
string theory is that it has
no fundamental
dimensionless parameters.
We'll spend a lot of
time talking about
the way that perimeters
show up in the theory.
But if you just look at the theory
on the string worldsheet itself,
there are no dials you can adjust.
There's one-dimensionful
full parameter,
which is the length of a string
or the tension of the string,
but that just sets the
units for the problem.
There is nothing else. This is
quite different from
quantum field theories.
I mean, you look at any quantum
field theory Lagrangian
and the textbook,
and there are lots of
coupling constants and
interaction strengths and masses
and stuff. There's none of that here.
So all the parameters that we see in
the low-energy world are determined,
actually, by the geometry
of the internal space.
The interesting thing about
that is that that means that
the parameters are determined
not by the Lagrangian,
but by the solution to
the equations that come
from that Lagrangian.
So all the parameters
in nature are set
by the expectation values of fields,
primarily those that dictate
the properties of the internal
geometry like it's size and shapes.
You can imagine stretching either
the overall size of this torus,
or you could distort
one of the cycles to be
pinched smaller or
something like that.
Or you could distort
the sizes of the cycles
in this complicated space.
As an example of a field,
we might think of
a real valued quantity.
It takes real values at
each point in x, y, z,
and t, and it might,
for example, represent
the volume of the six manifold.
More generally, we'll
study fields that involve
the volumes of cycles
in the manifold rather than
the whole thing itself.
Now, continuously
variable parameters,
parameters that you can
adjust with no energy cost.
So imagine you can change the size of
this thing without
changing the energy.
Those things correspond to
scalar fields in
our four-dimensional theories,
scalar fields with zero mass.
The name for a massless
scalar field is a modulus.
So moduli fields,
massless scalar fields,
are catastrophic in cosmology.
If you have moduli,
everything goes wrong,
the light elements that are formed in
Big-Bang nucleosynthesis
in the first few minutes
can be destroyed by
decays of the moduli.
The modulae can cause too much
dark matter to appear,
that can cause the universe
to re-collapse,
and perhaps simplest understand
that can cause fifth forces.
So we're very familiar with
a massless spin to field
the graviton that mediates gravity,
and we know about a
massless spin, one field,
the photon that mediates
electromagnetism.
These are both long-range forces.
But we've not seen any evidence for
long-range forces mediated by
massless spin zero fields.
So the theory better not predict
massless spin zero fields.
So this is what I meant
by realistic solutions.
We're not going to actually build
the standard model or anything.
By realistic, I just mean not ruled
out by the preceding considerations.
So a realistic solution
for today just
means a solution without continuous
deformations, without moduli.
These solutions we
call isolated vacua.
But since we always
mean the same thing,
we usually just call them vacua.
So if I say, I found
a vacuum of string theory,
in this context what
I mean is I found
a solution of the equations
of motion Ricci-flat six
manifold where the resulting spin
zero fields all have masses
bigger than some minimum value.
That's positive. Now, I
claimed that there are no fundamental
parameters in the theory,
and we're here by demanding
that there no continuous
parameters in the solution.
>> All that leaves our discrete
parameters in the solution.
When you've discarded the continuous
data describing a geometry,
you have molded out by
continuously variable things,
what's left is topology.
So all you have left is
the topology of the six manifold.
So what this brings us to is that
a cosmologically viable solution
of string theory is
specified by the
topological [inaudible]
of the internal
six-manifold. Yes David.
>> What about the
cosmological constant?
Do you want to view that as
some integer times something.
>> We'll talk about the cosmological
constant in this context
in just a minute.
Because I was invoking
super-symmetric theories,
there's no 10 DCC.
So there won't be 4 DCC,
except in as much as it
arises from these solutions.
So yeah we'll have a slide about
how the landscape presents a CC.
Yeah, good. Other questions?
>> Were you including fluxes
in your topological invariant?
>> Next slide. Yes. Thank you.
So this is this is the,
I should have really
said associated to here.
As you know it is true
that there are specified
by topological variants
associated to the six-manifold.
On this slide heuristically
they count holes and handles in
X but no really they
involve first of all.
Okay the topology backs like hodge
numbers of X, the six-manifold,
it's intersection numbers,
but also stuff you can
put on X. Okay, absolutely right.
The topology of extended objects with
some tension called D branes
on sub-manifolds in X.
What D branes depends on
which kind of string
theory you are describing?
But you'll always be putting
them on sub-manifold.
You need to specify what
the wrapping numbers are.
So how many times do you
wrap them on various cycles,
and then what bundles do
you put on top of them.
You have to specify a vector bundle
on top of the D brane.
There's a fancier description
which I won't get
into but roughly speaking
this is what you have to do.
Then finally, you have to
specify the number of units of
magnetic fluxes of various kinds,
threading various cycle.
So here I have Maxwell flux
threading through a little ring,
but you should think of three form,
five form fluxes etc. Yes, please.
>> Well can there be
other discrete symmetries like
just a Z2 symmetry of
your surface or something?
>> Oh, yes.
>> You said everything has
to be topological but there
are sometimes discrete automorphisms.
>> Yeah that's right. So
you can find solutions
where originally there modular
you go to a point in the modulized
space and at that point in
modulized space there are
enhanced symmetries that's correct.
So those can arise.
That's an example of something that
I'll be sketching a little
bit later which is that,
when you've put down all of
these data that I'm
talking about here,
the solutions become isolated
rather than being
continuously variable.
If one of them lands on a place with
an additional symmetry, then
it would be as you say.
So they can indeed
arrive in [inaudible].
Yes, thank you.
Other questions. Peter.
>> If we just think
about string theory
as rock data we're starting from
the beginning and only imposing
the minimum set of conditions
needed for stability.
I'll go for self-consistency.
Then you would start with
a four-dimensional theory,
crossed some internal theory
that has constraints on it.
To what extent do those constraints
require that six dimensional theory,
even to call it
six-dimensional to have
a geometric interpretation
as a compactification?
>> They do not.
>> So you're talking about
a class of theories.
>> Precisely.
>> For which there are geometric
rectifications but there are
other string theories that are
completely outside this class. Yes?
>> You're completely right
what I'm describing here is
a geometric part of the string.
Landscape the part in which
the part that's not
free plus one-dimensional
is in fact the space.
It could be a conformal
field theory for sure.
Yeah, absolutely.
In the fullness of time
we hope to understand
though is in roughly
the same level as here,
these ones however are
a little easier to
picture in certain respects.
So yes. Thank you.
Questions or corrections.
Okay. So now
when you've done all of this,
the invariance that you need to
keep track up to
specify this solution,
I meant to have finite
number of integers.
You should think and I'll
say a little bit more,
this number is usually hundreds
maybe thousands of integers.
But that's it, you don't
have to say anything else.
So okay, well I mean
it's fun to think
long ago we learned that
matter is quantized,
and then Descartes proposed
the corpuscular theory
which Newton also loved.
Einstein formulated it for
electromagnetism Millikan
showed he was right.
We learned that
force carriers are quantized,
and now string theory
is telling us actually
the parameters are quantized too.
There's really nothing
continuously variable in there.
However, don't get the wrong idea.
It's not that I'm saying that
the dimensionless parameters are
interesting rational numbers
or something far from it,
the measurable parameters like
the fine structure constant
are presumably solutions to
absolutely horrendous
transcendental problems.
But these are transcendental
problems that
take finite integer inputs,
and then produce
something observable.
So summary then
cosmologically viable quantum gravity
theories in three plus
one dimensions,
are specified by
finitely many integers,
and with regard to
keeps question I think.
I would expect them to come
out the same way if we were
studying non-geometric ones too,
there's still specified by finitely
many fundamental integer parameters.
So parameter vector in Z to the
N where N is a 100 or 1,000.
Now not all of these
points in this lattice
are consistent or allowed
by the theory itself.
Also to my mind there's
no compelling evidence for
any infinite family of
vacua of string theory,
isolated vacua of string theory
with minimum scalar mass
fixed above zero.
All the infinite families I
know of have the property that
as you continue in
the infinite family,
eventually the minimum mass
tends to zero in the limit,
and those become unviable.
For that reason I'm going to assume
the number of viable vacua
is just finite.
This finite set of vacua
is the string landscape.
Okay. With due apologies I'm studying
a particular piece of
the string landscape.
But if these parameters
are understood broadly,
this is what I mean at least
by the string landscape.
>> Even if it weren't actually
you'll see if you still have
a similar data discretion even if
that data isn't necessarily
a geometry there was [inaudible].
>> Most certainly I would still
expect to find a finite set of
solutions specified by a collection
of integers that's correct.
That's right.
Any questions about this?
So how big is it well
depends on where you're
looking in for example type to be
compactification on six manifolds
called Calabi-Yau three-folds
the number that shows
up is 100 to 500,
that's the typical number of three
cycles and two and four cycles.
For example will run into the
number 491 a lot later on.
The number of vacua is estimated
by [inaudible] Douglas
is say 10 to the 500 in this region.
Now, in F-theory and
Calabi-Yau fourfolds,
Wali has argued that
you might be able to
find some number like this 10
to the 272 thousand vacua,
because the number of
cycles is bigger it's up
to 1.8 million rough way.
So these are large discreet sets
that we'd like to explore,
and our goal is to connect the
solutions to the observed universe.
If we want to do that
what we need to know is,
what values can the parameters take?
They don't sell out
all of Z to the N,
but they fill out some subspace,
for some subset of Z to
the N. If I give you
an allowed set of values,
what would be observed in
the resulting universe?
By observations I just
mean pretty broad stuff
like what kinds of
matter fields are there?
How many generations of
quarks and leptons are there?
What's the gauge group?
What's the vacuum energy?
What is the dark sector look like?
What are the masses and
couplings of dark sector?
So we can conceive of
this is a map from
parameters to observables
where the parameters
are all the values of N integers
that are allowed by the theory,
and is the number of parameters.
So you think of it as 1,000.
The observations are
some finite precision you
could think of them as
real numbers if you want but
of course we only measure
stuff to finite number of
decimal places in experiments.
The observations are
the possible values
of some measurable parameters.
So like the things that go
in the particle data book,
the list of all
the numbers we've figured
out from all the particle
experiments today.
This map is too hard.
So we always have an intermediary
step where we convert
from fundamental parameters to
the date of a quantum field theory,
and then from there to observations.
What I mean by the data
of a quantum field theory
while it has lots,
but the part that we need to think
about now is includes the data
of a Riemannian manifold called
the field space with some metric,
and a function from
the field space to
the reals which is the potential.
This is the measure of
the energy density of the system,
if the fields find themselves at
a particular point
in the field space.
Now the vacua that we're
going to think about are
the local minimum of this potential.
So some of these red dots are our
maximum but inside you
can see some minima to.
Local minima so the gradient zero and
the Hessian matrix of double
derivatives is a positive matrix.
>> When you say field space,
that's space of field?
>> Correct. This is the-
>> [inaudible] infinite
dimensional manifold.
>> No this is the space
whose local coordinates,
are the finite number of
fields in the theory.
So for example if you had
a theory of five real scalars,
then the field space
would just be a five manifold
the Riemannian five manifold.
The number of
possible configurations is
infinite but what we're
talking about here are
situations in which the fields
are not right now non-trivial
functions of space, three space.
So we're imagining
rather than phi of X,
Y, Z, T, which would indeed
be infinite dimensional.
We're considering that
everywhere in space and time,
there's one value of phi.
So that reduces the problem
to simply being coordinatize
by points on a manifold
whose dimensionality is the
number of scalar fields.
>> That seems like rather large
simplification to singular [inaudible]
>> It is a large simplification for
many things, we have to do
better, but, for example,
right now, the Higgs has a VAV.
It has a vacuum expectation value
and we can describe things by that.
As far as we know in the theory
of inflation for example,
we start out with a field space.
Let's say a one-dimensional
field space,
and that describes what we think
of as the background evolution.
The average behavior of the universe
and then you study
fluctuations on top.
Indeed the fluctuations are
infinite dimensional but you
first studied the background.
So for describing backgrounds this is
what this is what we
do and then we have to
quantize the fluctuations
on top and then you
have a quantum field theory with
infinitely many degrees of freedom.
>> Okay, thank you.
>> Yeah. Okay, so now let's come
to the question of dark energy.
So one observable,
in the late nineties
people figured out that
the energy density of empty space
is about 10 to the minus 26
kilogram per cubic meter
rather than zero which was
what had been anticipated.
So that makes up most of
the energy density in
the universe now with
Adams a tiny fraction.
String theory does not have
a continuous parameter that you can
dial to match this observation.
Note that this number if written in
Planck units is a really small number
so it seems like a weird thing.
In a quantum gravity theory,
you would expect that most things
you compute come out order one in
Planck units or maybe order ten or
something but not some
very small number.
So this is a strange thing for
a quantum field theory
to predict. Yes please.
>> Can't I dial back continuously?
>> Well, that's a Vev but
there's not a parameter.
So what you need to do is
find an isolated solution in
which the Dilaton Vev is
as large as you want.
Because if the Dilaton Vev
can be dialed continuously,
then you're ruled out by all
of the math like problems.
Once you add a, for example,
fluxes that give
a potential to the dilaton
that has finitely many
possible allowed positions,
at each of those things change,
and the question is; is
there one that's good.
>> Yeah. So you're saying
you've got to fix the dilaton.
>>For sure. Yeah. But
if you don't do that.
>>If the dilaton was something
related to dark energy
then it might be
part of a quintessence thing.
Then it wouldn't be.
>> Yeah. So an alternative possibility
is that there's a runaway,
as you know the problems for
describing that are even harder.
So there's a thin hope
held by some that there is
an alternative explanation other
than the one I'm providing.
Correct. But even so
that wouldn't be a parameter,
that would be a field.
You have to be buying
into there being away
around the model I problem and then
you adjust the Vev of the field.
Now, since you can't adjust that,
the only remaining hope
would seem to be that,
if string theory does have vacua,
which predict lots
of different values
each vacuum predicts its own value of
the vacuum energy
which range from say
minus m Planck to the fourth,
to m Planck to the fourth.
So that's pictured here. Here's
energy and here is a collection of
vacuum with different values
of the energy density.
Let's say here's the observed
one and here is zero.
Well, if the theory
produces considerably
more than 10 to
the 120 vacua and the distribution
is uniform enough,
you should not be surprised to find
some that are within
the observational allowed region.
This is just saying you
should be surprised.
However, I told you
that string theory has
no fundamental parameters that can
be adjusted and the number of
vacua of string theory
is only 10 to the 20,
then you would expect this
problem cannot be accommodated.
There's no reason for
string theory to be able to
be compatible with
measurements of dark energy.
So this is called the anthropic
solution to the cosmological
constant problem,
not really a solution to the problem
but it's at least an accommodation.
Any theory that has
no adjustable parameters,
and many fewer than this many vacua
has a very serious problem to face.
Okay, so this is an example
of the thing that you
can try and solve by
counting solutions.
By counting topological things,
and that's something that I'll try
and describe in the remainder.
So now, having gone through
the problem we face let me
try and make it concrete.
In a particular case of
Tori hyper-surfaces,
and this is now work
some of which I've
been doing this is with
Cody Long who's here,
my current students Mahmet Demerits
and Andres Riss Taecom.
My colleague a mathematician
Mike Stillman, Mike Kernel.
So finding Ricci-flat six-manifolds
in the wild is super hard.
In the case of calabi-Yau
threefold hyper-surfaces,
a special category of them.
The problems combinatorial
and is very easy to automate.
So Toric variety, this is
the arena will be working in.
[inaudible] variety is
an algebraic variety
that you can make out
of simple pieces.
You can glue together
copies of algebraic Tori.
So C star is the complex plane
omit the origin,
and N here is going to
range from zero to four.
When n is zero, C starts at zero,
I'll call that a point.
So we're taking a point, C star,
C star squared, up to
C star to the fourth.
All I'm going to do is
glue those pieces together
according to instructions
encoded in a Lattice Polytope.
Polytope here is that
just the convex hull
of a set of Lattice points.
It is a vast class of
Toric varieties that are encoded in
four-dimensional polytopes that have
a certain property called reflexive,
which means that the only interior
lattice point is the origin,
both for the polytope
and for its dual.
Basically we're going
to think about certain
nice polytopes in z to the fourth.
Those give us instructions for
assembling eight manifolds.
Four complex dimensional, real
eight-dimensional manifolds.
Now four-dimensional reflexive
polytopes have been classified.
You can classify yourself
the two-dimensional ones,
they are just 16 and then Kreutzer
and Scarpa classified all
473,800,776 four-dimensional
reflexive polytopes.
They enumerated them. They didn't
just prove that this number exists,
but they figured them all out and
they listed them on their website.
Okay, so here comes some phrases
which I'll explain in
the following slide.
Each fine regular start
triangulation of such a polytope,
gives the instructions
to make a Toric variety
inside of which lives
a smooth calabi-Yau threefold.
So what's a triangulation said
subdivision of a polytope
into simplices.
So here's a triangulation
of a square.
Here's the other
triangulation of the square.
These other words, fine means
that just uses all the points.
Star means that all the simplices
contain the origin,
and regular means that descends from
a construction in one
dimension higher.
But basically, what you should
think of in this statement is
each sufficiently nice
triangulation of one of these
finitely many nice polytopes
yields a calabi-Yau threefold.
So we have a finite set of
instructions for assembling the data
of calabi-yau three-folds
and we'd like to
understand what observables
we can get from this.
This is a fruitful way of
going because we've reduced
an analysis problem solving
a second-order PDE,
the mangia amp-ere equation
that tells you that
the Ricci tensor vanishes
which is basically hopeless,
to instead of solving PDEs
our next step would be to
solve polynomial equations,
for us that's still too hard
but then we can reduce it to
integer manipulations which we
start to get some traction.
>> In other words, you have
all the solutions
because certainly not
finding the Ricci-flat manifolds.
>> Very interesting. So
within this category,
we can find all of them by
the combinatorial operation.
Up here, I think of
a very interesting question to
which I don't know the answer,
and I'm pretty sure the answer
is not known, is the following.
Are there any Ricci-flat
six manifolds that are
compact that don't
have reduced holonomy,
in which the holonomy is SO6?
So we're always using
the same tricks, special holonomy,
SU3 holonomy and then
you can usually map that
to some thing in
algebraic geometry and then in
some cases in toric hypersurfaces,
you can reduce through
the combinatorials.
We do know we lose something in
the restriction to
this particular toric variety case.
However, as I'll sketch at the end,
you might imagine we
sometimes dream that
this corner that we're looking at
does still have most of the vacua.
I think that's plausible
although it's not established.
But in this first case, are
we losing anything in going
to an algebraic description
from an analytic one?
I have no idea. It could
be that there's infinitely
many Ricci-flat six manifolds
with holonomy SO6
that can't be found by any of
these constructions but I've never
heard a hint of them either.
>> Is there any physical reason that
special holonomy would come
up or is it just there?
>> Supersymmetry.
>> Okay.
>> Yeah. So special holonomy
gives you supersymmetry.
If we were bought into having
supersymmetric solutions
of the theory,
which are awfully nice,
then we're actually
already okay studying these ones
with special holonomy.
Yeah. But the plain equations
of motion just say the
Ricci tensor vanishes.
So are there nontrivial solutions?
Very interesting question.
I don't know.
So let me just give you an example
of the discrete parameters,
just to one time list
out the stuff you would choose.
Apologies for the excerpts.
I'm slightly simplifying some of
the terms but just in
interested being concrete.
So what do you do? You
want to make a vacuum.
You pick a polytope. There's
473 million. Pick one.
Choose a triangulation of it.
That determines a hypersurface X in
V and now you choose
quantized fluxes which means you
choose some vectors in h3 of
XZ which is Z_2h_2,1 of X.
This is one of the hard numbers of X,
so you should think of
this as couple of 100.
Among other things, you
might think of choosing,
with apologies to the authors,
choosing D7-brane
configurations by choosing
to wrap D7-brane on various
four cycles in this manifold.
That amounts to the choice
of a collection of vectors.
A ranges from one up to
how many D7-branes you want,
a collection of vectors
in each four of XZ.
Which again is Z to
the some hundreds or something.
Then for each D-brane you
have to choose the bundles.
You have to specify a
quantized to form on that too.
There's more besides, but this
has that kind of stuff
you have to pick.
So you throw down those
choices of integers.
Then that allows you, in principle,
the compute the potential. Yes.
>> Is there any hope
or any interest in
choosing all these things
uniformly at random?
>> Yes, we'd love to.
In fact, I'll comment
on that in a minute.
Let me get back to that when I
discuss why you would
try and count things.
We would very much like as
a null hypothesis to be
able to do exactly that,
to sample them randomly from
the set of allowed values.
Yes. In a couple of minutes,
I'll return to that at some depth.
So then the task that we
face, the computational task,
you can get a triangulate
a polytope look at
a triangulation to compute
some intersection numbers.
How many points to
these polytopes have?
Well if you look in Kreutzer
and [inaudible] list,
the number of points is
the Hodge number H11 plus four.
The number of intersection
numbers of divisor,
triple intersections of
divisors in a threefold
is a H11 cube approximately.
In fact, H11 only goes up to 491.
It doesn't even sound that hard
because we're only talking about
triangulations of a set
of at most 495 points.
So surely, this is easy.
Well, sort of.
You can make it easy to find
one such triangulation.
The trouble is, first of all,
most triangulations that
you can easily find are
not going to be
fine regular and star.
If you just throw down our
triangulation and often won't be that.
But even that can be solved.
You can efficiently find
these triangulations.
Trouble is, how many are there?
So here's the arena
where we're looking
at is the famous shield plot.
This axis is H21,
the number of three cycles in X.
This is H11, the number
of four cycles and X.
Many of you who've worked on
this, many others as well.
Each point here is
a Hodge pair H11 comma H21.
There's only 30,000 of them.
There are many more polytopes.
So many polytopes have
the same Hodge numbers.
But still there's only
finitely many polytopes
and they're all represented here.
Here's the hardest one. H11 is 491.
That's the one with 495 points.
Still, does not sound too hard.
So how many triangulations
do these things have?
Well, we don't know.
But we can put a bound,
unfortunately an upper bound,
that's combinatorially big.
So we've shown this is work
with my students [inaudible] ,
the number of fine regular
star triangulations
is bounded above by
this binomial coefficient.
V is the volume of the polytope.
It's some lattice polytope in 4D.
So just compute its volume
and choose H11 plus three
and here's a plot of it.
So these are all of the points,
all of the polytopes in
the credulous crack list with
H11 from 250 to 491 and here is
log base ten of the number of
the number of triangulations
according to this bound and
you see it goes up to 10_900.
The winner appear the biggest
one H11 of 491 is
10_928 and the runner-up is
65 orders of magnitude below.
Now, to this question about
uniformity, we begin to see,
in some cases some parts
of the system may so
dominate combinatorially
that you might want to
sample it. Yes, Bobby.
>> Sorry quick question. Do you have
any sense of how the
number of different sets
of triple intersection numbers is
scaling with the number
of triangulations?
>> Would love to know. We're
going to be working on it,
but we don't even have the
preliminary result in that direction.
>> Is it possible that some of
these fine regular star
triangulations give rise to the same-
>> Yes.
>> - [inaudible] at the end?
So you might be overcounting?
>> We might be vastly overcounting.
Thank you. Let me remark.
First of all, even without that,
we are definitely
overcounting because we know
some constraints that we have
not put into this bound.
So this is definitely a bound.
But I'm certain that
the actual number lies
a bit below this at least.
Now, could there be
a gigantic reduction
via apparently different FRST
is giving equivalent physics?
Yes. To me, that
still seems pretty hard to handle
at these large Hodge numbers.
People in this room did a lot
of work to achieve this at
H11 up to six and it
seemed hard there.
I'm hoping that in the next couple
of years we'll get a handle on that,
but we don't have
any handle on it right now.
[inaudible].
>> So I also would pick
a couple of examples for
a fixed Hodge numbers,
you would have a lot of spread and
there's a lot of structure in,
I mean, beyond the upper bound?
>> Pardon me?
>> This is the upper bound
that you plotting, right?
>> Yeah.
>> To some extent. If you look at
the actual triangulations
for a particular polytope,
that might not saturate it.
Then there are lots of
like distributions and so
on that you can look at
for various ones that-
>> Right. In order to look
at one of these and see
how many degeneracies
does one see and so
how far below this
does the real thing lie, right?
>> Yes.
>> Yeah, I think that's something
we'd very much like to do.
We're in a certain sense
involved in it.
>> So the mean value might be
completely solved, for example?
>> First of all,
the mean value may not be
peaked at the largest H11.
It may be peak much lower.
It could be 800 orders
of magnitude below this.
Yes, definitely possible.
But this gives a sense.
Now we'll get to this in a minute
there of the targets we sort of face.
We'd like to figure out
in a system like this,
what really are the actual number
of triangulations that show up.
So why do we want to count solutions,
coming back to this question of why
would you want to sample things-
Sorry, can I ask?
Yes. Please.
I have a question. So
has there been any work
that computed these numbers
for small H11s and-
Yes.
-and see how like-
Yeah.
>> Yes. People here.
Brent for sure, Vishnu was
involved in this too, right?
Ian was involved in
some of this I think.
Various collaborations
have rather systematically
figured out everything up
to h11 up to six or so.
That's the place where even
if you're pretty clever with
modifying things like Sage
and other existing packages,
it still gets sort of hard.
We came up after awhile.
Rick and I, we came up with
some methods that allow us to
now easily go all the way up
to here very, very quickly.
However, the thing that they
were able to do down there,
h11 up to six was
fundamentally in principle
achievable because the number of
triangulations was
less than 10 to the 12.
So you could imagine doing it.
Up here, I'm not sure how
much you can envision really
doing if it turns out that
you can prove a lower bound
also and that's an unsample
of really large number,
then we'd be in trouble.
>> So this is a trend?
Given that by the plot
that you showed,
not this one but this one.
Is that reasonable?
If you I guess up to six
you can maybe say you can't
see any patterns for
h11 that's in six part.
>> Extrapolating from
the small h11 results
to here is pretty challenging.
I'm a big fan of trying to
do that because what you
might try and do is
use one over h11 is
the small parameter in
your model and build for
example a random matrix
model or something
where where n is h11
and you expand in one over
n. I would love to do that.
I've written a bunch of papers
trying but it's a little hard
until you also have a little bit
richer data up here to calibrate.
So that's one of the things
that we're driving at
is sampling here well
enough not just getting
like a bound which could be way
off as many of you who have easily
recognized get enough data
here that you can
calibrate such a model.
>> Just to help me follow the
discussion I hear you talk about h11
six and then your chart starts
the smallest number is 250.
>> Yeah.
>> What's going on?
>> Right so they
weren't worth looking
at at the small values
for this present purpose.
So h11 goes up from
one the quintic or in
the first non trivial one
is two up to 491.
>> Okay.
>> Okay. The thing is
the number of points in
the polytope is h11 plus
five, plus four rather.
So these are
just the richer and more
complicated polytopes
with more triangulations.
So this formula holds
for any h11 but it's
interesting to look at
when the numbers are very
large and these
dominate in accounting.
So to the question of
uniformally sampling,
if I were to sample
all triangulations
of polytopes in
the Kratzer Szarka list,
a reasonable guess given
present information.
With caveats that I've
already mentioned,
a reasonable guess is that
almost all of them are
in fact triangulations of
this biggest polytope.
Which is why I've zeroed in
on this large h11 regime.
Okay. Now, that would actually be
simplifying and
clarifying in a nice way.
It would mean you only have to study
one polytope and to be honest,
it's a pretty nice polytope.
I'll show you in a minute.
Right. So let me cut through.
This is just to say
if you study a lot of
solutions and you find that almost
all of them have some property,
then you might take that as
a null hypothesis the sort of
natural prediction of the theory.
Okay. But let's just
say it's reasonable.
Well, there's 10 minutes then I
can say two words about this.
So suppose the string theory has
some large finite number
of solutions that fulfills
some gross criteria like having
the Standard Model or
something like that.
Then if an overwhelming fraction
of those solutions have
some other property like say
they have two kinds of
dark matter not one,
then I would say you should view
that without any other information.
You should view that as
the prediction of your theory
that's a null hypothesis that
you should be trying to study.
This comes back now to the question
about uniform sampling.
This counting measure prediction
excludes selection effects.
It excludes the possibility
that maybe some of
these solutions
because universes that
recollapse quickly and others
don't. That's all left out.
All I'm attempting to do is to say
the set of possible quantum gravity
theories for these rules is
finite but large and all we want
to do is count occurrences of
various properties in
that finite set without
regard to selection effects
that favor one over another.
Okay. Now, if that's
what you're doing,
then it's reasonable to
focus on the place where
all the numbers are.
Most of the numbers here are in
the complicated polytopes
with big h11.
Okay. This is also
effectively what Wali
was emphasizing in the case
of F theory where if
there's one geometry that has 10
to the 272,000 vacua and that
dominates over everything else
you might as well just study
that one geometry and forget
about all the rest. Yes.
>> So fluxes sort of
push you to large h21.
>> Yeah.
>> Triangulations push
you to large h11.
Do you think maybe were actually
at that little points in the middle?
>> No, I don't 251 to 251.
>> Yeah.
>> I don't know. The trouble
is I don't know how to account
for the degeneracies
in these over counts.
>> Yeah, like maybe they are
lot of the same triple session.
>> Right and one
interesting question is
if you write out
all the things you get to
choose so back where I was saying
what what you get to pick,
and it's interesting to ask in
one regime of string theory
one of these islands.
If you multiply through
all the number of choices you
get at each point,
which number dominates?
Right? You're saying
maybe the fluxes and
the triangulation choices sort of
matched that could well
be in some corners.
Could be that there are
other corners where it's
either the triangulations or
the fluxes that dominate.
Yeah. I'd very much like to
know the answer to that.
Okay. So this polytope
it sounds like it's the end point of
this thing it's so complicated
it's not actually.
So the columns here are
the points in Z to the four
where its vertices lie.
It has five vertices
in Z to the four.
It's a simplex. So just
has like triangular faces.
It's a very, very simple space.
Here's its biggest face triangulated.
It's sort of a weird
patterned triangulation
but, that's all we're looking at.
We're looking at triangulations
of objects like this.
Okay. So now, let's try
and connect this to
things we might try and
do with machine learning.
So this is the target.
I think for machine learning.
I've carefully pruned away
continuous families of solutions
that are not cosmologically viable.
Solutions where
the internal spaces are
non-geometric where we don't
yet know how to set all
of the knobs correctly
and that zeros as seen
on this space the Kratzer Szarka list
where we think we do
know how to describe
the fundamental parameters in
terms of some integers
that you can choose.
The question is just what are
the values of those integers?
When you make a choice
what do you get?
So one thing you might try
and do is a count vacua.
You might like to estimate or bound
the number of triangulations
of a polytope
when this number is
so large that it's
impossible to do
a direct enumeration.
We talked about that a bit already.
Another thing you might try and do,
I'm particularly interested in
doing is finding desirable vacua.
So based on physics reasoning,
you figured out some desired feature
say something that gives
an interesting inflationary
cosmology something with
a bright polarization signature in
the cosmic microwave background
or something that would be
excluded by current dark matter
searches or whatever you like.
So concretely, something that could
give a bright primordial
gravitational wave signal is
a situation in which
the largest eigenvalue of
the metric on the field space is
bigger than some threshold value.
So the target is find
a triangulation in which
the metric on field
space is bigger than
some as top eigenvalue bigger
than some given number.
Now, I think there's an interesting
kind of problem because
this base that we'd like to
explore is an extremely
large discrete set.
There are some
correlations in it but we
don't know right now what
those correlations are.
So we'd like to somehow
find places that
optimize this property
without checking them all.
You might also just ask a more
intrinsically topological
question like
find a triangulation with
a particular pattern of
intersection number.
What I mean by with
the feature, you could say,
I want to find a triangulation
that definitely has some property.
It seems to me it might be easier to
find sort of a concentration
of interesting results.
In other words, to have
some machine learning algorithm
produce a collection of
triangulations that are worth
testing because
their probability of having
a desired feature is vastly increased
compared to the median probability in
the ensemble even though
there are no guarantees.
This is not that hard because
most of the features that
at least I care about we
can verify it in seconds.
So it's not that bad.
If you just do it a
guess-and-check as long as
it's an inspired guess.
Finally, it would be really nice
if you can make predictions
straight from the polytopes and
not triangulate in the meantime.
For example, given a polytope
predicts some features
that hold for all of
its triangulations or that hold
for all triangulations in
some new subclass that you pick out.
This would be a huge simplification.
Surely, this is not
the right number but
anyway it's some gigantic number.
This would be much simpler because
honestly studying a couple
of triangulations of
every polytope in the Kratzer
Szarka list is something we
could do in a year at most.
But studying the full space of
triangulations is
unmanageable at this point.
Okay. So to summarize the task,
we face a finite but
extremely large set
which is the Columbia threefold
hypersurface in torque variety
subset of the string landscape.
We want to evaluate
various fitness functions on
this discrete set and we want to
find local and global maxima.
It takes us milliseconds to
seconds per evaluation so we can
explore it but we need some help
figuring out where to look there.
There are tons of correlations
in the data but it's just
hopeless to try and find all
of them by human intelligence.
It would be great if
we could get some help
exploiting these patterns and
finding them via machine-learning.
Then sending this aside,
just this particular corner
that I focused on,
I should stress there's
an immense array of
very closely-related problems
here and in other regions of
the landscape and
it'd be great to get
input on how to solve
those too. Thank you.
>> Thank you very much. We're already
quite some questions but we
have time for one or two more.
>> I don't have
any real intuition for how
superposition works when
you're superimposing
universes but is it
possible that we shouldn't
be looking for a particular vacuum
but that we're in
some superposition of them?
>> I have not found yet
a manageable superposition
question here
just because the sizes
of the systems are so
big you would expect there to
be tremendous decoherence effects but
I don't know how to think
about that problem to be
honest. I think Mike does.
>> This is a regime where we
justify all these
calculations by saying that
the extra dimensions are
big enough that we can use
these geometric considerations
and that they don't super impose.
Now, there could be
some other non-geometric regime
where the solutions are
small and then it really
it would be some other basis
in which this is,
this is the basis and
then the real solution but
is not a regime we
have access to it now.
>> Okay.
>> All right then I just
say we break for coffee,
meet again at 10:15 and if you're
speaking next and they haven't
send us the slides please do so.
So we can make a smooth transition
between the short talks. Thank you.
