That's right, yeah, thank you very much.
So this joint work with Domenico Fiorenta, Hicham Saati, with John Oerta and also with
Vincent Schlegel in parts.
So as you may know, even though it's not always evident, this is a conference that is secretly
about string theory.
And as you may also notice, there is a notorious open problem of string theory, namely to figure
out what the theory actually is, the full non-perturbative theory.
So perturbative string theory is well defined.
The non-perturbative theory is open.
This was an active problem maybe in the 90s, and then somehow interest vanished.
In 2014, at a strings conference, the annual one, Greg Moore, who's maybe in the other
parallel session, kind of reminded the community that we still have no fundamental formulation
of M theory, but physical mathematics must return to this grand issue.
I will present some results that I think may be interpreted as being indicative of going
in this direction.
So I'll start with something very basic, what might even be the principle of that would-be
theory.
And I'll start with two very cherished, established principles, the gauge principle and Pauli
exclusion.
That's essentially all I'll start with.
So the gauge principle in full beauty says, you may never ask if two things, two field
configurations are the same.
You should always ask, is there a gauge transformation that connects them?
Similarly, if you have two such gauge transformations, you should never ask, are they the same?
You have to ask, is there a gauge of gauge transformation that connects them?
This principle has a beautiful manifestation in mathematics.
This is the principle of homotopy theory.
It says, you may never ask if two things are the same, two objects.
You should ask, is there a homotopy between them?
If you have two homotopies, you should never ask, are they the same?
You have to ask if there are higher homotopy and so forth.
So for instance, when physicists model infinitesimal gauge transformations by what they call the
BRC complex, this is in a precise sense, it will appear in the next slide, an infinitesimal
homotopy n-type for n, the maximum number of, you know, n-fold ghosts of ghosts that
appear in the BRC complex.
Similarly, Pauli exclusion has a beautiful, well-established mathematical inclination.
This is the principle of supergeometry.
Pauli exclusion says that no two fermions may ever be in the same state, and mathematically
this is the statement that a fermionic field is an odd-graded, you know, function on a
super manifold, on a super space.
Now on the mathematical side, these principles or these theories of homotopy theory and supergeometry,
they have an evident, beautiful unification to what might be called superhomotopy theory.
So this means something that I will not have time to really discuss in these 20 minutes.
This is really the theory of what we call the infinity topos of infinity stacks over
the side of super manifolds.
This doesn't fit into 20 minutes, but a toy version of this does, it's not a toy version,
but a simplified version, in a controlled way, a simplified version, namely the rational
version.
So recall, to that end, homotopy theory is a very rich, extremely rich theory with loads
of aspects to it.
There's a plain classical aspect, which everybody knows.
This is what you see in the top right corner here, the homotopy theory of spaces, the classical
homotopy theory.
We've also seen, yesterday at least, in Enrique Tillman's talk, there's stable homotopy theory
concerned with spectra, which are generalizations of the concept of infinite loop spaces.
And what should be amplified, because it's very nice and beautiful, is there's a unification
of these two aspects to the theory called permeterized spectra.
We fix one space, think of it as a permeter space, and over this you have a spectrum that
varies over that space, a bundle of spectra.
So where plain spectra are coefficients for generalized homology theories, we'll see this
in action in a moment, these permeterized spectra are coefficients for twisted generalized
homology theories, which are just the kind of things that we expect to see in string
theory, say if we do D brain charge and twisted K theory.
Now it's an old ancient result due to Quillen and 69 and popularized in a simplified way
by Sullivan and 77, that if you disregard torsion information, or if you invert all
primes as the topologist would say, the homotopy theory of spaces simplifies drastically and
becomes equivalent to the homotopy theory of differential graded commutative algebras.
So I'm cutting some corners here, experts will know that the statement holds only for
nilpotent spaces, and I should say coalgebras instead of algebras here, but for the sake
of time constraints, and since it's accurate enough, we can think of this as being DGC
algebras.
Now, since we want to understand what might be super homotopy theory, there's an evident
super geometric or super algebraic version of DGC algebras, namely super DGC algebras,
where you just, on top of the chromological grading, you throw in one more Z2 grading
with the obvious signs to make it super graded commutative.
And the first indication that something might be going on here regarding merging of math
and physics is that, despite almost complete lack of knowledge for motopy theory and the
ordinary physics literature, it turns out that the co-fibrant such super DGC algebras,
namely the super Sullivan algebras, are well known in the physics literature under a different
name.
They're called FDA's, a term that has been maybe introduced by von Neuenhausen and greatly
used by Dowry and Frey in a big program that they set up where they showed that FDA served
to efficiently construct higher dimensional super gravity theories.
In the original article from 82, a very remarkable article, they reconstruct 11 dimensional super
gravity in an elegant and beautiful way, using a certain focal cycle that they found
in the FDA of 11D super space.
That focal cycle will appear on its own in just a moment.
So similarly, a bit later, it was observed, which is actually maybe a simpler statement,
that if a rational and unstable motopy theory is equivalent to the motopy theory of chain
complexes.
And since I omitted a co here, I can put a co there, but don't worry about these coes
too much.
And evidently, there's a super version of co-chain complexes where you just have super
modules of super vector space in each degree.
You can merge these things, this might appear at the very end of the talk, if I may make
it in time.
So if you have a space, or which a spectrum is parameterized, and it's encoded by some
DGC algebra, then the rationalization of that spectrum will be by the left column, the co-chain
complex, but it's not also sitting over the space, so it will actually be a DG module.
Oh, that thing.
So this is the rational super motopy theory that we're going to use until the very last
slide when I give a lightning outlook on the non-rational theory, which is of course what
we're really interested in.
So one beautiful homotopy theory is that it subsumes cohomology.
Maps, cohomology is about mapping spaces in a motopy theory.
If you do have a space and map it to some spectrum, that is the same thing as a co-cycle
on that space with coefficients in the cohomology theory classified by that spectrum.
The important point for us here is that for every map, we can ask for its homotopy fiber,
and that homotopy fiber is the higher extension that is classified by this co-cycle.
You're all familiar with how two co-cycles, say on a group, classifies the central extension
of that group.
If one phrases this in a motopy theory, it turns out that central extension is but the
homotopy fiber of the two co-cycle.
Now, if you have an n co-cycle for n greater than two, there's no established notion of
what the corresponding central extension would be.
You can always form the homotopy fiber, and it does behave just like a higher central
extension should be.
So that's what we're going to look at.
We're going to consider a tower, a whitehead tower of ever higher extensions emanating
from something.
There's a twisted version of all of this where you map to a permeterized spectrum.
When you have an underlying map to the permeter space, that's a twist.
Another co-cycle is a twisted co-cycle.
Okay.
So we now work out in rational super homotopy theory a tower of extensions each invariant
with respect to automorphism's modular R symmetries.
So beware everything in the following happens in super rational homotopy theory.
At some points, I'll identify some spaces that might seem surprising, which does work
in rational homotopy theory.
Okay.
So we start at the very beginning, the most simple non-trivial object in super homotopy
theory is the super point, which I call T01.
It's the thing that is an ordinary space, it's just zero dimensional, just the point,
but it does have this one odd super direction.
So we can ask, it's easy to see that on the super point, there's a one dimensional space
on two co-cycles.
So we can extend by these two co-cycles.
What do we find?
Simple computation shows that the result is the super line or really the super circle
if we're doing it properly, the object that underlies the supersymmetry of supersymmetric
quantum mechanics, one dimensional supersymmetry.
So this is the thing that is dual to the differential graded commutative algebra that is free on
a single generator in degree one odd, by degree one odd and closed.
So this is a super DGC-A and under duality on a sort of the defining duality that makes
DGC-algebra's models for homotopy types, this is the super point.
It has one odd coordinate.
If you did the same thing in degree even, you would get T1, the rational model for the
circle.
So to make this more interesting, this is what we're going to do in the following, we
will always also double the fermions.
So you can just as well consider, of course, the super point with n super directions.
Let's just do it for two super directions.
Now the first little miracle, I might call it, happens.
You compute what are the two co-cycles on this thing by which is the maximal central extension.
You find a three dimensional space of two co-cycles.
So the maximal extension will be something that is classically three dimensional and
has two fermionic directions.
If you work out what the bracket is, that thing is D equals three n equals one super
Minkowski spacetime.
The way this works, this has to do with the relation between supersymmetry and the division
algebras.
These three co-cycles parameterize two by two Hermitian matrices with the interest in
the real numbers and they just so happen to be the Pauli matrices for real spinners in
D equals three Lorentz geometry.
So we keep playing this game now.
We double the fermions to go to n equals two D equals three super Minkowski and we ask
what are the invariant co-cycles on this.
So on everything, there's all the automorphisms of these group acting and it turns out if
you always take the semi-simple part of these automorphisms, modulo r symmetries, modulo
the stuff that just transforms the fermions, one always finds the spin group.
So we even this, you see, I didn't put anything Lorentz in here.
It all, you know, it just comes out.
It's the same as this comes just out of the extension theory and we can compute the automorphisms
and discover the spin cover of the Lorentz group.
So we can double and there's one invariant co-cycle up to scaling.
And if you compute the extension, it turns out to be four D n equals one super spacetime.
You can double again.
Next one finds two invariant co-cycles, the two-dimensional space.
So we find the 60 theory for n equals one.
Now this has two doublings.
It's a type 2B doubling where you take the same representation or the type A doubling.
The 2A doubling has, it turns out, so all this needs to be computed.
You know, this comes out of the math.
So this happens to have a four-dimensional space, of course, I guess.
The extension of that turns out to be n equals one D equals 10 super Minkowski spacetime.
Those who know about this stuff will recognize these dimensions, even though I didn't put
this in, these are precisely the dimensions in which Green-Schwarz superstrings exist.
But I didn't put that in.
It came out.
Again we have doubling.
There's famously the 2B doubling and the 2A doubling.
Now we find one more co-cycle that is invariant under the spin group and that gives an extension
to 11D super spacetime.
I'm not claiming that there is no further invariant extensions at this point, but there
is no further extensions that do give Lorentzian spacetimes.
I'm not sure what to make of these further extensions.
Also down here you can ask what happens if you take the super point with three guys.
So then if you play this game, you get something that is given by three by three matrices to
the entries in the octonions, which is something that has always been conjectured to be related
to heterotic M theory, but we don't know yet.
So in summary, this is the theorem of a proofless generator.
The existed diagram is shown of maximal torus extensions at each stage invariant with respect
to the semi-simple part of automophism's modular R symmetry, and these automophism always happen
to be the spin groups.
Okay.
So next we ask for higher invariant extensions, right, in a process that the homotopy theorem
is called the whitehead tower.
We exhausted all the two co-cycles.
Next we ask for the invariant three co-cycles.
There's an old theorem that in the physics literature is known as the old brain scan
that there is precisely one, you know, one immeasurable space of extensions here on these
10D super spacetimes, which are called string to B, string head and string to A. Why do
I call them such?
Because these co-cycles, so I'm writing now as an example once, you know, remember all
these extensions come from extending, forming homotopy fibers of co-cycles.
The three co-cycle is the WZW term of the Green-Schwarz string.
So Green-Schwarz originally wrote down some extra functions to make it manifest super
string.
It had a complicated term, which they inserted to enforce something called kappa symmetry
and a little later, so it looked complicated, a little later I know, and Vincent Sescu observed
that this complicated looking term is nothing but the, you know, the standard WZW term as
known from the WZW model of an ordinary group, if you take the target to be the super Poincare
group and take this three co-cycle, which, you know, I'm not explaining these symbols.
These are super field binds, you know, contracted with gamma matrices.
That's the way you find them in the literature.
Those who have seen it will recognize it.
For others, these 20 minutes are not enough to explain this.
It keeps going.
Next, we ask for the next higher co-cycles.
Which extensions are there of the string to A extension, which is a Gerber extension of
this space?
If you want to think of this, this is the total space, rationally, of the B field Gerber
sitting on this space.
It turns out it has a whole zoo of higher co-cycles corresponding precisely to the type
to A D brains.
I'll say in a moment how it does correspond precisely.
Similarly, one finds extensions corresponding precisely to the type to B brains over this
extension.
So what do I mean by this?
Again, the co-cycles, to classify these extensions, which now exist in every degree, every even
degree here, every odd degree here, they, you know, have some expression, which experts
will immediately recognize as being the famous WSW in terms of this, you know, the super
P brain models for the super D brains.
So the Green-Schwarz models for the super D brains.
This is the chain-patent field here, the Chan character, really, of the chain-patent bundle,
and this is the super, you know, the bifurmonic part of the RR forms coupling to the forms.
The D0 is special.
I just want to mention this.
It does happen to sit with the 11D space-time and the 10D type to A space-time in an homotopy
fiber product in a pullback diagram, which I think we expand a bit on this in our article,
which you can think of being a mathematical incarnation of the fork-loss statement that
the zero-brain condensation is what grows the 11th dimension in type to A space-time.
Now let's concentrate on this type, on this 11D space-time.
It itself has higher extensions.
So it fights a one-dimensional space of extensions, two Gerbe extensions, which are labeled by
M2-brain, and this happens to have one more higher extension by six Gerbe, which I write
the M5-brain thing.
I'll be speaking about this in more detail in a moment, but just to recapitulate what
we have so far, so space-time and M-brains have emerged in some sense from the super
point as iterated higher invariant extensions.
Let's focus on this M-brain sector, and notice the strategy will be, we're going to push
this down again by mathematical process of cyclification that turns out to be modeling
double-dimensional reduction.
We push it down again and see what we discover from the M-brains down in lower dimensions.
First of all, to show again that these terms are justified, so it turns out this extension
here, this invariant co-cycle is the one that describes the Green-Schwarz M2-brain model,
which wasn't written down by Green-Schwarz, but by Bergsoff, Seskin, and Townsend.
Actually, the co-cycle itself appears already in dowry-afre, they just didn't talk about
M-brains, they didn't know about them.
They were just constructing supergravity, and they have a process that constructs a leaveny
supergravity as a classical X-function from just this thing.
Similarly, the M5-brain extension here is classified by seven co-cycles, so it looks
more complicated, but nevertheless, so it exists.
This was also already stated in dowry-afre, they already saw this.
You need a quite sophisticated fierce identities to see that this is a co-cycle, which I think
is due to dowry-afre, when it seems to have been forgotten, it was rediscovered and understood
to be the WWE term of the M5-brain by Pacis O'Rourke, and they don't cite dowry-afre,
it seems they didn't know.
What we're going to do now, what we want to do is, you see, we have this funny co-cycle
sitting on this extension, what we do want is co-cycles on space-time.
I'm going to do an equivirant descent now in homotopy theory, down of the M5-brain
co-cycle down here, and we will find a single co-cycle in twisted cohomology.
Turns out by a general principle of homotopy theory, which we just explained a bit in this
article here, what we have to do to make such a descent work is, you first of all have to
arrange that the coefficients sit in a similar homotopy fiber sequence as the space-times
do.
A little computation shows that this can be achieved, this is solved, this problem is
solved by putting the quaternionic hopfibration here.
So you ask, how can I find a homotopy fiber sequence like so with something here, which
will have an interpretation of being the homotopy quotient of B7Q by B3Q, and it turns out the
solution to this is the 4-sphere.
The 7-sphere sits over this as an SU2 bundle, this SU2 bundle has a second shine class,
and rationally that's what one sees here.
So this is rational, remember here, so I'm exchanging, you see, I'm exchanging an Einberg-McLain
space by an odd sphere, so this works in rational homotopy theory, and I'm exchanging something
more complicated to an even 4-sphere.
Second part of the statement is that the COSYCLE does descend, there happens to be a map that
factors this diagram, this commutes up to a homotopy, this commutes up to a homotopy,
and this is now the descended by some theorem, the descent of this COSYCLE.
So we have a unified version of the M5, M2 COSYCLE, and I can't explain the full computation
here, but just to give you a plausibility argument for why this, so why this is not
going entirely crazy, why this is related to something you have seen before and you
know, so what is this S4 coefficient, so the DGC model for it is by classical result
of Sullivan's, or maybe even, yeah, I guess that was written down by Sullivan, it's modeled
by the DGC algebra that's generated by a guy in degree 4, a guy in degree 7, the guy
in degree 4 is closed, the guy in degree 7 is differential proportional to the square
of the guy in degree 4.
This equation, see we interpret this now as coefficients for these COSYCLE, so this will
be interpreted now as the background charge for the M2 brain, which is the G4 field, and
this is the background charge for the M5 frame, which is the G7 field, and this equation,
which is really this 4 sphere equation, so of course the equation of motion of the C
field and its dual in 11 dimensional supergravity, so this is not the proof of the statement,
but this is one way to see quickly how it's plausible.
Okay, let's focus on this M-brain COSYCLE, but remember the left hand side was really
the extension of the type 2a space term, you see I'm working us now towards finding the
double dimensional reduction of this, we want to re-express the stuff we found in 11D in
terms of structure that we can see in 10D, so we write this explicitly again as the extension
of the 10D space somewhere we had, and for symmetry reasons we do the same thing for
the 4 sphere on the right, now the 4 sphere is a CIRCLE extension of something, namely
of the amount we quotient by some CIRCLE action, so the natural solution is to take the regard
of 4 sphere as being the unit sphere in the space that is the reals plus the quaternions
and let S1 act in the canonical way as a subgroup of SU2 on the quaternions and form
the homotopy quotient, so this is an orbifold here, this action is not free, it has two
fixed points, and then S4 is the homotopy extension of this, notice that this orbifold
that appears here from just our extension business is the orbifold that happens to appear
in the NEORIZON geometry if you put a black M5 brain as an A-type singularity, so in conclusion
our unified M2 and M5 COSYCLE is really of this form, it goes from an extension of 10D
super spacetime to an extension of this orbifold S4 mod S1, and now let's think a bit more
about what we're doing here with our extensions, it turns out, so forming these extensions
is a drive functor from super homotopy types equipped with a two COSYCLE, so with a COSYCLE
extensions which one writes like the slicing notation, this is a super homotopy types equipped
with a map to BS1 and our extension process forming homotopy fibers is this functor here,
this turns out to have a derived rider joint and the derived rider joint one can work out
acts by cyclification, it forms free loop spaces homotopy quotiented by rigid rotation
of loops which in the string theory literature is often called the twisted loop space, on
cohomology this means takes a space, some cohomology to the corresponding cyclic cohomology,
this is a very general statement about base change in infinity topos here, there's also
a left adjoint here, the base change for those who know is that induced by the map from the
point to BS1, but anyway so this exists and so we can make use, whenever we have an adjunction
it means it's telling us something about the inner structure of the things we're considering
so let's apply this, so I apply cyclification to the diagram we had before, so there's now
the cyclified M2 M5 brain COSYCLE, apply free loops and homotopy quotient odd rotations
and since we have an adjunction and now here's a right adjoint applied to a left adjoint,
there's the adjunction units that go from 10E space, so these canonical maps that homotopy
theory gives us and from this orbifold to this thing, the composite that appears here
if you go from here to here is called the adjunct under our adjunction between the M2
M5 brain COSYCLE to this one, you can work out what this is, turns out that is the COSYCLE
for the unified F1, D0, D2, D4 and NS5 COSYCLES down in 10D, so what does this mean, so to
make this plausible first and foremost what is the cyclic space of the force sphere that
is sitting here, there's an old formula by Sullivan and some student of his name I forget
now back from the 70s who worked out Sullivan models for the cyclic loop spaces and it turns
out if you apply to the force sphere, you get the DGC algebra that is generated precisely
by the Bianchi identities for the field strength of these brains, you see the NS field strength
here which is closed, you see the beginning of the twisted around complex here for the
RR charges of these brains up to D4 and you see the more complicated Bianchi identity
for the NS5 brain and then what I'm not showing is one has to compute that this thing taking
coefficients in here really does assign these green towards COSYCLES for these brains.
This gives rise to an evident question, if you want to continue can we actually complete
this diagram, that's the mathematical question and the physical question is where's the rest
of the brains, but recall that physics folklore says no, the D6 and higher is not supposed
to rise from double the measure reduction of the M5 and M2, it's supposed to rise by
some mysterious process gauge enhancement, so it's okay that we don't see it from this
double the measure reduction, but the question is still does it appear somehow, okay it does
but let's first make some room, we need to be a bit more sophisticated, so it turns out
a direct lift it doesn't exist but if we linearize the problem in homotopy theory it does, so
if we go to remember stable homotopy theory was like homological algebra which is about
linear stuff, if we kind of linearize in this sense, if you take the Goodwill linearization
in that I take these spaces here which all sit over S3 and I take the fiber by suspension
spectra and then the corresponding infinite loop space, so this is making the problem
fiber-wise linear, yeah just if you haven't seen this just accept it maybe, so we're
asking I'm asking how can I find this linearized lift, the good can I solve this obstruction
problem sort of in Goodwill linearization and there's a curious fact that made this
work in 2000, the Ruygens-Arilig, they had a method for computing minimal models for
such parametrized spectra in rational homotopy theory and they compute that in this fiber-wise
suspension of S4 over S1 sitting over S3 there sits a copy rationally of twisted connective
K theory, the way this works roughly is you know this has two fixed points so this homotopy
coefficient looks like an S3 with two copies of BS1 attached to the poles and as we're
forming fiber-wise suspensions performing the suspension spectra of BU1, so there's
some rational version of Snay's theorem going on that picks out a copy of KU there and
now it does work, there is now a lift of the original doubly dimensionally reduced co-cycle
that factors like so, so the rational version of this twisted K theory is just the twisted
around complex that you all know and this way we find that all the d-brain charges find
their home rationally as they should in the twisted around complex, okay let me need to
come to an end I suppose, I have five minutes left or something, one minute, okay so we
found the double dimension reduction of the unified M to something to break, so we want
to repeat this process, look at the untranquated KU now, so we do the extension process, we
do the double dimension reduction one more dimension, we came from 11 to 10, let's go
to nine, so we apply this equivalence, we apply a cyclification, we pre-compose with
our junction unit to again produce the adjunct now producing a co-cycle in 90, this is compactifying
the 10-D theory on a circle, now remember that the 90 theory also had another extension
the type 2B extension which I'm writing here, so we can write that down and whatever co-cycle
the type 2B you know, if we can find a co-cycle here on type 2B spacetime we can doubly dimensionally
reduce it by applying cyclification, pre-composing with our junction and forming the adjunct
and if I can now find an equivalence here such that the co-cycle is Lorentz invariant
is an invariant co-cycle then by junction I will have defined from the 2A co-cycle to
B co-cycle, turns out this exists, this is the main theorem we have with this T duality
article, so you have to put you know K theory plus shifted K theory then this equivalence
exists such that everything is invariant which is our working the rule of the game
and one finds this co-cycle which happens to be precisely the one for the type 2B brains,
so this is the inclination of T duality on this co-cycle level in fact one can analyze
this in more detail the commutativity of this diagram which is now really fast but let me
just claim this, this is equivalent to what's called the Buscher Rules of RR fields usually
attributed to an article by Kentaro Hori who may be in the parallel session, we have some
quotient here over some thing called the T duality 2 group, if we forget the K theory
part we get another computing diagram which turns out to be precisely the rational inclination
of what's called topological T duality introduced by Bachleit et al and made precise by Bünker
Rumpf and Schick, we can find coming back to our story of taking any co-cycle that we
find we should form the extension if we form the extension that's specified with this total
co-cycle it turns out to be the double space time in the sense of physicists use it where
you know these two circle fibers the type 2A and the type 2B fiber both appear and there's
a job sitting over it which is this head which makes it generalized geometry we can still
pull that back now to 11D since this is in 70 to get the 12D thing sitting over 11D space
time and that turns out to have all the properties rationally to be a candidate for you know
the rational model for an F theory or vibration particular if you do rational hence GL2R rotations
on this on this double copy here it corresponds to S duality on these brain co-cycles so in
conclusion a fair bit of the expected structure of M theory emerges out of the super point in a
rational super model theory maybe the main impact of this is that it leads to an evident conjecture
you want to say well that is true then maybe the full theory emerges since we were just working at
this rational toy approximation to the full story but we know what this full super geometrical model
theory is so maybe we should do the whole thing there this needs to be worked out but there's one
thing I'm a analyst as long as nobody kicks me from this stage here it turns out that if you do go
to full super geometrical model theory there's a precise sense in which the super point itself
emerges from the empty object from the initial object by some sequence of mysterious steps that
I can't explain here but which I explained and in this chapter to book on the foundations of mass
that will appear later this year I thank you for your attention