 I'm happy to introduce our last speaker, Timo Weigand from the University of Hamburg. Hello, thank you very much, good morning everybody and first of all thanks to the organizers for having me for the kind and general communication. I'm very happy to be here, enjoying the interaction and I'm very happy to talk about this project which is kind of a longer program that as far my involvement of concerns started together with Wolfgang LeAsier who is with us now online and Song Julie at CERN, then now in Korea and I'd like to give an update and the more recent developments in this work with my post-work CISAR Quota and Alisson from the University of Hamburg and with Mike Lee, definitely. And since we are all on a busy schedule and may have to leave I'd like to give a punchline first and since we are two audiences, one in physics and one in mathematics and if you get two punchlines. So first to the maybe more mathematically inclined part of the audience, I'd like to argue or give an example of how GV and DT invariants play an important role in physics not only in general but in particular in verifying certain physics conjectures about quantum gravity and especially the tower weak gravity conjecture will be relying on this mathematical structure of DT invariants and their particles. And this is one example of the many profound indications of the Florida program called quantum program which makes predictions about quantum gravity from a physics point of view and to the extent that quantum gravity can be realized in examples by spring theory this makes statements about compact string geometry and compact is important because this is the relation to the gravitational. We're not going to talk about non-compact geometries which are relevant for gauge theories but really the compactness comes from the gravitation. And the punchline for the more physics-oriented part of the audience would be that we prove the asymptotic tower weak gravity conjecture in 5G n equal 1 m theory compactification and this is one example of the many well-known to you applications of tower geometry and also enumerative geometry. So let me introduce the object of interest in this talk. It's the weak gravity conjecture. This is well known to the physicists in the room. I'm not sure how well known it is to the mathematicians who haven't followed this so let me remind us all of the statement. It's one of the general statements about what the quantum gravity theory should have irrespective of whether this theory comes from a spring theory or some other theory. And according to these authors every quantum gravity theory should have the following property. If you have a gauge sector coupled to quantum gravity so for example one gauge theory or young mill gauge theory more generally and I'm for simplicity I'm only focusing on the delian part and you have a certain chart letters associated with this gauge theory so charged object charge state for representations of the gauge proof. Then every ray in the chart letters must support a tower of super extreme of state. That's the claim I'll define in a second what I mean by this and the significance is from the physics point of view this conjecture predicts the existence of certain states with a certain ratio of charge to mass. That's the claim in every quantum gravity theory there should be certain states that infinitely many with a suitable ratio of charge to mass will be more precise in a second and from the mathematics point of view why this is interesting from mathematics because we can use string theory to realize quantum gravity theory if we believe that this is the consistent quantum gravity theory and then the existence of these states can be translated into a very precise statement about the non-vanishing of certain enumerative invariant on compact alabias which are the compactification states of the string or anchor. So this is the logic and this is of course what we'd like to show in the in the first and I need to be more precise for what is meant by super extreme and what is meant by power super extremal. So this is now the physics motivation of course or the physics statements there are first of all two ways of talking about super extremality and so that we don't confuse let me recall them. So first of all super extremal means the particle super extremal if the ratio of its charge i.e the weight of its representation per this week to its mass it's bigger than this ratio of charge to mass of every black hole in the theory. So depending on your theory you have black holes you can classify the black hole you compute the charge to mass ratio of these black holes for these classical solutions of gr you take the maximal q to m of the black holes these are usually these are the extremal black holes and you and you say a particle the super extremal if its charge to mass ratio is bigger than charge to mass of every black hole in particular. So there must be particles with this property that's the prediction from quantum gravity and another way of phrasing a related way of phrasing that and this explains the name we gravity conjecture would be giving two particles so two states of a certain representation of certain weights in a gauge group then they must be self repulsive this means the sum of all the repulsive forces or the repulsive force from the gauge group must be bigger than or equal to the attractive forces and there can be two types of attractive forces one from gravity obviously important gravity but if there are scalar fields they can also be in addition attractive forces that go to carbon and this would mean that for these particles gravity or more precisely gravity plus your carbons are everything that is attractive should be weaker than the repulsive force from the gauge sector and this was said to be the slogan gravity should be the weakest force acting on those particles in that sense it's gravity the weakest force acting on those particles and in general we have since we are also talking about modularity with modularity we have mass of scalar fields modularity is one of the topic of the workshop of course in physics this means mass of states mass of states leads to the yukawa interactions and tends to a modification so these are the two versions of super extremality they are not equivalent in general because of these extra yukawa terms but in the limit of weak gauge coupling and certain asymptotic limits of the theory we'll discuss within multiple detail they are equivalent hence we will mostly be talking about this which is also closer to the topic which is the name weak gravity conjecture but we have to be careful that inside the modularity space they are not exactly so this is the super extremality and we said there should not only be some super extreme state but there must be a tower what do we mean by this this means that there must be exist a super extremal particle of charge n times q for every charge q where n should be some infinite so it could be for example for every q there must be infinitely many particles this would then be a lattice version of the weak gravity conjecture or the whole sub a whole sub level should be populated but more generally it would be enough if some multiple nq for n and infinite for every q is populated by so this is a lot of states infinitely many states must be there in such a week in such a credit so i'm not going to talk about the motivation behind this this of course and what led to this also a long time ago to propose this there were considerations of black hole physics and of remnants in cosmology and things like that there's a purely motivated conjecture but the nice thing is that one can translate this into statements about geometry whenever we take examples of the quantum gravities from m theory or for string theory and translate those into statements about the numeracy invariance that we can then prove for this proof so in fact um which types of states could occur first of all one knows that every bps state is automatically super extremal or extremal so if you have a bps state in your theory then these states are automatically of the types of states satisfy this weak gravity conjecture so in particular if you have infinitely many bps states in your theory and along a certain charge direction of charge letters you have infinitely many such states then you're done then for that direction in charge letters the weak gravity conjecture is sadness but what happens if no bps tower exists in a certain direction in charge length this can happen and in fact this is the generic situation the generic situation first of all because in most theories there does not exist such a thing as bps state in non-super symmetric theories there are no bps states and even in theories with minimal supersymmetry there are no bps states in six-dimensional theory for example with minimal supersymmetry there are no bps particles there are bps strings but not bps particles and the same is true in 40 n equal 1 theory with minimal supersymmetry so bps very nice but it's the non-generic situation it's the luxury and in fact even in theories where you would have bps states in principle because you have eight super charges and so well for example 5d n equal 1 theory there you do have bps particles and in some cases this is enough to prove the weak gravity conjecture in certain directions because we can show a tower but not in all and this is what i'm going to discuss about but what happens if the bps states are not there and this is the main result formulated a bit more technically compared to the third slide so let's specify let's take a laboratory we have physicists we make an experiment our laboratory is now m theory compactified on a calabria 3 to be very specific you get a theory in five dimensions i'll discuss this for those who don't work on this in a second what would mean for this gyro mean and the statement is suppose there is a direction in the charge that does not admit a tower of bps particle this can happen then one of the following is either there exists a super extremal tower of non bps states so we can find a tower of states super extremals are charged to mouth based on the condition but the particles are not bps or there is no weak coupling limits for that gate and we argue that this is then a matter of debate we strongly believe that this means that the gravity conjecture is not even necessary to hold but this is a different story but the fact is that whenever there is no bps tower at all either there is non bps tower which we identify explicitly in full generality or one cannot take a recoupling limit meaning that the gauge proof is always strongly coupled the coupling is always at plank scale so it is debatable if the original argument leading to the gravity conjecture holds at all and in this sense we are going to establish the asymptotic tower we gravity conjecture meaning we gravity means particle could charge a mass ratio bigger than that of a blank hole tower mean infinitely many and asymptotic means in theories with the recoupling limit so those theories that we as physicists usually think of as controllable perturbatively controllable gauge proof the asymptotic power with gravity conjecture and this can be shown in in general and to do this we need to address more quantitatively these two specifications here first we need to think about excuse me we need to think about the asymptotic part so theories with the recoupling limit this means we need to go need to consider infinite distance limits and modular spaces in this case modular spaces of kala many folks kala kala we are people so this will be the first part to understand which in which regions of modular space we have to look and then we have to do the state counting them and this will be related to certain gb odc invariant which by means of certain physical realities count also non-vps state in the five-dimensional that we will discuss so that's the part for the talk we first have to understand these recoupling limits and then talk about the state counting and then see if this physics conjecture is actually okay so um i am trying to either avoid jargon or this i will not succeed but i would at least i'd like to explain the jargon to those who do not work on this topic a little bit and if i fail to do so please tell me right away so that i can explain what the jargon was supposed to mean so i start first of all with the physical theory that we want to that we want to conclude so we said it's the compactification of m theory to five dimensions what was meant by this the physicists know all this apologies for those of course but just to set the notation so we start with this with the theory in 11 dimensions that's the super gravity theory in a long way thanks limit so gravity theory in 11 dimensions and the important input is that there is a three-form gauge field in this 11-dimensional theory um so this is like a gauge field but with three-form three-form indices so generalization for the jargon or however you want to do in the balance this is where our gauge degrees of freedom will come from in the seconds and what will be the particles so what will be the charged object there are two types of charged objects in this 11-dimensional theory already first there are the famous membranes these are two plus one-dimensional objects so they have two spatial dimensions one time dimension and they couple to the three-form simply by pulling back the three-form to the world volume of the membrane hence they are electrically charged under the three that's the jargon and that's also a hot stool to the three-form so more precisely the derivative of the three-form has a hot stool in 11 dimensions that's the derivative of the six-form the object that couples to this in the analogous fashion is called the five-spatial and one time direction and both of these will be important so the logic will be this will give rise to the bps sector and this will essentially be giving rise to the non-bps sector in a way that we will do okay so that was the starting point in 11 dimensions and now we so that's the unique gravity theory or effective field theory of what one believes to be a unique gravity theory in 11 dimensions and now the usual gain we compactified it on a compact calaverial three so we think of our base time to be the direct product of one plus four dimensions times the compact calaverial three direction and in this way we arrive at by looking at the low energy at a theory in five dimensions so in this r1 form the effective field theory it's super symmetric so it has eight supercharges let's not worry about this and this is a gravity theory and it's a gauge theory where does the gauge decrease the freedom where does the gauge bundle come from its connection is obtained by simply splitting the three form that we have in 11 dimensions in this fashion so we take a certain basis of the two comma of the one comma one form on the calaverial three and we simply write c3 as the sum of one form in r1 form times two form in x3 and this way we get the one form which is which inherits the gauge degree the freedom of three so this is the gauge potential yes that is well because i'm on the calaverial three that's all i can do so i can i can i can consider the three form in five dimensions but that would be a different object i can't consider it and the second possibility would be i take one leg in five and two legs in x3 because if i want you to think about harmonic form so masters degrees of freedom there are no one i mean one comma zero eight of x3 vanishes because it's calaverial three so that's the only option so i want to so that's also the three form but this place no role for us now i want to think about how do the usual gauge bundles or the connections on the gauge bundle come about in this language and this is how it how it all is okay and let me explain the second parts again apologies to the physicists who know this very well if we have this theory now in five dimensions so we inherit the gravitational part and we also inherit the gauge degree freedom in the sense that you now have gauge gauge groups here and the usual yang-milz theory and or or the one gauge theory and then the coupling and this is important the coupling i mean the gauge coupling this is determined now by dimensional reduction it is essentially given by the structure of two form wedge star of the three form internally and this is a certain combination of certain volumes so the cala geometry of the calaverial three fold controls the coupling what stands here is one over the coupling squared so when this is large then the coupling is small so perturbative and recoupling mean large f alpha beta quote unquote in a certain way and this means large volume and which volumes so these are volumes of certain devices and volumes of certain polymer of certain curves on x so that's the only message here controlling the couplings mean controlling the cala moduli of the calaver of the calaverial okay good and um what are we want let's just remember remember what we want to argue we want to show we want to check are their states are their particles under these you once that i get of certain charges q alpha we don't know yet so you want to we want to check this and we want to check if that charge to mass ratio satisfies a certain inequality which was motivated by physics this is the inequality you don't need to look at the details i just want to tell you that once the charges are known this inequality is again a modular dependent state because the masses of the state whatever they are they will also depend on the moduli we will see in a second why so um that's um repulsive that repulsiveness or super extremality condition other context we can translate into uh what it means in this effective field theory and this simply will give us the certain inequality that this modular dependentism that we have under control if we if this means okay so um yeah good but this was just to show that this can all be done specifically um the details we don't need to know okay so now which so we we talked about the gauge potentials what about now the particle which type of states could that be so the first idea and first source of potential states that would satisfy the constraint would be the bps particle the status already bps particles are known to always be extremely also big and which which are candidates for bps particles in the five-dimensional theory they come from these m2 brains recall two slides ago we had an 11 dimension two-thousand one-dimensional object with membrane which d3 coupled now how we do we get a particle of them in five dimensions again by splitting the world volume in a similar way we'd say the time direction d in r14 sorry r14 not r15 and the uh the remaining spatial two-direction along the internal manifold so the jargon is being wrapped the membrane on the curves you all know this because you keep you all know how to compute g gv invariant and some of it in invariant and this counts at the states and these are the states that you know how to think and so these are particles and they have charges why because the original object in membrane was coupling to the preform the gauge field is just the descendant of this in five-dimension so the charge is just computed by um um um um flagging this into here this decomposition very simple of course the charge is then given in terms of an intersection product between a certain device and the curve so this is a logical number the mass of these particles turns out is given by the volume of the curve because the original object had a certain tension. You put it on a curve, and then the volume gives the mass. And the volume of the curve is something that is controlled by Kehler geometry. And hence, we are again in this business that the Kehler geometry tells us everything about not only the charges, either the intersection numbers, but also the mass. And in particular, if the curve is holomorphic, then the particle is PPS. And then as we said, it's either super-extremal, and we are invisible. Therefore, the first question is, does there, in every direction of charge letters, exist infinitely many a tower of PPS particles? Or put differently, if you take a fixed holomorphic curve class, is it true that there exist infinitely many multiples of this curve class, such that the Gromov-Wittman variant, or rather the Gruppler-Kumar-Baffin variant for this curve class is non-zero? That is a well-defined question to ask. And the answer is no, this is not true, as you know very well. However, what is true is that certain curve classes have this property. Namely, if you take curve classes in the movable cone of your Kehler-V-O-3, then it was conjectured by Murat and friends that for those, the following is true, the genus, excuse me, the genus 0, the genus 0, Gruppler-Kumar-Baffin variant, for any multiple n times this homomorphic curve class in the movable cone is non-vanagic. That is a conjecture. And let me recall for you, or let me recall, you know this better than I do, that the movable curve cone is the cone of curves ruled to the cone of effective devices. So on this Kehler-V-O-3, we have the cone of effective devices, the dual cone, if the movable cone, movable cone, movable because they essentially can move freely in the Kehler-V-O-3 or you can have a family that feels out to the Kehler-V-O-3. So this is not a proven statement, but it's a conjecture. It has been confirmed in an overwhelming number of examples of complex Kehler-V-O-3 fold in future papers. And there's also a good reason why this should hold, because as also the Kehler-V-O-3 friend for those curve classes, the associated particles are such that they are BPS. And the BPS condition is the same as the black hole extremality condition. It turns out this is a non-trivial statement. They show that by BPS flow, they show that for these curves in the movable cone, there always exists a nice endpoint of the flow and that in fact, the BPS condition equals the BPS extremality condition, which is not always true when you have scalar flow. So therefore, it is expected that also since the black hole exists, also the particle should exist and hence the invariant should be non-trivial. So this is an interesting project by itself to try and prove this from a purely mathematical point of view while this is true. We are going to ask a different question. The remaining challenge, therefore, is what if there are no BPS cones? And generically, this is the case, because there are many examples. Of course, the movable cone does not exhaust the class of, does not exhaust the, not every element in H2 is inside the movable cone, obviously. So there are many directions where this simple solution or this solution could be gravity conjecture of no cone. And the simplest example would be the cone default. If you have a cone default, a two-title that you just slowed down in the cone default condition, then clearly the GV invariant for one time to describe as one, but for any positive multiple thereof is zero. So in this direction, the charge letters do not have a BPS power symbol. So the question is, why is this? And does this invalidate the tower with gravity conjecture or what's the situation? That's what we would like. That's the question. And the main result, I repeat it again. Whenever there is no such BPS tower, for example, as for the cone default or many other cases, then either one can instead find a non-BPS tower from a different sort of particle and this is related to a different type of invariant. We'll come to this. Or the U1 is always strongly coupled, meaning that we can never take an asymptotic limit in which this U1 gauge group is a nice weakly coupled theory. And hence, we would claim it is plausible that the original physics arguments don't work. But this is another point. But these two are possible. And again, let me repeat so that you remember what we want to do. First, therefore, need to consider all recoupling limits by looking at the telegeometry. And we are going to show that in all recoupling limits, we will find the super-general tower, either of the BPS or of non-BPS. So we're going to prove this just coming from the other side. Everything that has a coupling limit will be proven to satisfy. OK, question so far? Exactly, exactly. So you pick a certain linear combination of U1 and or put differently, if a curve is charged under, has a charge under certain linear combination of U1, such that a multiple times this charge does not admit these BPS states, then this linear combination can never become recoupled. Or instead, you can find a non-BPS curve that does the job. That's the claim. OK, so now I need to do two things. I really want to get here, but I first have to start there because otherwise, I guess, this is how we have to affect the problem. We affect the problem in the recoupling limit. So we need to understand what the recoupling limit is. But this we can do. Why? Because let me recall, recoupling limit means recoupling means. So we have our Lagrangian. We have a gauge coupling. This is inverse gauge coupling squared, 1 over G squared. So recoupling means G small. This means F large. F is essentially a linear combination of volume. So we need to consider the limits in which volumes under the calorie will become large. However, we have to do so at fixed overall volume. The total volume of the Calabria we must remain constant because, to be more precise, the gauge coupling is a normalized volume. So it's volume of devices divided by the total powers of the volume of the total Calabria. So scaling everything up does not help at all. If I'm just making Calabria larger, this makes these volumes larger, but also this volume larger so this doesn't help. So we are mathematically interested in the infinite distance limits in Calabria-like space at fixed overall Calabria volume. Full stop. That's what we need to do. That's what we need to do. And indeed, that's what I just said. But that's only a necessary condition for the decoupling limit. These entries here have to go to infinity because this is 1 over coupling squared. Because these infinite distance limits in Calabria-like space at fixed overall volume. But this is now something that only probably the fitness will appreciate. This is only necessary, it's not sufficient. Why is it not sufficient? It's a bit harder to explain. The point is just because the coupling goes to zero, this does not yet mean that the theory really becomes weakly coupled relative to gravity. So we need to consider the relation between essentially the coupling strength for the gauge sector and the coupling strength of gravity. And that should be small. That is the precise criterion. And our proposal is that we must consider the following object. This is a physics statement, but it's very important because otherwise we would not be able to control this theorem. The gauge coupling squared times the plan scale, this is what tells us about the gravitational coupling. This has to be small compared to the cutoff scale of the quantum gravity. The cutoff scale of the quantum gravity is the scale where the one loop corrections to the Einstein-Hilbert term in gravity become comparable to the tree level. So where the perturbative quantum gravity, the perturbative effective field theory of gravity breaks down. And this is a quantity that has been discussed in great detail in the physics literature. It goes back to first to Twally and has been a lot of papers studying this because it's so important and it's called decision. So for the mathematician, just think about we have a certain recoupling limit, a certain asymptotic limit in tele-modulized states plus another constraint to check that comes from the physics. And for the physicists, it's the statement that we have to relate the gauge coupling to actually the cutoff of the quantum gravity. And this is very important because this will go to zero as well in the limit. And hence we really need to make sure that the ratio goes to zero and not just the numerator. Okay. So let's characterize first the recoupling limits and then we come to the state count which recoupling is going to occur. So as we said, a necessary condition is that we look at these infinite distance limits in tele-modulized space at fixed volume, fixed overall, overall. And these one can classify, we classified them together with Wolfgang and Soju some time ago and they can be classified in this way. Such limits can be obtained either if your Calabria has a T2 vibration, a genus one vibration, need not be elliptic but at least genus one vibration or it admits a K3 vibration or an abelian surface vibration, a T4 vibration. This should be a T4, not a T3. K3 or T4. And so this is a topological criterion for a Calabria 3-2, it emits an infinite distance limit in tele-modulized space at fixed volume. And when this topological condition is satisfied then the limits always look like this. In the first case, the volume of the base goes to infinity, let's call the scaling lambda and the volume of the fiber goes to zero, just inverse so that the total volume stays finite. And similarly on the K3 or T4 vibration side, the volume of the base goes to infinity and the one fiber goes to infinity. So the only Calabria 3 with an interesting recoupling limit or with an interesting infinite distance limit in tele-modulized space must admit these vibration. And this is very nice because many people here in the room are interested in vibration of either of these two types for completely different reasons. And the statement is, these are the only relevant cases if you ever want to study a weekly couple gauge theory. All the other ones that do not enjoy this are not relevant. Second, these are the ones that have been studied for ages and spring theory because they admit nice duality frame. And this is why these dualities are relevant because they automatically come when we want to think about weekly couple gauge theory. This is a very general statement that I think puts the importance of these vibrations into a very physical context and explains why they are. What the feature, so general fiber in these mathematically or physically? No special feature. An arbitrary gene is one vibration. Need not be elliptic. And an arbitrary K3 or Q4 vibration. Need not have sections or all color be our vibration. Do the job, exactly. All the color be our vibration. The fiber dimension, you mean Ramney's fiber double cover? No, no, no, no. No, no, thank you. No, absolutely not. The dimension must be positive. Yeah, thank you, thank you, absolutely. No, this is just a discrete structure. Here I'm interested in really taking an asymptotic limit in the smooth modulation, in the model. Yeah, thank you, it's only these. Okay, so the claim is these are the only ones that admit a recoupling limit. And now for these, something interesting happens that again it's clear to the physicists. Supposedly we take such a limit. Let's look at this first object. How much time do I need? 15 more minutes, okay, good. I thought I would do much. Anyway, let's discuss this. What happens in this limit? The base becomes large, the fiber becomes small. So what happens now in physics? In physics, we said we have these two brain, two plus one dimension brain. We can wrap an M2 brain on this fiber. This gives a particle. When the fiber becomes small, the particle becomes light. We can wrap it infinitely many times because the gramophitin invariant of any positive copy of the torus fiber is non-zero. This is the general result that you can show with the lecture theory or other means. So this is always gives rise to infinitely many particles, wrapped once, wrapped twice, so forth, resulting with every multiple of the torus while we get the GD invariant. It gives a particle of volume the torus when this goes to zero, lambda goes to infinity and the limit you get infinitely many masses of particles. They are BPS. What is the role of these particles? From a physics point of view, all you see that you get many, many light particles when you switch on your collider, you will find many, many particles. You don't know where they come from, but they come at accurate distance in the mass. The mass of the second particle is twice the mass of the torus. And this means that these particles behave like calypsox line states. This means that effectively a new dimension in physics opens up. This is the decompactification limit from five dimensions to six dimensions. And you end up with a theory in six dimensions in which the torus is gone and this defines your compactification. This is of course the well-known F theory limit. And the quantum gravity scale here for the physicists is of course just the punk scale in one dimension, which goes to zero compared to the original time scale. So this has to be taken into account. More interesting is the second type and the nice thing is that that's the only thing. The second type, what happens now? Now it's surface fiber goes to zero volume. What can happen? In particular, recall, we also have these five frames. These are five plus one dimensional object. So I can now take four of these dimensions along the surface fiber, then I have two dimensions left, space and time. These two dimensions I put into my space time, non-compactified, and I get a string. I get a string which becomes light. The tension of that string is the volume of the fiber, it's small, it goes to zero. And since this is the Calabriault, since the fiber is Calabriault, the string that I get is either a critical heterotic string or type two string. I've also been known for ages in the physics literature. If this were a P2 fiber or a non-Calabriault fiber, it would not be true. But in this way, I get a new string, heterotic or type two string. The tension goes to zero, it becomes smaller than any other scale. So my emergent theory at infinite distance is that of a new type of string theory, heterotic or type two string. So I started with an M theory, the word string has not been used, but in this limit, I end up with a string. So this is the reason why string theories are important from that point of view, because they naturally appear at one of the two possible ways how to go to the coupling and that intimacy. And the quantum gravity scale, in this case, is the species scale of solids with that string, and we have to compare everything to that. And by the way, there's a conjecture. So this is an example where this happens, and the conjecture is this emergent string conjecture, that is always completely general, that every infinite distance limit in every modular state of every Calabriault always either leads to something like this, the decomplication or to an emergent new string theory, new compared to the original theory that we found. Okay, so now we've understood the recoupling limit, the possible recoupling, the possible infinite distance limit. What now remains to understand is which gauge groups in this five-dimensional theory become weakly coupled in these limits, because for these we want to check the regravity condition. And this is what we did more recently, we classified them, and the statement is very simple again. The only you want, which can ever become weakly coupled, are one of the following two types. Namely, when you take a linear combination of you want, you can think of this as being associated with the curve flow. Why? I mean, the easiest way would be, think of a three-form, we had a three-form C3, I can integrate the three-form over a curve, then I have one leg left, left, and that's my one form. So to every you want, to every linear combination of you want, I get a curve. And the curves associated with the you want, which can become weakly coupled in one of these two limits are of the following type. Either it's the torus fiber itself, and only the full torus fiber. So no degenerates components, no exceptional curves inside the torus fiber, only the generic fiber, or the curve is the curve inside a generic K3 or K4 fiber, meaning those fibers that can occur at finite distance in the complex structure of the fiber. So what I do not want is curves that are localized only in say type two or type three hulikop type fibers, K3 fibers that hulikop. These will not lead to the repeat couple of H-theories, but all the other ones, the ones that I can deform everywhere, these will lead to weakly coupled you want. So we have now complete control over all the potentially weakly coupled gatefruits. They are either always associated with the torus fiber or with essentially a generic curve in a K3 or K4 fiber. That's the claim. And to show this one has to look at all possible degeneration and look at the intersection form in these degenerate cases, but that's the other thing. Okay, so now I'm done. Now I want to check the weak gravity connection. Now I want to ask, for these you want, are there BPS towers? And if not, are there at least non-BPS towers? We could start with the first case. The first case, are there BPS towers? And this is very simple. The answer is yes. Of course, there are BPS particles charged under the C1. Namely, the BPS towers are just given by M2 brains wrapping the torus fiber themselves. These are particles. There's infinitely many because we already said that the torus fiber has, any multiple of the torus fiber has non-zero GD invariant. Hence there's a tower. And these particles are precisely the ones that are charged under this corresponding U1. The U1 is the dual device. So it would be the section or the multi-section which you always have. So this is precisely the combination of the two. And everything else, every other curve will be charged under U1 that cannot become weakly coupled. And hence we do not care if there's a BPS or non-BPS. So those that are weakly coupled that can become charged under something weakly coupled. For this, there is BPS towers. So that's the boring case. And I'm skipping the technical for the middle case. The interesting case is now the one of the vibration, of the circle vibration. And I'm only going to discuss the K3 case for again, believe me. So what's the remaining question? The political WR3 admits the K3 vibration. Then we already said the only U1 which can undergo a weak coupling limit is of the type that it corresponds to compactifying to reducing the three forms C3 over a curve which is part of the generic K3 fiber or can be deformed into a generic K3 fiber. And indeed one can check that these curves that the associated U1 satisfy the statement also the more complicated physics statement that the gauge coupling becomes small compared to compactive. So now I need to check finally, what are the, do I have BPS towers? Do I have super extreme? So now we are in the realm that many people here have been working on extensively. Namely, we need to think about the letters, the letters of curves inside here, the generic K2 fiber. And as you all know very well, those letters over the wheels can be decomposed into self-fuel and enter self-fuel parts of positive and negative self-interfection. And this is a letters of rank 1, R, where R is not larger than 90 in the generic K. So I now wonder, are there particles that I can get by wrapping M2 brains on other uncertain curves inside here? Any curve that would satisfy the regretting and then you want it. And the statement is this depends on the self-interfection of the curve that I look at. If the self-interfection is non-negative, then a BPS tower exists. Meaning the GV invariant for any multiple of that curve is non-zero. First of all, as a check, such curves are indeed movable. They are movable inside the K3. If the self-interfection is non-negative, they can move inside the K3. And they can also, since the K3 moves over the P1, they move therefore in the entire X, in the entire Calabria scheme. So they satisfy the conjecture of more random friends. They are in the movable cone, hence it would detect them to admit the power of BPS particles. And this is well known, has been known for a long time because the BPS index is counted by a modular form. More precisely, the number of the GV invariant such in the zero of the curve C is given by this coefficient, the N where N is T squared over two. So as long as this is non-negative, this is the coefficient of non-zero. And not only is it non-zero for one time the curve, but also for any positive multiple of the curve as well. So here we have a BPS tower and the regravity conjecture is done. However, generically, we have these curves C squared of negative self-interfection as well. These curves are rigid in the K3 fiber. They are not in the movable cone. Therefore, we do not expect the BPS tower and there is also no BPS tower. But this is now the claim. In this case, a tower of non-BPS takes over. And these states come not from the M2 range, but from this other source of states that we could potentially look at, namely from the M5 frame. More precisely, we can now consider the following object. Once more, we wrap the M5 frame now on the fiber. This gives us a string in five dimensions, not just any string, but since it's the K3, it's a heterotic string. So it's a very special string, a critical string. This critical string has excitations. The excitations of this string are charged under the gauge group. And the claim is there are always some states that are super extrema that we need to show and that's infinitely many. So this is what we need to show. And in order to show this, we have to translate the counting problem of counting the excitations of this string that you get by wrapping in five-dimensional curve into a counting of BPS invariance of Donald Thomas type of on the conduct of type A string theory on the same color beyond. And this was discussed in detail by people in the room in particular, let me point out the table. So more precisely, I'm translating now this problem of counting string excitations to counting BPS invariance. Those of you who have worked in this program very well, let me just review how this works. We have this M theory on Calabria III. This was the theory in five dimensions. Now we put this theory on another circle. So I go effectively to four dimensions. What I can now consider is the following state. I can consider a bound state of this M5 frame wrapping the fiber, the physical string. The string can wrap many times on the F1 and I can also have what the physical KK momentum. So waves along the string. And I can bound it to the particles that I got by wrapping M2 brains on any curve, any curve in the letters of the case. And this is known to be equivalent under certain string theory duality to the object counted by DT invariance and before DT we do a bound state or object counted by coherent sheets with these charges, R times R for the wrapping number times the class of the K3, Q of the class of the curve and N the class of H0, N copies of H0, so to speak, that would be the zero bound state. This is a well-known reality. And equivalently, you can think of these states being the winding modes, winding and KK modes of the headlocked string that you get by compactifying M5 brain on the string at the KK level N and with charge vectors. And, okay, so far so good. And now the important point is in the special case where R is one, so where you wrap only one, you can identify this KK number for the state in four dimension with the excitation level of the headlocked string. This is because the KK and the excitation level in string theory always appear in the same fashion. So when I know that a certain state in four dimensions like this at certain KK level, then this means that if I go to five dimensions, I don't wrap the string, but the same five dimensions that the same states, same charges exist at a certain excitation level of the string. This is only true for the headlocked string with R equal one because this is a critical string. In general, this would not be true. These states in five dimensions are non-BPS. And hence, they must be non-BPS because we've already shown there are no BPS states that would do the job. So these states have a chance to be two-dimensional and they are counted by BPS states in one-dimensional. So these can be counted by these invariants. This way we get non-BPS string excitation one-dimension higher and we now need to do things. First, we need to show that there is a tower. For these, a tower of a certain type with a relation between the excitation level or KK number depending on how you want to do it and the charge. We call this is two-squared negative, so this is how we do it. And then we need to show that they are super extreme. And then finally, we have. So let's first, quickly, I'm being fast enough, but let me speed up. First, we have to show that these states exist. So the claim is there are special states at KK number N and where the KK number is equal to this charge. Or if you think about the gene variance where the N is minus one-half G squared for this gene variance, these always exist. That's the claim. And they are super extreme. That they always exist. We know by looking at the counting problem. The same statement that I just made more formally just means that the elliptic genus of the z-rotic string is related to can be written as the linear combination of a generating function for the s-number and the z-type forms here. The bbs-numbers, the 40 bbs-numbers are counted by this gene variance. And we now need to show that precisely when N plus Q spread over 2 is zero, that then the corresponding omega gamma is non-zero. Then we have those things. And this follows from the observation that was made very clear in this paper in particular. And I'm looking at a special case. The paper is much more general. That's the generating function for the DT invariance has this nice form. In the specific case, K3 with R equal 1 of this eta minus 24 times vector value model of form that counts essentially all whose coefficients are related to the net elliptic numbers. And you can show that precisely when N plus Q spread over 2 is zero, so that when the corresponding netherlatches number would have to be zero as well, that then the bbs-number is non-zero and the state. So this counting problem has been solved already. We just need to put it together. And now what's left for us to do is to show that the state, this is the tower of states. So tower meaning, if I take the charge Q negative, then I can always find an n such that a state exists. And of course, any multiple of Q also exists because I just multiply then n by the corresponding number. Hence, I have a tower. And what we need now to show that this tower is, in fact, super extremally satisfied with this projection. So we have to look at this form of self-repartitionist condition. We have to look at this specific state. And this we can do with the Hearth of Spring Theory. We know what the masses of states are at a certain excitation level. There are a number of technical complications because we work in five dimensions, so we're taking the consular parameters, et cetera. But when the dust settles, one can show that in the asymptotic limit for recoupling, everything just cancels that was causing trouble and miraculously the state just satisfies the risk-reality projection. So everything falls into places, but only for those states. And these are the only ones generally that do the job. Shall we break up with the final statement? Yes. Here it is. So we've shown the asymptotic version of the gravity conjecture of the tower with gravity conjecture in five dimensions. Similar statements can be made in other theories without non-DQF states, namely in six dimensions and in four dimensions, particularly for the minimally supersymmetric ones here. Quantum corrections, same important role that we also looked at. I discussed the K3 story. The G4 story will go the same way, but it would be very nice to understand better from the countering point of view. That's maybe something you can think about later. And now for the physicists, a nagging conceptual question, what now if there's no tower in a certain direction? Well, what we know now is this is only then the case when the U1 cannot become recoupled. But you could still ask, should the Greek gravity conjecture not be satisfied nonetheless? This is an ongoing debate. We have some opinion on this and hopefully contribute to the discussion. And for now, thank you very much. That's thanks for the question. Is there any quick question for this speaker? Yes. OK, any good team. So for example, heterotic on T4. No, I would assume that the same. I would assume that the heterotic on T4. Yes, I think everything would be OK because this is heterotic on T4, right? No, 5G, 5G, any good team. On heterotic on T4. I'll have to think about it. I'll have to think about it. I would think so. I would guess yet because there would always be interpretable, there would always be VPS tower that could be a KK tower or string excitation tower. So my guess would be that it always works. Yes. Yes, yes, yes. I would think so. I don't want to make a definite statement. My guess would be that it works. Perfect. So let me take the opportunity before thanking the speaker also to thank for this very nice, wonderful conference to Simone, Chuck, and Johannes. Let's thank the organizer. Let's also give a quick thanks to Sebastian. Where is he? He's not the rich person. Yeah, thanks, Peter. That's great. There is a lunch. Yes, it's already cooling.