 All right, so I'd like to first thank the organizers for the invitation to speak here. It's really a great honor. I will be talking about inflation in strength theory. And I don't think I need to remind this audience that inflation is a remarkably successful effective theory. It's generic predictions of an almost scale invariance, almost Gaussian primordial spectrum, continue to be an excellent agreement with data. And one of the founding fathers of inflation who is here should be very pleased with how well his theory is doing. Now effective as it might be, inflation actually is sensitive to plan scale physics. And one way to see this is to note that the type of potentials suitable for inflation is very sensitive to its UV completion. Even a small dimension six Planck-suppressed Operators can give an all-the-one corrections to one of the slower parameters and stop inflation. So without knowing anything about its UV completion, we can't say for sure that we have a viable model. Now turning this argument around, inflation is an excellent opportunity for strength theory. With a concrete quantum theory of gravity, we can assess whether these quantum corrections are going to spoil the nice properties that we want for inflation. In some cases, we can argue that these quantum corrections are small. And in some other cases, we were hoped to even argue that these corrections are absent. Now this UV sensitivity that I mentioned is general for any models of inflation. But as I now argue, models that give rise to a detectable level of gravitational waves are even more sensitive to UV physics. Now that was, unfortunately, a false alarm last year about a possible detection of primordial gravitational waves. By now, it's pretty clear that a claim of discovery is premature. But with many, many experiments going for these observables, many of them are ongoing. Some of them are going to be launched in the near future. It's not inconceivable that primordial gravitational waves with a tensor to scalar ratio of the order of 10 to the minus 2, or maybe even 10 to the minus 3, may be detectable in the not so distant future. So while many of us are disappointed that it has not been detected yet, it's still worthwhile to explore what the consequences would be if a detection is made by any of these future experiments. Now a detection of primordial gravitational wave at this level not only point us to high-scale inflation with an energy scale of the order of gut scale, not far from where we expect string theory or quantum gravity to come into play, a very simple argument due to David Lyfe suggests that the Infroton field range was superplunking. Now this strong UV sensitivity is what motivates us to consider inflation within string theory. And I should mention that the Lyfe bound is not very sensitive to the precise value of the tensor to scalar ratio. Even if R is eventually measured to be a tenth of the value allegedly reported by BISAP, we would have come to the same conclusion. The important point is to detect this primordial gravitational wave. If the gravitational wave is at the level of 10 to the minus 2, or maybe 10 to the minus 3, we would have come to the same conclusion. Now we will refer to models where the Infroton field range is bigger than the plant scale to be large-field inflation models. And over the years, many excellent ideas have been proposed, including a very simple chaotic M-square-5-square inflation model or the so-called natural inflation model which involve axioms. Now chaotic inflation, M-square-5-square inflation or natural inflation are radioactively stable. They are stable against quantum corrections. If gravity and Infroton are all we have, there's nothing wrong about that. But the UV completion of general relativity typically include many new degrees of freedom. In the case of string theory, this UV degrees of freedom includes the clues of client states, the string states, they are very heavy. But when you couple the Infroton field to this UV degrees of freedom, typically the Infroton potential would receive an infinite number of corrections suppressed by this heavy energy scale. It would have been okay if the Infroton field range is a Plankin, but if the Infroton field range is bigger than the plant scale, as you integrate out this heavy degrees of freedom, after the Infroton moves a super Plankin distance, it will see a very big difference from the type of potential that you start with at the origin when inflation begins. So the message that I would like to emphasize here is that while a low energy effective field theories could claim victory by tuning the parameters in the Lagrangian to be small, or by postulating a global symmetry, a string theory has to meet a higher level of rigor before we can make similar claims. Now we can be more specific and consider axions. As you know very well, axions are natural Infroton candidates because they enjoy a shift symmetry, a shift symmetry that is perturbatively exact and is only broken by non-perturbative effects. So naturally the energy scale associated with this non-perturbative effects is suppressed by the instanton action. Moreover, the continuous symmetry that you start with is broken to a discrete one. The potential generated by this instanton effects is a periodic potential with a periodicity that is set by the axion decay constant. Now the very popular model, natural inflation, amongst to keeping only the leading instanton terms. So you have a very simple cosine potential and it's quite obvious that the Infroton potential is flat enough for slow row if the axion decay constant is bigger than the prong scale. Now you can justify dropping the higher instanton terms if you can show that the instanton action is sufficiently large. So the leading instanton effects will suffice in that case. The question is whether both conditions can be satisfied simultaneously, namely that the product of the axion decay constant and the instanton action that I call M over here is bigger than the prong scale. So right away we won't expect the product of the instanton action and the decay constant to be arbitrarily large because in the limit when the product becomes infinite, we will recover a global symmetry and quantum gravity does not like this to happen. So to get a handle on how the product can be bounded, we turn to a concrete quantum theory of gravity, namely string theory in this case. Now in string theory, there are many axion candidates. The axions arise from higher-form fields. Just like the usual one-form gauge fields, this higher-form fields enjoy a gauge symmetry. Now when you dimensionally reduce this form fields to four dimensions, let's say you have a two-form gauge field, you reduce it on a two-cycle, you get the pseudo-scaler in four dimensions. And the shift symmetry that you are looking for comes from a gauge symmetry in a higher-dimensional theory. So it looks like we are in pretty good shape. There are many, many form fields in string theory. There are many, many internal cycles. We have lots of candidates for axion inflation and they all enjoy the shift symmetry that we are going after. Unfortunately, some careful studies surveying different formulations of string theory done about 10 years ago have concluded that axions with the properties that we want, namely superplunk and decay constants, do not seem to arise in any control limits of the theory. It's not that we cannot get a large axion decay constant, but if the axion decay constant is large, you are in a regime that the higher-instantane effects could become important, or there could be some additional states that become precisely in this region of the marginalized space. That seems to be a rather general property of axions that arise from this concrete quantum theory of gravity. Now, there are two broad mechanisms that have been proposed to get around this problem. One is what you might call the communist approach. If one axion does not get you to superplunk in distance, many of them together can, and each of them contributes to the common good. Now, earlier ideas of this type include inflation and axion alignment. I would say more about this shortly, what these mechanisms mean, but by now there are many, many more variants of ideas along this line, and in each of these cases, different field directions were identified for the collective efforts of the axions to go superplunking. The kinematic field range of the axions is enhanced, there's no question about that, but there's no guarantee that the UV corrections are under control. But this needs, of course, to be checked, but putting this aside, we find that there is a whole zoo of models with large axion, effectively large axion decay constant. Another way to get around this problem is axion monogamy, and this class of model is fundamentally different from the previous type in two ways. One is that only a single axion is involved, you don't need to invoke thousands of axions, and the axion has already a mass at a perturbative level. So the axions in axion monogamy is not really an axion in a traditional sense, its field space is non-compact. So as you go around the circle, you don't return to the same point, and in other words, the axions already pick up a mass at the perturbative level. Now, this property that the axion has a perturbative mass turns out to be very important later on, as some of the arguments that I would make on the multi-axion system, as you will see, do not apply to such non-periodic axions. So let us take a closer look at some of the multi-axion scenarios that have been proposed. Let's start with a simple example with two axions. Let's call the axions phi and rho. If you have two axions, and if you have two instanton-generated potentials, generically, both axions would pick up a non-perturbative mass. However, if the axion decay constants can be tuned, such that only a linear combination of these axions is massive while the orthogonal direction remains a flat direction, you could ask what is the field range in the orthogonal direction. The field range in the orthogonal direction is infinite because there's no potential. Now, this is not what we want, but you can easily convince yourself that if you perturb this relation slightly, you can obtain an effective field range that is much bigger than the Planck scale. So in the limit when these decay constants are so fine-tuned, you have flat directions, the field range is infinite, but you can just, depending on how big the decay constant you want, you can just perturb this relation, and if they are approximately equal to each other, you find that the effective field range can be much bigger than the Planck scale. Now, in the case of inflation, you don't tune the model parameters, but rather you choose the direction in field space to inflate. In the case when you have n-axion, you can go along the diagonal direction, and in the case of two axions, you see that the field range is enhanced by a factor of square root of two. If you have n-axions in the n-dimensional axion space, you can enhance this effective decay constant by a factor of square root of n if you go along the diagonal direction. Now, if you imagine that you have thousands of axions, the effective decay constants can be bigger than the Planck scale, even though each individual decay constant is of the other gut scale. And now, you can generalize these ideas by combining them, and including all possible effects that you can think of, for instance, kinetic mixings. The axions do not have to be... The kinetic terms for the axions do not have to be in a diagonal basis where the potential is diagonal. So you can include these kinetic mixing effects. You can combine the two previous effects that I mentioned, and you can also invoke properties of large n and carry out some random matrix study and one finds that by doing so, you can enhance the field range by an even stronger factor with the number of fields. In some cases, the effective field range can even... depends on the number of fields exponentially. So it looks like, even though the individual decay constants are a Planckian, seems possible that's the effective field range for a multi axion system can become super Planckian. So the question that we would like to ask is whether there is a fundamental reason that in the single axion case, we seem to find from exhaustive surges that the product of the axion decay constant and the instanton actions to be bounded by the Planck scale. And if there's a reason why this is the case, we would like to know whether multi axion models or models with monodramy can evade this reasoning and how do we evade this reasoning. So this is where we bring the weak gravity conjecture into the discussion. So there are reasons why the weak gravity conjecture is referred to as a conjecture rather than a theorem because like the folk laws that global symmetries cannot exist in quantum gravity is not proven, the weak gravity conjecture is also one of those folk laws that are not proven. Nonetheless, the conjecture is motivated from black hole physics. It is consistent with what is known and there seems to be no counter example that has been found. Now we are going to take an agnostic view on whether the weak gravity conjecture is truly a consequence of quantum gravity. Instead we will take the weak gravity conjecture as a starting point and try to derive its consequences on inflation. Now the weak gravity conjecture was first made by Galileo. The conjecture was that because gravity was so weak the piezo tower is still there after 800 years. But this is not the weak gravity conjecture that I would like to refer to. The weak gravity conjecture that I refer to is a statement that gravity is the weakest force. Namely that for every long range interactions there must exist a particle with a charge that is bigger than its mass. In plant units. Now this order one factor was left rather vague in the conjecture. But as you will see we have made this order one factor more precise in our work. So how do we understand this conjecture? Well a heuristic way to understand this conjecture is to consider a U1 gauge theory that violates this conjecture. Namely if you have a U1 gauge theory that has only particles with a charge that is smaller than its mass what would happen? If that's the case you will find that the gravitational attractions between these particles is stronger than the gauge repulsion. So these particles would form bound states. These bound states are stable because there's nothing lighter for them to decay into. And you can convince yourself that you can form infinitely many of these bound states. Now the presence of infinitely many stable bound states may lead to problems with remnants. I will say more about that later on. But from this heuristic argument you can understand what this one in the weak gravity conjecture corresponds to. The one in the weak gravity conjecture corresponds to the ratio of the charge and the mass of an extremal black hole in the theory under consideration. So the weak gravity conjecture basically said that you must be able to find a state with a charge to mass ratio that is bigger than the charge to mass ratio of an extremal black hole. Otherwise this infinitely many stable bound states have no channels to decay to. Now as I said the conjecture is a conjecture. The statement is still being formulated. There's still an ongoing discussions of how to make this statement more precise and there are several versions of the weak gravity conjecture that have been put forward. The strong form of the weak gravity conjecture requires these states that satisfy the inequality to be the lightest charge particle in the theory. And this is motivated by the spectrum of states in perturbative string theory. Now the weak form of the weak gravity conjecture only requires the existence of such a state. Doesn't matter who these charge particles are as long as you find a charged particle that satisfies this inequality then there's nothing wrong with coupling this theory to gravity. And the reason why one might expect a stronger form of the weak gravity conjecture is that quantum effects are important for making a state super extremal meaning that the particles mass is smaller than the charge. And the lighter the states the more important are the quantum effects. And in the case of perturbative string theory this quantum effect is the cashmere energy. So when you quantize the string when you look at the oscillators the commutators would give you a negative number and that's precisely what makes the state super extremal. Okay so what does the weak gravity conjecture has to do with axions? Naively if you can generalize this conjecture to p-dimensional objects coupled to p plus one form then with the replacement of mass by the tension of this object it seems possible to explain the difficulty in finding superpunking decay constant in control limits of the theory. And the reasoning goes as follows you can loosely speaking think of the inverse of the decay constant as the charge and the instanton action as loosely the tension. And if you make that extrapolation starting from the weak gravity conjecture you arrive at this inequality which seems to be consistent with what was known from the exhaustive search of superpunking decay constants in string theory. However this argument is harder to mate with instantons rather than particles because it involves bound states and the decay. And so that's the reason why we sell out to find evidence for the instanton versions of this weak gravity conjecture. So is there a way to derive something like this with the instantons? Now we can get some handle on this question using t-duality. So t-duality is a symmetry of string theory it maps a theory with a given size to a different theory with inverse of size and not only is it a symmetry of the spectrum it's also a symmetry of the interactions. So to be concrete you can say you start with an axion in type 2b string theory that coupled to the instantons this is what gives rise to its non-properative potential. You can take one of this non-compact direction compactify it on a circle then t-dualize and decompactify. So under this chain duality which is known as the Z-map you are mapped to a type 2a string theory and the axions becomes the u1 gauge field whereas the instantons becomes the particles. Now we see from this analysis that the naive expectation of how the weak gravity conjecture applies for instantons holds up. The instanton action and the inverse of the decay constants are mapped to the mass in the charge up to these factors which are hard to guess without doing a calculation. But naively there's a problem because under this Z-map the coupling constants become infinite. So starting from a weak coupling theory you end up with a strong coupling theory in type 2a but this is not the end of the world we know how to handle this in string theory the strong coupling limit of type 2a is m theory so instead of formulating the weak gravity conjecture with type 2a string theory we can do so with m theory and ask what is the charge to mass ratio of an extremal black hole in m theory compactifications down to five dimensions and this is what we find at least for this particular case of axions. So the statement we could make is that for an axions that come from a Raman-Raman II form in type 2b string theory we find that the product of the axion decay constant and the instanton actions to be bounded by this precise factors of the plan scale. Now we can also work out the corresponding bound for axions coming from other form views in other formulations of string theory there they give you different all the one factor but they all seem to fit into this inequality of this form. So in a concrete sense we have made the weak gravity conjecture precise. Now you can generalize these discussions to multiple axions and again it's useful to go to the dual picture with u1s and charged particles and let's consider two u1s or two axions so there must exist at least two charged particles for this black holes to decay if you have two u1s and as suggested by the single axion case we are instructed to consider the following charged factors in a two-dimensional space. The extremal black holes lies on a sphere with a given radius and to simplify the formula we normalize the radius of this extremal ball to be one and all the charged black holes lie within this sphere. Now naively it looks like we only need to find a state with a charge to mass ratio bigger than one in each of these charged directions and we are done but this is not the case because if you consider a black hole with a charge in some other directions in this two-dimensional charged space there are some black holes that still cannot decay. So you need to find states with a bigger charge to mass ratio to account for these other black holes and the more precise statement is that the convex hull generated by these charged vectors should contain the extremal ball. The convex hull is the shaded region here. Now these charged vectors do not need to be orthogonal as I have shown on this slide. The only thing that is needed is that the extremal ball is contained within the convex hull. So you can find other examples if you were able to find states with this particular charge to mass ratio all the black holes under considerations can decay. So in this picture the aligned axiom scenario corresponds to choosing charged vectors that almost align but the extremal ball is not contained inside the convex hull and in the case of inflation the charged vectors need to be enlarged by a factor of square root of n in order to contain this bigger it is an extremal ball and this basically undo what this basically undo what you gain the factor of square root of n that you gain by going diagonally. And our discussion is not limited to the special cases. We have shown quite generally that given a set of instantons that give us a superplunking diameter in the axion field space no matter how complicated the kinetic terms are the extremal ball is not contained inside the convex hull generated by these instantons. Now the grid boundary conjecture is a conjecture if you don't want to use a conjecture to constrain a model you can ask whether gravitational instantons can induce sizable higher harmonics to the axion potential and spoil the properties of superplunking field range. This was recently done by the Madrid group and the concretion basically agrees with ours. So in the remaining five minutes or so let's see whether there's a way out. Now we were very careful in choosing the word fence in our paper and not walls because they are loopholes they are interesting loopholes that we find that could potentially help us evade the problem that I mentioned. Now first of all the weak gravity conjecture requires only the extremal ball to be contained inside a convex hull it doesn't care who these charge factors are so you could imagine that in addition to the instantons that give you the infertile potential suppose there's an additional instanton that gives the negligible contributions to the potential but with a very small decay constant such that this criteria that we find earlier is satisfied. So in the presence of this spectator instantons the convex hull conditions can be satisfied so the weak gravity conjecture is not violated. And notice that the instantons being heavily this instantons are mapped to a very massive particles in the particle picture because the instanton action is very large in plant units so they do not correspond this spectator instantons do not correspond to the lighter states in the theory so the strong form of the weak gravity conjecture is still violated. So which form of the weak gravity conjecture should be satisfied in a consistent quantum theory of gravity? So we call that the weak gravity conjecture is motivated by black hole physics and in this context it was argued long ago that having infinitely many stable remnants in an finite mass range could lead to pathologies basically you can go to an accelerator frame where you find that the you have a thermal bath of infinitely many species and the entropy diverges. The weak gravity conjecture is a stronger requirement. The weak form of the weak gravity conjecture states that having infinitely many stable remnants even within an infinite mass range are still or not acceptable. And the weak the strong form of the weak gravity conjecture requires this remnants to decay to the lighter state. Now the loophole that I mentioned amounts to hiding our ignorance about the spectrum of states at or above the Planck scale. So a priori there's nothing wrong with that but let me remind you that the whole purpose of realizing large free inflation in string theory is to tame the UV sensitivity and this sort of defies the purpose. So I think the burden of proof is on those who claim to find a viable model of large free inflation in string theory. So let me also mention that and this should be obvious to you that the constraints that we find on the multi axion systems do not immediately apply to axion monotony because the axion in this case is mapped to a very massive is mapped to a massive gauge field. However, there are non-pertuative processes that could take us to a different branch with the potential even though inflation takes place on one of these branches there are non-pertuative effects that could take us from the inflationary branch to some other branch and suppressing these tunnelings could lead to potentially an interesting bound on the infertile field range or in other words the tensitude scalar ratio. So let me just conclude I have illustrated the UV sensitivity of inflation especially for models with detectable level of gravitational waves and we have made the weak gravity conjecture quantitatively precise for axions as long as you can dualize the axions to U1 gauge fields our diagnosis would apply and this allows us to put constraints on large field inflation models with axions in terms of the convex hull. Now although the weak gravity conjecture is motivated by black hole physics both the strong and the weak form are not derived from first principles so it would be interesting to firm up the statements and if the weak gravity conjecture in a strong form is truly a consequence of quantum gravity then several large field inflation models are ruled out if it can be violated we also point out some interesting loop holes to get large effective field range and what I find quite exciting is that there is a surprising interface between black holes in quantum gravities and inflation and perhaps understanding this connection could teach us something interesting about quantum gravity so thank you for your attention. Okay, questions? Hi, I had a probably a dumb question did I understand you correctly that the T-dual of a weakly coupled type 2B gave a strongly coupled type 2A? So this is not a T-duality in a traditional sense because this is T-duality in a non-compact direction however there has been checks about the validity of this duality it's called a C-map just to distinguish it from T-duality so and there have been work done in checking this correspondence for instance you can check the instanton calculations on one side matches with particle loops on another side so this way of dualizing the theory maps you to strong coupling but are there more mundane type of T-duality which involved the internal direction does not have this feature because basically we want to recover a non-compact direction in this process and the non-compact direction basically give us a strong coupling in the type 2A side which would become the M-theory direction This is just a comment and if we not UV complete the inflation there are very basic problems with that and actually it doesn't do what it is supposed to do for example you need to also reduce the plank mass effectively by the same fraction so it doesn't work and also it will lead to eternal inflation because the quantum fluctuations dominate over the classical rolling of the potential so you need to have a UV completed version otherwise it doesn't work that's right so if you just take inflation by itself the the four-dimensional plank mass is also scared with square root of N so we basically did not gain anything and you can sort of see this from this picture of convex hull but a lot has been done afterwards and there have been attempts to gain a bigger power with N for instance there are examples where the enhancement is exponential with the number of fields it's harder to argue why these models are problematic but in light of this regrettable conjecture we can say that well no matter what you do you are going to run into the same kind of problem any more questions? okay let's thank Gary again thank you