 OK, so this is the last of my lectures here. And so there's several more things I want to tell you about. Here is sort of an outline of what I'm hoping to get through in the next hour and a half. I want to first talk about colluzic line theory, which is what happens if we take a theory of gravity and then compactify it to a spacetime of lower dimension. And the reason I want to talk about colluzic line theory is it provides a nice illustrative example, not only of the weak gravity conjecture, but of various other swampland conjectures. And so if you've already started working on the problems that I posted, you may already have figured some of this out for yourself. But if you haven't, it's worth working through the details on your own. It turns out that a lot of the features of this example actually generalize to basically all theories of quantum gravity that we know. After I discuss colluzic line theory, I'm going to talk about the swampland distance conjecture, which is a swampland conjecture that was formulated by Hiroshi Oguri and Kamran Vafa back in 2006, which has recently had a there's been a revival of interest in this conjecture. And it's been realized that it can apply to lots of theories like theories of inflation. And we'll also see that it connects very nicely with the colluzic line example. So it generalizes features of colluzic line theory. Then I'll talk about some stronger formulations of the weak gravity conjecture. Last time I told you that the minimal version of the weak gravity conjecture is not very phenomenologically useful on its own, because it could just be telling us that there are small corrections to big black holes that make them slightly less massive. And that's not a very useful statement. But we'll see that there are these stronger conjectures that come closer to saying useful things for real world physics. And at the end, I want to give you at least one application, which is an application to the question of whether the photon can have a mass. If it turns out that I have a bit more time, I will also talk about applications to inflation. But we'll see whether that fits in the time or not. So what is colluzic line theory? What we're going to do is consider Einstein gravity, so GR in d dimensions, d spacetime dimensions. But we're going to dimensionally reduce by considering the theory on a space, which is d minus 1 dimensional Minkowski space, times a circle. So we're going to view this as a theory in little d, which is big d minus 1 spacetime dimensions. So this is the simplest example of how to take your gravitational theory and compactify it. We're going to take the theory to live on a spacetime where one of the spatial directions is a circle. And so if you look at physics at distances that are much longer than the size of the circle, you should be able to treat the physics as some effective field theory that lives not in d spacetime dimensions, but in little d spacetime dimensions, in one lower dimension. And the way to do this is to think about decomposing your higher dimensional metric in terms of fields that transform under the lower dimensional Lorentz group. So we have a spacetime metric in the full theory. But we can write this metric in terms of some metric in d dimensions plus a metric for the other direction, for the circle. And since it's a circle, I'm going to give it a coordinate that's some angular variable theta. So this is some coordinate that goes from 0 to 2 pi. But in fact, the most general metric is a little bit more complicated than this. The most general metric can also have off diagonal terms that relate the d dimensional directions with the extra compact direction. And because of the structure of these terms, that factor that kind of mixes these two is something that has a mu index, a vector index in the lower dimensions. And so from the lower dimensional point of view, we have a theory that has multiple fields. It has a metric, e mu nu. It has a scalar L that tells you the length scale of the extra dimension, how big it is. And it has a gauge field. So this was kind of the big insight of Kulutza and Klein, not long after general relativity was first formulated, that you could obtain a theory that has a gauge field like the electromagnetic field by starting with a theory that has only pure gravity in higher dimensions and reducing the theory on some internal space. So everything we're doing so far, this is really just effective field theory. We're starting with an effective field theory, namely Einstein gravity in the higher dimensional theory. And we're reducing and getting a new effective field theory. But because Einstein gravity is kind of a subsector of complete quantum gravity theories like string theory, there will be concrete string compactifications that are actual honest theories of quantum gravity where some subset of the fields behaves just in the way that we're talking about. Well, they does an actual physical dimension. It's just small. It has finite size. So if you could do experiments at very small length scales, you would see that it exists. If you can only do experiments at very long distances, you don't notice. And so our universe that we live in could have more than four spacetime dimensions. It's just the others are small enough that we haven't seen them yet. And in fact, gravity has only been tested at something like micron distances. So at very short distances, we don't really know if the gravitational force actually looks like it comes from a higher dimensional theory or not. OK. So these fields also have names. The scalar is sometimes called a radion, because it's a scalar that controls the radius of the circle. More generally, scalar fields that parameterize these compact dimensions are often called moduli fields. This is one example of a scalar field that we might call a modulus. And the gauge field that arises in this way is sometimes called a gravite photon. That is a photon field that somehow arises from gravity. OK. So we have an onsots for a metric. And you can just plug this onsots in to the Einstein-Hilbert action. And if you do, sorry, let me further specify, I'm going to make an assumption that g mu nu l and a mu are independent of theta. Now that's not the most general assumption. The most general metric will have these fields depending on all of the coordinates, including x mu and theta. But it turns out if there's non-trivial dependence on theta, we get modes that look heavier from the lower dimensional point of view. So we'll come back to that. But for now, we're just going to look at what we call the zero modes, where these things only depend on the lower dimensional coordinates, not the higher dimensional ones. OK. So when we do that, we find that the action when we plug in this onsots has the following form. You may not write everything. There are terms involving a mu. But the action contains the length scale of the extra dimension, multiplying the lower dimensional Ritchie curvature. And the reason for that is just that, yeah. So what I've done here is actually to already integrate over the other direction. Right. So your d-dimensional measure has the d dx, but it also has d theta. And so what we've done here is to say, well, we assumed our fields were independent of theta, which makes the theta integral trivial. And the theta integral is just going to give us a 2 pi. But we also have a square root of minus g in the d-dimensional theory, which is going to turn into a square root of minus g in the lower dimensional theory times a square root of this factor that comes from the other components of g. And that's going to give us an L. OK. So basically, L is the size of the circle. And so when we integrate over the extra direction, we get a factor of the size of the circle, which is L, and we get a factor of 2 pi, which is the range of theta. OK. So this is what you've got if you just take that on socks and plug it in. And you see that what we have is an Einstein-Hilbert action in the lower dimensions, but multiplied by a scalar field L. This is sometimes called a metric in the Jordan frame. Because we have this non-trivial kinetic mixing between the scalar and gravity. So that's what you've got if you just start with the simplest on socks and plug it in. Now, you're free to work in this frame. But in this frame, your kinetic terms are kind of weird because L times R schematically has things that look like L times 2 derivatives of the metric perturbation, plus L times things like h squared and so on. And so if you work in this frame, your theory is a little bit awkward. So because of this term, you'll have Feynman rules, for example, where you have a propagator that kind of is a scalar on one end, L and a graviton on the other end, h. So this is not the most convenient frame to work in. L has no kinetic term of its own. It kind of only has the ability to propagate by mixing with the graviton. So the trick to resolve this problem, well, there are two tricks. Either you could have picked a better on-sots to begin with if you were really clever. Or you take this on-sots, and then once you have it, you try to figure out, how do I get rid of this L and the pre-factor? And the way to get rid of L is to do a field redefinition. So we're going to change variables. So you're always allowed to do this in quantum field theory. You can reparameterize your degrees of freedom, whatever you like, whatever is most convenient to work with. And so we're going to do that here. We're going to replace the metric, g mu nu, with a nu metric, which is g mu nu multiplied by the scalar field L divided by some constant reference value, L naught. Or I think the notation I used on the problem set was angle brackets around L. But just kind of by dimensional analysis, we're going to pick some reference value. And we're going to raise this to some power. And I forget what I called that power in the problem set, maybe beta. But the claim is that if we pick beta intelligently and we do that field redefinition, we're going to get a nicer action that just has an Einstein-Hilbert term and also has an independently propagating scalar L. And I'm not going to tell you exactly what this constant is because that's part of what you should work out on the problem set. But the upshot is in the new frame, we call Einstein frame. And again, what I mean by frame is really just what onsots we pick to relate the D dimensional metric to the D minus 1 dimensional metric. In the new frame, we have a nicer action, which has the usual Einstein-Hilbert form for gravity. Plus some constant times a scalar kinetic term, which involves derivatives of the log of L. So that's something you'll just find when you do this calculation. And when you do this scaling, this field redefinition, which is an example of what we call a vial rescaling, you find that the kinetic term you get for the scalar has this logarithmic form. This is not a conformal transformation. This is a field redefinition. So a conformal transformation would be a transformation of my spacetime. Here I haven't touched the spacetime. I've just redefined my field. They're closely related in some ways. But it's really a field redefinition and not a conformal transformation. Was there another question somewhere? Yeah, what I've done is just to absorb some of these factors. 2 pi times the reference or expectation value of L. I've absorbed that into the definition of what I'm calling the lower dimensional Newton's constant. Yeah, thanks. And then again, there were terms that involved the gauge field, which I didn't bother to write before. But now let me write what that actually looks like. So in terms of the gauge field, I get, let's see. I think again, I'm not going to write all the constants very carefully, but I get some other constant. And I get a constant times a power of L. Let me just, again, I think I used some notation and the problem said I don't quite remember what it was. Let me call it gamma times F mu nu F mu nu. Where again, F mu nu is our usual field string that we compute from a gauge field appropriately anti-symmetrized. OK, so again, you can work out all the details of this. The main things I haven't told you are just what constants appear. The problem set tells you the answer for how the Ritchie scalar changes when you do this field redefinition. The first problem is to work that out for yourself. Once you know that formula, though, the rest of this calculation is not actually very complicated. And the main physical things I want to emphasize, though, are that the kinetic term for the scalar depends on the log of the size of the dimension. And the pre-factor in front of the gauge coupling depends on a power of the size of the dimension. So the canonically normalized scalar proportional to the log of L, but the gauge kinetic term is proportional to a power. So that's the first thing I want to emphasize. It will turn out that even though this looks like something that depended on the details of this calculation, this actually reflects a very generic fact about theories of quantum gravity. So it's actually very common that we find these scalars where the gauge coupling, another way to say this is the gauge coupling depends exponentially on the canonically normalized scalar. Depends on a power of L, but the canonically normalized thing is the log of L. And so if we rewrite it in terms of the canonically normalized field, it's an exponential difference. The other thing to say here is that we started out by assuming that all these fields were independent of theta, of this extra-dimensional coordinate. But of course, they don't have to be. And what happens if we have a field that is not independent of theta is that we get modes that when we reduce from the higher-dimensional theory to the lower-dimensional theory appear massive. And the reason is that dependence on theta corresponds to momentum in the circular direction. And if we have a particle that has momentum in that direction, that means it has some energy associated with that momentum. But from the lower-dimensional viewpoint, we don't see that as momentum because we don't know this direction even exists. But we still see that the particle has some energy. And so it looks like rest energy. And so from the lower-dimensional viewpoint, that looks like a contribution to the mass of the particle. So what happens includes a Klein theory is we can decompose any field in the higher-dimensional theory in terms of modes with different momentum around the circle. And because it's a circle, because it's periodic, that momentum is actually quantized. So what we can say is our higher-dimensional field, g of x theta, can be written as a sum of modes labeled by some integer n, which are fields that are functions of only x, multiplied by e to the im theta. So we're just going to do a Fourier decomposition of the field. And so what we obtain is an infinite tower of Kuhl-Sakline modes with masses in Einstein frame. Yeah, let me not specify Einstein frame. The masses are associated with couplings to the physical scalar L. And exactly what I mean by those couplings could be frame-dependent. But let me just talk about the physical masses rather than the couplings. The physical masses are just n squared divided by the expectation value of the size of the extra dimension squared. So if I have n units of momentum around the circle, I have a wave function, which fits n periods inside that distance 2 pi L. And that corresponds to an energy of n divided by L, divided by the size of the circle. So this is the generic feature that happens when you have extra dimensions. Whenever you have extra dimensions, you have infinitely many fields, which correspond to all the different ways you can excite the higher dimensional field within the internal dimension. So starting with the single field and the higher dimensional theory, you obtain infinitely many fields in your lower dimensional theory. Not only that, but all of these Kuhl-Sakline modes have charge under the gauge field, the gravifoton a mu. And that charge from the higher dimensional point of view is just the momentum around the circle. This takes a little bit more work, but if you plug in this onsets into our metric and you then work out the kinetic terms of the fields in a lower number of dimensions, you will see that they pick up the right kind of couplings to look like a mu is gauging this conserved charge, which is just how much momentum they have in the higher dimensional theory. Any questions about this? If you can back to find one? Good. Yeah. So if you can back to find a different manifold, you will always still have Kuhl-Sakline towers. And these towers correspond to different eigenvalues of the Laplacian on whatever that manifold is. So there will always be infinitely many different modes because the higher dimensional field has more degrees of freedom than the lower dimensional field. And so all the different ways you can excite it in the internal dimensions correspond to different modes. What will not be true in general is that all of those modes are associated with some kind of gauge charge like happened here. This happened because the circle has an isometry. And so the momentum around that circle is a conserved quantity. If you can back to find some more general manifold that's some complicated thing that doesn't have a lot of symmetry, you will still get lots of modes from all the different eigenvalues, but they won't necessarily be associated with conserved charges in the same way. Is it problematic? It's not problematic as long as, okay, if you're asking questions about length skills that are big compared to the size of the extra dimensions, then you can't excite these modes because they're too heavy. And so you can do lower dimensional effective field theory without taking them into account and you'll get consistent answers. Once you get to energies that are high enough that you start to excite these modes, then it's difficult to do effective field theory in your lower dimensional picture because these modes are not parametrically far apart from each other. By the time you see the first one, the second one is not very far away and the third one is not very far above that. So yeah, it's difficult to do calculations from the lower dimensional viewpoint once you reach those energies. It's better to just use the higher dimensional theory at that point. In principle, you should be able to use the lower dimensional theory and sum over all of these modes. In practice, that's sometimes difficult because there's sometimes tricky kind of regulator related issues that there can be, if you truncate your sum at a finite point, you can sometimes find that it looks like that explicitly broke some kind of symmetry that was there because of your high dimensional theory. And so as always, if there are symmetries in the problem, you have to be careful about how you regulate calculations. But I guess the real answer is that if you really know that your theory is a higher dimensional theory about these energies, you should just work in that higher dimensional theory. Okay, yeah, let's see. So I think the right answer to that question is that, so in this theory, we have a tower of these massive modes that I wrote here and these were modes of a spin to field, right? So a massive spin to field in four dimensions, let's say, has five physical propagating degrees of freedom. And you can think of it as a massless spin to field that ate a massless gauge field and a massless scalar. So what happens here is if I give these things theta dependence, I will see other modes of those fields, but those modes are really sort of gauge equivalent to modes of the massive spin to. And so it's sort of a gauge choice whether I want to put all of the massive stuff into this field or whether I leave some of it in these. Sorry, what was the question? Good, so at the level that we're working so far at tree level, theta has no potential. And so the circle can be any size. If we compute loop corrections in a general theory that doesn't have supersymmetry, we will find that a potential is generated. And so if you have fields in your extra dimension, you can compute some kind of Casimir potential that depends on how they propagate around the circle. And so at one loop, you will find that there is a potential for theta depending on the exact field content of your theory it might or might not have a minimum. And so in some theories, what we're doing here may not be very well defined because this theta may be some dynamical degree of freedom that just wants to run away to infinite size or something like that. But what we're doing here makes sense and in supersymmetric theories where the potential cancels because bosons and fermions propagating around the extra dimension have opposite sign contributions. It also makes sense if the potential has a minimum somewhere and then we can go to that minimum and just fix that size. So let me make a few observations now because the reason I went through all of this, well first it's a good thing to know something about colloquial decline theory. It's certainly a possibility that our universe has extra dimensions in string theory. It's almost a necessity. But the real reason I went through all of this is because I want to tell you about these other ideas of the Swampland distance conjecture and the tower and sub lattice week gravity conjectures. And this turns out to be a perfect example of phenomena that we'll see there. Another question? In some sense, yeah, when you do Wilsonian RG, the question was, is the set of modes here related to how when you do Wilsonian RG you kind of integrate out an infinite set of modes. When you do Wilsonian RG, you integrate out all momental modes above some scale and you keep the ones below that scale, right? And so what we're seeing here is that it's possible that there's some scale above which you have momental modes that go around other dimensions. And so if you go below that scale, you get a lower dimensional theory but you have to integrate out all those infinitely many different excitations. So yeah, in some sense, there is a relationship because we're really just talking about all the different momental modes or sort of generalized momental modes when we have some complicated internal space. Okay, good. So a few more observations. So I already mentioned this exponential relationship between the gauge coupling and the size of the circle. The lower dimensional Planck scale is given by, so maybe I should tell you when I say Planck scale in a theory in D dimensions, what appears in front of the Einstein-Hilbert term is something that has mass dimension, mass to the D minus two, because the Riemann tensor always has two derivatives of the metric, the metric is dimensionless and so to get the dimensions right, we need D minus two. And I'm defining what I mean by the Planck scale so that this is one over eight pi times the D dimensional Newton's constant. There are different conventions for what we mean by Planck scale and four dimensions, this agrees with what everyone would refer to as the reduced Planck mass. That's 2.4 times 10 to the 18 GeV. Okay, so our lower dimensional Planck mass is given by the appropriate power of our higher dimensional Planck mass multiplied by two pi L. So again, that's just because we're integrating over the theta direction when we reduce our higher dimensional action to a lower dimensional action so we get the size of the circle appearing. What that means is if my size of my circle is big in higher dimensional Planck units, which it has to be for me to have control of my effective field theory, my lower dimensional Planck scale is much bigger than my higher dimensional Planck scale. So in particular, if our universe has extra dimensions that are relatively big compared to the Planck scale, the higher dimensional Planck scale would be much lower than what we call the Planck scale. And so we measure Newton's constant at low energies. We say the Planck scale is 10 to the 18 GeV and if we scatter things at that energy, we should start making black holes and we should see strongly coupled gravitational physics. But if our universe has big extra dimensions, then actually the true gravitational cutoff will be much lower than that because when you do scattering experiments at the higher dimensional Planck scale, you'll start making higher dimensional black holes. That's why around 20 years ago, Arkhani, Hamad, Damopoulos, and Diwali proposed that our universe could actually have millimeter size extra dimensions and then experiments at the LHC could have started making black holes. It didn't actually happen, but it was a logical possibility. Yeah, that's right. I'm assuming we have an Einstein-Hilbert term in the higher dimensional theory. Okay, good. And the other thing to say is, again, I told you there's some power of length that appears in front of the gauge, in front of the gauge theory. Remember that when I have a gauge field, what appears in front of it is one over the coupling constant squared. If I use a normalization where the gauge coupling couples to the charge current with no extra factors, and so what we're seeing is one over the coupling squared goes like this power of L, which in fact is a positive power of L. Okay, I haven't told you what it was because that's one of the things I want you to work out on the problem set, but it is a positive number. And so when I go to large radius, I get very weak gauge coupling. So the weak gauge coupling is related to large radius. Large radius is also related to light collucicline modes. Okay, so the KK modes are these massive excitations that are momentum around the extra direction. When that size is big, their mass is small. So we have three things that are related to each other. One is a weak gauge coupling. One is a tower of modes becoming light, and the other is a large size of some extra dimension, which also makes yet another thing true, which is weak gravity in the lower dimensional theory, the sense that the lower dimensional theory's Planck scale is much bigger than the higher dimensional theory's Planck scale. Okay, so now maybe you start to see why I wanted to talk about this in relation to everything else I've talked about. We've said that there are no global symmetries in theories of quantum gravity, and we talked about the weak gravity conjecture, which said that if I have a very weak gauge coupling, interesting things should happen, like I should have light particles. And so here we see some connection between weak gravity, weak gauge coupling, and light particles that carry charge. And in fact, these KK mode masses can be rewritten in terms of the D dimensional Planck scale, and the coupling, I've been calling it ED, the coupling of the gravity photon. And so this is just the D dimensional version of the weak gravity conjecture. Their mass is their charge times their coupling constant times the appropriate power of the Planck scale. The KK modes saturate weak gravity conjecture. They all obey the inequality. In fact, they obey it with an equal sign instead of a less than. And that's because in a supersymmetric theory, this turns out to be a BPS bound, and there's a reason for it to be exact. In a non supersymmetric theory, the scalar field, the radion will acquire a mass, and it won't affect the black hole solutions. It turns out the black hole solutions have different extremality bounds with a massless radion and a massive one. And so if the radion is massless, as you expect from Susie, they saturate the bound. If the radion is massive, they obey the bound with room to spare. So that's an example of how a theory of quantum gravity can obey the weak gravity conjecture. We have a gauge field that because of the origin of this gauge field and the extra dimension, it's automatically associated with all these different charged particles that correspond to momentum in that direction. And the constants are all related to each other in such a way that the weak gravity conjecture is obeyed. Yeah. In the non supersymmetric case, what happens is good. Right, so let's talk about like the 4D case, just for clarity. The weak gravity conjecture with a massless radion in four dimensions says the mass is less than or equal to charge times the gauge coupling divided by the square root of two times the Planck scale. But the WGC with a massive radion says the mass is less than or equal to the square root of two times the charge times the coupling times the Planck scale. And the reason for the difference is that in the case with a massive radion, we're thinking about just Reissner Northstrom black holes. So we have gravitons and photons turned on and those are the only fields. In the case with the massless radion, the radion couples to the gauge field. And so once you turn on your electric field outside the black hole, you source the scalar and you can't avoid turning it on. So you get a non-trivial scalar field and that scalar field carries energy and that changes the extremality bound. So in figuring out if you obey the bound or not, you need to be careful about what is the extremality bound and the extremality bound depends on the way that the scalar fields couple to the gauge fields. And so with the massless radion, you get this bound and your modes just saturate it. They exactly hit the inequality. Now if you don't have super symmetry and you start computing loops, two things will happen. First, you'll get corrections to this formula. But second, your radion will pick up a mass. It'll acquire a potential. And because the radion acquires a potential, that means what you really should be comparing to is this inequality. And this inequality, these modes obey by a factor of two. So as long as you're in a regime where loop corrections are under control, where they're giving you corrections that are small and not order one, you will still obey that. Okay, so I think this is all the things that I want to say about Kalutza-Klein theory. Are there any questions? That's right. What you need is that, right. So it wouldn't be this picture where we just have a circle and all the fields propagate in the extra dimension. What would be allowed is if we have some extra dimension where some fields are kind of localized to a lower dimensional surface. So the kind of large extra dimension scenarios people have discussed is you have some 5D theory where your standard model fields kind of live on some kind of brain within the extra dimension, but gravity can propagate across the extra dimension. And so then you don't actually get Kalutza-Klein modes of the photon, say going around the circle. You only get Kalutza-Klein modes of gravitons. And so as you said, we know a lot about how electromagnetism works below micron distances, but here we're not changing electromagnetism at those distances, we're only changing gravity at those distances. That's much harder to study. Well, yeah, that's in, well. In effective field theory, you certainly are. You can just say you're going to define your action to have fields that live here and fields that live there and that's a healthy thing to do. In string theory, you can actually find UV complete examples where there are fields that live only on some brains. But it's not necessarily completely trivial to engineer something like the standard model with chiral fermions and everything living on the localized thing that you want. It's not impossible either, but it's a little more complicated. There are kind of more rules about what you can put on brains in string theory than if you just kind of naively write down an effective field theory. But yeah, nothing is really wrong with this idea of large extra dimensions. That's what Arkanihama, Demopolis and Diwali got a lot of excitement about this idea generated 20 years ago or so when they pointed this out, pointed out that this was not ruled out. And it's a little less discussed these days because the LHC has not produced dramatic signals of higher dimensional black holes or something like that. But it's still true that we just don't have tests of gravity to very short distances. Okay, good, so I should jump ahead and tell you about the conjectures I wanted to mention because we're running a bit a long time. Okay, so there's something called the Swamplan distance conjecture which again is due to Agourian Vafa from 2006. And what they say is quantum gravity theories have moduli, scalar fields that control couplings. Certainly true in string theory. Coupling constants are never just parameters that you can adjust. They're always vebs of scalar fields. There's always some field whose value could change and it would change, say, the fine structure constant. It's a little more, it's kind of going out on a limb to say that all quantum gravity theories have moduli but certainly all the well understood theories do. All of these moduli have infinite distance, loci regions of the field space that are infinitely far away in field space. And the claim is that if I have some space of scalar fields and I have some point P, then there exists some other point Q where the distance between P and Q is bigger than any arbitrary number you picked. So you can always find a point that's far away. And there is a tower of particles, some sort of modes, sometimes strings or brains, but maybe just particles with masses of order e to the minus alpha t, some alpha bigger than zero. That's basically what Agourian Vafa suggested as a generic feature of quantum gravity. And they guessed this, there's no argument for this. I can't convince you of this by telling you some story about evaporating black holes or something like that. But it's at least an observation about a large class of understood theories of quantum gravity, they have this property. And the easiest way to understand how to think about this property is to think about the Kluge-Sekline example. So in the Kluge-Sekline example, our modulus is the scalar field L that controls the size of the extra dimension. And the infinite distance region is when I make L infinitely big. So I make the circle bigger and bigger. And when I make the circle bigger and bigger, I have this tower of Kluge-Sekline modes, whose masses become smaller and smaller, like one over L. But we also know that the canonically normalized scalar is not L, it's the log of L. And so these modes are becoming light exponentially in terms of the canonically normalized scalar field, which is what gives us the distance in scalar field space. So by distance, we mean the distance in field space as measured with the metric on the scalar field that you read off from this kinetic term. Metric is one over L squared DL squared. And so the distance goes like log of L. So Kluge-Sekline theory at least obeys this conjecture. There's an infinite distance region. When you go to that infinite distance region, you get a tower of things that become light. And not only do they become light, they do so in an exponential manner. What Aguri and Vaffa are suggesting is that this is very general in quantum gravity. Yeah, that's a good point. You can go the other direction. You can go to infinitely small L. And as you said, in just Kluge-Sekline theory, that doesn't give you a tower of modes becoming light. However, what happens in string theory is every time you can do that, when you can take L to be small, there's some sort of winding modes, like strings that you wrap around the circle. And in the limit when you make the circle small, those become light. So in fact, you're right. If you really want this conjecture to be true, that Kluge-Sekline theory, you can't get away with just the ingredients you have in field theory. You need some kind of extended object that you can wrap around the circle. But that happens in every theory of quantum gravity that we know. Okay, so that's their conjecture. There are further refinements of this conjecture. You can ask how big is this number alpha? And in known examples, what happens is that alpha is an order one number in Planck units. And so if you have a field that goes a distance bigger than in Planck, you get all these modes that are becoming light. And that's problematic for some models of inflation that wanna have the inflaton change its value by more than in Planck during the course of inflation. Yeah, yeah, that's right. So yeah, it's possible that these modes can decay. So there's no statement in this that these have to be stable modes. But maybe what you can say is that somehow the density of states in the theory below some mass scale is growing exponentially as you're going along this distance. I think that's kind of a well-defined statement independent of whether the things decay or not. Yeah, thanks. In a weekly coupled theory, usually you can talk about an infinite tower of modes because even if they decay, they have narrow width. And so it kind of makes sense to talk about them as particles. But you're right that if they're decaying, if they have a big decay width, then you have to be a little more careful about what you mean. Yeah, good. In fact, I think I must not have written down the exact phrasing they have in the paper because I think what they claim in the paper is that in any infinite distance direction, this happens, any such direction. There are directions you can move in where you can't go infinite distance. Examples are axions. So these are just periodic fields. So you just can't go infinitely far away. But the claim is there always are some directions in which you can go infinitely far and whenever you can go infinitely far, there should be some associated tower that becomes light. Well, what do you mean by infinite distance then? I would say infinite distance. We're talking about infinite distance in some space of scalar fields. And I'm calling these scalar fields moduloid, but what? Right, right. Yeah, I think the original conjecture was sort of stated only for really massless moduloid fields that have flat potentials. It seems to be true more generally and in fact there is some evidence that it's true for axion monotomy where you might say that you just have a periodic axion, but the point is there's some sort of effective field that is not periodic because you can keep winding around forever and you're changing some other quantity every time. And people have studied, there are various papers, I can't remember any exact references offhand, but if you search for instance for some recent papers by Rene Valenzuela, she has studied some of these examples of axion monotomy models and argued that this actually does happen in axion monotomy. And anytime that you can go super plonky in distances by winding many times around, you do get some kind of back reaction effect that starts to bring down some modes. But yeah, there are definitely aspects of the conjecture that are maybe not stated as precisely as we might like them to be. Okay, but this is a claim that Aguri and Vafa made and they provided various bits of evidence for this in different corners of string theory. And so what I want to emphasize here is just that all of the kind of ingredients here you can understand by thinking about Kuhl-Sekline theory. And so in some sense, one way to think about the Swampland conjecture is that all corners of quantum gravity kind of look like Kuhl-Sekline theory. So the intuition that you can build in Kuhl-Sekline theory actually is valid even for theories that are much more complicated. And now I want to come back to the wheat gravity conjecture because what we saw in the Kuhl-Sekline example is that we had this tower of particles. They have this relationship between mass and charge. And so they do what the Swampland distance conjecture asks. They also do what the wheat gravity conjecture asks. They're particles whose mass is less than or equal to their charge in plonk units. And if you think about these two things together, you might guess that there should be some kind of new statement that combines the Swampland distance conjecture and the wheat gravity conjecture. I'm not saying we can derive this, but you might guess the Swampland distance conjecture plus wheat gravity conjecture might lead you to guess that something stronger than the wheat gravity conjecture is actually true. Namely, any gauge theory could have an infinite tower of charged particles with different charges q, the qn such that the mass of the n-th particle is less than, well, let me write it this way. The mass to charge ratio of the n-th particle is less than the mass to charge ratio of an extremal black hole. So this is a guess. It's a guess that is satisfied by Kowitsa Klein theory. We have infinitely many particles, different charges, every one of them obeys the wheat gravity conjecture. This kind of statement has been called a tower wheat gravity conjecture. One place where this was written down was a paper by Andriolo Junghans, 2018. In fact, initially my collaborators and I guessed that an even stronger statement was true, which we called the lattice wheat gravity conjecture. And the lattice wheat gravity was the conjecture that for every charge q allowed in the theory. As I mentioned in one of the earlier lectures, I'm assuming throughout the charge is quantized, which means there's a lattice of allowed charges. For every charge we guessed that there's a particle obeying the wheat gravity conjecture. Again, that is true in Kowitsa Klein theory. You have modes with every possible momentum around the circle. However, this is false. And so what we did a year later was to formulate a slightly weaker conjecture, which is the sub lattice wheat gravity conjecture, 2016, which is that for every charge q allowed by the theory in some sub lattice of the full charge lattice, there's a particle that obeys the wheat gravity conjecture. So all of these statements, tower, lattice, sub lattice, they all have a similar flavor. They're all saying you're not just going to find one particle that does the job, you're going to find infinitely many particles of different charges. And the differences between these different conjectures is just about exactly which charges you can expect to find. In some sub lattice of the full lattice of the same dimension as the full lattice. In fact, you could have asked the question about the original wheat gravity conjecture. Could it be satisfied by a particle of charge q equals zero? In which case, maybe you would say the graviton always obeys the bound. But yeah, somehow all of these need some disclaimer that we require to be true for some non-empty set of things. Okay, so all of these are postulating infinitely many particles, each of which have a mass below some bound. As I've told you, one motivation for this could just be a guess. We already have a different conjecture that there's some evidence for that requires infinitely many particles to appear in long distance limits. We've seen that in some examples, long distance limits in field space, like large radius, are the same as wheat coupling limits. And the wheat gravity conjecture is telling us mode should appear in wheat coupling limits. Also we know that when we send a gauge coupling to zero, we would be restoring a global symmetry. And we know that theories of quantum gravity are not allowed to have global symmetries. And if all that happened was the minimal wheat gravity conjecture, that tells us some particle has to appear that's light. But that doesn't mean the theory has a problem. We could make the gauge coupling really small. All that says is there's a really light charged particle and nothing is wrong with that. But with these conjectures, if we try to make the gauge coupling really small, we have an infinite tower of things that's becoming light. And that is a problem for your effective field theory. Because it means any effective field theory that only knows about finitely many fields is going to break down at some point when you start making the coupling small. So these versions of the conjecture kind of enforce this idea of no global symmetries. They sharpen the no global symmetry statement by telling you concretely what happens when you try to violate it by having a really tiny gauge coupling. What happens is this tower of modes becomes light. Yeah, there's an example that I could give, but since I'm running low on time, I think I would prefer to say other things in the rest of the time. But there's an example that basically, you slightly generalize the Clutes of Klein example. You compactify on a torus, but you do some orbifold. And the orbifold basically removes half of the charged states that you wanna have, but half of them still obey the bound. And you might think that you could then generalize that to remove a large fraction of all the states, but we actually haven't found any example where the gravity conjecture is violated on more than about, let's see. Maybe we should say, I should say, every example we know the weak gravity conjecture is satisfied by at least a third of the sites on the charged lattice. So we don't know anything where it's just like a super sparse set of things that obey it. There's another question. There, what was the question? Oh, the lattice is a statement about any consistent gauge theory coupled to quantum gravity. The claim is that charge is quantized, and so there are only a discrete set of charges that are allowed in the first place, okay? So you know that that's true in the standard model, right? Everything has electric charge that's a multiple of one third. You don't find square root of two, or pi, or anything like that. You just find these quantized charges. And there are arguments that that should be true in any theory of quantum gravity. So the lattice is not really an assumption, I think. That's just sort of built into the structure of how gauge theories should work. Okay, so in the last 15 minutes or so, I just want to briefly say a few things, which I'm not going to have time to fully explain, but I just want to tell you a few other things about kind of the status of this research direction. And then I want to briefly tell you why I think that this kind of argument, if we could make it more rigorous, could actually tell us something about the photon being massless. Okay, but let me first try to give you a better motivation for this. Why should these stronger statements be true? The first reason that I can give you, the minimal weak gravity conjecture is internally inconsistent. And again, this was shown in the paper by Ben Heidenreich, me and Tom Rudilius in 2015, where we formulated the lattice weak gravity conjecture. The reason we formulated lattice was not the guess that I motivated it with earlier. It was this argument that we could take a theory that obeys the weak gravity conjecture. So it has some gauge field, it has some particle that obeys the bound that's mass is less than its charge. We can compactify on a circle and we get a new theory with a bigger gauge group. Because we get a new gauge charge that comes from the collucian charge around the circle. And one way of framing the original weak gravity conjecture was that charged black holes should be able to shed their charge by emitting a charged particle. If you take a theory where that's true, compactify on a circle, you have a new family of charged black holes, which can have charge under the original gauge group, but also charge under the collucian gauge group. And it turns out that in general, there can be some combinations of these charges for which a black hole with those charges cannot shed its charge. Even though there are particles that obey the weak gravity bound for the original U1, there are particles that obey the weak gravity bound for the collucian U1. It doesn't mean that you obey the bound for all possible combinations of the two. So fully explaining that would take more time than I have, but it's actually not that difficult to understand. This also depends on the formulation of what the weak gravity conjecture means for these theories of multiple U1s that was spelled out by Cliff Chung and Grant Revin in a nice paper in 2014. But the tower lattice, sub-lattice formulations are consistent. If you have infinitely many particles available and you compactify, you also have all the cluts of client modes of all those different particles. And it turns out you have enough options that any set of charges you give a black hole, you can find some particle that it can radiate to discharge. The second thing is sub-lattice, the sub-lattice WGC is true in perturbative string theory. This was shown in my paper with an Ivan Reck and Tom Ridilis in 2016 and also in another paper by Miguel Montero, Gary Xu and Pablo Soler at the same time. So you can just directly prove this in perturbative string theory using something called modular invariance. And more recently, there's been further evidence that sub-lattice WGC is true in certain corners of F theory where you have a weakly coupled gauge group. This has been shown by Lee, Lerch and Weigand in a series of papers over the last year or so. And again, the argument relies heavily on a different version of modular invariance. So I can't tell you an argument that follows from general principles that proves that this is true, but I can tell you it's true in many kind of well understood controlled corners of quantum gravity. So what I spent a lot of the first two lectures trying to convince you of was there's statements that we have good reason to believe are true, like there are no global symmetries in quantum gravity or like the minimal version of the weak gravity conjecture is true. But these statements are pretty useless for phenomenology because they're just too weak. They only tell you things about, say, tiny corrections that are related to black holes. What we really want are statements that get us closer to particle physics. Things that say that light particles have to exist with certain properties. And these statements are not necessarily there yet. Sublattice doesn't actually tell you how sparse the sublattice is, but what I can tell you is in the examples where we can check it, it's never very sparse. And so there are always particles with charge one or charge two or charge three that obey the bound. And if you have a really weak gauge coupling, those particles have to be really light. So we're moving in the direction of finding sharper statements that are more useful. We're not necessarily all the way there yet. Now in the last, let's see, I have about 10 minutes, right? Because we started a little bit late. Okay. In the last 10 minutes, I want to sort of very briefly sketch one application of these ideas which is to the question of the photon mass. There are also some very interesting applications to inflation. So if I had another hour, I would tell you more about that. But let me talk about the photon mass. Does anybody in this room really believe the photon has a non-zero mass? You're going to volunteer? Why not? Good. Anybody have a good answer to why not? Most of you did not say you believe the photon has a mass. So if you believe the photon doesn't have a mass, why do you believe it doesn't have a mass? What's your most convincing reason that you know the photon doesn't have a mass? What was that? Gauge invariance. Gauge invariance. Why do you believe that physics should be gauge invariant? Because it describes the real world. Okay. That's a good answer. The standard model describes the real world really well. The standard model has a gauge invariance. That gauge invariance requires the photon to be massless. That's a reasonable answer. Describe the real world. But the question then is, could we also describe the real world with a different theory that didn't have gauge invariance and had a photon mass? What would go wrong? As opposed to we tried to describe the real world with a theory where we just say the hypercharged gauge boson has a tiny mass term that we just added to the Lagrangian by hand. Can anybody tell me something bad that happens? The electric force would have a range cut off. Good. That's true. And so you can measure that, right? You can say how far do we know the electric force has to extend. But what that'll get you is just a bound. That'll say the photon mass has to be smaller than some number. But it doesn't tell you it has to be exactly zero unless you can really keep testing at kind of infinitely long distance, right? So the claim is a tiny photon mass allowed by data. And how tiny, well, there are various bounds that have different assumptions that go into them. But let's say below 10 to the minus 18 electron volts is probably safe, yeah. Yeah, the photon would have a rest frame then, that's true. And it's weird, I agree, but is it obviously false is the question? No, it's not, right? So the speed of light as a property of space time, there's the speed that appears in special relativity that relates space and time, that's a fixed constant. But actual photons would not move at that speed, they would go slower. That's true, and in fact, that's where some of the strongest bounds come from, which come from looking at fast radio bursts and looking for differences in the arrival time of photons of different wavelengths. So yeah, you can get very strong bounds, but the point is they're always bound. They don't tell you it's exactly zero, they just tell you it's really small. And this is just a property of effective field theory. In effective field theory, you can add a photon mass, and nothing bad happens. Okay, let me contrast this with other examples you might know. If we had a gluon mass, we could calculate scattering amplitudes. Two gluons in, two gluons out with longitudinal polarization, and we would get an amplitude that grows with energy, right, constant, something like this. And so you get some cutoff at photon energies of order. The gluon mass divided by the coupling you put in, where above that you can't make any predictions because your theory is strongly coupled. If we have a graviton mass, similar story. We can scatter longitudinally polarized gravitons, see the scalar polarization of a massive graviton. We get an amplitude, it goes like one over m-plonk squared energy to the check if I wrote this down so I don't have to think through it on the spot. Energy to the 10th divided by m-grabbiton to the eighth. And so there's a cutoff at a maximum energy, which is some funny power. Graviton to the fourth fifths, and plonk to the one fifth. Some of you might have studied some massive gravity theories where you play a lot of games and try to make this better behaved, but even then there's always some kind of bound. That goes to zero when you send the mass to zero. The massive gluons and massive gravitons have the property that if you give them a small mass your theory breaks down at some low energy. And it breaks down because you scatter the longitudinal modes and there are these interactions that make the amplitudes blow up. But with photons, the photon doesn't couple to itself unlike the gluon and the graviton. You can add the longitudinal mode to your theory but it doesn't actually show up in the amplitudes. And you don't get any new cutoff associated with it. Is there a question? That's exactly right. And that's part of why it's so hard to constrain the photon mass because you could say, well the photon has three degrees of freedom instead of two. So let me just get some hot system and just count how many degrees of freedom there are. Study black body radiation or something like that. And the problem is that that last longitudinal mode is really hard to actually activate. So it doesn't thermalize and you can't easily count it. So it's really hard to set a bound in that way. There's a nice discussion of this in Sidney Coleman's quantum field theory lectures that were recently published as a book. He says, you know, the way to figure out if the photon has a mass is just turn on your oven and look at the black body radiation. And then he says, this is garbage because that oven will take 30 trillion years to reach thermal equilibrium if the photon mass is around the kind of balance that the experiment puts on it. Okay. So the claim in the last minute or two is just that photon masses are perfectly healthy in effective field theory. But what I want to suggest to you is that photon masses may not be healthy in quantum gravity. And I'm not going to have time to do this argument justice. But the point is that a Lagrangian for a photon at a mass term is equivalent to a different Lagrangian. And this different Lagrangian is what's called a BF theory where I add a two-form gauge field B, a mu nu with field strength H mu nu rho. So you can think of the photon as getting a mass from eating some scalar degree of freedom in four dimensions that scalar can be dualized to a two-form and you can rewrite your theory in this way. But now this is a theory where we have two gauge fields. We have our photon and we have this new field B mu nu. And it turns out the only way to make the photon mass small is to either make the coupling of the usual gauge field small or to make the coupling, which is a dimensionful quantity of the two-form gauge field small. The reason is that the term that couples them together is quantized. The theory only makes sense if this number is an integer. And so we can't make that number small. Our only option is to make this small or to make that small. But either of those is making a gauge coupling small. It's either making an ordinary gauge coupling small or it's making a two-form gauge coupling small. And the claim is that when you make a gauge coupling small in a degree of quantum gravity, your theory should break down because that's always this limit where you're restoring a global symmetry. And then the weak gravity conjecture or the swampland distance conjecture tells you that when you try to take that limit, your effective field theory's not gonna work anymore. Okay, so there are details to the argument that I don't have time to fill in, but that's the gist of the argument. And my claim is that in the context of every massive photon theory we know of in quantum gravity, either it gets amassed from the ordinary Higgs mechanism or it gets amassed in this way. And there's a bound on how small you can make that mass. Okay, so I guess I've gone slightly over time. I don't know if there's time to take a question or two. Thank you all for the interview.