 In the last hour of these lectures, I want to try to talk about a couple of connections of the weak gravity conjecture to applications to inflation into the photon mass. Before I do that, let me briefly mention one more topic that's a little bit more on the formal side, which I'm not going to try to explain in any detail, but I think it's really an interesting development, and I want to just mention it, which is a connection that's been emerging over the last few years between the weak gravity conjecture and the idea of cosmic censorship. And this is discussed in four papers over the last few years by different combinations of these four authors, Christopher Horowitz, Santos, and Wey. And let me just briefly sketch what they've found. They claim that Einstein-Maxwell theory and ADS4, so the theory of just photons and gravitons, violates cosmic censorship. And the way they find these solutions that violate cosmic censorship is they turn on a boundary gauge field, which has some radial profile that falls off faster than 1 over r. This is localized in some region of space on the boundary, but it's time dependent, and they gradually increase the amplitude of this boundary gauge field and see how the bulk solution responds. And what they find is that above some critical value, a max, they can't find any well-behaved static solutions. And when they search for this time dependent solution, they find that curvatures grow arbitrarily large. So these are numerical calculations they're doing to solve general relativity. And so they find that cosmic censorship is violated in a four-dimensional theory. And not only that, it can be violated in a big region of space, which is interesting, because there are older violations of cosmic censorship. For instance, in the Gregory LaFlamme instability of black strings and five dimensions. But those are always things sort of very localized, whereas this can happen over a big region. But then they study a different theory where they add charged scalars. And they find that scalars can have instability. So first, they compute quasi-normal modes, and they find the imaginary part of the frequency can become positive. So the modes are growing exponentially in time. And then they study the time dependent solutions, and they find that these scalars start to turn on. And basically, they screen the deep interior of the bulk against the effects of this boundary gauge field. But only if they're sufficiently light. So they've quantitatively tried to find what the sufficiently light mean. What they find is the condition for this to happen is equivalent to the weak gravity conjecture. In three examples, first the case with just the U1, then a case with the U1 with the dilaton, and finally in a U1 times U1. Example where they check the convex hole conjecture. And in the case with the dilaton, what they're finding again is that these scalars need to be super-extremal. So it's the thing I was referring to as the weak gravity conjecture rather than the alternative conjecture, which would be a repulsive force conjecture. So I find this really interesting, because it's taking another long-standing conjecture about gravitational theories, the cosmic censorship conjecture, and showing that there's some close relationship between that and the weak gravity conjecture. And I don't think it's fully understood why this is true. But to me, it's very interesting that not only is it true in the simplest example, but that when they start exploring these more complicated examples with multiple fields, they're still finding that exactly the right condition to avoid these singularities in the bulk is the thing that we've been calling the weak gravity conjecture. So I think this is a really interesting thing to keep an eye on. It's maybe giving us some hints about why the weak gravity conjecture might be true, or at least connecting it to other interesting gravitational physics. So that's all I want to say about that topic. And now I want to talk for a while about inflation. And the reason for talking about inflation is that it's one possible application of the weak gravity conjecture. It's also really the reason, I think, why this topic was revived. So you probably noticed I gave you some references from back around 2005 and 2006. That's when Vafa first started discussing this idea of the Swamplan program, the paper that introduced the weak gravity conjecture by Irkani Hamid and collaborators was from 2006. And then I didn't give you any other references after 2006 until something like 2014. So there was kind of a gap when people were not talking much about these ideas. And then about five years ago, there was a big revival, and there was a lot of interest in these ideas. And the reason why a lot of people got interested, certainly the reason why I started working on these topics, was because of the bicep experiment. So you might remember the news in 2014. I think it was March 2014. The bicep two collaboration made a big announcement that they had seen primordial gravitational waves in the CMB. They did not. It turned out they were seeing effects of dust in the galaxy that was producing a signal that mimicked the effects of primordial gravitational waves. Nonetheless, there was a brief interval where it appeared that primordial gravitational waves had been seen, and so the natural thing to do was to think about what the implications were. And I still think that's an important thing to do, even though the signal turned out not to be real. There's still a chance that in the coming years we will get such a signal, especially because the experiments are expected to improve by more than an order of magnitude in the coming years. So the question is, what would it mean such a signal? And some answers to this question were given already 20 years ago, paper by Leith. So let me tell you a bit about the physics of inflation. We've seen scalar perturbations, the temperature perturbations in the CMB, which are the things that seeded the formation of galaxies in our universe. And one way to talk about what these things mean is to think of them as being part of the metric. So we could sort of parameterize a general metric in some way. The metric contains both scalar perturbations, i, or this field psi, tensor perturbations. But you probably know that in flat space, in Kovsky space, Einstein gravity, the graviton just has two propagating modes, which are tensor modes. In Kovsky, we're in decider, there are only tensor modes. And one way to think about these other modes is they arise when you have spacetimes with less symmetry. And you can think of the scalar perturbations as being like goldstone modes of broken time translation symmetry that have been eaten by the metric. So there's sort of a gravitational version of the Higgs mechanism. When we break time translation, the graviton sort of eats a scalar mode. Another way to say that is you can work in different gauges. And some of them, there's no scalar perturbation in the metric. There are just perturbations in the density of stuff in space from point to point. But in some gauges, you can kind of absorb fluctuations in the density into these scalar modes, which you can think of kind of like the Newtonian potential of in the non-relativistic limit. Not really. Yeah, because we're talking here about something that's breaking Lorentz invariance. So this doesn't look like a Lorentz invariant master. But yeah, it's sort of specific to this Lorentz breaking phenomenon where we're treating time differently from space. So why am I telling you this? It's because the interesting quantity that you could imagine measuring is the ratio of how much power is in tensor perturbations to how much power is in scalar perturbations. The scalars have already been measured. Quantity, like p subscript s, the power in the scalars is measured to be something like a few times 10 to the minus 9. I would have to tell you exactly how I'm normalizing things to completely make sense of this. But if you ask, what does this tell us about inflation? This is related to the Hubble scale, the expansion rate of the universe during inflation, according to a formula that looks something like this. Another way to write that, if I have a slow rolling inflaton field, so the usual kinds of inflation models that people discuss is Hubble scale to the fourth divided by phi dot squared. And another way to write that is h squared over m Planck squared epsilon, where epsilon is the first slow roll parameter. So what's going on with all of these formulas is that you see there's some quantity in the denominator that depends on a time derivative. So in the limit, when I go to pure decider space and h isn't changing, this would go to infinity. And that might look pathological at first. But what it really means is just that the way that we're absorbing those perturbations into the metric depended on breaking time translation symmetry. And so if we're in the limit where there's no such breaking, it doesn't make sense to talk about the scalar perturbations being part of the metric. And so this is not a well-defined quantity. So that's one kind of source of intuition about why these things that have time derivatives show up in the denominators. Another source of intuition is that the scalar perturbation and really given a name like zeta and one way to write it in a gauge invariant way, something like that. So that's showing you that it's a combination of something that appears in the metric and something that depends on density perturbations. But it turns out that with some assumptions that I'm not going to try to explain too carefully, zeta is basically the same as I think of delta n, which is the fractional, well not fractional, but the difference in the number of e-folds of inflation that have elapsed from point to point. So if you have more density in some places than others, if your inflaton has moved a little bit further in some places than others, inflation will end in some of those places earlier than in other places. And that time difference measured in units of e-folds, how rapidly the universe is expanding by a factor of e, is what the scalar perturbations are measuring. And so again, in the limit where you have just kind of complete symmetry and nothing is changing in time, smaller phi dot means longer time delays. And again, this is sort of becoming ill-defined. So those are just kind of some intuition about why this denominator looks the way that it does. By contrast, the tensor perturbations, the gravitational waves, have a power that just goes like h squared over and plonk squared. And so both of these have the h squared over and plonk squared in front. That's basically just coming from the fact that the h is just the fact that you're in decider space. It has a characteristic couple scale and everything is kind of fluctuating by that amount. All the fields have Hubble-sized fluctuations. The implonk is just kind of from normalizing everything through the graviton kinetic term. And the difference between tensors and scalars is this factor of epsilon. Ratio of tensor power to scalar power is proportional to epsilon. Where epsilon, basically how fast the Hubble scale during inflation was changing in a Hubble time. So first, it's not surprising that we've seen scalar power before we saw tensor power. Tensors power is smaller in slow roll inflation by this small number epsilon. And we don't know how big epsilon is because we haven't measured this yet. So what we know about inflation is the energy scale during inflation went like h squared and plonk squared. And this is like epsilon times the scalar power, which we've measured some number of order 10 to the minus 9 in plonk units. But until we know what epsilon is, we don't really know what the energy scale during inflation was. So why was bicep exciting? Well, first, if you measured epsilon, if you measured the tensor to scalar ratio, you know how big epsilon is, and then you know the energy scale during inflation. So if we saw tensor modes, the first thing we learn is the energy scale during inflation is something like the gut scale to the fourth power. The Hubble scale during inflation is something like 10 to the 14 GeV. You can see V goes like, well, V to the 1 fourth, the characteristic energy scale goes like epsilon to the 1 fourth. And so the statement that it's around the gut scale would have been true if the bicep signal had really been hormonal gravitational waves, but it'll still be true even if we see a signal in order of magnitude smaller because 10 to the 1 fourth is not a very big number. This changes a little faster because it goes like the square root of epsilon. So that's one thing you would have learned. Inflation happened at really high energies if you saw such a signal. But the other thing you would have learned is maybe even more interesting. And the way to see this is to use the fact that epsilon depends on phi dot. So a bigger epsilon means a relatively big value of H dot, which means a relatively big value of phi dot, which means the inflaton changed by a relatively big amount during inflation. And in fact, the formula for that can use the number of E-folds as our time variable, but d phi dn by dt times one over the Hubble scale. See from this, d phi dt is related to epsilon. So this ends up going like the square root of epsilon times n-plank times the number of E-folds of inflation. And so what you would learn that during inflation, at least during the part of inflation from the time that the modes that are currently crossing our horizon were produced until the end of inflation, the inflaton changed by an amount that's in order one number times n-plank and something scaling like the square root of, I guess I didn't use the name R yet, the tensor to scalar ratio, I'll call it R, normalized by something like 0.01. So this is what, this is what Leith pointed out 20 years ago, so this referred to as his bound. And what it means is if you see primordial gravitational waves, you know that some field during inflation changed by an amount of order n-plank. And that is why there was this revival of thinking about the Swampland ideas at the time that Bysub claims that they saw the signal because if a field is changing my amounts of order n-plank, then these statements about quantum gravity start to really matter. So in particular, the Swampland distance conjecture that I mentioned earlier said that if you go amounts that are large compared to n-plank in field space, you expect towers of modes to start becoming light. And so you should worry about whether your models are going to be viable if you're thinking about a model of inflation where a field moves a Planckian distance. So if you tried to build a model where a field moved a Planckian distance, the first thing you see is you can't just write down a generic effective field theory because if I start writing down v of phi is one half m squared phi squared plus lambda phi to the four plus something over m-plank squared phi to the sixth plus something over m-plank to the fourth phi to the eighth and so on. If all my coefficients are order one numbers, this cannot really be an inflation model because I'm just going to have when my field starts traversing distances of order m-plank, all of these different terms are going to be important and my potential is not going to be a nice smooth thing. It's going to be kind of fluctuating on scales of m-plank and I don't have a flat region that I can use to inflate. And so what had been appreciated for a long time is if you want to build a model of inflation that can produce gravitational waves, you need some kind of underlying symmetry that explains why your potential is flat enough over such a big field range that it's viable for inflation. And this led to various models. There's an idea called natural inflation due to Freeman, Frieze and Alento in 1990 where they said, suppose the inflaton is an axion. So it's a periodic field. Phi is equivalent to five plus two pi F. Then your potential has to be built out of periodic things and we kind of know how this works from the QCD axion. You might expect some potential that depends on some kind of characteristic of the refinement scale to the fourth power times the cosine of your field and then maybe some corrections that go like cosine of twice your field. And as long as the coefficients decrease fast enough, this can give you a nice smooth kind of periodic potential that can be used to have a model of inflation. And indeed it gives you a model of inflation if this period is big in plonk units, which is what you would have wanted to see a gravitational wave signal. So from the viewpoint of effective field theory, there's nothing really wrong with these super plonky and field ranges. You need a symmetry to protect things, but if you can tell me that there is some underlying symmetry like this kind of approximate shift symmetry that comes from the period of the axion, then everything can be under control. People started looking for these kinds of models in string theory. And the first thing to say is there's a really nice way to get axion-like fields that have this periodic structure out of higher-dimensional gauge theories by integrating, if I have a p-formed gauge field integrated over a p-cycle, I get a periodic scalar. So maybe the easiest one to explain is if I just have a circle as my extra dimension and I integrate along the coordinate of that circle, the corresponding component of my gauge field, then if I do a gauge transformation, where I add to this the derivative of something where this gauge function alpha winds around the circle, then this gauge transformation will just shift my scalar field by two pi, okay? And a similar thing happens for these higher-dimensional cycles. And so there's a natural way to get periodic scalar variables from higher-dimensional gauge theories. And this has been appreciated for a long time, so it's been known that string theory has axions. Witton pointed this out already in 1984. There were papers by Choi and Kim, and by Bar 85. So this kind of thing was already appreciated for people thinking about axions and the strong CP problem all the way back from the very early days of super-string theory. And in the context of inflation, this was emphasized, well, there were various models over the years, but there was a simple model that used just the five-dimensional circle idea that got a lot of attention from Arkani, Hamad, Chung, Criminelli, and Randall in 2003, okay? So all the ingredients are there in higher-dimensional gauge theories to give you periodic scalars that are good potential candidates for inflation. And so from the point of view of just building models and doing effective field theory, these are fantastic models, there's nothing wrong with them. However, when people tried to find models like this in string theory, it proved to be very difficult, and the difficult part was getting the period to be big in plonk units. One paper pointing that out appeared already in 2003, and already in the first paper about the weak gravity conjecture, Hani Hamad, Matul, Nicholas, and Bafa, tried to give an explanation for why this might be true. And the explanation was if we start with a higher-dimensional gauge field that we're getting an axion field from in this way, we can ask what gives rise to the mass of that axion? And the answer is that the mass is coming from charged particles in the higher-dimensional theory that run in loops, the particles in their collude secline modes run in loops and correct the mass of the lower-dimensional axion field. And there's a nice way to think about this calculation, at least in the limit when those particles are heavy, which is that the potential comes from wrapping a world line of these massive particles around the circle that you can back to fight on. Or in the string theory context, where we have some higher-dimensional cycle, wrapping a brain, a Euclidean brain of the appropriate dimension around the cycle. And because of that, the mass of the potential for the axion depends on the spectrum of particles that are charged under the gauge field that we started with. But that's where it becomes interesting to connect this to the weak gravity conjecture because the weak gravity conjecture is telling us something about the spectrum of charged particles in the higher-dimensional gauge theory. Okay? So let me sketch how that argument goes. In the case where I just compactify from 5D to 4D, I get some expression that looks kind of like this. If I have a very massive particle in the higher-dimensional theory, its contribution is exponentially suppressed. It gives me a cosine of the axion field. I can think of this exponent as the action for the Euclidean world line wrapped on the circle. And so then there'll be higher-order terms that are more exponentially suppressed. If I wrap the particle n times, I get something that has an e to the minus n times this instanton action. But what they observed is the weak gravity conjecture in five dimensions tells me that there's some particle whose mass squared is less than the gauge coupling in five dimensions times charge squared times the 5D Planck scale cubed. I've mostly been writing formulas like this in four dimensions, so I've been writing it squared, but it generalizes to the D minus 2 power in D dimensions. But the period of the axion, F, you can work out just by dimensionally reducing the kinetic term of the gauge field in five dimensions, the kinetic term for theta in four dimensions. And you find that F depends inversely on the radius of your compactification and on E5 squared. And your 4D Planck scale squared is your 5D Planck scale cubed times the radius times 2 pi. So what they observed was, if I just multiply this inequality on both sides by R squared, I learned that M squared R squared is less than E5 squared R, which is like one over F squared times Q squared times M5 cubed R, 4D Planck scale squared. If I want F bigger than M Planck, I find MR is less than one, and the instanton action is not large, and I can't trust the leading answer anymore. So this argument was given in the original paper on the weak gravity conjecture. And what it shows you is that you should at least be careful about building models of inflation that come from higher dimensional gauge fields where you're trying to get a super plonky and field range. Because they're generically going to be some set of corrections in these theories that are hard to control. Now that doesn't mean that it's impossible. And in particular, in this 5D example, Dela Fuente, Saraswat and Sundrum, 2014, pointed out that if you just do the loop calculation, the nth order term in the limit of very small masses and less than one over R, where these exponentials are not suppressed at all, goes like one over end of the fifth. And that's already enough to give you a smooth potential. So they said you don't actually need to suppress all these instantons to have a nice smooth potential that you can inflate with. You just need some really light particles. So the story's more complicated than that. There's been a lot of back and forth over the last five years between people trying to cook up clean Swampland arguments against inflation models and other people trying to build explicit models that get around the balance. And I'm not going to go into that in any more detail because in the last 15 minutes or so, I wanna change topics and tell you about a different application. But let me just say that I think the question is still open. Whether these Swampland ideas can exclude large field inflation or not, there are various constructions and string theory like axiomynodermy that claim to build things that have very big field ranges, but then you have to start worrying about back reaction problems. And it's still a very active discussion of what the constraints are. It's also true that if you wanna fit the data, if we see a tensor to scalar ratio anytime soon, it'll mean that the Inflaton field went a distance that's big in Planck units, but it doesn't mean very big. It might be four times in Planck, five times in Planck, not parametrically big, not a thousand in Planck. And so it's right around the border always. And these arguments always have kind of squiggles in them. And so it's not so clear that they're good enough to tell you that you can rule out theories where an Inflaton field goes four or five times in Planck. But I think if nothing else, the virtue of these kinds of arguments is they're making people look at the string constructions of inflation models with more scrutiny and be more careful about back reaction. And I think it's leading to progress in thinking about what realistic models would actually look like. In the last 15 minutes, I want to change topics completely and tell you about another application, Wompland Ideas. And that is to the question of whether the photon, if you have the standard model photon or maybe some dark photon in the hidden sector, you have a mass. So why is this an interesting question? If we ask about non-abelian gauge fields, if I give gluons a mass, because they're self-interacting, I can then compute some scattering amplitudes. I can scatter longitudinally polarized massive spin one particles. What I find is an amplitude that goes like the gauge coupling squared times the energy squared divided by the mass of the particle squared. And then I can apply some partial wave unitarity estimate that tells me M shouldn't be too big. And what I learned is there's a UV cut off on my theory at energies below. The theory breaks down at high energies, but the scale at which that breakdown appears is no larger than something like square root of eight pi M over G. And that's precisely the kind of argument that was important for building the LHC. We knew that the W and Z bosons had mass. We knew that if we scattered them in the standard model without a Higgs boson, the amplitudes would blow up at some energy around a TEV. And so something interesting had to happen by that UV cut off energy. It turns out the interesting thing that happened is the Higgs. If you add the Higgs to this picture, the amplitudes no longer grow and everything is healthy. There's a similar story for graviton masses. If I give gravitons a mass and I scatter the scalar mode that they've eaten, I find that the amplitude grows like the energy to the 10th power divided by the mass to the eighth times the plonk squared. And I have a cut off below some energy scale somewhere in between the graviton mass and the plonk scale, but it's a lot closer to the scale of the graviton mass itself. So we can't just add masses to gluons and gravitons without new ingredients that fix the theory at these energies. Different thing about photons is that in effective field theory, photon masses are harmless. If I just add a mass to the photon and do nothing else, make no other changes to the theory, I could make changes that would be harmful. I could add an a squared squared. I could add a Richie scalar times a squared. I could do various things. But if I just add the mass and nothing else and then I compute scattering amplitudes, absolutely nothing goes wrong. There's no associated UV cut off. I get a perfectly healthy field theory. And so we always assume that the standard model photon is massless. But in field theory, there's no reason it has to be. I could just add a mass to it. And nothing goes wrong with my theory. There's no U1 gauge invariance, but we don't need gauge invariance. If we don't have massless, it's been one particles. And so for the standard model photon, the best argument I can give you for why you should believe the photon mass is zero is that experiment does the masses very small. One bound that I can tell you comes from fast radio bursts. And there are a lot of authors of these papers. So let me not write all the names, but paper by Wu, Zhang and various collaborators in 2016 and a paper by Bonetti Ellis and collaborators also in 2016. Give this argument. A fast radio burst produces radio waves. And radio waves have relatively low frequency. And if you want to look for a photon mass, it's best to look at low frequency photons because the dispersion relation has a bigger influence from the mass if the wavelength is very long. And so what they do is they just look at the arrival time of different frequencies of radio waves from the same burst and say if the photon had a mass, the lower frequencies would be moving slower because of that mass and they would arrive later. So if you look for radio waves coming from very far away and you know that they're arriving at about the same time, you can put a constraint on the mass and the constraint that they put is something like 10 to the minus 14 electron volts. There are claims of stronger constraints on photon masses in the literature. So this is not the strongest one, but this is kind of the easiest to explain. It's very kinematic. Others depend on magnetic fields and then you have to ask questions about would the theory that allows a photon mass allow vortices and there are all kinds of debates about exactly how strong the bounds are, but this one is a very solid bound because it doesn't depend on anything except kinematics. So if the standard model photon has a mass, the mass has to be ridiculously small compared to all the other scales in particle physics. And that's the best field theory argument I know for why you should believe the photon is massless. It's just that if it had a mass, I wouldn't know how to explain that number. It would be a small number that came out of nowhere. But still, if some experimentalist came to me tomorrow and said, I measured the photon mass to be 10 to the minus 19 electron volts to five sigma, I wouldn't have a good argument. I could tell them for why I believe they're wrong just based on field theory. It would be really weird. But I couldn't say that I know it contradicts anything that we know about quantum field theory. So, what did you say? You're saying that there are other bounds that are more precise? But not from FRBs from some other source or? Okay, you're asking if there are more precise bounds than this one, is that the question? Yeah, so there are various claims of stronger bounds. Both laboratory constraints and things based on like the magnetic field of Jupiter or the magnetic field of the galaxy. So yeah, if you look up review articles on this or if you look in the PDG, you'll find stronger claims. I think 10 to the minus 18 is sort of the strongest claim that is commonly accepted. But again, a lot of those others have more interpretational difficulties. So I don't, they're probably reliable but there's more physics involved in making sense of them. The reason I'm quoting this one is just that it depends only on kinematics. It depends only on the fact that if I have a mass, then things of longer wavelength move more slowly than things of shorter wavelength. And so I think this one is the one that's sort of unambiguously reliable and easy to understand and that's the only reason I quote this one. But yeah, there are stronger bounds that are quoted in the literature. Okay, so I just have a few minutes left but what I wanna jump to is why I think that effective field theory cannot tell us the photon is massless. But I wanna claim that Swampland ideas could tell us the photon is massless. And I don't think these arguments are completely rigorous yet but at the level that we sort of understand the ideas about the weak gravity conjecture and the Swampland distance conjecture, I think this is pretty convincing. So I wrote a paper on this last summer and the argument is the following. One way to give a photon a mass would be through the Higgs mechanism. But if you wanna obey a bound like this and give the photon a mass through the Higgs mechanism, whatever field is playing the role of the Higgs would have to have a super tiny charge compared to the electron charge. So it's, what do I wanna say? What we're usually thinking about in the context of talking about photon masses is not giving them a mass through the Higgs mechanism. It's giving them a mass through what are called Stuckelberg masses. And a Stuckelberg mass is where I take the photon mass. Well, first there's sort of the Proca mass is just writing down a mass term like this. But the Stuckelberg mass sort of upgrades this to something that has a fictitious gauge invariance. So I introduce a real scalar theta which shifts under a gauge transformation to make the gauge invariant. And this is a sort of a fictitious gauge invariant. But if you look at examples of Stuckelberg masses that come from string theory, it seems like there always is some underlying actual gauge invariance. There's another interesting thing that happens in the string theory examples. So one way to distinguish this from a mass that comes from the Higgs is to say in the Higgs mechanism you get not only this real scalar theta, you get the other real scalar that's the actual Higgs boson where if you change the value of that scalar you're changing the scale of this mass in front. So your first instinct might be to say that the difference between the Stuckelberg mechanism and the Higgs mechanism is the existence of that sort of radial mode that changes the mass. But then if you look at examples of what people call Stuckelberg masses in string theory, they usually have the radial mode anyway. And the reason they have the radial mode is that supersymmetry doesn't let you have just a single real scalar, it forces you to have the additional radial mode too. So what I wanna claim is the right way to distinguish Higgs from Stuckelberg. Certainly in the context of string theory examples that I know, but I would say more generally in the quantum gravity context, distinction between Stuckelberg and Higgs is whether there's a point in field space where I can send the photon mass to zero which is at finite or infinite distance. So in the Higgs context, I have a kinetic term for my Higgs that looks ordinary and it's showing up in the mass term in some form like this. So if I send phi to the origin, that's not the minimum of the potential but it's still a point in field space and the kinetic term is perfectly healthy there. But if you look at examples in string theory of Stuckelberg masses, they usually look more like this. They have a kinetic term that is not canonical but blows up when phi goes to zero. And so the point where the mass goes to zero is infinitely far from any finite. And this brings us back to the Swampland distance conjecture of Aguri and Vafa where they say that if you go infinitely far in the space of some scalar field, there's always going to be some tower of modes that become light. And in fact, that's true if you look at any example of a Stuckelberg mass for a U1 gauge field in string theory, any example I'm familiar with at least. If you can't just interpret it as a Higgs mechanism, if it's really distinct, it's this kind of case where the point where the mass goes to zero is not really part of your field space. It's infinitely far away. And if you try to go there, you find a tower of modes that becomes light. And I'm really out of time but let me just briefly give you the punch line. One way to make this claim even sharper, I was just talking about the Aguri Vafa statement but we can also relate this to the weak gravity conjecture by saying that the Stuckelberg masses can always be rewritten in the form of what's called a BF theory where B is a two-form gauge field. The objects charged under that two-form gauge field are strings. The relationship to this picture is the scalar field theta can be dualized to the two-form. If I have a string that's charged under B, as I wind around it, theta winds around from zero to two pi. But now the weak gravity conjecture can be applied to either the original gauge field or the two-form gauge field. And it says E goes to zero or F goes to zero. Both cause my EFT to break down. So I'm restoring some sort of global symmetry in either limit. But the photon mass is just E times F. And so the only way to make it small is to make either that small and run into problems with the weak gravity conjecture for the two-form or to make that small and run into problems with the weak gravity conjecture for the original gauge field. So that's the summary. You can find more explanation and attempts to connect it to phenomenology in the paper I wrote last summer. But I think it's kind of fun that these Swampland ideas can tell us about something about our universe as basic as the question of why is the photon massless. Thanks.