 Thanks for the invitation to be here, and please do ask questions if you have any. Can everyone hear me? Okay. Is it Okay, is that better? Can you hear me now? Is that better? I'm just not speaking loudly enough, I guess. Okay, is that louder? Seems to be. Yeah. Okay, so I'm going to talk to you about the weak gravity conjecture. As we'll see this is really sort of a family of related conjectures, and there's some broader context to this which is what's sometimes called the Swampland program. The name the Swampland program was given to the set of ideas by Cameron Vafa back in 2005, and the general idea of the Swampland program is that we know there are many infinite families in fact of consistent quantum field theories, and it also appears that there are likely to be many consistent theories of quantum gravity. But the idea of the Swampland is that the set of quantum gravity theories is actually in some sense much smaller than the sense of quantum field theories that make sense. So to sort of sketch the kind of thing we have in mind, there's some large space of consistent effective quantum field theories that can be perturbedively coupled to gravity. They have a stress energy tensor. But the idea is that not all of these actually can be completed into full quantum theories of gravity, and in fact, the quantum gravity theories that we know are rather special. So we know some examples of continuous modulite spaces of theories of quantum gravity, various families of these theories, and we know isolated vacua. For instance, certain ADS vacua seem to be isolated points in the space. They have no continuous deformation. And so the set of these quantum gravity theories is called the landscape. And the idea of the Swampland conjecture is we might be able to identify some sort of boundaries in the space of effective field theories where we can say that if a theory is on the wrong side of some sort of inequality, it has no hope of being in the landscape. It just can't be completed into a consistent theory of quantum gravity. And so there might be some regions in this space that we can rule out in various ways, and these are called the Swampland. So effective field theories that can't be completed into consistent theories of quantum gravity. Okay. So why might we care about this? One reason is just to better understand quantum gravity itself, if we can identify ways that we can exclude effective field theories from being consistent theories of quantum gravity, we'll get a better understanding of quantum gravity. But another reason, and the reason that I got into this, is that you might hope to connect to the real world. So you might hope that there are theories that, as theories of particle physics, as phenomenological theories look perfectly healthy, but that we know can't really be the answer for our universe because they're located in the Swampland, and so they can't be UV completed. Okay, so the goal of this program is to identify better, sharper ways to delineate what could be in the landscape from what could be in the Swampland. And there are a few different ways that we approach this question. One is to try to start with some set of general principles, maybe holography, maybe aspects of black hole physics, and argue that certain kinds of effective field theories are inconsistent with these principles, and so could not be completed into theories of quantum gravity. Another approach is to study explicit examples of theories of quantum gravity, things that we can actually construct in string theory, and notice if they have common features and try to generalize. If we notice that certain things just never happen in examples that we have within string theory, you might guess that those sorts of phenomena lie in the Swampland. The danger of that approach is you might be looking under the lamppost. We might only know the string vacua that are easiest to construct, and so if you're too hasty, you might claim that certain things are in the Swampland when actually they're not, just because we haven't been clever enough yet to build the right theories of quantum gravity that have those properties. And so what's been happening in practice over the last several years and work on this this program is a lot of back-and-forth between these two directions. There are attempts to derive things from general principles. There are studies of explicit examples and guesses about what might characterize the landscape versus the Swampland, and then those feedback, there are attempts to prove those guesses or to construct new examples that might violate them. And so in this way the kinds of conjectures that we're making are getting sharper and crisper, and there's more evidence for them from both of these directions as time goes on. Okay, so as you may have noticed, there's been a lot of activity in the area of the Swampland over the last five or so years, and it's really much more that I'm going to have time to tell you about in this set of lectures. So let me refer you to a couple of examples of places where you might read more. There's a set of TASI lecture notes that came out in 2017 by Brennan, Karda, and Vafa. That's one place where you might read more about the Swampland program in general. There's also a very recent review article by Palti that just came out a week or two ago. There's a 200 page long review of many different aspects of these topics. So if you want to learn more than what I'm going to have time to tell you about, I would recommend that you read these. In particular, they have a lot of discussion of some recent ideas that have gotten a lot of attention where Ugrie and Vafa and various of others have made some pretty bold conjectures that in fact there are no desider vacua in the landscape that all desider vacua are in the Swampland. That's a claim that I'm going to say very little about in this set of lectures. So if you want to learn more about it, I would recommend that you read some of these lecture notes. In this set of lectures, I'm going to try to be a little bit more focused and talk about topics that are pretty closely related to the weak gravity conjecture and not all of the other aspects of the Swampland program. So let me summarize some of the things I'm going to tell you about. There's a claim that theories of quantum gravity cannot contain global symmetries. I'll try to explain some reasons why we believe that's likely to be true. There's a statement that in some ways sharpens that claim, makes it more precise, which is the weak gravity conjecture. I'll tell you what that conjecture is. I'll talk about various generalizations of it. For instance, the theories with multiple gauge fields. I'll give you some examples of consistent string theories and how they satisfy the conjecture. I'll talk a little bit about some potential applications. One set of potential applications is the theories of inflation in the early universe. Another is to the question of whether the standard model photon or photon-like particles in hidden sectors could have masses and along the way, I'm going to connect to a different conjecture that goes by the name of the Swampland distance conjecture, which was formulated by Aguri and Bafa in 2006. Among all of the other set of Swampland ideas that have come out over the last several years, I think the Swampland distance conjecture is the one that's most closely related to the weak gravity conjecture. In fact, it sort of leads us to stronger formulations of the weak gravity conjecture that seem to be true in examples, and so I'll discuss that. Okay, so that's kind of the big picture outline of where we're headed. There are several other references. I'll try to give you references as I go along, but I'll say that a lot of what I've been, what I'll be talking about is based on several papers I've written over the last few years with Ben Haydnreich, who is a professor at UMass Amherst. And Tom Rudelius, who is a postdoc at the Institute for Advanced Study in Princeton. So if you look up our three names, you'll find several papers that we've written that I'm going to be drawing on heavily in the lectures. So that's the outline. And now the first topic that I want to delve into in more detail is this claim that there are no global symmetries in theories of quantum gravity. So as you know in quantum field theory, global symmetries occur very commonly, and they're very useful, and there are several different kinds of global symmetries that are interesting. Maybe the most familiar kind are simple, continuous symmetries that act on local fields, and they have conserved currents that are vectors. So for instance, the familiar SUN left times SUN right flavor symmetry that shows up in QCD-like theories, but in fact there are other kinds of global symmetries. There could be discrete symmetries as well as continuous symmetries, and both of these could be generalized to some higher form symmetries, p-form symmetries, that act on extended objects, and these kinds of symmetries are very common in gauge theories. For example, if I give you a quantum field theory with just a free U1 gauge field, just a single photon, this theory actually contains two conserved currents, which are nothing other than F mu nu and its dual F tilde mu nu, thinking about the four-dimensional case. And these are conserved currents which act on Wilson and Tift loops. So the charged objects in the theory are not localized at a point, they're extended along a line, and there are generalizations of such p-form symmetries that act on surfaces of higher dimension. These kinds of global symmetries have gotten relatively little attention until pretty recently, but one place that you can read about them is in a paper from 2014 by Galloto and collaborators. So these are various kinds of global symmetries that appear in quantum field theory, and the claim, one of the claims of the Swampland program, in fact one of the earliest Swampland type claims that was discussed in the literature at all, is that there are no global symmetries in quantum gravity, and that includes any of these. No continuous symmetries, no discrete symmetries, no symmetries acting on local objects, no symmetries acting on extended objects. There are absolutely no global symmetries in theories of quantum gravity. So this is a claim that's been around for at least 30 years, and it's one of the most well-established claims in the Swampland program. It long predates the idea of the Swampland itself, and I'm going to summarize a few different arguments for why you might believe this is true. The most rigorous ones have been recently given by Harlow and Oguri, but that's specifically in the context of asymptotically ADS quantum gravity, but it's really believed to be true of any consistent theory of quantum gravity. So why might you believe this to be true? One of the oldest arguments was given by Tom Banks and Lance Dixon in 1988, where they observed that if I had a perturbative string theory, and I imagine that there's some global symmetry in the spacetime, that would be associated with some current operator on the world sheet of the string, and given such a current operator, we can dress it with a factor of e to the i k x to make a vertex operator that creates a massless spin-1 particle in the spacetime. Yeah, good. What do I want to do? Yeah, I guess Yeah, you're saying it like that. Yeah Thanks Okay So the claim was that if I told you that I had a spacetime global symmetry, you could look at the world sheet theory and say actually that symmetry is gauged because we can just explicitly create the gauge field that that couples to it. Okay, there's a very similar sort of argument in the context of ADS-CFT. If I hand you a gauge theory that has a conserved current J mu, then in the dual description, there's a massless gauge field A mu in the bulk. So again, there's a similar sort of flavor. Given a conserved current, the gauge field just explicitly has to appear through the holographic dictionary. Okay, so these are arguments that apply to particular forms of quantum gravity theories, but you might not be happy with those because you might ask what if there are other kinds of quantum gravity, they're not perturbative string theory or not asymptotically ADS. So there's another argument, which is maybe a little more vague in some ways, but I think pretty convincing, which is based on black hole physics. And I think that some version of this argument has kind of been around for a while, but one place where I know that it's written down in a pretty clear way, the paper by Bankson-Cyberg about eight years ago. And this argument is that if I have a theory that has a global symmetry, then everything in the theory has to respect that global symmetry, even black holes. Right? If black holes explicitly violate the symmetry, then it was never really a symmetry in the first place. And so somehow, if I throw things into a black hole that have global symmetry charge, the black hole state has to know what its charge is, and that charge has to be conserved as the black hole evaporates. But for a global symmetry charge, there's nothing outside the black hole that tells me what the charge is. There's nothing I can measure, there's no electric field. And so if I trust effective field theory, the black hole will Hawking radiate, it'll gradually shrink, but it won't lose its charge. It might change its charge, it might happen to radiate particles that are charged, and so its charge might kind of random walk as Hawking evaporates. But it has no reason to systematically prefer to emit charged particles over particles that are not charged. And so in the space of black holes labeled by their mass and their global symmetry charge, there's some minimal mass for which I trust effective field theory well enough to talk about, to talk about these things of semi-classical black holes at all. So in strength theory, this would be when the radius of the black hole is at the strength scale. So below some line in this plane, I don't trust effective field theory, and I don't know how to calculate anything. But above this line I do, and I can make black holes of arbitrarily big charge. If I have anything in the theory that carries this global symmetry charge, I can just throw those things into black holes and make arbitrarily big black holes with arbitrarily large amounts of charge. And these black holes will Hawking radiate until they shrink down to the size at which I stop trusting my effective field theory. And by starting with black holes of different charges and different masses, I can eventually create a collection of black holes whose mass is always around this minimum mass at which I trust effective field theory, but which have any charge that I want. So the conclusion of this argument is that there are in fact infinitely many different kinds of black holes that all have a mass and some narrow range around this minimum mass at which I trust my semi-classical calculation. And the claim is that that should bother you. You shouldn't trust that this is a consistent theory. Certainly any kind of thermodynamic ensemble isn't going to make sense because I have infinitely many different states with similar energy. And these kinds of arguments appeared in the context of discussions about black hole remnants and the black hole information problem already decades earlier than this paper. This also violates, if you're looking for something more precise, it violates Rousseau's covariant entropy bound, which is again sort of believed to be a property of general theories of quantum gravity. Although I think like many of these Swampland ideas, we could say that it's not necessarily rigorously proven that this is true either. But at least this establishes a link between this idea of no global symmetries and another widely believed property that theories of quantum gravity should have. I should mention these arguments are not necessarily completely airtight. So one paper that objects to this argument was written by Diwali and Guzman years ago where they claim that there are some theories where black holes actually carry some kind of skirmionic air that can be detected at long distances and could potentially alter these kinds of semi-classical arguments. So I think none of these arguments are completely rigorous and none of them should convince you that this has been definitively established in all theories of quantum gravity, but at least at least these arguments are strongly suggestive. Good. So the question, if I heard the question correctly, it was does this argument apply to discrete symmetries? Is that right? Okay, that is a very good question and that's something I should have mentioned. All of the arguments I've given you so far are really just arguments against continuous global symmetries. In these cases, I explicitly assumed that we had some kind of conserved current. And here I assume that there were infinitely many different possible charges. If we talked about discrete symmetries, there's actually some discussion of that in the Bankston-Cyberd paper. If you had a really big discrete symmetry, you could try to make a similar argument just by saying there's a finite but gigantic number of black hole states and that might still violate the bound. But it's certainly a less crisp argument than for continuous symmetries. So at least my understanding is that until fairly recently there were no strong arguments that had appeared supporting the claim that there are no discrete symmetries in quantum gravity, except that every place that anyone found a discrete symmetry in string theory, it turned out that it was actually gauged. Now, that has recently changed and I'm going to briefly mention a recent paper by Harlow and Ogury, a paper that came out just a few months ago. There are two papers. The one with the smaller number is four or five pages long. The one with the longer number is 150 pages long or something like that. But the argument, so there's a lot of content in the long paper and I'm not going to do it justice. I'm just going to briefly mention it to you today. But they claim to give a general argument against global symmetries in asymptotically ADS theories of quantum gravity using calligraphy, using ADS CFT. To do so, they have to give, yeah, writing bigger, okay, yeah, thanks. So in order to give their argument, they actually have to give quite a bit of discussion about what is a global symmetry. Is that more visible? And the reason they have to do this is there are certain pathological cases that you have to exclude. And so for them, a global symmetry is sort of somewhat local and that involves some operators that depend on an element of the gauge group and also some region, some spatial region. So they say that to really define a global symmetry and local quantum field theory, we should be able to construct operators that kind of only act with that symmetry within some bounded region. And in cases that have a conserved current, this just follows from the existence of having a local current, but they want to claim that this should be sort of an axiom of what we mean by global symmetry. And I'm not going to try to give you all the arguments to convince you of that, but I think it's a reasonable claim because if you don't make this kind of assumption, there are sort of pathological examples of symmetries that you wouldn't really want to think of as symmetries. Right, yeah, this can also be generalized to the p-form symmetries that act on extended objects. Yeah, so they give a general discussion where they explain what this means, both for the usual zero-form symmetries that have one-form conserved currents and for the more general things that act on extended objects. Yeah, so in fact, their argument is meant to apply, when I say it's a general argument against global symmetries, it's meant to apply to all the different kinds of global symmetries I mentioned before. Continuous, discrete, acting on local operators, acting on extended objects. But it really hinges on sort of a detailed discussion of what they mean by global symmetry, which is a bit technical and I'm not going to try to explain it to you. But I will just tell you the one sort of fact that plays a role in their argument is something that Harlow has worked on quite a bit called entanglement wedge reconstruction. And the claim is just that if I allow you to work only with data from some boundary region R and ADS, that I can only access some portion of the bulk theory defined by some surface, which they call an HRT surface, which your next lecture can tell you more about than I can. But the idea of their argument is basically just to kind of tile the boundary with some set of regions that completely cover the boundary, but that individually don't allow you to access the entire bulk. And then derive some kind of contradiction by showing that this means there can't be localized charged bulk states because if I tried to put such a state in a region that doesn't overlap any of these regions accessible from parts of the boundary, the state would commute with all the different symmetry generators and wouldn't actually be charged. Okay, so that's a very brief sketch. I can't do justice to their full argument without spending a lot more time, but that's the general sort of idea. And the structure of the argument doesn't really care about whether this is continuous or discrete or whether it acts on zero forms or whether it's a zero form symmetry or a higher P form symmetry. And so they claim that this is sort of a completely universal argument that given there are axioms for what a global symmetry actually is, there's no such thing in a theory of quantum gravity. That is asymptotically ADS. And so the weakness of this argument is that it still requires that assumption about asymptotically ADS. So this idea of no global symmetries in quantum gravity has been around for about 30 years and we see that various arguments supporting it starting from very different assumptions, whether we're making specific assumptions about the form of the quantum gravity theory being a perturbative world-sheet string theory or about the asymptotics or about black holes. These various arguments all lead to the same conclusion that theories of quantum gravity cannot have global symmetries. And that's probably the most well-established of the Swampland claims. So compared to everything else I'm going to talk about in these lectures. This is relatively well-established. The downside of being well-established is that it's relatively useless. If I want to use this to make statements about the real world, I really don't care if my theory has an exact global symmetry. I often care quite a lot about whether my theory has a good approximate global symmetry. But this doesn't tell us anything about approximate symmetries. It's a precise statement that actual symmetries that map one state of the theory to another do not exist. But that doesn't tell us how close we can get without sort of violating the spirit of this claim. So the direction we'll be going is trying to have statements that are more useful and that they take us away from this limit of exact symmetries and allow us to talk about approximate symmetries. On the other hand, the further we go from the less well-established the claims are going to be and the more sort of circumstantial the arguments are going to be. So while I'm on the subject of no global symmetries, let me mention one other Swampland conjecture which again predates the name Swampland. And this is called the completeness conjecture. Again, I think it's an idea that was sort of discussed for a while, but it was written down in a clear way by Polchinski in 2003. And the claim of the completeness conjecture is if I give you a quantum gravity theory that has some gauge fields in it, there will be some lattice of allowed charges under those gauge fields. These could be electric charges. They could be magnetic charges. But there's some set of allowed consistent charges. Then some state of the theory exists that carries that charge for each possible charge. We probably have to be a little bit careful about boundary conditions when I say this. Certainly, this is something that seems to be true in asymptotically flat space theories of quantum gravity. In the ADS context, I could have boundary conditions that allow a state of electric charge but not magnetic charge. And then there could also be magnetically charged objects, but I would either need them to come in a pair with something with an anti-charge or I would need to change my boundary conditions. So there are some subtleties about boundary conditions. But it's still kind of morally true in these ADS theories that the statement holds. And just to give you examples of what this implies, some sort of magnetic monopole should exist. Or at the very least, some sort of dions that if I put them together in a multi-particle state with things with electric charge, I can make things that have purely magnetic charge. One reason why Polchinski was a good person to suggest this was that if you had this idea before D-brains were discovered, you would have known there were these Ramon-Ramon gauge fields in string theory and concluded that something charged under them must exist. And of course, D-brains are the things that are charged under those gauge fields. So that's a case where this proved to be true. It wasn't kind of initially obvious that a charged object exists, but they turned out to exist. Okay, so this is another statement that is pretty generally believed to be true, pretty well established. Again, not super useful for the real world because it doesn't tell you how heavy these objects are. We could say with confidence that we believe that magnetic monopoles should exist in the real world. But if you can't tell an experimentalist what its mass is, that's not a super useful claim to make. But something that I think has been appreciated only relatively recently is that this completeness conjecture actually follows from the statement that there are no global symmetries. And the reason is that if you don't have objects of all allowed charges, you will then find that you'll have these kind of higher form symmetries that act on the Wilson loops or Tuft loops that correspond to the missing charges. To give you an example, if I told you I had a U1 gauge theory where the only charged objects I find have charges that are multiples of three, this gauge theory actually has a Z mod 3 one form symmetry that acts on the Wilson loops that's a global symmetry. And so if we rule out all global symmetries including the discrete kinds and including the kinds that act on extended objects, then the completeness conjecture actually follows as a consequence of that. And I learned this fact from Thomas de Matrescu. I'm not actually sure where it was first stated in the literature, but it was also discussed in the paper of Harlow and Aguri. Does it go the other way? I don't think so. I don't see any reason why every global symmetry would have to be associated with some sort of missing charges and a charged lattice. I think the answer is no. If it was true, it would be not obvious that it was true. So these are kind of classic ideas about theories of quantum gravity that have been around for a long time, and that seem to be true, but they're weak enough statements that they're not super useful. If you want to draw conclusions about the real world, they tell you that things exist, but they don't tell you what their mass is, they don't tell you where to find them. And so the weak gravity conjecture is one attempt to take these ideas and sharpen them into something a little more quantitative that actually tells us that not only that objects exist, but that they exist below some sort of threshold. And so what I want to do next is tell you what the weak gravity conjecture is. But before I do that, let me just recap some facts about black holes that are probably familiar to many of you, but are worth reminding you of. Okay, so I told you before that if we had a black hole with a global symmetry charge, there's no way to measure that global symmetry from outside the black hole. That's not true for gauge charges, at least for continuous gauge charges. If I have a black hole charged under a continuous gauge symmetry, it has some external electric field that I can just measure from outside the black hole. And associated with this external measurable electric field is a lower bound on the black hole mass for a given charge. The black hole mass is bigger than some multiple of its charge. And there's nothing really mysterious about this. It's just following from the fact that there's an electric field outside the black hole, the electric field stores energy and that energy contributes to the mass of the black hole. And so parametrically you can get the right form of the bound just by saying that the mass of the black hole is bigger than the integral of the electric field squared outside the black hole. So the mass is bigger than the charge of the black hole squared times the gauge coupling squared divided by the black hole radius. And the fact that it's a black hole means that its radius is in order its mass and plonk units. And so when you put these together, you get the mass is bigger than charge times the gauge coupling times the plonk scale. So that's parametrically the right conclusion. Quantitatively you can work out what the exact bound is by just finding the classical black hole solutions. And so to give you one quantitative example, for Rice to Northstrom black holes in four dimensions, so these are black holes in Einstein Maxwell theory. We just have gravitons and photons. And then we find that this bound is true with the factor of the square root of two set of conventions that I can specify. So in plonk is the reduced plonk mass square root of 8 pi g Newton. Sometimes this product of E and Q is just called the charge. But I'm going to use conventions where the charge is a quantized quantity. So Q takes values only in integers. And then this number E, the gauge coupling can be read off if I tell you that the gauge field kinetic term has a factor of 1 over E squared in front. In the convention that a particle of charge Q couples to the gauge field by integrating A over the gauge fields world line and multiplying by Q. So those conventions tell you how I ended up with the square root of 2. This is how I'm normalizing everything. And the reason to be a little bit fussy about normalization is that the precise number that we get here will depend on the details of what theory we study. So in particular, here we had only gravitons and photons. I'll tell you a bit about theories where we also have scalars. And if the scalars coupled with the gauge fields, you get different solutions and this number actually changes. And so when I tell you what the weak gravity conjecture is, it's going to be important to try to be quantitative about that number. Because it's going to turn out that in the presence of scalars, there's more than one thing that you might mean by the weak gravity conjecture. And to figure out which one is the right one, you have to be careful about it. That's right. And that's another good point, which I was going to come to a bit later, but I might as well say it now since you asked. The question was, does this bound mean that gravity dominates over the electric force? And that is exactly right. They called this an extremality bound. An extremal black hole is one that saturates the bound. And the claim is that two extremal black holes of the same charge have no net long range force between them. So the gravitational attraction balances the electromagnetic repulsion. And consequently, exactly as you said, one interpretation of this extremality bound is that four black holes, gravity is a stronger force than electromagnetism. So any two black holes that are not both extremal, both saturating the bound, will be attracted to each other because gravity will overcome the electromagnetic repulsion. Good. So that's one important consequence of this extremality bound. This session ends in five minutes, right? So the statement about extremality generalizes to a larger class of theories. So for instance, we could consider Einstein Maxwell dilaton gravity, where the dilaton part means that I have a massless scalar, really stronger than massless, a scalar with a flat potential. And the way that scalar interacts with the theory is going to be not only by coupling to gravity, but by coupling to the gauge field with the factor of e to the minus some constant times phi in front of the gauge field kinetic term. So this is familiar, for instance, from the 10-D dilaton in string theory, but it also happens with lots of other things. If I study collutes of Klein theory, for instance, as we're going to see, we get similar couplings where phi is the radion field that tells you how big the circle is that you compactified on. So these couplings show up pretty often. And the claim is that in this theory, you can again find some set of classical black hole solutions that carry charge. But now in these solutions, the scalar field is turned on, you can't avoid turning it on because you have an electric field, so there's a non-zero value of f mu nu squared, so that sources phi and the classical solution. And so you get a different solution, and in fact you get a different extremality bound, and that bound turns out to be of the same form as the previous bound with a different constant in front. It differs by a factor of the square root of one plus alpha squared. Okay, so this is quite general. There are lots of different variations you can study on how scalar fields couple to gauge fields, but you will typically still find an extremality bound in any of those cases. The alpha is the constant that appeared in this coupling of the scalar to the gauge field. Let me just make one other brief comment before ending this session, which is that one thing I haven't really said so far, but I've kind of tacitly been assuming is that charges are quantized. And so if you like, you could think of that as another Swampland conjecture. In quantum gravity, if I hand you an ability in gauge field, the gauge group is always u1. It's always compact, and charges always quantized. The gauge group is never r. And in fact, that claim can be argued with very similar arguments to the one about no global symmetries. So if we had a theory where I had two particles whose charges were irrational multiples of each other, one in the square root of two, I could throw those charged particles into black holes and make black holes with any combination of those charges that I want and let them evaporate. And I would end up again with infinitely many states in a given range of charges. So that argument also appears in the Banks and Cyberg paper. And so if you found the black hole argument for no global symmetries convincing, you should also find it convincing that charge is always going to be quantized. And so I've really been assuming that in everything that I said. Like when I said there is a charge lattice and magnetic monopoles are part of it. All of that relied crucially on the assumption of quantized charge. We'll pick back up after the break.