I'd like to have this cool maturist from the University of Harvard who will teach us about formal developments in high-energy physics. Please. Thank you. So yeah, so the official title listed on the schedule is formal developments in high-energy physics, but that's a little broad. So specifically, the things I'm going to be talking about will be mostly things related to the weak gravity conjecture and a little more broadly what's called the Swampland program. And my name is Matt. I'm the second Matt you've had teaching you. And I guess there'll be another one coming next week. So the organizers seem to have worked really hard to get lots of people named Matt or Matthew to talk to you. OK, so this name, the Swampland program, comes from a paper that was written by my colleague Cameron Bafa back in 2005. And the idea of the Swampland program is that we know that there are infinitely many different quantum field theories. Quantum field theories can have continuous parameters like coupling constants. So there are continuous infinities of consistent quantum field theories, but it seems much harder to find consistent theories of quantum gravity than it does to find consistent quantum field theories. Any effective quantum field theory that has a stress energy tensor can be coupled minimally to gravity as an effective field theory. You can just take the Einstein-Hilbert action and add to your quantum field theory. And you get a consistent effective field theory, but it may not actually be a consistent theory of quantum gravity. It may not be a theory that can consistently make predictions at all energies. Good question? Yeah, good. So the question was, what does the word minimally mean when I said minimally coupled gravity to a quantum field theory? What I mean is, take your quantum field theory and just promote the metric that appears there to a dynamical metric. So if you haven't studied general relativity, what I'm saying might not be that familiar, but in general relativity, the metric is a dynamical degree of freedom. And this is the Ritchie curvature scalar that you compute from the metric. And everywhere else, there was a metric inside this Lagrangian. You might have had something like a g mu nu d mu phi d nu phi. And to get a consistent effective theory of gravity, we're just going to treat this metric as something that can fluctuate. So that's the minimal coupling. Can I write bigger? Is that the question? So we can have lots of effective field theories coupled to gravity. But the idea of the swamp land is that not all of these will be things we can complete into consistent gravitational theories. So people have tried to find consistent theories of quantum gravity for a long time. And there are not that many successes. In particular, we know consistent theories of quantum gravity that come from string theory, but they seem to have lots of restrictions that effective field theories in general don't have. So the idea of the swamp land is we have this big space of effective theories. Within this space, there are certain theories that we know are consistent. Some of them are just kind of isolated points in the space. Some of them are kind of continuous families inside this space, so things that have what we call modulized spaces or these isolated points, isolated vacua. But also, there are big regions of the space of effective field theories that just don't seem to be consistent theories of quantum gravity. So no consistent quantum gravity, say, for anything in some region of this space. So the idea that Vafa advocated in 2005 was we should try to identify what are some criteria where if I hand you an effective field theory, you'll know if it has a chance of being over here where it might be a real theory of quantum gravity, or could it be over here where there's just no hope of finding a consistent quantum gravity theory that fits that effective theory? Yeah, that's a very good question. So if you didn't hear the question was, when I say no consistent theory of quantum gravity, do I just mean no consistent string theory or do I really mean no consistent theory of quantum gravity even if it's something other than string theory? That's a good question, and that's something to keep an eye on as I go through my lectures because a lot of the evidence that we have for what might distinguish things in the Swampland for things that aren't is based on string theory because string theory is the corner of quantum gravity that's relatively well understood. But of course, there may be other theories of quantum gravity that are not so well understood that we just don't know about yet. And so I think the aim of the Swampland program is really to say things about quantum gravity in general. And so there are a few cases where we'll be able to do that using arguments that are based on things like black hole physics that we think are universal to any theory that has gravity. But some of the arguments will be more based on things that we've observed in string theory. And you should keep an eye on which arguments are which because the arguments that are just based on examples in string theory are probably less reliable. Things that are based on long distance properties of GR and black hole physics are probably more reliable. And one of the goals of this program is to try to take those observations we make about string theory and figure out how we might put them on a more solid footing and see if there are really statements that we believe should be true about any theory of quantum gravity. But you're right. That's an important distinction. It's possible that all consistent theories of quantum gravity are somehow related to string theory, some limits of string theory, but we don't actually know that. And so some string theorists might kind of tacitly assume that, but you shouldn't let them get away with it. You should keep asking them that question. OK, so this region where there are no consistent theories of quantum gravity is what we call the swampland. And these theories over here that are consistent theories of quantum gravity, we call the landscape. So the landscape are these things that are consistent. And our goal is to try to develop some tools for distinguishing effective field theories that are part of the landscape to those that are part of the swampland. Oh, there's a volume control there? OK. But it sounds like I'm louder now than I was. OK. Good. OK, so how do we go about working on this question? There are a few different ways to make progress. One way to try to make progress is to try to deduce things from general principles. OK, so as I was just mentioning, these might be properties of black holes. These might be ideas like holography, at least for quantum gravity and asymptotically ADS-based, can tell us a lot. Another way to make progress is to find explicit examples of quantum gravity theories, for example, in string theory, and try to generalize. So you might ask if those examples all have common properties that might be universal or quantum gravity, maybe there are things that never happen in those examples, and that might be a hint that there are certain things that are just not consistent. In practice, a lot of the progress has been made by starting with number two here, looking at explicit examples, trying to generalize from them, and then going back, and once you have noticed something about the examples, see if you can figure out if there might be a general principle behind it. And so there's kind of a lot of iterating between these two different ways of working. And as I mentioned, Bafa wrote this paper back in 2005, kind of proposing that this is a useful thing to do. In fact, as I'll be reviewing, there are a lot of these ideas that are actually much older than this. They just didn't have the name Swampland attached to them when people first started thinking about them. But there's also been, in particular in the last five years, a lot of new interest in the subject. And I will probably have a bit more to say about that maybe in the third lecture. One reason why the subject got revived was there was a claim of an experimental result from the bicep collaboration. They claimed to see primordial gravitational waves in the cosmic microwave background, which would have been evidence for certain theories of inflation that some people had postulated were in the Swampland. And so there was actually a concrete experimental observation that seemed like it had something to do with some of these hypothetical boundaries that you shouldn't cross. If you haven't heard of the bicep experiment, the reason you haven't heard of it is that it turned out they were wrong. They didn't see primordial gravitational waves. They saw patterns in the CMB that came from dust in the galaxy that happened to mimic primordial gravitational waves. Nonetheless, there was a time period when it appeared they had made a discovery and that got many of us interested in these questions because it at least showed that these questions could be linked to observations you could imagine making with existing experiments. So there's been a lot of activity, especially since 2014. And I'm just going to be able to tell you a little bit about some of what's happening. But if you're interested in learning more, let me point you to two references. First are some TASI lecture notes based on lectures given by Vafa and written up by Brennan, Karta and Vafa in 2017. And the second is a very recent review article by Aaron Palti that just came out in March of this year. So there's a lot more detail in these references than what I will have time to explain in the lectures here. In particular, these references both discuss an idea that's gotten a lot of attention which is a claim that quantum gravity may be inconsistent with the sitter vacua. So a sitter vacuum is a space that has a positive cosmological constant which causes the universe to expand at an accelerated rate which is what seems to be happening to our universe now and the universe is expanding in a dark energy dominated way that looks a lot like a cosmological constant. If the claims that that cannot happen in string theory are true then that is a very interesting claim about our universe. Personally, I think the evidence for that conjecture about string theory is pretty weak. And so I'm not going to spend much time talking about it but if you're curious about it you can learn more about that from these references. But in these lectures I'm going to be a little bit more focused, I'm going to talk about a few of the Swampland ideas that I think are the most solid, the most well-established. The first idea that I want to focus on is a claim that there are no global symmetries in theories of quantum gravity. Second idea that I want to talk about something you can think of as a refinement of that is the weak gravity conjecture. And the third idea I want to talk about something that of the Swampland distance conjecture and which is closely related to some stronger versions of the weak gravity conjecture. So I'll explain what all of these things mean as we go along but these all are statements that I think have relatively solid evidence behind them. And then in the last lecture I'll also try to talk a little bit about some applications of these ideas to phenomenologically relevant questions. So I'll talk a little bit about how certain theories of inflation can be in tension with these conjectures. And I also want to talk a little bit in the last lecture about the question of whether the photon can have a mass. That's something maybe you can think about before we get to that point. In quantum field theory it's possible to give a mass to an ability and gauge goes on like the photon without really hurting anything. And so it's worth thinking about what have you been told in quantum field theory classes about why the photon is massless? Is it convincing? And I would claim that there's no really convincing argument in quantum field theory that tells us the photon has to be exactly massless. Phenomenologically it has to be really light but really light is not the same thing as massless. But I think maybe some of these ideas about the swampland could really tell us that the photon has to be exactly massless. So I will explain that in the last lecture. So that's sort of the set of ideas that we will be talking about over the next few days. If I say things that are unclear please stop me and ask me questions. What is the motivation? There isn't really a motivation but the question is just, is it possible? So in the standard model we describe our universe in a theory that has a massless photon. But the question is how reliably do we know that it's really massless? An experiment can't really tell you that it's exactly massless. Experiment can just tell you it's really light. And you can look this up in Yuval's favorite book, the PDG. You can find bounds on the photon mass which tell you that it has to be extremely light. Let's see. Some of the bounds are something like 10. It has to have a mass below 10 to the minus 18 electron volts. So that's super light for all practical purposes that's massless. But the question I want to ask is just do we really know that it is massless or not? I'm not saying there's a motivation for it not to be I'm just saying how confident can we be that it really is? Yeah. If there were cosmological implications of the photon having a mass, you could use them to try to set stronger experimental bounds. I don't think I really know any such conditions. There are bounds on the photon mass from things like the magnetic field in the galaxy. So there are certainly astrophysical bounds. I don't think I know any bounds that are really cosmological as opposed to astrophysical. But there may be that I just don't know about. Any other questions about what I've said so far before we start going into details on some of these things? Okay, so the first thing I want to try to talk about is the first claim here that there are no global symmetries in theories of quantum gravity. Okay. So before I say something about the claim, maybe I can just remind you that global symmetries are perfectly okay in quantum field theory. In fact, they're very useful tools for analyzing quantum field theories. In QFT, without gravity, global symmetries can exist. So one example, if I have QCD with some number of massless quarks, NF, massless flavors, I have a global symmetry and that global symmetry, the screw. So the global symmetry can rotate the NF left-handed quarks among each other with the general, general special unitary transformation. It can do the same thing to the right-handed quarks. And there's also this U1 very unnumbered. They just count how many quarks there are. So I have a global symmetry and a theory of massless, a theory of QCD with massless quarks. And you know that this is an interesting symmetry. For instance, the spontaneously broken by the dynamics that generates a QQ bar condensate. And we learn a lot from that, like the existence of approximate Goldstone bosons, which are the pions. So knowing about the symmetry is really important. The symmetry is also associated with conserved currents, Mu. So I have one current operator, Jmu, for every generator of my symmetry. But in the real world, we also know that this is not a symmetry. In the real world, we have QCD and we have quarks. But this global symmetry is completely broken. It's broken in part by the fact that the quarks have masses. It's also broken by the gauging of hypercharge. The barion number in the standard model turns out to be anomalous. So none of these things are really symmetries of the standard model, even though they might be good approximate symmetries. So in the real world, these are just approximate. There are other kinds of global symmetries that arise in quantum field theory. They may be less familiar to you. I'm going to mention some of these less familiar symmetries. I'm not going to spend a lot of time to explain them. So if they're unfamiliar, I just want to make you aware that they exist and then you can learn more about them elsewhere. Suppose I have a quantum field theory with a U1 gauge field, but no charged matter. So we have a photon, but it's not interacting with electrons or anything. Then the symmetry, sorry, this field theory has a symmetry, which is a less familiar kind of symmetry. It's what's called a higher form symmetry. Okay, so we started with a U1 gauge field, but this theory actually has a U1 times U1 global symmetry. And just like in the other cases we mentioned, there are conserved currents that generate the symmetry. But here the conserved currents are f mu nu, the field strength of the gauge field. And f mu nu tilde, the dual field strength that you got by combining your field strength with an epsilon symbol. And when I say these are conserved, what I mean is just similar to what we usually mean for a conserved current.  And these are true equations of motion in this theory because what would ordinarily show up on the right-hand side of this in QED is the current of the electron. But here we said there are no charged particles. So these are conserved currents, but they're conserved currents that have two indices, unlike the ones that you're used to that have one index. And the charged operators are Wilson or Tuft lines. So for the usual global symmetries, they're charged local operators. But for these symmetries, the charged objects are not local operators. They're things that live on the line. Yeah. So the action of the symmetry is related to, I'm going to try to explain this very briefly and I probably won't do a very good job, but let me first write down the reference. But I kind of want everybody to be aware that these things exist, but I also don't want to spend a lot of time on them because I could easily fill up a whole lecture with this and not get to the other things I want to talk about. So I'll try to give you a very quick answer to your question, but let me just tell you the reference for this idea of these higher form symmetries. There's a paper by Dayoto, the Pustin, Seiberg, and Willett that came out in 2014. And the quick way to try to answer your question is to say that for an ordinary global symmetry, we have a charge which comes from integrating the current over space. And this charge operator has the property that it can act on local operators. And one way to think about that is there's a sort of commutation relation where if I associate the charge with some spatial slice, say this is my time direction, if I then move my spatial slice past this charge, I pick up a phase, which is e to the i times the charge living on the operators that was in between. When I have these higher form currents, what happens is there are charges that are associated with integrals over higher co-dimension surfaces. Okay, so the charge of objects like Wilson-Lions, a charged point I can surround by a sphere with dimension one less than the space that I'm living in, but a charged line I can surround by a sphere with dimension two less. So I surround it with something like this. And so the charges are associated with these lower dimensional surfaces and their action is associated again with some sort of commutation relation that if you kind of move the surface through the charged objects, you pick up phases that are associated with their charges. So that's the very short summary of how these things act. There's a much more elaborate story and I'll refer you to this paper if you wanna learn about it. But the main thing I wanna emphasize for the purposes of these lectures is just that quantum field theories can have global symmetries as you know, but they actually have lots of different kinds of global symmetries and you may not have seen some of those before because actually some of these have only really been discussed in the literature pretty recently. So that's one just comment. There are lots of different kinds of global symmetries in quantum field theories. The second comment is this particular kind of global symmetry is associated with the absence of charged things. And that's going to be important for us when we talk a little more about the claims about quantum gravity, which are going to require charged objects to exist. All of the global symmetries I've mentioned so far are continuous symmetries. They're things like U1, so there's a circle, there's a general phase we can have, or they're things like SUM, so there's some non-Abelian continuous symmetry. There can also be global discrete symmetries in quantum field theory. And this was an example where the conserved current was a two form. You could also have examples where it's a three form or four form. And again, those are associated with even higher dimensional charged objects. So the first of these swampland claims that can distinguish consistent gravitational theories from those that aren't is quantum gravity theories have no global symmetries. And to elaborate on that, they have no continuous global symmetries. None of these P form symmetries for any P. So not just no symmetry like these chiral symmetries of a QCD like theory with massless quarks, but also no symmetries like these extended symmetries that a theory would have if it had a photon but no charge matter. And they also have no discrete global symmetries. In fact, it turns out you can define discrete P form symmetries. And so again, there are none of these for any P. And these claims have been around for a long time. Especially the claim about continuous global symmetries has been around for a very long time. But even the claim about discrete symmetries has been around for quite a while. But long before Bafa came up with the name Swampland, people were making this claim about theories of quantum gravity. And there are various arguments for this. Yeah. Okay, good. I should qualify this. If we're careful, we can probably understand CPT as a gauge symmetry. But yeah, just to let me just qualify this by saying internal symmetries. You're right, discrete space-time symmetries are a little harder to figure out how to think about from this perspective. Another thing that people sometimes ask about is if you have a gauge symmetry, you can talk about the global part of the gauge symmetry, the part of the gauge symmetry kind of at infinity, which is sometimes thought of as a global symmetry. That's one place where this reference is useful because they make their definition of global symmetry more precise. And for their definition of global symmetry, you need not just something that kind of acts on the whole space-time at once, but you can sort of act on localized subregions within the space. So that's sort of saying that the symmetries we're talking about are things that have a notar current associated with them if they're continuous, if they're discreet it's slightly more subtle, but there's still some sort of localized version of the action of the symmetry. And if you act with the global part of a gauge group or you act with something like CPT, you're sort of acting on the whole space-time at once and there's no way to kind of localize it to a subregion. And so it's not really what we mean. But that's, yeah, that's a technical subtlety where you have to be careful about exactly how you define a symmetry to make the statement really precise. So there are various arguments for how we know this is true in different contexts. One of the oldest arguments made all the way back in 1988 by Tom Banks and Lance Dixon. This is an argument purely about perturbative string theory on the world sheet. And the argument is if you look at a theory on a string world sheet and it has a current that corresponds to a global symmetry, then you can use that current, you can dress it with some other factor, make what's called a vertex operator that creates a gauge field. So these look something like e to the ikx times something that has a vector index, like derivative of x, okay? If you're not familiar with world sheet string theory, don't worry too much about the details. The point is just that in that context there's an argument that if I hand you a theory and tell you it has a global symmetry, you can give me back a gauge boson that gauges that symmetry. And so it was never really a global symmetry to begin with. It was always a gauge symmetry. There is a gauge field associated with it. So this is an argument that shouldn't convince you of anything about quantum gravity in general because this is one of those arguments that's very specific to a certain context in string theory. But within that context, it's just a fact that you can't find any global symmetries. There's a sort of similar argument in the context of ADS CFT. I hand you a conformal field theory that has a conserved current, GMU. The ADS CFT dictionary tells you that there are massless gauge fields, ADS spacetime. Okay, so both of these are sort of constructive procedures that if you're given a current, you can find a gauge field and see that your symmetry was always a gauge symmetry. But I'm not sure I heard the question. Operators can be dressed, okay, good. So what I mean is if you knew about this operator, for instance, you can attach this factor in front to it, this e to the ikx. And what that factor does when you work through the dictionary that tells you how the theory on the string world sheet corresponds to the theory in spacetime is anything that has this e to the ikx factor attached to it corresponds to a propagating field with momentum k in the spacetime. So by dressing, I mean you can sort of attach this factor to something. And that will allow that operator on the world sheet to now create a spacetime state that is a gauge field that has momentum k. And so you can just construct those particles. Right, so I think what you're asking is I've kind of cheated a little bit by saying if I have a conserved current in the CFT, there's a gauge field in the ADS. You can say what if there's a conserved current in the ADS itself, right? Yeah, I think that's a fair objection. You could sort of say that any conserved current in the ADS must somehow act on local operators in ADS, but those operators must correspond to something in the CFT and so it would be surprising if there wasn't also a current just kind of restricted to the CFT. But that might not be a completely rigorous argument. I'll come back to ADS CFT in a minute and tell you something that's a little more convincing, although also something that I won't have time to fully explain. There isn't really, no. So the question was, this is an argument about ADS CFT. We don't seem to live in ADS space. We seem to live in decider space or something closer to decider space. And that's fair. So what I should say is these arguments are really theoretical arguments about examples of quantum gravity theories we have and looking at their common features. You know, the unfortunate fact is that most of the quantum gravity theories we understand well don't look very much like the universe that we actually live in, right? And so you are completely justified in asking if we know whether these arguments extend to our universe or not. And I think it's fair to say that most of the arguments that are currently known for things like this don't obviously apply to our universe. On the other hand, if you find that there are many different corners of quantum gravity where these arguments always apply, you might start to think that there's a deep reason why this is true that would apply also to our universe. But yeah, these statements by themselves only prove things in particular contexts. And what we're trying to do is just look at a lot of those contexts and see if we can generalize something. So yeah, that's a completely fair objection to make. The next argument I wanna talk about though does apply a little bit more directly because it's an argument that's based on black holes. So let me move on to that argument now. Even this argument, I think you can find things to object to. So we can talk about that. But I think the argument I'm about to give you is kind of the gold standard for the best kinds of arguments people have been able to make for these Swampland kinds of statements. Because they're really based on just properties of semi-classical GR, which should be things that we think are true for any gravitational theory. Before I say that, let me say one other thing, which is that both of these arguments verify, these are arguments against continuous global symmetries. In both of these cases, I was talking about conserved currents. And so these arguments don't really seem to have anything to do with the discrete case. So now I'm about to give you another argument. And at least in the minimal form, this is really just an argument against U1 global symmetries. I think there are ways to try to make it an argument against any continuous global symmetry, but the starting point is really about U1.  Yeah, I think you're asking the same question that came up before, which was if I have a continuous global symmetry in an ADS quantum gravity theory. Yeah, so again, the question already came up. There's sort of a missing step here, which is we assume the conserved current in the CFT. If we had assumed a conserved current in ADS, and then argued that there's a gauge field that couples to it, then we would have shown that it wasn't really a global symmetry of ADS, it's a gauge symmetry of ADS. But here we assumed a global symmetry of the CFT, and then argued there was a gauge symmetry in the ADS. And so you can ask, why does that tell us anything about global symmetries in ADS? And I think you have to work harder to try to make that into an argument. I think the rough idea is that if you have a global symmetry in ADS, you should be able to sort of restrict it to the boundary. And so there should be some restriction of the current in ADS to a current on the boundary, and then you can try to apply this argument. But I agree, there's a missing piece, and I haven't tried to make it precise. And one reason for not making it too precise is there are better arguments that I'm going to try to give you now. Okay, so now I want to give you an argument against you one global symmetries that's based on black hole physics. And I think that some version of this argument has been kind of known for a long time. But the first place I'm aware of that tried to actually write down this argument in a reasonably careful way, was a paper by Banks and Cyberg that came out 2010. And the argument is the following. Suppose I have a gravitational theory and it has a U1 global symmetry, okay? So there are some objects in the theory that carry charge under that U1 symmetry. By throwing these charged things into a black hole, I can make black holes that have big charges under the U1, okay? So that's the claim. You can make big black holes with any U1 charge that I want.  So that's claim one, claim B. Claim B is that these black holes will evaporate through Hawking radiation, but that Hawking radiation is not preferentially going to make the black hole lose charge. And so this is really the claim that if I have a black hole that carries a global symmetry charge, there's no way to measure the charge from outside.   But if it's a global symmetry charge, the black hole solutions in GR don't care about global symmetries. There's no electric field. There's nothing you can measure. So from outside the black hole, it just looks like any other Schwarzschild black hole. There's no way to know that it has global symmetry charge, yeah. Okay, let me see if I can clarify, because I may have said something unclear a minute ago. So you said that if you put electrons in a black hole, there will be some potential and there's a force mediated by the photon and that's true, but that's a charge under a gauge symmetry, right? The electrons are carrying electric charge, which is a gauge charge. Oh, you're asking how can there be a potential outside the black hole? Right, it's not that the photons are escaping the black hole, they're not coming from behind the event horizon. Rather, there just is an electric field outside the black hole horizon. Okay, so because charge is conserved, if you throw charge things into a black hole, they have some electric field and that electric field can't just disappear because you have to respect Gauss's law, so it can't just vanish. And so what you find is there are electrically charged black hole solutions in GR, they're called Reister and Nordstrom solutions and they have an electric field that's turned on outside the horizon. And so what would happen if you're throwing electrically charged particles into a black hole, is there'll be some time dependence that eventually makes the final state look like one of these Reister and Nordstrom solutions where the electric field is turned on outside. It's not that the field somehow got inside and then leaked out through the horizon, it's just that there's some time dependence and there's always some field outside of the horizon and at late times it's still there because it can't disappear. Okay, so that's what happens with gauge interactions. Black holes charged under gauge interactions have electric fields, you can measure them. And it turns out those electric fields affect the way the black holes decay, the black holes actually preferentially discharge. So if I have an electrically charged black hole, it's going to want to emit charged particles until it loses its charge. But if I have a global symmetry charge, nothing like that happens. As far as GR is concerned, it's an uncharged Schwarzschild black hole. And what that means is it's going to evaporate, it's going to Hawking radiate. Its charge might change as it radiates, it might happen to radiate charged particles, but it doesn't have a reason to do that more often than anything else.  Sometimes it gets bigger, sometimes it gets smaller, but it doesn't have a preference to do one over the other. Yeah, good. Yeah, that's an important question. So let me say there's a premise behind this argument, which is that I'm assuming that talking about a global symmetry and a theory with black holes makes sense in the first place, okay? So one thing that you could say, and that some people do say, is that black holes just destroy global symmetry charge because in the presence of a black hole, there's no obvious way to define what you mean by the global symmetry charge. You know, maybe particles carrying global symmetry fall under the singularity and we don't know what happens and maybe there's no way to even define what we mean by a global symmetry in that context. And I think that's a thing you can consider, but what we're going to try to do here is try to assume that the global symmetry is a well-defined thing and then sort of lead to some contradiction that makes us think it wasn't in the first place. So what I really have in mind here, one way to talk about it is to assume that the black hole will eventually evaporate completely into lighter particles, okay? And then I could measure the global symmetry of my initial state that had no black holes. I could measure the global symmetry of my final state that had no black holes. And then I don't run into the question that you asked of exactly what happens at intermediate times, but I'm going to assume that the charge is conserved so that whatever happened in the middle, I end up with the same charge coming out that I had going in. I agree that if I don't do that, there are questions to ask about how do I draw my spatial slice and what happens with the singularity and things like that, but let's just assume that the global symmetry makes sense and see what happens.   And as it evaporates, the charge kind of fluctuates, it might go up, it might go down. But the point is that if I repeat this experiment enough times, because the black hole has no reason to preferentially lose its charge, what I should expect to happen is I can just generate lots of black holes with lots of different charges until the point where I stopped trusting the calculation of Hawking radiation. Okay, so Hawking does some semi-classical calculation. It's based on effective field theory. Once the black hole gets down to size, sort of, of order of the Planck scale, I don't really know how to calculate anything anymore.  Conclusion is, if I'm just assuming that charge is still a well-defined notion, I have these charged particles coming out of my black hole. There must be charge left behind in this object once it gets to that size. Okay, once it gets to that size, maybe it decays in some way that I can't calculate, but let's just talk about the object of that size. The conclusion is there are black holes of radius R, which is roughly whatever my cutoff is, my UV cutoff on my EFT, with all possible charges, because I could have started off with whatever charge I wanted, 10 to the 100, 10 to the 52,000, anything I want, I let it evaporate, I'm going to get something whose charge is comparable to that. If I do this with sufficiently big, sufficiently highly charged black holes to begin with, I ought to be able to get things of the size where I stopped trusting my calculation with any charge that I want.  I evaporate, as I evaporate, the mass gets smaller, the charge might get bigger, it might get smaller, but it doesn't preferentially do one or the other, so I get some kind of process like this, until I hit the minimal mass, of which I trust that Hawking's calculation tells me what happens, and then I get some black hole of that mass and some final charge. And I just repeat this many times with many different black holes or many different initial charges, different random walks, and I can just populate lots of different states along this line. Yeah, I'm actually assuming it's discrete, and that's a good question. I'm going to come back to that later, but whenever I say, where did I say it? Whenever I say a U1 symmetry, I assume that the charge is discrete. I'll come back and try to justify that more later, but yeah. So all the particles have discrete charges. Similarly, all the black holes have discrete charges. It's only losing charge in some quantized unit. So it's not a continuous infinity, but it's a discrete infinity, okay? So all possible charges that are integers. So that's the claim, yeah. So I mentioned earlier that I think we could improve this argument to apply to anything. So yeah, in fact, I think, yeah. For this argument, you're right. We can just always identify some U1 inside of anything else, and we can run this argument. Later, when I talk about things that are electrically charged, there'll be a difference, because if I have a different group here, it might confine, and then the electric field might not be meaningful far away, but since this is not a gauge symmetry, it's a global symmetry, you're right.  No, not necessarily, because if there's a particle that carries charge, there's also an anti-particle that carries the other charge. And when the black hole evaporates, it could emit one or the other. So the charge could go up or it could go down, depending on whether it's emitting a charged particle or an anti-charged particle. Well, baryon number is an example of a U1 symmetry. The charge is discrete, but the group is U1. Yeah, so certainly baryon number, if baryon number were not anomalous, it would be an example of such a symmetry. Okay, so I've given you the argument that I haven't given you the punchline. There's another question, I guess? It isn't, and that's the interesting thing about this claim b, okay? So if I have a black hole that's electrically charged, so it has a gauge charge, then what you say is exactly right. It prefers to emit things that carry the same charge that it has so that its charge gradually gets smaller. And you can see that in the calculation of Hawking radiation. If you do the Hawking radiation calculation for a charged black hole, what happens is the fact that it's charged and there's an electric field outside the horizon means effectively Hawking radiation is thermal, but in the presence of a charge, it's thermal with a chemical potential that favors one charge over the other charge. But if you have a global symmetry charge, that charge just doesn't show up in the calculation of Hawking radiation. There's no electric field, there's nothing associated with it.  And it's just going to emit all possible particles with the thermal spectrum. So that's an important assumption. You do have to trust the semi-classical calculation. Okay, so that's one place where you could say that you don't buy this argument, but if you say you don't buy this argument, it means that you believe that in gravitational theories, there are places where effective field theory breaks down at long distances, far from the cut-off where you wouldn't have thought that it did, okay? Which is a possible thing, but it would be a pretty strong statement to make. So, okay, so various assumptions went into this as we've been discussing, but given these assumptions, the output is that this theory that has a global symmetry also has infinitely many states of different charge under the global symmetry with the same mass. And now that's the point where there's a claim that this is a problem. And this is another point where you can stop me and tell me you're not convinced, but this contradicts what's called the covariant entropy bound. So the covariant entropy bound is a claim of Raphael Busso, is building on older work of Beckenstein, dates back to the 70s. And I'm not going to try to give you the complete rigorous definition of what this is, but the claim is basically a region in a quantum gravity theory cannot contain more states. Okay, this is a not very covariant way of saying the covariant here is kind of where all the details are that Busso filled in, but the region shouldn't contain more states than the area of the region's boundary and plonk units.  The black holes kind of are the maximal number of states you can fill in a given region of space. But what we saw here is, if I just ask how many states are there of size set by this cutoff, they're actually infinitely many. So it's violating this bound, it's violating it badly because the bounces, there should be finitely many possible states. And we're not just saying, oh, we got twice that many, or we got 10 times that many. We're saying we found infinitely many. Now, you could ask, why should I believe the covariant entropy bound? And then I would have to try to give you an argument for that, too. And I'm not sure how sharp we can make that argument. It's sort of expected, it fits with all these ideas about holography that seem to be true in certain contexts in gravitational theories. But I think that's another place you can make an objection to say, maybe there are quantum gravity theories that just don't respect that bound at all and look very different from theories that we know. And again, that could be possible, just like it could be possible that there are quantum gravity theories that are effective field theory breaks down very badly. But both of those are sort of radical assumptions. And if you trust the usual assumptions people make about quantum gravity, then this argument tells you that you shouldn't have global symmetries. Are there more questions about this argument? The finite amount of matter. I see what you mean, right, right, right. I guess that's not completely clear. You know, this is not an argument that requires you to actually make all these things. It's an argument about sort of what states are possible in the theory. So if you have a universe that kind of only contains finitely many degrees of freedom, but that somehow would allow you to start off with lots of different initial states, maybe you could still make this argument. But I guess the best version of this argument is just we imagine a universe that's asymptotically flat and infinitely big. And then it's sort of clear that in principle we could set up this experiment with sufficiently many particles. But yeah, if you were in the sitter space or something and you believe there are only finitely many degrees of freedom in the first place, maybe I couldn't actually do this thought experiment. So I think it's fair to, I think that's a fair objection. The argument is really most convincing for asymptotically flat gravity theories. Well here we've assumed that the charge is always conserved.  And that's the thing we're drawing here, right? Now it's true, as you said, we could just assume that the symmetry is violated in the first place. But then that gets us to the same conclusion we wanted anyway, right? So we wanted to sort of assume that the symmetry works, which means the black hole, even though you can't measure the charge from outside, it carries some memory of that charge. And then what we're arguing is then you get all these different states that remember that they have different charges and that's the problem. Let's see, we don't have all that much time left today, right? Are there other questions people have now? Yeah. Is there a good argument for claim B? Claim B is just a claim about the semi-classical calculation of Hawking radiation. That's right, but that means that effective field theory is broken down very badly. Because if you trust effective field theory, you can just calculate what the black hole radiates.  So that's right, that's the loophole that effective field theory could just break down in some dramatic way in the presence of global symmetries. And that's an interesting thing to consider, but we usually assume effective field theory works. Good, that's exactly the other thing I wanted to say in the remaining time today. Which is, actually before I say that, let me say one other thing. But then I'll get to your question. The one other thing I wanna say is, so this was the third argument I gave, right? The first was string world sheet, the second was ADS CFT. This was the third, let me just mention a fourth, which is a more general ADS CFT argument by Harlow and Aguri that came out last year. They wrote two papers, one ends in seven, one ends in eight. One of them is five pages long, one of them is 150 pages long. The five page paper gives you a summary of the argument. The 150 page paper gives you a lot of details where they're very careful about exactly what they mean by global symmetries and some of these subtleties I mentioned earlier of global symmetries acting in the spatial regions and what distinguishes things like CPT from the symmetries we're talking about. But they give an argument that applies to all global symmetries. They discreet continuous NEP form. The only catch is it's only for asymptotically ADS spacetimes. And I'm not going to try to summarize their arguments since we're low on time, but I want you to be aware of it. This is certainly the most kind of rigorous version of this argument that anyone has ever given. And the only shortcoming is it's only works in ADS. Good, but now we come to the question. Okay, so what I've tried to convince you is if I haven't convinced you that quantum gravity theories don't have global symmetries, I hope I've at least convinced you that it's a plausible thing to conjecture that they don't because there are several different lines of evidence that point in the same direction. But we could sort of plot these conjectures on two axes. One is usefulness and the other is convincingness. And this no global symmetry statement kind of occupies a corner somewhere over here. It's pretty convincing. It's also pretty useless. And what we're going to do over the course of the next couple lectures is put a couple of other points on this map, like the weak gravity conjecture, which is more useful but less convincing. And a lot of the work in the Swampland business is trying to either think of more useful things to say or take some of the things that have already been said and make them more convincing. Okay, but why is the no global symmetry statement not useful? It's not useful for the reason you said. It tells us that there are not exact symmetries. But what it doesn't tell us is there can't be really, really good approximate symmetries. And in the real world, that's all we care about anyway, right? So just to remind you, for instance, in Baryon number, it's really important for us that Baryon number is a good approximate symmetry. If Baryon number was badly broken, then we would have all decayed depositrons and pions in the course of this lecture. And we would be pretty unhappy about it. Baryon number in the standard model, a good approximate symmetry. But we can just write down operators that violate it that are gauge invariant, right? Things like one over lambda squared times three left-handed quark fields times a left-handed lepton field or one over lambda squared times UUDE, right? So there are these various dimensions, six operators that violate the symmetry explicitly. And we know experimentally that the proton lifetime is bigger than 10 to the 34 years.  But that seems perfectly compatible with what we've said. We've said here there couldn't be some exact symmetry because if there was, we could imagine throwing lots of protons into black holes and carry out this thought experiment. In fact, there's an interesting exercise you could do which is to think about, as I start throwing more and more charge into this black hole, how long does the black hole evaporation process take if I'm starting with a really big black hole because I threw a huge number of protons in and how does it compare to the time scale on which the symmetry breaks. But it's not obvious that there are any problems here, right? In particular, this scale is below the Planck scale. So if I told you gravity violated the symmetry, you might naively think we would write down things suppressed by the Planck scale. Well, that's compatible with what we know. And so to apply this kind of thing to phenomenology, we need to come up with a sharper statement. It's not enough to say there are no exact symmetries because if I have a symmetry that's not exact but violated just by a really, really tiny amount for all practical purposes and all experiments you could imagine doing, that's just as good as having the symmetry in the first place. So this is an interesting statement because it's one of the few statements about quantum gravity theories that are kind of widely agreed upon as being likely to be true of all theories of quantum gravity. But my own background is in particle phenomenology even though I've been thinking about these quantum gravity questions in the last few years and as a phenomenologist, this is just not a statement that I care about. So that's where we really wanna go is to take these kinds of statements and move them up the usefulness scale. Well, hopefully also not making them much less convincing than they were to begin with. Okay, so next time I will tell you what the weak gravity conjecture is and in what sense it's kind of a stronger version of no global symmetries. Are there any quick questions I could answer now in the last few minutes? Particles? I'm not sure I heard the- Sorry, you mentioned the theory with the U1 without charged particles. Yes. That had this- They had the higher form symmetry. Right. Yeah. Since these symmetries are not supposed to be there, it's then the theory itself is claimed to be in the Swampland. The theory with no charged particles would be claimed to be in the Swampland. Yes. Then the question is, what is the real pathology of this theory? Is there, I mean by itself, even if not referring to symmetries or- The real pathology of this theory by itself. I'm not sure I have a great answer to that question. You know, it has a global symmetry, so we could try to run these black hole arguments, for example, but these arguments involve throwing things into the black hole that carry charges. So you would have to assume that you had these Wilson line-like things that you could throw into the black hole. It's actually a little unclear how that argument would work for the extended objects instead of point particles. Harlow and Aguri claim to prove things that apply to these one form symmetries, but I can't quickly summarize their argument. It's a complicated argument. So I'm not sure I have a great answer for you offhand. I guess one possible answer is you can find, charge black hole solutions. If I have a theory that has a gauge field, even if it doesn't have any charged particles, at least in classical GR, there are charged black holes. And so it would be kind of odd if those objects didn't actually exist in the theory. In particular, you could imagine set up some space where you have boundary conditions such that there's an electric field, and then in GR, in semi-classical GR, you could imagine a pair producing a charged black hole and an anti-charged black hole. And so it seems like there are physical situations you can set up where the theory would tell you you make a charged object, even if you thought you didn't have any charged particles. But I'm not sure that's a sharp argument. So I don't know that I have a completely convincing answer for your question. I didn't understand how do you use the pattern number symmetry as an argument because the lifetime of the proton is larger than the age of the universe and it's kind of you are in the, I don't know, in which limit it could be an approximate symmetry, but for me, I don't understand the argument. Let's see, I'm not completely sure if I understood the question. I guess one thing to say is the bound on the lifetime is much longer than the age of the universe, but that doesn't mean that you couldn't see protons decay, right? The reason we have such a strong bound is that we can have a large number of protons. And so even if each one is not likely to decay at an age of the universe, one out of 10 to the 23 of them, good. The other thing I guess I would say is I'm just, yeah, what was I trying to argue by bringing this up? I was just trying to say that even if you're completely convinced that there are no global symmetries in a theory of quantum gravity, all that would tell you about Baryon number is that it has to be broken a little bit. But it doesn't tell you- But you don't know. But how you know that, how you know that which is an approximate symmetry? What I'm saying is if you're convinced that theories of quantum gravity cannot have exact global symmetries, then we know that Baryon number in our universe is not an exact global symmetry, so it must be a little bit broken. So all I was trying to say here was that's perfectly consistent with what we know. Even though it's a really good symmetry to good approximation, it's perfectly consistent to say that it's not exact. Any more questions? Let's take a minute.