 OK, so the last time I was telling you about this idea that there are no global symmetries and there is a quantum gravity. And today I want to tell you about something called the weak gravity conjecture. But let me first mention just one kind of intermediate idea which is called the completeness conjecture, which was stated by Polchinski in 2003. Again, I think it's an idea that was kind of discussed earlier, but he gave a nice, clear discussion of it in this paper. And the claim of the completeness conjecture is that if you have a gravitational theory, there was a volume control somewhere, right? OK, is that better? Good, OK. OK, so the claim of the completeness conjecture is that if you have a quantum gravity theory and it contains a gauge field, you should find things that are charged under that gauge field. And in particular, in theories of quantum gravity, there's a further claim, which I may come back to and discuss a bit more, but I may not have time, that gauge charges are always discrete. OK, so with a U1 symmetry that there's going to be some quantized charge rather than any real number being allowed as a charge. And again, there are arguments based on black hole physics for why we should think that this is true. That means that if we have some set of gauge fields, we have some lattice of allowed charges, some discrete set of possible charges. And the claim of the completeness conjecture is that for any of those charges, you should be able to find some state in the theory that has that charge. It may not be a single particle, it may be some collection of particles, but there should be some way to realize all the possible charges. In particular, that means that if we have a U1 gauge theory, we should be able to find magnetic monopoles, because one of the allowed charges is magnetic charge. And this seems to be true any time we have a gravitational theory, we find that the U1 gauge fields do have magnetic monopoles that exist associated with them. If the U1 is embedded in a non-Avelian group, these are sort of the Tuft-Polyakov monopoles that you may have learned about. If the U1 gauge field comes from a Kalutsa Klein reduction, these are what are called Kalutsa Klein monopoles. But in general, some sort of magnetic monopoles always exist. And Polchinsky himself is famous for discovering charged objects. So it was known for a long time that there were theories of supergravity that contained gauge fields that were anti-symmetric tensors of some higher rank. But it was not known if there were things that were charged under those gauge fields. And one of the things that Polchinsky is famous for is that he showed that there are objects in string theory called d-brains that are charged under those gauge fields. So that's one example of this completeness conjecture. If you find the gauge field, there must be something charged under it, even if initially it's not obvious what the charged object is. And the completeness conjecture is related to the wheat gravity, sorry, is related to the no-global symmetry statement that we talked about yesterday. Through the kind of example that I mentioned, where yesterday I discussed that if you had a U1 gauge field and you had no charged particles, there would be a global symmetry associated with the absence of those charged particles where the conserved current is just f mu nu where it's dual. And so these are some higher form symmetries. But the absence of any global symmetries, including higher form symmetries, then implies that there must be charged objects. And that was really the statement of the completeness conjecture. So these two ideas of no-global symmetries and having objects of every possible charge are very closely linked to each other. But they also are both in the diagram I drew yesterday of how convincing these conjectures are versus how useful they are. Both of these are in the corner where they're pretty convincing and not very useful. And again, the reason the completeness conjecture is not very useful is it tells you that something exists, but it doesn't tell you anything else about the thing. It tells you something that is charged exists, but it doesn't tell you how heavy it is. If you're an experimentalist looking for magnetic monopoles and I tell you I'm absolutely sure they exist, that doesn't help you very much. If you don't know what are their masses, how do you make them? So these are in the corner where they're reliable statements, but they're not very powerful statements. And so the weak gravity conjecture is starting to move in the direction of making a more powerful statement because it's not just saying that charged objects exist, it's telling you something about how heavy they are. So the weak gravity conjecture is starting to take these kind of existence statements and turn them into something a little bit more quantitative. So before I tell you what the weak gravity conjecture actually says, I should tell you a little bit about charged black holes. So if I have general relativity and I also have a gauge field, it turns out there are black hole solutions where the black hole carries charge. We mentioned this a little bit yesterday, coupled to Maxwell theory. Charged black holes exist. When I say they're charged, I mean they have an electric field outside the horizon. There are different charged black hole solutions, the kind that we're going to be most interested in are static, spherically symmetric black holes, charged black holes. They're called Reister-Nordstrom solutions. Reister-Nordstrom black holes. And I'm not going to write down the details of the solution. All you really need to know is that these things exist. They have charge and they're spherically symmetric. There are also rotating solutions, rotating charged black holes called Kerr Newman black holes. But for our purposes, mostly just knowing that Reister-Nordstrom black holes exist is going to be enough. The black hole solutions exist and they can have different mass and they can have different charge. But it turns out there's an inequality that for a given charge, there's a minimum mass. Reister-Nordstrom black holes obey something that we call an extremality bound that the mass of the black hole bigger than or equal to charge of the black hole from appropriate units. The proportionality constant here that I'll come back to. Yeah. Yes. This inequality is valid for the classical solutions. So as you said, in a quantum theory, if I have a black hole, it can evaporate, it can shrink. And, right. So maybe the one thing that I should tell you is for Reister-Nordstrom black holes create this. Things that have their mass equal to their charge in the appropriate units. Again, I'll explain what I mean by appropriate units a little bit later. We call these extremal black holes. The Hawking temperature goes to zero. So extremal black holes will stop Hawking radiating. They can't admit things like gravitons anymore. And so as a result, the answer to your question is yes. The evaporation of black holes continues to respect this inequality because anytime you're in the semi-classical regime, first of all, if we're in the semi-classical regime and the black hole is slowly evaporating, you can still approximate it by the classical solution at any given time. And the classical solutions obey this bound. And secondly, if you say, well, what if I'm saturating that bound? Why can't I just admit a little bit of mass and violate the bound? The reason is that you stop emitting things because the temperature goes to zero in that limit. So there's no there's no known way to kind of push things past this bound in the context of semi-classical gravity. This is related to another set of ideas called cosmic censorship. It turns out if you try to make a black hole solution that violates this bound, if you try to find a solution to GR where the charge is bigger than the mass, this solution has a naked singularity in it. It has a singularity that is not hidden behind an event horizon. And there's no known way to take a space time that does not have such a singularity and kind of perturb it or scatter things or do some physical process and create something that does have a naked singularity. So that's loosely what's referred to as the weak cosmic censorship conjecture in general relativity. There are some known counter examples to the weak cosmic censorship story and it's kind of a long story to get into. But at least in this context of Reissner-Nordstrom black holes, the upshot is nobody knows a way to violate this bound by taking a solution and trying to perturb it somehow or thinking about some physical process. So this seems to be kind of a robust physical property of these space times, yeah. I'll say more about that in a minute, but you're right that that black holes, if there are charged particles that the black hole can emit, then the black hole charge will change as the Hawking evaporates. And in fact, the black hole will preferentially emit particles that have the same charge that the black hole has so the charge becomes smaller. In Hawking's calculation, that looks like a chemical potential that shows up for the black hole thermodynamics. In fact, even in the extremal limit, the Hawking temperature goes to zero, but if there are charged particles that can be produced, they can be swinger pair produced in the electric field outside the black hole. So even if Hawking radiation has sort of formally shut off, the black hole could still lose its charge. So in general, we do expect that the charged black holes, if they can emit charged particles, will tend to not become more extremal over time. They'll tend to become less extremal over time because they'll be losing their charge to particles that they emit. Okay. So before I say more about that, let me just tell you roughly why this inequality is true. And there's nothing very mysterious about it. You can find that it's true by finding the classical solutions, but you can understand why it's true in a simpler way, which is just that we're talking about some black holes and they're charged. And so they have an electric field. An electric field outside their horizon. And the statement that there's a minimal mass for the black hole is really just the statement that there's energy stored in this electric field. So if you have a charge, you have an electric field. If you have an electric field, you have some energy. And so there's some minimal amount of mass in the form of that energy. And so parametrically, you can understand this inequality just by saying the mass of the black hole bigger than the integral over space outside the black hole horizon, the energy stored in the electric field. And this is roughly, if the black hole has charged q, then this integral is roughly this. The electric field falls off like one over r squared just by dimensional analysis. And we're integrating the square of that over the space outside the black hole. So we have an integral r squared dr over r to the fourth. And so that goes like one over r. But the definition of a black hole is that the mass is the same as the radius in Planck units as it's a black hole is times in Planck squared. And so if I plug that estimate of the radius into this, the mass is bigger than around, bigger than or around this quantity divided by the radius. But that means the mass bigger than or around e times q times the Planck mass. Okay, so that's the parametrics of this. There are squiggles all over the place here. I'm not claiming that this is right to within order one factors, but at least kind of order of magnitude, you can understand where the extremality bound came from just by calculating the energy. And what you see is that also the coupling constant of the gauge theory showed up here. So when I write an e into q, sometimes in the context of black hole physics, people will just call this product e times q, q. But what I have in mind here is that q is a quantized charge whereas e is a continuous coupling constant. So in the standard model, q for the electron is minus one, e is something like 0.3 is the square root of four pi alpha. Okay, so black holes that are charged obey some inequality that looks like this. And if you have a specific black hole, you can check the order one numbers. So it turns out for Reiss-Neuer-Nordstrom black holes, four dimensions, the inequality looks like this. It has a square root of two in it. In the conventions where my gauge field kinetic term has a one over e squared in front where m Planck squared is one over eight pi g Newton. Reduce block scale. So that sets the conventions and then you find the numerical factor is the square root of two. But the square root of two was not super important for our discussion. What you should keep in mind is just there's some order one number here and the black hole mass is bigger than its charge in block units. E dimensions, our last version of this. Again, there's some order one number that I'm not going to write down. And if you're in a different number of dimensions from four, the gauge coupling constant will not be a dimensionless number anymore. Also, g Newton will have different dimensions. But when you put them together in this combination, it works out to have the right units to give you a bound on the mass. So this is a general fact, not just about 40 gravity, but about any number of dimensions. The black holes will satisfy some externality bound. Turns out that extremal black holes are interesting objects. I mentioned already that their temperature goes to zero, so they're not Hawking evaporating. It also turns out that the force between two extremal black holes, like charge, same sign charge, at least the leading order long range force. So if I take two massive objects that are far apart from each other, they'll have a one over R force pulling them together from gravity. But if they're charged and their charges have the same sign, they will also have a one over R force pushing them apart from electromagnetism. And it turns out that for extremal black holes, these two forces exactly balance each other. So that if I have two of these extremal black holes that are far apart, there's no long range one over R potential between them. Statements generalize to a big class of theories that also have massless scalar fields. So to give you one example, we can consider what's called Einstein-Maxwell-Dillaton theory. This is the theory where Lagrangian contains gravity through the Einstein-Hilbert action, but also contains scalar field by and where that scalar field couples to the gauge field. So the gauge field kinetic term is not just f mu nu squared, but has this extra factor of e to the minus something times phi in front. In this case, charged black holes have a scalar profile. So the black hole has an electric field outside its horizon, but this term in the Lagrangian tells us that an electric field sources the scalar phi. So if you have a trivial electric field, you can't avoid having the field phi also turned on. So there's some solution for phi of R. And there's also energy stored in the gradient of that scalar field. And so the extremality bound changes, but parametrically, it has the same form. The mass of the black hole bigger than the square root of two over one plus alpha squared times e times the charge of the black hole. So when alpha goes to zero, we turn this coupling off then the scalar doesn't do anything and we get back the bound that we had for Weisner-Nurzstrom. But when alpha's non-zero, you get a different bound. But qualitatively, the properties are the same. It's still true that there is a bound. It's also still true that if you take two of these black holes and they're far away from each other, the net force between them is zero. There's now a force mediated by the scalar. It's an attractive force, but the sum of the scalar force and the gravitational force, again, exactly balance the electromagnetic repulsion. And this is much more general. So you can show that this is true for a wide variety of different ways of coupling scalar fields to the gauge theory. So these are kind of generic properties of charged black holes. And now we can ask what happens to charged black holes as they evaporate. Okay, we've already discussed this a little bit, but let's collect what we know. So evaporating charged black holes, they prefer to discharge if possible. Again, the distinction between yesterday and today is yesterday we were talking about black holes charged under global symmetries. There was no electric field. And so they didn't preferentially discharge. Now we're talking about black holes charged under gauge symmetries. There is an electric field and they do prefer to discharge. Question. Yeah, good. So the claim that the long range force goes to zero generalizes to many kinds of scalar couplings. The claim that the Hawking temperature goes to zero also generalizes to lots of relatively well-behaved scalar couplings. It turns out this dilaton case that I wrote down is kind of special. There's some range of alpha for which the temperature goes to zero. And there's another range of alpha for which the temperature goes to infinity. And the case where it goes to infinity is a little bit hard to think about what that physically means. In particular, when you get close to the horizon of such a black hole, you would not expect that your effective field theory is valid anymore. So yeah, there's something slightly pathological about that case. And I'm forgetting exactly what the condition is. It's for alpha bigger than some critical value of the temperature goes to infinity. But for lots of other kinds of couplings, the temperature does go to zero still. Okay. So these black holes prefer to discharge if possible and there are kind of two mechanisms for that. One is the E field implies that there's a chemical potential. So if you're thinking about this from the viewpoint of black hole thermodynamics, these things have a chemical potential. And the other thing, as I mentioned, is the Schwinger pair production can happen. You can make a charged particle and it's anti-particle E field. That's true even at zero temperature. Okay. So what that means is black holes tend to become less extremal if possible. In particular, we don't expect to find charged extremal black holes in the real world. For one thing, because if they were charged, they would tend to attract oppositely charged things to them and neutralize themselves. For another thing, because charged black holes tend to not keep their charge. And so in the real world, if you could somehow make a black hole with a big charge, which would be a hard thing to do in the first place. But if you could, it would very quickly start emitting electrons and discharge. Everything I've been saying has this caveat attached to it that they prefer to discharge if possible. So you could say, what if there are no charged particles? There are no light charged particles. Then the black holes hocking radiate uncharged particles, the black hole mass decreases, charge doesn't change, and the black hole will approach extremality. But that's where the comment about the hocking temperature becomes important. Black holes will radiate, they'll get smaller and smaller and keep their charge. But as they get closer and closer to hitting the bound, their temperature gets closer and closer to zero. And so the rate at which they're radiating gradually turns off. And you don't get any sort of paradox where the radiation pushes them past the bound. Hocking temperature goes to zero, black holes stop evaporating. So yesterday we talked about how if we had a global symmetry, then black holes could evaporate. And we could end up with black holes of any charge with the same fixed mass. And so there were infinitely many different things of different charge in the same, that you could fit in the same spatial region. And so you would badly violate some sort of entropy balance. Here the argument doesn't really work the same way because we can have these black holes and they can evaporate, but for a given charge they can only evaporate so far before they hit the extremality bound and they stop evaporating. So if we were to plot, again as we did yesterday, mass versus charge, and again there's some kind of minimum mass in which we trust the whole effective field theory set up to talk about what black holes do. But now there's also this separate extremality bound. This says the black holes always lie above this line. And so if I start with a big black hole of some charge, first if there are light charge particles it can emit, it will tend to evaporate in some way like this, where it loses both charge and mass because it's emitting lots of charged particles. If there are no charged particles it can emit, it will evaporate like this, but it'll stop when it hits that line. So we don't get the same kind of paradox because we're not populating things that all have the same mass. So that's the story of what happens if we have charged black holes in a theory with or without charged particles. And I would say that in the things I've told you there's no obvious problem. It's not like the case of a global symmetry where we could talk to ourselves into thinking there's some kind of paradox or something has to be violated about our intuition about how physics works. Then the less, in 2006, our Connie Hamid model, Nicholas and Vafa lectured that this second case with no charged particles never actually happens. We call the wheat gravity conjecture. The wheat gravity conjecture says efficiently light charged particles will always exist such that extremal black holes can shed their charge. So if we have an extremal black hole, appropriate units of mass is equal to its charge. There's some process by which this black hole can turn into a smaller black hole mass bigger than its charge. So this black hole is not extremal. Then just kinematically, whatever else came out of this process has to be some object whose mass is smaller than its charge. This was an extremal black hole. This is a black hole that's not extremal. We also call that sub extremal. Whatever the other thing is that came out of this must have been super extremal in an object that violates the extremality dump. So another way to say that is that the wheat gravity conjecture is the conjecture that something exists mass divided by its charge less than or equal to the mass divided by charge of extremal black hole. So that is what the wheat gravity conjecture says. So for example, for four dimensional Einstein Maxwell theory, if we just have gravity in photons, this would be the conjecture that there's some particle whose mass is less than or equal to the square root of two times the gauge coupling times its charge times the plong mass. The first thing we can do is check is this true in our universe? So maybe I should ask you, do you know a particle whose mass divided by its charge is less than the square root of two times the plong mass? So what is the mass divided by charge in plonk units for the electron? Quite that big. But yeah, the reduced plonk mass that appeared in the formula is something like two times 10 to the 18 GeV. The electron mass is five times 10 to the minus four GeV. And so the electron obeys this bound with 20 something 22 orders of magnitude to spare or something like that. So this is easily true in our universe for you want electromagnetism, but the conjecture is that it's true for any theory of quantum gravity. Anytime you have a gauge field, you will find something charged under the gauge field whose mass obeys this bound. So one thing to notice is this is strictly stronger than the completeness conjecture. The completeness conjecture just said things exist that are charged. This says things exist that are charged, but they can't be too heavy. Actually, sorry, as stated, this is not strictly stronger because the completeness conjecture requires all possible charges to exist. You could say, what if my minimum charge here is a thousand or something and I just don't have anything with smaller charge? So it's not really strictly stronger than the completeness conjecture, but it's stronger than the completeness conjecture in the sense that it's making the existence into a quantitative statement about the mass range where something exists. That's a good question. Let me come back to that question in a minute. Okay, the next thing I wanna say is that is another way to formulate this conjecture, which in the context of Einstein-Maxwell theory is equivalent. Question? The best rationale I can give you is to go back to no global symmetry. So if we have charged objects, but we don't have charged objects of every possible charge, then you'll have some sublattice of the complete charged lattice that's occupied by charged states. And the quotient of the full charged lattice by that sublattice will be some discrete group. And the claim is that that discrete group would be a global symmetry of the theory that would act on extended objects like Wilson lines. So if you do not complete the full charge lattice, you will have a global symmetry. It'll be one of these funny higher form symmetries that'll be discrete. It won't be the kind of global symmetry that I can give you like the black hole argument for why it shouldn't exist. But if you really believe the strong version of the statement that no global symmetries exist, then that will require you to fill the whole charge lattice. You could gauge, right, right. You could say that discrete symmetry is gauged, but if that discrete symmetry is gauged, it means that you're really not allowed to have things in those sites on the charged lattice in the first place. So in that case, I would say you just sort of misidentified what your charged lattice was. And if you identify it correctly, there really is something in every allowed site. Okay, so the weak gravity conjecture, as I've stated it here, is a statement about extremality. It's a statement that there are particles whose mass to charge ratio is smaller than that of extremal black holes. But there's an equivalent statement in the context of Reister-Nordstrom black holes. So for Einstein-Maxwell theory, equivalent conjecture, let me rephrase this, there exist large particles such that if I take two of the particle and put them far apart, they will repel each other. Okay, again, the electron easily satisfies that. You know that if you have two electrons, the electromagnetic force between them will cause them to repel each other, and that's much stronger than the gravitational force. And this is the reason why the conjecture was originally referred to, or why we commonly refer to it as the weak gravity conjecture. Because in some sense what it's saying is there must be some particle for which gravity is a weaker force than electromagnetism. If my theory contains an electromagnetic force, there must be some object for which that force is a stronger force than the gravitational force. And again, of course, this is easily true in our universe, electromagnetism is a much stronger force than gravity. But I think it's worth keeping in mind that these statements are equivalent for Reissner-Nordstrom black holes, or for Einstein-Maxwell theory, where the only forces are gravity and electromagnetism. Because we said that extremal black holes are things for which those forces balance each other perfectly, and so the conjecture that there's something super extremal is the conjecture that there's something for which the gravitational force is weaker. However, if I have a theory that also has other kinds of forces, like things mediated by massless scalar fields, these two conjectures are not necessarily the same anymore, because there could be particles that interact differently with the scalars than the way that black holes interact with the scalars. And so it's not necessarily obvious, and it's not necessarily true in every example, that checking whether particles obey this bound on their mass to charge ratio is the same as checking whether they repel each other. Their charge could be bigger than their mass, so electromagnetism wants to repel them, but if they interact really strongly with some scalar, that could attract them more. And so recently with collaborators Ben Heidenreich and Tom Rodelius, I wrote a paper where we suggested that it's useful to distinguish these two things, and we call this one the repulsive force conjecture, because it doesn't actually refer to gravity and statement. And so in principle, in theories with massless scalars, these are two different conjectures. So far, we have no evidence that either of them is false. They both seem to be true in every example we know in string theory, but in some cases they are different. Question? Okay, so the question, if some of you didn't hear it, was what about for non-abiliy engaged theories? So for instance, if there's a confining gauge theory, there's no long range force associated with it. So what does this conjecture tell us? And what I would say is the kind of minimal version of the weak gravity conjecture that we've been talking about doesn't really have anything to say about non-abiliy engaged theories. One reason for that is, again, if the theory is confining or not actually associated with long range forces, then there's no such thing as an extreme old black hole. You can't make a black hole that has an electric field outside its horizon if the force is short ranged. And again, there's also no long range force to talk about this other repulsive force conjecture. So the minimal version of this conjecture, the original version of this conjecture doesn't have anything useful to say about non-abiliy in theories. Even if you just take this inequality to be a defining statement of the conjecture, this will be trivially satisfied in non-abiliy in theories because there is a charged object, namely the gluon itself, which has zero mass. Nonetheless, I think that there are versions of the weak gravity conjecture that do have useful things to say about non-abiliy in theories and I'll try to come back to that a bit later. But one way to give you some intuition for why that should be true is if I take, say, an S-U-N gauge theory and I compactify that theory on a circle. So I go to a theory in lower dimensions. I can turn on a Wilson loop for that theory around the circle, okay? So I can turn on a background gauge field for which e to the i integral a around the circle is not zero. And in the presence of such a Wilson loop, generically, S-U-N breaks to u1 to the n minus one. One way to think about that is that this, from the viewpoint of the lower dimensional theory, this looks like some adjoint scalar field. And so effectively it can higgs the gauge group down to the maximal torus that lives inside the gauge group. And so if you believe that the weak gravity conjecture has something to say about u1 gauge theories and you're interested in a theory that has some other gauge group, you can imagine compactifying that theory on a circle and getting a theory that has u1 gauge groups and then thinking about what the weak gravity conjecture tells you about that theory. And so that's at least one reason to think that the existence of a conjecture about u1 gauge theories might actually have something to tell us about other gauge theories. And I'll try to come back to this point later on when I say some things about some stronger versions of this conjecture. Yeah, great. That is a good question. Yeah, that is a good question. I would say that you should not find this argument convincing because, right, what they say in the original paper is sort of that, a theory that violates this would have infinitely many different stable objects and that seems weird or I guess to some people that might seem pathological, but it's not actually obvious that anything really bad happens. So as I was saying, unlike the case of global symmetries, you can't argue that you get many different states of the same mass. Okay, that you could argue is a pathology. Here you get many different stable states, but as you go up in charge, they're getting heavier and heavier. And so it's not so obvious that you're violating some bound or anything. There are various words you can say about why this might be a problem, but I don't find any of them super convincing. So there's something called the third law of thermodynamics. Which has various statements. One version of it is that you shouldn't find zero-temperature systems that have finite entropy. Zero-temperature things should just have one ground state. Extremal black holes violate that. Their Hawking temperature is going to zero, but they're still big high entropy objects. And if you have a theory that violates the weak gravity conjecture, you will have lots of these black hole states that sort of spontaneously as they're evaporating, they get closer and closer to zero-temperature. But again, I don't know why this third law of thermodynamics should be true. I don't even know what the correct statement of it is in general. If you try to look it up, you'll find versions of it that say things like you shouldn't be able to reach zero-temperature in finitely many steps. I don't know what a step is. Yeah, it's hard to find really sharp things to say that make you believe this is a problem. So I don't think you should be convinced by the argument. In the last few years, there have been several different attempts to prove the weak gravity conjecture in different ways. And I would say none of them have been fully convincing, in my opinion. Nonetheless, there's a lot of non-trivial evidence that this conjecture is true in all the examples we know of quantum gravity theories. But I think it's still an open problem to give a totally sharp, crisp argument for why you should believe it has to be true. Is that a question? What does ADS-CFT say? Good question. There have been attempts to prove this using ADS-CFT. Maybe I can give you one name. I don't remember the, I don't have the archive number written down. But there's a paper by Miguel Montero that came out, I believe. I think it was last year. It may have been the year before, but within the last year or two. It tries to prove this using ADS-CFT and some theorems in quantum information theory. So what Miguel does in that paper is basically to try to argue that extremal black holes are not compatible. If you have an extremal black hole in ADS, and then you try to interpret that as some state in the CFT, he wants to argue that these states are incompatible with things we know about quantum information. And the way he does that is to compute some entanglement entropies in the presence of these stable extremal black holes, and argue that the entanglement entropies scale with the area of the region you're computing and the entanglement over in a way that's incompatible with some theorems. So that's one attempt that I'm aware of to use ADS-CFT to make a really sharp statement. I think it's a very interesting paper. It has a lot of technical subtleties in it because these theorems about quantum information are really theorems about lattice systems with sort of discrete sites and low dimensions. And what Miguel has to do in the paper is take this kind of continuum theory and try to argue that those theorems can be applied to it. And so I can't say that I've been completely convinced that I understand all the details well enough to know that I believe his proof, but I also can't tell you that I know any flaw in the proof. It's just, it's really technical and full of subtleties. So I think there's still room to try to use the ADS-CFT correspondence to give a proof that's kind of clear enough that everybody can agree that it's definitely correct. And I'm not saying that Miguel hasn't done that, but I think at least we need more kind of exposition and more thinking about all these details to be sure that we really understand them all. Okay. So what I would say about the weak gravity conjecture is that I called it the weak gravity conjecture. I didn't call it the weak gravity theorem because I don't think it has been proven in complete generality, but there have been a lot of proofs that with certain assumptions can argue that this follows from some set of general principles. And I wanna mention one of those attempts or one direction in which to try to attempt that, which goes back to a question that I kind of told you earlier I would come back to which was the question of how big can this charge be? So in the weak gravity conjecture we said there's some charged particle that obeys this bound, but I didn't tell you that it had to be a particle of charge one or even a particle of small charge. I just said there's some charge. And so if you really take that version of the statement you could ask what if the charge is just enormous? What if the charge is 10 to the 20? And in that case you might wonder if I think of objects with really big charge maybe those objects just are black holes. And you could ask, could a black hole itself obey this bound? And your first thought would be well, this bound is the opposite of the extremality bound and black holes obey the extremality bounds so surely a black hole should not be able to obey this bound. But then we get to the question of this less than or equal to that I wrote. I didn't write strictly less than I wrote less than or equal. So the first thing to say is why less than or equal and not less than? Because if it's really less than or equal you might say well, extremal black holes just obey the bound anyway. So isn't this trivially true? And the reason why we don't conjecture less than is that in some super symmetric theories, theories that have 40 and equals two or higher, Susie, there are what are called BPS bounds, which in some cases can imply that the mass is bigger than the charge in appropriate units for all states in the theory. So in those examples, you're never going to find something that obeys the weak gravity conjecture with a strictly less than because it's just incompatible with super symmetry. Instead what you find are these BPS objects that exactly saturate the bound. BPS objects exactly saturate the weak gravity conjecture. So that was recognized already in the original paper on the weak gravity conjecture. They realized that there were cases where the bound was just going to be exactly saturated and they wrote a less than or equal to because of that. I should also mention that there's been a lot of confusion about the relationship between BPS bounds and extremality bounds in the literature. So there are cases where extremal black holes are BPS there are also theories that have BPS bounds and that have extremal black holes but where the extremal black holes are not BPS the two inequalities are different. And there's a paper that's coming out sometime soon by Murad Alim and Haydn Reich, Tom Rodelius, that's going to explain in detail exactly when BPS bounds and extremality bounds agree. And so if you're interested in that issue I'll just refer you to that paper which I think should be out by the end of the summer. But for the purposes of this discussion all you need to know is that there are these supersymmetric cases where it's going to be an exactly equal to rather than a less than. There's also an expectation that the only case where it's exactly equal is in the supersymmetric case because otherwise there's no deep reason why any particle should just saturate the bound. There's no symmetry to protect an exact equality but the fact that there's no symmetry also applies to black holes. So I've told you that there are black holes there are black hole solutions in classical GR that have any mass bigger than or equal to their charge. That in fact is a statement, existence black holes with any m bigger than or equal to q statement about the two derivative classical theory. So what do I mean by two derivative? I mean the action for gravity, Weinstein-Hilbert and the action for gauge theory is F me a new squared. But we're not talking about classical gravity when we talk about the weak gravity conjecture we're talking about quantum gravity. And in the context of a quantum theory we should expect corrections. We should expect corrections from quantum mechanical effects from loops. We should also expect that our action is a completely general non-renormalizable action that includes higher derivative terms. And so one concrete question you can ask is what happens to black holes higher derivative terms in our action? So for instance we can add four derivative terms to our action. These can involve things like some coefficient times F me a new squared squared plus some coefficient times F me a new F new row F sigma, F sigma mu plus some coefficient times Riemann tensor R me a new row sigma F me a new F row sigma and so on. There are several different operators you can write here. In four dimensions it turns out that these are the only three you really need to care about because you can use you can use equations of motion and fieldery definitions to reduce other things to these. In higher dimensions you would also add Gauss-Benet. Okay but in general there are just all these things that have four derivatives in them and we can add them to our Lagrangian. And these change the black hole solutions. They change the corrections to our black hole solutions that come from these corrections to our Lagrangian and they not only change the solutions they change the extramality bound. And what they do to the extramality bound is they make it look something like this. For Reister-Nortzstrom we previously had this inequality but now we're going to get this inequality times one plus some coefficient C divided by the black hole charge squared plus things that are higher order where the C is some linear combination of the coefficients that appeared in our Lagrangian. So we've added higher derivative terms and if you think about a big enough black hole your curvature is really low near the horizon and so these higher derivative terms are going to be suppressed. Basically they're going to come with extra powers of one over the radius of the black hole and for an extramal charge black hole the radius is equal to the charge in Planck units. And so that's why we have this expansion of the extramality bound and inverse powers of charge because we're doing a derivative expansion and so this is really inverse powers of radius. And so for a really big black hole these things are not going to make much difference and the bound is going to look like it did before. But as your black hole gets smaller and smaller the corrections are going to be bigger and bigger up to the point where you just lose control of your effective field theory. So the claim is that if I have a theory without some symmetry like supersymmetry that protects the form of the extramality bound I'm going to get corrections that make the extramality bound different than the usual one. And so if the usual extramality bound was this line what's going to happen is for really big black holes the bound is the same but as you move away from it the bound is going to curve away from that bound. And either we're going to get a stronger bound. In case one the black holes have to be even heavier than they did before and the bound is going to be that they're above some curve like this one. For case two the bound is going to be weaker and the black holes can be a little bit lighter than they were in the tutor of the action. Now in case two say that black holes themselves obey the weak gravity conjecture. So we define the conjecture relative to the mass to charge ratio of the really big black holes. And now if we look at smaller black holes they can have a smaller mass. The big black holes would be allowed to at that charge if the curve bends down. And this possibility was again already pointed out back in 2006 in the original paper. This calculation of the change in the extramality bound was done in a paper by Yevgeny Katz, Kubash Matal and Megapati also in 2006. And what's happened in the last couple of years is various people have presented claims of proofs that in fact case two is the only thing that ever happens in a consistent theory. Aim that consistency requires case two. And there are various arguments for this. There's an argument based on black hole thermodynamics and a paper by Cliff Chung and Grant Rehman. I wanna say 2017 but I don't remember exactly when the paper came out. There's a claim that you can derive this from Unitarity in a paper by Miradad Mirababayi who works here. So you can try to talk to him at some point if you're interested. There's a distinct claim, a different Unitarity argument in a paper by Artani Hamad, Khawam and third author whose name I'm forgetting at the moment. See if I can remember it later in Philoden. But that paper has not been published yet but various talks have been given on the result. I would say that none of these arguments are completely general. They all involve various assumptions. They involve assuming some kind of separation of scales before you integrate out some particles. Actually, the Artani Hamad argument is probably the most general looking although it's specific to four dimensional theories whereas you might think this should be true more generally. But in any case, I think these arguments are very suggestive and they strongly suggest that any consistent theory of quantum gravity is going to have this property. That if you don't have supersymmetry protecting your black holes, your black holes are going to get corrected such that they become a little bit lighter than they would have been at leading order. And because of that, the minimal version of the weak gravity conjecture is again, not a very useful statement because you might have hoped that it's telling us about the existence of light particles in our effective field theory. And those particles would be interesting to know about but if what it's really telling us is just about tiny corrections to big black holes, then it's a much less useful statement. So gravity conjecture can be satisfied by small corrections to big black holes. I've told you about the no global symmetries conjecture, I've told you about the completeness conjecture and I've told you about the weak gravity conjecture and I've told you that so far all of them are pretty useless if your goal is to make statements about phenomenology. But what I want to tell you now is that there are stronger versions, wrong forms of the weak gravity conjecture that are more useful for phenomenology and I have just a few minutes left for today so let me just very briefly give you an argument that was already in the original weak gravity paper starts to hint at why this might be a useful statement and this is what's called the magnetic weak gravity conjecture. So again, this was already in the original paper, they said, well we've postulated the charged objects exist whose charge is less than that of an extremal black hole. We could also postulate that magnetically charged objects exist whose mass is smaller than the mass of a magnetically charged black hole and so some sort of magnetic monopole should exist whose mass is less than its coupling one over E times its magnetic charge. Magnetic monopoles couple with a coupling that's one over the coupling that electric particles couple with. Okay, so that by itself is just kind of applying the weak gravity conjecture to the magnetic charge, that's nothing new. But then they made a second statement. That's claim one and now we have claim two which is the classical radius of a monopole is a UV cut off. So what do I mean by classical radius? Same as the classical radius of the electron, what I mean is integrate the energy stored in the field down to whatever radius makes that energy equal to the mass. Classical radius are classical find the energy in the key field equal to the mass of the monopole and from this you can calculate the classical radius is roughly the charge of the monopole squared, the magnetic charge squared divided by E squared times the mass of the monopole. The radius at which if you integrate the B field, the energy in the B field down to that radius you get the total mass and the claim was that our classical inverse can be identified with some UV cut off on the theory because when you get into the core of the monopole you need to understand the high energy completion of the theory to know what happens inside the core and now they just take the statement, plug it into this and derive that the magnetic WGC implies that the UV cut off is less than E times M plot. So we're just replacing the monopole mass by Q squared over E squared times R and then comparing the left-hand side to the right-hand side and we find one over R is less than times Q times M Planck sorry, is less than E divided by Q times M Planck. Q is always bigger than one so if we take Q equals one we get the weakest bound which is just the lambda is less than A times M Planck. Okay? So this is a much more interesting statement because now we're claiming that if I send the gauge coupling to zero the energy up to which I trust my theory is getting smaller and smaller. So my effective theory is somehow not going to be good if I start making the gauge coupling really small. So that claim was in the original paper and what we've learned in recent years is there are other stronger versions of the weak gravity conjecture that helped to explain what happens at this energy scale and in particular the claim will be that at this energy scale we always get not just a single particle but a tower of different particles with different masses. That's what I'll explain in the next lecture but in the last couple of minutes are there any questions about what I've said today that I can answer? Yeah.