 Okay, thanks. So we discussed how if we had black holes characterized by their mass and their charge, if we had black holes with global symmetry charge, they would Hawking evaporate down to some minimum size while maintaining most of their charge and we would get infinitely many states of a given charge. So what we've seen is that for black holes that carry charge under a continuous gauge symmetry, there's some diagonal line in the space and black holes only live above this line. So this is the extremality bound. And so if we have black holes in a theory with gauge charges and they Hawking evaporate, if there is no way for them to shed their charge, if there are no light particles with charge that they can emit, then their Hawking evaporation will cause them to lose mass but they will stop when they hit the extremality bound and they can't evaporate anymore. If there are light charges, then in fact the black holes will tend to preferentially reduce their charge as they evaporate to follow some curve like this. But the important point is that if we have gauge charge, unlike global symmetry charge, we have only finitely many black hole states in a given range of mass for which we trust effective field theory. And so we don't reach the kind of sharp conclusion that we reached before. On the other hand, coming back to the question that Ateesh asked at the end of the last lecture, you could say if we have a very, very weakly coupled gauge theory, that starts to look a lot like a global symmetry. The limit where I send the gauge coupling to zero, it's hard to distinguish between a global symmetry and a gauge symmetry. And so if theories of quantum gravity really don't like to have global symmetries, you might think they also really don't like to have very weakly coupled gauge symmetries. Now maybe if we had precise enough arguments about the density of states, we would still run into a problem here because this number might be finite but large. But I don't really know any successful attempt to make that argument completely precise. Nonetheless, this kind of thinking that having a very weakly coupled gauge symmetry should start to look like a global symmetry and that should be forbidden is what led to the weak gravity conjecture. So the weak gravity conjecture originated in a paper in 2006 by Nima Arkani-Hamed Lubash Matal, Alberto Nicolas, and Cameron Vafa. And their statement of the weak gravity conjecture was that if I have a consistent theory of quantum gravity that contains a U1 gauge field, then my theory should contain at least one charged particle. Maybe particle is a little too precise. Some sort of charged object whose mass to charge ratio is smaller than that of an extremal black hole. So we saw that for black holes, there's a minimum mass to charge ratio and the claim is there should be something in the theory that violates that bound. Something whose mass is smaller than it would be allowed to be if that object were a black hole. And maybe at this point it's important to qualify one point about the previous argument. So we said that black holes had a minimal mass because they have an external electric field and we can integrate that electric field down to the black hole radius and that stores some energy which contributes to the mass. The distinction between light particles and black holes is that when we make this kind of argument that the mass should be at least as much as the energy stored in the electric field, for a light particle we would only make this argument down to the Compton radius rather than the Schwarzschild radius. So black holes are heavy in Planck units which means if you're integrating n from infinity you hit their Schwarzschild radius before you get to their Compton radius. Whereas for a particle like the electron if you're integrating n from infinity you hit its Compton radius long before you reach its Schwarzschild radius. And for that reason these are different calculations. So what this inequality ends up looking like is a bound on the charge relative to the mass which depends on the theory. So again in Einstein Maxwell theory in four dimensions for instance you would claim there should be a particle of mass m in charge q which obeys this inequality. The first thing you might notice is this is true in our universe. If our u1 is the u1 of electromagnetism the electron mass is 511 keV whereas the square root of two times e times m Planck is something like 10 to the 18 keV and so this bound is not even close to being saturated it's satisfied by more than 20 orders of magnitude by the electron in our universe. So if you want to draw a phenomenological conclusion from this conjecture you're going to have to look for something a little bit further if you'll then just electromagnetism. It's necessary that the charged particle is stable. It's not really necessary but if the particle can decay to something else that's lighter than one of the lighter things it decays to will always also obey this inequality and so I think you could keep iterating until you eventually hit something stable. Okay good that in fact that statement about decay leads me to one motivation for this conjecture which is that the weak gravity conjecture try to stop that wrestling. The weak gravity conjecture allows for extremal black holes to discharge so an extremal black hole in appropriate units has mass equal to its charge. If it decays to two lighter objects then just kinematically one of them must have mass bigger than the charge and one of them must have mass smaller than the charge and so the weak gravity conjecture can be thought of as the kinematic condition that allows for extremal black holes to shed their charge. The other thing to notice about the weak gravity conjecture as we were discussing before the extremality bound for black holes tells you that for black holes the gravitational force always overcomes the electromagnetic force. This is precisely the opposite. If I have two particles that satisfy the weak gravity conjecture and I take them far apart their electromagnetic repulsion which scales like E squared Q squared is stronger than their gravitational attraction in the limit that they're non-relativistic which goes like their mass squared over M Planck squared. So this is the origin of the name weak gravity conjecture. This is a conjecture that there must exist some particle for which gravity is a weaker force than electromagnetism. The original conjecture was a statement about a single U1 gauge field in a theory with gravity. It didn't really refer to massless scalars or other possible particles in the theory. And as I mentioned before the form of the extremality bound really depends on the additional content of the theory. So if we have a theory with a dilaton for example that couples to the gauge field the coefficient in this inequality will change. And what we have now are in fact two different statements that you might imagine generalizing the weak gravity conjecture to in a theory that contains scalars. So I wanna distinguish those two statements. One of these I will refer to as the repulsive force conjecture. And this is the statement that there exists some particle which at long distances has a repulsive force between two copies of itself. And the other I will still refer to as the weak gravity conjecture which is that there exists a super extremal particle that is one that violates the extremality bound. In the case of Einstein Maxwell theory where I just have a photon and a graviton these are equivalent statements. But if I have a theory with scalar fields for instance the weak gravity conjecture only makes reference to how those scalar fields couple to the gauge fields. When I find a solution in a theory where my gauge field kinetic term depends on some scalars I'll get some extremality bound that depends on that scalar coupling. Whereas this conjecture depends on how the particle that I'm talking about couples to the scalars because the scalars will mediate some attractive force between two copies of that particle. And depending on whether that particle couples more or less strongly to the scalars in the black holes do this could be a different statement. So these two statements because they kind of collapse into each other in the original case of just Einstein Maxwell theory these statements haven't always been carefully separated from each other in the literature but it's important when you're reading about cases that have scalar fields as many supersymmetric theories do they have massless moduli fields. It's important to keep track of which one of these two things is meant by weak gravity conjecture. One general comment to make about these conjectures is that they all have the form there exists a particle. It's not a statement that every particle in the theory has to obey this bound it's just a statement that there must be some particle in the theory that obeys this bound. And the fact that it has that form sort of signals that these kinds of statements are not going to be extremely easy to prove. So there can't just be any general argument about taking two of these particles and doing a scattering experiment or something and applying general principles like unitarity that are going to immediately tell you this result because this is a result that applies to some particles and not to other particles. So that in some ways makes it a bit harder to imagine how you would prove this as a general statement. Another aspect of this definition that's a little bit fuzzy is what do I mean by particle? I said particle or object. Because in fact, if I don't specify that this particle has to be light, it turns out that the weak gravity conjecture could actually be satisfied by black holes themselves. And that sounds at first glance like a contradiction because I told you that black holes always obey the extremality bound and this inequality is the opposite of the extremality bound. But the first thing to say is there are certain theories where there are BPS extremal black holes. So in some theories there are black holes for which BPS bounds, which are a consequence of supersymmetry and extremality bounds coincide. There are cases where these are the same bound and in those cases BPS black holes saturate the bound. And so I wrote a less than or equal to sign over here for a reason. The reason is that we know theories where there's nothing strictly less than because anything strictly less than is just forbidden by supersymmetry. And so in those cases, for the weak gravity conjecture to have any hope of being true at all, we have to allow for the case where these are actually equal. And in that case, BPS black holes can be the things that obey the bound. But if I look at a more general theory where the BPS condition is not the same as the extremality condition, then supersymmetry doesn't protect the form of the extremality bound. And as a result, the extremality bound gets corrections. So I told you that the extremality bound looks like the mass being bigger than some multiple of the charge. And that's true if I find classical black hole solutions in the simplest theory with only a two derivative. But more generally, we expect that any theory is going to contain higher dimension operators suppressed by some ultraviolet scale like the string scale or the Planck scale. And if I find black hole solutions in those theories, they will differ from the black hole solutions in a theory with just a two derivative action. And so we could write down various four derivative operators in the Lagrangian, like F mu nu to the fourth power or a Riemann tensor times two f's or other terms, gaspinae. And these operators modify the black hole solutions. But because they're higher dimension operators, they don't modify big black holes very much. They have a bigger effect on smaller black holes. And if you compute the black hole solutions in the presence of these higher dimension operators, you find that there's still an extremality bound. And it has the form that the mass is bigger than the charge times some constant as we saw before. But now times some corrections that look like one plus some other constant divided by the black hole charge squared, more sub leading corrections, where this coefficient C, C1, some linear combination of coefficients of the four derivative operators. So the extremality bound that we're used to is really a statement about Einstein Maxwell theory with just the Einstein Hilbert action and the minimal kinetic term for the photons. As soon as we start adding these more complicated terms, we get different results. And the calculation of the bound in the presence of these operators was done in a paper by Yevgeny Katz, Thuvash Mantel and Megapati in 2006. So if you wanna know exactly how this coefficient C relates to the coefficients of these various operators, you can find it in that paper. And so depending on the sign of this linear combination of operators, we're either going to find that black hole states are not allowed to be quite on the extremal line, but have to be a bit heavier in the case that C1 is positive or in the case that C1 is negative, they're now allowed to lie a little bit below the extremality line. And there's some evidence that in consistent theories, this linear combination is always negative. And so in fact, black holes themselves can have masses a little bit below what the two derivative extremality bound would have allowed. I don't think this has been rigorously proven in general, but there's a suggestive argument in a paper by Chung, Liu and Riman from last year. Also, I think there's an argument that I think is specific to the case of four dimensions, but I'm not completely sure, which are Connie Hamed and some set of collaborators have not yet published, but have given some talks on that prove that given certain assumptions, this number is negative. The Chung, Liu and Riman paper tries to make an argument based on black hole entropy. And their argument makes a lot of assumptions. There are some reasons to think it really only applies to kind of tree level UV completions. The R. Connie Hamed argument is about looking at the renormalization group running of these operators and arguing that eventually this one, you know, squared squared has the biggest effect and that that one is known to have a positive coefficient for other reasons related to uniterity. So I think the status of this claim is still a little not quite rigorous, but at least there are enough different suggestive arguments that you might believe that it's just true that black holes in consistent theories of quantum gravity actually can exist slightly below the traditional extremality bound. So what I told you before was that the wheat gravity conjecture says that there should be particles whose mass to charge ratio is smaller than that of an extremal black hole. But if extremal black holes don't actually have a definite mass to charge ratio, if we get this curve that bends because of higher derivative operators, it's not quite clear what I mean by this. And so what I think I should mean by this, at least a statement that seems to be an interesting statement for which there's some evidence, is that the mass to charge ratio should be below the asymptotic extremality bound. Meaning I should read off the linear part of this bound at large q and define that to be what I mean by the wheat gravity conjecture. And if that's what I mean by the wheat gravity conjecture then in the case that this coefficient is negative, in fact, even black holes themselves could be the things that fulfill the wheat gravity conjecture because they have a mass to charge ratio smaller than the asymptotic slope of that line. So what does this mean? I would say that this minimal version of the wheat gravity conjecture, again, is not a very useful statement. If it's not telling us about the existence of light particles, it's just telling us that big black holes get small corrections. So the minimal wheat gravity conjecture, I think, is not quite sharp enough yet to get a lot of phenomenological mileage out of because it's too vague. It doesn't tell us these objects have to be light. It doesn't tell us they have to be small charge. It just tells us something has to exist that obeys this bound. And that thing could just be a black hole that gets a small correction. Okay, so where I want to head with us is to try to present to you some variations on the wheat gravity conjecture that actually do have more power, that are more useful statements that are getting closer to being relevant to phenomenology in the real world. But before I do that, I want to detour a little bit and just give you some examples of settings in the quantum gravity context where we can calculate the spectra of particles and see that there are particles that obey this bound. So I want to give you a quick review of Kaluza Klein theory, and also talk a bit about charge particles in the heterotic string spectrum. But maybe before I do that, I should ask if there are any questions about kind of the meaning of the conjecture itself or the things that I've been saying about it. In this modified theories, how do you define the charge? How do you find the charge? How do you define? How do you define the charge? Good, so both the mass and the charge here are just read off from the asymptotics of the solution. So the charge is just how quickly the electric field is falling off an infinity and the mass is the ADM mass, which you can read off from how the metric is falling off. That's a good question, right? So if you didn't hear the question, the question was, is there evidence of the same behavior when there is a dilaton coupling? As far as I know, those calculations have not been done, and that would be really interesting to do, I think. It gets a little bit more complicated because if I start writing down these four derivative operators, I get couple dilatons through them in many different ways. So I think it's not quite clear what choices I would want to make, but yeah, certainly it seems to be a well-defined exercise to make some sort of onsets here, try to compute how it changes the black hole solution, see what linear combination of things it depends on, and then see if we have reasons to think that linear combination is positive or negative. So I think that would be very interesting and as far as I know, no one has done it and I don't know anyone currently doing it either, but it's worth doing. More general gauge groups, I will have more to say about that in a little while. So I'll come back to that question. Okay, so I guess now I'll move ahead by talking about some examples where we can calculate spectra. And the reason that I want to do this is because when you start looking at concrete theories of quantum gravity, you rapidly see that not only do they have a particle that obeys the weak gravity conjecture, they tend to have infinitely many particles that obey the weak gravity conjecture. And in fact, they have a lot of other common features. That will be part of the story when I talk about generalizing the weak gravity conjecture and also about the swamp land distance conjecture. Okay, so the first thing I want to talk about is collucian theory. So I want to start with a theory that contains Einstein gravity and D dimensions and compactify it on a circle. So we're going to study it in a space of one lower dimension times a circle from the lower dimensional viewpoint. This has a U one gauge symmetry where the gauge field is just the graviton with one leg pointing along the circle direction. Okay, so this may be familiar to many of you but I think it's worth reviewing to sort of emphasize the features that are going to show up when we talk about some of these other conjectures. Also, I'm starting with Einstein gravity and compactifying. What I'm going to tell you about is just a statement about classical field theory but it's worth saying that we know that collucian theory appears in sort of a subsector of consistent theories of quantum gravity like string theory and so all of the statements I'm going to make can actually be found in real explicit UV complete examples. Okay, so we're studying the D dimensional theory where the D dimensional metric is given by some lower dimensional metric plus term which I'm going to write as R squared times some scalar phi of X squared times D theta plus a mu dx mu squared where theta, the periodic variable is our coordinate on the circle that we compactified on. And so the size of the internal dimension is R times the VEV of the field phi. Okay, what was the question? Okay, times two pi, yeah. And so this field phi, the radion field that controls the size of the circle is sort of our prototype of a modulus and we're going to be talking a lot more about moduli fields and how they're related to the weak gravity conjecture as we go along. So this is the simplest setting in which such a field appears and there's an exercise that if you've never done it before, you should do which is to take this metric and just plug it into the Einstein-Hilbert action and see what the action looks like in lower dimensions. And the action looks like this. It inherits a pre-factor that depends on the higher dimensional and Newton's constant. Then you get the size of the extra dimension two pi R phi. Then you get, I've tried to write this as a curlier R to distinguish it from R the radius. This is the Ritchie scalar of the D-dimensional theory. And then there are more terms involving the gauge field AMU that I'm not going to write at the moment. You will find that there is no kinetic term for phi. There's no term involving derivative of phi squared with the ansatz as I wrote it here. But phi mixes, has kinetic mixing with the graviton through its coupling to the Ritchie scalar. And we can define the lower dimensional Newton's constant as the higher dimensional Newton's constant divided by two pi R phi. Phi doesn't have a potential so it could really, its VEV could be anything. But here I'm just imagining that we look at solutions where phi is asymptotically approaching some constant value. And that's what I'm calling the VEV. Okay, so that's the action that we got at the beginning for collude secline theory. But now we should, we're free to work with that action if we want. But it's often convenient to do what's called going to Einstein frame, which means eliminating the kinetic mixing between the scalar and the metric. So we can do a field redefinition where we replace our metric with this power of the scalar, multiplying the metric. This is a vial rescaling. Notice it is a field redefinition. It is not a diffeomorphism. It's not a change of coordinates. And there's a general formula for what happens to the Ritchie scalar when you do such a vial transformation that if we rescale a field, then the Ritchie scalar changes in this way. This is another thing that if you've never calculated it, it's a good exercise to do. And in particular, this fact that it depends on the logarithm of the thing we rescaled by will be important for our discussion. Okay, so using this fact and doing this field redefinition, we find that our action becomes a different action where now we no longer have a scalar coupling to the Ritchie scalar. We just have its VEV and we now have a kinetic term which depends on the log of our radion field. And there was also a boundary term that I dropped. Now you could also have gotten to this just by directly starting with the different onsets, which looks like this. But that would not have been an obvious guess probably if you didn't start with this and then try to figure out how to eliminate the kinetic mixing. And now once we've done this field redefinition, we find that the gauge field couples to the metric, sorry, the gauge field couples to the scalar. Kinetic term looks like that. So our gauge field as a pre-factor, that's a power of phi. But the kinetic term depends on the log of phi. And so in terms of the canonically normalized field, this is an exponential coupling, similar to the dilaton coupling that I wrote earlier. In addition to the massless fields that we've been working with, we have the colluzicline modes, the excitations around the circle, and a mode of colluzicline charge N, N units of momentum around the circle has a mass given by N squared divided by the thing that's setting the size the extra dimension, R squared times the value of phi squared. And we have infinitely many of these modes from all the different momentum modes. So why did I go through all these details? It's because I wanna emphasize some features of this that turn out to be pretty universal and that sort of formed the foundation for several of the Swampland conjectures. So some observations. The matching of the Newton's constants depended on the radius. And if we assume that radius was big in fundamental Planck units of the higher dimensional theory, then the lower dimensional theory has a higher Planck scale than the higher dimensional theory, which means that if you measure Newton's constant in the lower dimensional theory, and then you do scattering experiments and see when gravity becomes strongly interacting, it happens much earlier than you would have estimated from your low energy measurement of Newton's constant. So theories that have these large internal dimensions have very low UV cutoffs relative to what you might naively expect. Quantum gravity cutoff is much lower than the D dimensional Planck scale. The next observation is that the limit where I make the gauge coupling small by making the pre-factor of this term big is the limit when the charged particles are light. So in fact, the masses of these charged particles when written in units of the D dimensional Planck scale are just n squared times the gauge coupling squared times the higher dimensional Planck scale to the appropriate power times one half. And these masses saturate the weak gravity conjecture. If we were to work out what the extreme low black holes look like, we would find this coefficient of one half is exactly what we want for extreme low black holes. And that's not an accident. It's because if this theory were supersymmetric, then this is also a BPS condition because our U1 gauge symmetry is just part of the higher dimensional space time symmetries. And so supersymmetry will protect the corresponding extremality bound. The next observation is the kinetic term as logarithmic in the scalar field phi. That also means if I were to write it as a function of the gauge coupling E, it would be logarithmic as a function of that too. So the canonically normalized field has this form. And if I rewrite things, sorry, log log phi. And if I rewrite the coupling of phi to the gauge field in terms of the canonically normalized field, it's exponential. So it's E to the minus some constant times phi hat. New sport, the final related fact to point out is that because of that, the number of colluciclinemodes below the cutoff scale is exponentially large as a function of this canonically normalized field. So we have a modulus field. As it goes to large values, we find exponentially many modes appearing carrying different charges under the gauge group all of which obey the weak gravity conjecture. So that's one example where we find that not only is the weak gravity conjecture obeyed, it's obeyed in a way that's much stronger than what it had to be just a few more minutes, right? Let me just briefly mention one other example. It's handy heterotic string for concreteness. I'll talk about the SO 32 string. So we haven't talked yet about what the weak gravity conjecture means for non-abelian gauge groups, but there's a U1 to the 16 sitting inside SO 32. So let's just look at the U1 charges, the charges under the carton generators of the algebra. And with this gauge group, we can find particles whose charges are labeled by these 16 numbers, U1 through Q16 with the QI, either all integers or all half integers and the sum of the QIs and even integer. That's the charge lattice of SO 32, okay? And we can work out the spectrum of particles that carry these charges and the perturbative string spectrum. And let me just jump to the answer since we don't have much more time today. The lightest charged particle for charge Q not equal to zero turns out to have a mass given by this formula two over alpha prime times Q squared minus two. So this spectrum is asymptotically linear. If I were to plot mass versus charge, it stays below the diagonal line. And in fact, asymptotically, this matches onto the black hole extremality bound. So our gauge coupling E squared is related to alpha prime and G string in this way. Our 10 dimensional Planck scale to the minus eight power is related to G string and alpha prime in this way. And G string is E to the phi where phi has a nonical kinetic term. Well, I should put some constant factors in, but it has a D phi squared kinetic term with some constant in front. In other words, it's not logarithmic. So again, as in the previous case, we find a coupling which is exponential as a function of some modules. We find a set of states of different charges whose masses are bounded in some way by the charge. And in fact, you can work out the black hole extremality bound in this case. And it looks like this. The factor of one in front or in the 40 Rice-Ring-Wardstrom case we had two. But if I rewrite this in terms of alpha prime, it's just this. So this linear relationship just is the extremality bound. So the heterotic string spectrum has infinitely many charged particles, all of which obey the weak gravity conjecture and which are asymptotically approaching the black hole spectrum. Okay, so we have our two examples, Kuhl-Sakline theory and the heterotic string. And we see that in both cases, not only did a particle exist as the weak gravity conjecture told us, but infinitely many particles exist. And tomorrow I'll try to explain to you how we were led by this to some more powerful conjectures that extend the weak gravity conjecture.