 than the minimal wheat gravity conjecture that we discussed yesterday, and the second lecture will probably be mostly about applications to inflation and photon masses, but we'll see how I do on time. Okay, so to remind you yesterday we talked about the minimal version of the wheat gravity conjecture, which was that if I have a U1 gauge theory interacting with gravity, there should be some particle whose mass to charge ratio is smaller than that of large, extremal black holes. And one question that one of you already asked yesterday, which is a very good question, is what happens if you have a larger gauge group than just U1? And an answer to that question was given in a paper by Cliff Chang and Grant Ruhman about five years ago, and the argument that they gave is based on assuming that the interpretation of the wheat gravity conjecture should be that any extremal black hole should be able to shut its charge by decaying to lighter things. So we saw that if you only had a single U1 gauge field, this condition was required in order for an extremal black hole to be able to decay into two things, one of which would have a larger and one of which would have a smaller mass to charge ratio. It turns out that it's not quite so simple if I have more than one gauge group. So let's think about, for instance, the case of a U1 times U1. You might think that to satisfy the wheat gravity conjecture for a gauge group like this, I just need two charged particles, one of which obeys the bound for this U1 and one of which obeys the bound for that U1. But it turns out that's not really sufficient. And the reason is that the form of the extremality bound in this case, in the simplest case where these gauge fields are not mixing with each other, looks like this. So if I have a black hole that carries charges under both gauge groups, there's an extremality bound that depends on the square root of the sum of the squares of the charges under the individual gauge groups. And again, this is just because there are electric fields for each of these gauge groups outside the black hole, each of which are storing some energy. But now the claim is that if I just satisfy the wheat gravity conjecture for each U1 individually, I don't necessarily allow all the black holes to saturate this bound to discharge. OK, so that's easy to see. Suppose let's just look at one sort of toy example. Suppose I give you two charged particles, one of which has charged one under the first gauge group and has mass chosen to saturate the wheat gravity bound. And the second of which similarly has charged one under the second gauge group. So if I were just looking at each U1 individually, this would be enough. But the claim is that this is not enough if I have a black hole that carries both charges. And so we can see that. Suppose I have a black hole charged under both U1s, which is extremal. And we can just ask what would happen to that black hole if it emitted one of these particles? How would its mass change and how would its charge change? So suppose I can just emit this particle and lose mass M1. And similarly, its charge under the first gauge group will decrease by one. It might lose more mass depending on how much momentum this particle carries away, but this is the minimum amount of mass that it would lose. And then we can just work out what happens to the extremality bound after I do this. What do I get if I just evaluate this quantity? Is it positive or not? And so if I expand out the square and subtract everything, what you will find that this looks something like this, which is negative. So this decay is not kinematically possible because the black hole, by emitting this particle, would turn into another black hole that just isn't a good classical solution. It doesn't obey the extremality bound. And so adding these two particles is not sufficient how extremal black hole is to discharge. So this tells us that whatever the right condition is for a theory with more than one gauge group, it's going to be more complicated than just saying that you have to individually obey the bound for each gauge group independently. And it turns out that there's a simple geometric criterion to understand what is the correct condition to allow these black holes to discharge. And Schengen-Rehmann formulated this condition in this way. They said, consider the charge to mass ratio of a particle. So we're going to have a set of variables, let me call them Zij, which are given by the ratio of the charge of a particle in plonk units divided by the mass. And so the index i labels which charge we're talking about, which one of the gauge groups. And the index j labels which particle we're talking about. So for a given j, for a given particle, we can think of this, the collection of Z i's for that j as a vector in some space. And the black hole extremality bound says that if I give you such a vector for a black hole, then when I dot this vector into itself, I get something smaller than one. Its charge can't be too big for a given mass. So in the space of these perched to mass ratios, there's some region in which black holes can live. So this might be a space of many dimensions if we have many gauge groups. But there's some sort of ball inside this region that contains all of the black holes. So let me call that the black hole region. Extremal black holes would live on the boundary of this region, if we compute using the two derivative action. As I mentioned yesterday, if you have higher dimension operators, the black holes could move slightly outside or slightly inside this region. Unless there's some other physics like a BPS bound that follows from supersymmetry that could force them to live exactly on the boundary of the region. But what Chung and Rehman pointed out is that a condition that allows for all of the black holes to be able to shed their charge is to have some set of charged particles in the space. Yeah, let me not try too hard to draw a three dimensional picture because it's going to be difficult, but have some set of charged particles such that their convex hole contains the black hole region. So the convex hole meaning I connect up each of these particles and their antiparticles in such a way as to build a convex space. And this convex hole should contain the black hole region. So the example I gave you before where I just had two particles and we saw that not all black holes could discharge is an example where we had two U1s. So the black hole region is the interior of some circle and a plane. And I was choosing to have a particle here and here. They would also have antiparticles here and here. And their convex hole would look like this. And what we were seeing is that if I go in a general direction that doesn't point along one of the axes. The convex hole does not fully contain the black hole region in that direction. So there are these regions that are not contained and that means the weak gravity conjecture fails. On the other hand, if I had chosen those two particles to have smaller masses, sufficiently far away from the weak gravity bound in a single direction. Then the convex hole, it fully contained the black hole region and everything would be able to discharge. Or we could have a more complicated spectrum. We could have particles carrying both charges. Maybe I could have a picture of something like this. So the minimal version of the weak gravity conjecture doesn't tell you exactly what charges the particles have to have. It just says that whatever those charges are, the masses should be light enough that the convex hole contains the black hole region. So we saw yesterday that in fact there are examples like the Colutzicline theory that obey much stronger properties. If we look at Colutzicline theory, we would just find charged particles living on the boundary of this region in every direction. And there's a reason for that. It's because in this case the extremality bound coincides with the BPS bound. If you study theories that have scalars, you'll find that the black hole extremality bound can sometimes take a different form. So for one example, if I take a string theory and compactify it, I will have both U1s that correspond to Colutzicline charge or momentum around the circle. But also U1s that correspond to winding. How many times a string winds around the circle? And if you work out what black holes look like in that theory, you'll find that the formula I gave you before with the square root of sum of squares is not the right one. That works for the simple Reissner-Nordstrom case. But in this case you get a more interesting story where you have these linear regions, so you get kind of a diamond shape. But again the convex hole condition is the right criterion to impose to ask that everything into K. And in this case again, because these are BPS charges, you find that you just have actual states carrying all of these charges in all directions along the boundary. So in everything that I'm telling you, what I'm going to take is my definition of the wheat gravity conjecture for a general gauge group is going to be this condition that we have particles carrying charges under the gauge group such that their convex hole contains all possible black holes and allows extremal black holes to discharge. There's another way to formulate the wheat gravity conjecture in theories that have scalars that I want to mention to contrast it with this. So Aaron Palti defined something that he called the wheat gravity conjecture with scalar fields, which is that a particle should exist that obeys disinquality. The electromagnetic force should be bigger than the gravitational force coming from the mass plus the scalar force that depends on how the mass changes if I vary the value of the scalar field. And this is a version of what I would call the repulsive force conjecture. So the statement is that the repulsion of electromagnetism should overcome the combined attraction of gravity in scalars. And this is well defined for a single U1, but again it's not so obvious how to define this if you have more than one gauge group. There's no clear definition of this in the case of more than one gauge group. There's work in progress that I'm doing with Ben Heidenreich and Tom Rudelius, where we're trying to work that out. And the interesting thing is that it seems like this alternative conjecture and this conjecture might both be true. But if the particles' masses depend on the scalar in a different way than the masses of black holes depend on the scalar, these are two different statements. Another question you might ask is what happens for non-abelian gauge groups? And we sort of mentioned this yesterday when I showed you the example of charged states in the Adoratic String Theory. We can take the Qs to be charges under the Cartan generators of the Lie algebra. So the maximal torus, the U1s that are contained inside the gauge group. And then we can still apply this condition. But because the gluons themselves are charged in massless, they're infinitely far out in the space. And so this becomes sort of a trivial condition for a non-abelian gauge theory to satisfy. Right, so what I'm saying is that I would apply this to any gauge group. But what I mean by charge in the non-abelian case was not necessarily obvious. So what I mean by it is take the U1s that live inside the non-abelian gauge group and make this the charge under those. But yeah, I would require this condition of any gauge group. And we'll see that that becomes more interesting when I talk about a different version of the wheat gravity conjecture in a few minutes. But yeah, one way to see that there should be something related to non-abelian gauge groups is if I give you a non-abelian gauge theory, you can compactify that theory on a circle and turn on a Wilson line for the gauge group around the circle and break it down to the U1s that live inside the non-abelian gauge group. And so if the wheat gravity conjecture applies in that lower dimensional theory, it should tell you that something had to apply in the higher dimensional theory you started with. So so far, these are all kind of minimal versions of the wheat gravity conjecture. This is kind of the right minimal version of the conjecture in the case of a gauge group that's more complicated than U1. But as we were discussing yesterday, the minimal conjectures are maybe not as useful as we would like because they could be satisfied just by saying that black holes get small corrections that push them out of the black hole region defined by the extremality bound on asymptotically big black holes. So what I want to talk about for the rest of this lecture are some extensions of this idea that are much more powerful and which are motivated by the examples that I told you yesterday. So we talked about Kuhl-Sekline theory and we talked about the heterotic string and we saw that in both cases we had not just a single charged particle that obeyed the wheat gravity conjecture, we had an infinite family of charged particles of different charges, all of which obeyed the conjecture. And this is closely related to another idea which is called the swampland distance conjecture which came from Aguri and Vapa 2006. And so they had some set of conjectures, all of which were related to the fact that known theories of quantum gravity have moduli spaces, so spaces of scalar fields with flat potentials that are protected by supersymmetry. But in fact similar statements apply even away from the Susie context. Although it becomes harder to make these statements sharp, we saw in Kuhl-Sekline theory that we had this radion mode that controlled the size of the circle. Even in a non-super symmetric theory that field can be very light compared to the UV cutoff. But for now let's talk as if the moduli space exists and the scalars are massless because it's easier to formulate everything in that context. So their first conjecture about moduli spaces is that if I give you a moduli space, there exist points that are infinitely far away. So given a point in the moduli space and some positive number, you can find some other point in the moduli space whose distance is bigger than that number. So moduli spaces allow you to go infinitely far away. And the distance is defined by the scalar field kinetic terms. So there's some metric gij of phi e phi j. I guess I didn't say it before but it's the same metric that appeared in Paul T's statement of his weak gravity conjecture with scalar fields when you compute the attractive force on charged objects due to scalars. You need to know the metric on that space. Now the next statement that a gradient of alpha made is where things really become interesting and make a connection to the weak gravity conjecture. I say that if I go far away some distance t away from the point where I started, I will find an infinite tower of light particles whose masses are exponentially small. Masses of order e to the minus alpha t for some parameter alpha which is bigger than zero. And it's been further noted, for instance by Claver and Paul T, that this number alpha in known examples is always order one in plonk units. So what does this mean? This means that in quantum gravity, first there are these big moduli spaces. There are parameters that you can, there's scalar fields that let you vary the parameters of the theory and you can vary them by arbitrarily large amounts. But whenever you try to vary them too much, you find that your effective field theory starts to break down because lots of particles are starting to become light. And so if you don't incorporate those particles in your theory, you don't have a good understanding of the physics anymore. So effective field theories seem to break down for super plonky and field values. And we saw an example yesterday in Kulitsa Klein theory where we had a scalar with a kinetic term that loosely looked like in plonk squared d log phi, d log phi, where the masses of the KK modes went like 1 over phi. So the kinetic term tells us that if we vary this number phi by an order one amount, that we're going an order in plonk distance in the field space. And if I write this mass in terms of the canonically normalized field or the field that's actually measuring that distance, we see that the mass is decreasing exponentially. So that's an example. Similarly for the heterotic string, we saw that our masses roughly went like n divided by the square root of alpha prime or n times the string mass scale, but that's exponentially small in the distance reversed by the dilaton. So alpha prime in plonk units depends on g string, which is exponential in the value of the dilaton. So these examples that we saw yesterday are illustrations that kind of capture all of the features that Aguri and Vafa say are general features of modulized spaces in theories of quantum gravity, right, alpha as well, yeah, independent of distance, right. So the claim should be taken to be an asymptotic claim about large t. So the claim is at large t, there's some number alpha for which the masses are decreasing like e to the minus alpha t. It won't be exactly this, there will be corrections, but if we go to large enough t, then it'll be this where alpha is just some constant, okay. And there are many more examples in string theory. So for example, let's say I build a three plus one dimensional gauge theory and type 2b strings with the seven brains wrapped on four cycles. So something that might be phenomenologically relevant if you're trying to construct something like the standard model within string theory, then your gauge coupling one over g squared will go like the volume of these four dimensional cycles and those volumes can be parameterized by some scalar fields, some scalar moduli. And again, if I go a long distance in field space, if I go to very small coupling g, I'm making this volume big and I'm bringing down some coulisacline modes. Not as simple as these coulisacline modes that live on a circle, but still some set of modes are becoming light as I make the volume big. So that's the Swampland distance conjecture. And it's already a very interesting conjecture from the viewpoint of thinking about things like theories of inflation where you might want fields to travel long distances over time as inflation is happening. One distinction I should point out in that context is the distance that appears here is the geodesic distance in field space that depends only on the kinetic terms, right? So I have some space of scalar fields. I know a metric on that space. I give you two points and ask you to measure the distance. You do it in the usual geometric way just using the metric. But if I have fields that are actually evolving in time and we're not in this supersymmetric limit where I have a perfectly flat modular space, then the fields might not actually be following geodesics in field space because there could be a potential and the potential could change how the fields are evolving. And this might be an important distinction if we're trying to apply this to realistic theories of the real world that have Susie breaking, where the potential can be very important for figuring out what the fields are doing. And so let me just mention two papers that discuss the swampland distance conjecture in this context and how this distinction might be important. One of these papers is by Hebecker and collaborators. Another is by Landete and Chu. And there's been various other recent work along these lines. So it's not completely straightforward to say that just because Agoury and Vaffa tell us that if you go along geodesic distance in field space, you'll bring down a tower of modes that inflationary theories with long field ranges are excluded. You have to be a little bit careful in trying to make these kinds of arguments because these statements are really statements about the limit of exact moduli spaces, which is not the limit that's useful for the real world. So what I want to do now is circle back to the weak gravity conjecture. So you'll notice this conjecture made no reference to gauge fields. We were just talking about theories of quantum gravity that contain scalar fields that act as moduli. But in string theory or collude secline theory, as we've seen in examples, the values of gauge couplings are determined by the values of moduli. They're not just fixed parameters that you can't change. They depend on the values of some scalar fields. And so this starts to bring us to a link between these ideas in the weak gravity conjecture. The weak gravity conjecture grew out of the statement of no global symmetries, which was a statement that quantum gravity doesn't want you to have exact global symmetries. And therefore, if I have a gauge coupling and I try to make it very, very, very small, something bad should happen. The theory should somehow break down. Well, in a lot of context, what happens when you try to make a gauge coupling very small is precisely that you're taking a long distance limit in moduli space. The small gauge coupling corresponds to a very big value of some scalar field. Just as happened here includes a client theory, the gauge coupling was set by the value of phi. And so if you want to make a small gauge coupling, you have to go a long distance in field space. And then what Agoury involved to tell us is that some tower of modes should start to become light. And you should worry that you're going to lose control of your field theory because you don't know about all of these modes that are coming down from the UV cutoff and entering the energy scales that you would like to calculate in. Okay, so given these ideas together with the wheat gravity conjecture that I discussed before, it's tempting to conjecture something that's a little sharper than the wheat gravity conjecture, which is that in the case where I try to send a gauge coupling to zero, a tower of modes is going to become light. And so in some cases that tower of modes, as we've seen, is exactly the same tower of modes that the degree involved to tell us will become light. Okay, so just sort of trying to put these pieces together, it's tempting to guess that if I have a complete theory of gravity coupled to a gauge field with a very small gauge coupling, there must exist infinite tower of modes of different charge q, each one of which obeys something like the wheat gravity conjecture, the mass is less than its charge in plonk units. So I'm not claiming that the statement follows from other statements that I've made. I'm just saying that given the examples that we have and given the other statements, it's tempting to guess that this is a general phenomenon. And so the statement in the sort of loose form that I've written it as here is something we might call a tower wheat gravity conjecture. And that particular name was used in a paper by Andriolo Junghan's shoe last year. We could guess something even sharper than this. And the statement that's true in all examples that I'm aware of is we call a sublattice, wheat gravity conjecture. And this statement is, so there's some lattice of charges of gauge, charges under the gauge group that is allowed. And the suggestion is there's a sublattice of the full charge lattice and of the same dimension. So it's not like you could have a five dimensional charge lattice and I'm picking a lattice that only goes in one of the directions. It's got to be the full dimensionality for which every site in the sublattice contains at least one super extremal. So that's along the lines of the tower conjecture, but it's a little bit sharper. It's saying that these can't just have any random set of charges. There should at least be some sublattice. So maybe it's all multiples of charge two. Maybe it's all multiples of charge three. But there's some sublattice for which you can find an example of a particle obeying the wheat gravity conjecture for every site in the sublattice. Okay, so this conjecture was formulated by Ben Haydn-Rike, Tom Ridilius, and myself. Actually, we first wrote a paper. I seem to be missing the exact archive number, but we wrote a paper in 2015 where we guessed that this was true for the full charge lattice. We didn't have the sublattice part. It turned out there was a counter example, which I can briefly mention. And so we refined this to the sublattice statement in a paper in 2016. And there was a simultaneous paper by Miguel Montero, Gary Shue, and Pablo Soler with the same formulation. And we were led to guess this, not by the logic that I was giving you here that started from thinking about the swampland distance conjecture and examples. But rather just by thinking about the wheat gravity conjecture itself and trying to sort of stress test it and see if it was consistent. And the claim that we made in the 2015 paper is that there's sort of a flaw in the convex hull condition, the minimal version of the wheat gravity conjecture, which is that I could give you a theory that satisfies it and you could compactify that theory on a circle and get a new theory that doesn't satisfy it anymore. And the reason is the gauge group gets bigger. So in my compactified theory, I now have the collude decline charge along the circle. And as we saw, the convex hull condition tells us that when you have a bigger gauge group, it's harder to satisfy the conjecture than when you have a smaller one. You need to pull the charges farther away in order for the convex hull to enclose the black hole region. And so if I just had some particle with a sufficiently big charge to mass ratio to satisfy the conjecture, in the one-dimensional case, for instance, where my black holes live on this line segment, when I compactify the theory, I'm going to get a bunch of collude decline modes of this particle. So there's going to be a mode with charge 1 comma 1. There's going to be a mode with charge 1 comma 2. There's going to be a bunch of other collude decline modes that sort of accumulate close to the axis of pure collude decline charge. And so they will have some convex hull. And then there's some black hole region. And as I've drawn it, this example looks fine. But what you find is as you vary the radius of the compactification, there will always be some radii for which collude decline modes fail to fully enclose the black hole region and the weak gravity conjecture can fail. So the claim is that the minimal weak gravity conjecture, the original version of it, generalized to multiple gauge groups in the natural way, as Chung and Rehmann did, is not a completely satisfying conjecture because it's somehow not sufficient. If I give you a theory that obeys it, you can change that theory only in the infrared by compactifying. And get a theory that doesn't obey it anymore. So we noticed this in 2015 and then we went looking for examples in string theory so we could just understand how is the weak gravity conjecture satisfied in actual theories of quantum gravity? And what we found was that in all the examples that we could check, there were these towers of particles. And the towers of particles rescued the statement because now we have many different particles of different charges. And so they have collude decline modes in different locations. And when you start putting them all together, you always find that after you compactify, the theory still obeys the conjecture. So the claim is that these tower statements are more robust under compactification than the original conjecture. So that's one reason to think that these might be plausible statements. And another reason to think they're plausible, again, is they connect nicely to these distance conjecture ideas of Agourian Bafa. And the final reason, and this was really the main technical development in the 2016 paper that we wrote and that Montero and collaborators wrote, is that in perturbative string theory, we can actually just prove that the statement is true. And in fact, it follows from modular invariance. So it's possible to show just by studying a modular invariant partition function that has a chemical potential turned on, that charge states exist within a sub lattice and they obey a bound that looks like the weak gravity conjecture bound. Okay, so at least in perturbative string theory, this is a well-established fact. Of course, perturbative string theory is not all of quantum gravity and you can ask what about particles that are charged under gauge fields that live on D-brains, for instance. And we can't really say anything about that. Although there have been some recent interesting papers showing that a class of F theory compactifications also contain states that obey the sub lattice weak gravity conjecture. These papers are by Lee, Lisch, and Bygant. So we have a lot of examples where this is just a known fact about about actual theories of quantum gravity that you can construct. Let's see, so just have a few minutes left. Let me not go into detail, but just tell you the reason why it has to be a sub lattice instead of a full lattice is you can just construct counter examples where it's not the full lattice and the flavor of these counter examples is that you start with something that has multiple charges, like compactify on a torus, and then take an orbit fold where one of the directions gets the gauge field projected out. So you no longer have a gauge field corresponding to some of the charges, but states are still labeled by some kind of momentum in that direction. It's not really conserved, but a state that had charge in that direction before you orbit fold it will still have a bigger mass because of that extra momentum. And so what you can do is cook up a theory in which, for certain sites in your charge lattice, the lightest state that is not projected out by the orbit fold is a state that had momentum in that extra direction, and that adds to its mass but doesn't add to its charge because we removed the charge associated with that direction. That's the flavor, you can find the details in the papers. But the interesting thing is that in the examples that we know, where you can check what states obey the weak gravity conjecture, it's never a very sparse sub lattice, so it's maybe at most half the sites in the charge lattice fail to obey the conjecture. It's never like every 10th site or every 17th site. And whether that's just an artifact of looking at examples that we know how to construct, or whether it's a deeper statement, I'm not really sure. To the extent that it seems to be true that this applies to sub lattices that are not very sparse, this avoids the problem with the original weak gravity conjecture that you could obey it just with black holes. So the original conjecture, I said you could have a small correction to an extremal black hole, and that obeyed the conjecture. But that extremal black hole has an enormous charge in order to be in the regime in which we trust semi-classical gravity. Whereas if we require this to be true for sub lattices whose sparseness is only sort of an order one number, then we're going to find states of small charge that obey the conjecture, and those states are going to be lighter than the Planck scale. And so this is now telling us about actual particles, not just about black holes. So again, I don't have a proof that there are no examples where this is satisfied with the sparse sub lattice, even in the perturbative string context, but we've looked at lots of different examples of heterotic string orbifolds that we understand, and in those examples it never was. Okay, I guess that's all for this lecture.