Mariana Grana - EFTs with symmetric moduli spaces: the landscape and the swampland

Okay. Well, thanks uh for this invitation and I'm going to talk about this work I did with Dan Waldram, my former students, Veronica and Bernardo who are now both in Madrid and Stephanie who's Waldram students and came in these papers and we're still working on future directions which will be clear when we discuss. So um the first one of the swamp plan session first before going into this that I'm I'm sure you all saw this uh this slide is introduction and spoiler and uh and everything will be here the idea and then it's the spoiler of the results. So I'll go very slowly. If you get this then uh they will be then I mean I'll be happy you have a take-home message and then the rest I mean the how how we got to those results is very nice uh math I think and physics and it will be great if you hear it but if you're not interested at all in the details just bear with me on this on this slide. So everybody saw this picture, right? Is there anybody who hasn't seen this picture? No. Okay, good. This picture. So what part of the landscape swabla? So I'm going we are going to talk about effective effective field theories and what effective field theories I'm going to talk about and these are the theories with symmetric modulized spaces. For example, what you get in tooidal compactifications, the NS of the NSNs sector, OKKR divided by OKR * OKR quotient by the duality group OK KZ. So this is a symmetric model spaces and this is this is the kind of effective field theories that we're interested in and then the question is uh which ones are on the sample and which ones are on the landscape. We will not let me say from the beginning we will not assume any super symmetry even though this is a typical modulate space that you get in half super symmetric uh in in in tooidal toidal compactification let's say toidal compactifications of Taiwan or or um or things um preserving 16 supercharges we're not assuming at all super symmetry in this in this talk which just we just require the modulate space to be symmetric and there are the swan constraints so some conjectures so what are the properties that effective field theories have to have in order to be consistently coupled with quantum gravity and the one we'll be discussing is the so plan distance conjecture which I imagine everybody heard but let me it's important because That's the whole subject of the talk. So the this some plan distance conjecture says that when you go when you move to the boundary of the modulate space so modulate space will be the symmetric modulate spaces there's always a tower of states that be that becomes massless at least in this exponentially massless in in this form with a where d is a proper distance in modulate space and n is some charge. So there's a tower with different n with some integer n and the masses go exponential. So that's a that's the conjecture and we pro we proved it under my assumptions that uh I'll tell you what they are. So for symmetric modulate spaces using group theory one can prove the this swamp plan distance conjecture. Then there is the immersion string conjecture that says that the towers all these towers is always one frame in which these towers are either kaluta climb modes of a compactifying direction or towers of oscillation towers of strings of a tensionless strings. So the so this conjecture says that so both of these are conjectures. This conjecture says that these towers are either of this form or of this form. And then there is the sharpened swamp plan distance conjecture that says that the rate the these rates alpha they have to be at least or the the exponential decay has to be at least of this uh of this form where D is the space-time dimension of the effective effective theory. So this decay this tower should decay at least as as fast as this. >> Not the same. >> It's not the same D. Yeah, thank you. Yeah, we will use for this D we will use T. Then here I put distance but we will use D and in this Yeah, I should change it here by T. So this D is this D was the proper distance. This D is the space-time dimension of the of the effective theory. Um and then there is a so this is a conjecture. So the names of the conjecture never mind the the the contents. So um there is so this is a conjecture and then what has been observed in all the studies of compactifications and it seems like a also universal behavior from uh llian at least for for llian of space that the behavior that the exponentially exponential decay of kk towers goes like this. So it's has this piece and then a one overn and then it measures the number of directions that one is the compactifying. So for kuta line of a single direction the compact defying this is plus one for two this is plus one half and so these two things have been proven in in all examples worked out and what we will demand here is that so we proved the sland distance conjecture in general and then we looked at the the alphas and we will demand that one has so what we call emergence string conjecture rates. So the emergence string conjecture so we will demand that the the towers are of attention string of kkos and that the rates are this. So we look at what is in all the set of effective theories with symmetric modulate space which of with symmetric mod spaces which one satisfy uh which ones satisfy this uh this uh decay rate and demanding this we get a finite list of theories. would think and there are very very few like 33 and by theories what I mean is the group because all the symmetric modulate spaces uh they they are of the form C over K as as we will see a group a representation of particles so the particles are in a given so for this type of uh of modulate spaces what we get is that the particles are in the fundamental representation So the so that will come out. So for a given group there is in general a single representation where this such that this happens and a single dimension spacetime dimension where this happens. So there's a final list up to some uh up to some redundancy that I will explain. But um so from this point of view the the this uh immersion string conjecture rates this this this conjecture it's really predictive. So it tells you out of all these theories then only these ones can be in the landscape or only these one satisfy this. Are there any questions so far? Because as I said this is the main. So do you ask him anything about the asic geometry of the uh uh >> yeah so if I so what was the question exactly if I >> of the of the space time of the is it flat or in the >> you mean of space time and not of mod space no no we don't uh we just assume modulate space modulate space is this and then group theory will will let us study the boundaries of modulate space and this is what I will to and then once we know this but we just yeah there's no potential this >> no potential it's a real it's a actual mod space yeah yeah yeah thank you for the yes no potential yes >> sorry I just want to make sure I understood the emergence strain conjecture rates are those formulas right there >> okay >> yeah yeah so we exactly so we require the to have a um to have either. So any at any infin any infinite distance point should be associated with a tower that has this this behavior or this behavior. >> I suppose you will tell us how many of them are known and how many are not out of this 33. >> Yeah. Yeah. Yeah. >> And you insist on n being an integer too. I insist on n being an integer and in particular also d being an integer which that's not I mean that's something that one has to um yeah that one has to impose the n being in an in an integer will be natural in the um in the construction because it it's related to the number of directions here the compactifying and that has to I mean it's a group theory thing But the the D being integer that's something that you have to impose and that definitely restricts what you can do and actually what I'm going to say is um well let me say it let me say it >> yes >> sorry can you repeat what is m and n on the first >> ah sorry okay so uh yeah the sland distance conjecture is so good I I I assume this was seen by everybody and the sland distance conjecture as well. So the sland distance conjecture says that when you move to the boundaries of modulate space there's always a tower level by some number n. So this n levels the charge of a tower um such that the mass so m is the mass of this tower is proportional to n and decays exponentially with a distance. So it becomes light asotically light as you go to infinite distance. And in this very particular behaviors are exponentially decaying and exponentials then they were conjectured to be all greater than this. And then uh and then in particular given I mean this is the conjecture is that the string uh the string um the the I mean well this is very easy to show that this is the that the the tensionless string satisfies this is very easy to show and um and then this is something so the conjecture is that alpha is greater than this alpha string and alpha string is this that's how a tensionless a tower of tensionless string oscillators. This is how it behaves and then this is a universal behavior. It's not a conjecture but this is what's been observed in in all the on all the compactification studies. So um so we will prove this that there is a tower of states becoming exponentially light and then we will require this and then out of out of this requirement. So as I said this um this requirement is predictive in the sense that you can single out which effective theory satisfy that. Okay. Okay. So uh so this is what I wanted to say but I will say later so we assume locally symmetric spaces but maybe what we are doing is much more general and you'll see why. So these uh non-compact locally symmetric spaces are always of the form can be put always of the form of a coset of a le group a non non-compact le group are the ones that interest us divided by the maximal compact subgroup this group we assume to be red so the product of simply groups and some factors of r like for example in compactifications of the nsns sector there is this piece that we we talked about and r would the dilaton if you if you include the dilaton that's factor and then gamma is the duality group in this case it's okay cases but in general will be the duality group that that has to satisfy certain properties and as as we see so because it can be written like this it inherits all the the structure of the group C so the symmetric in the modulate space which is given by the gilling form on G and the what so what we are interested in are the shodics and the boundary because the in the in the swamp plan distance conure one has to move a shic distance. So the shicsics are very easy are these exponential maps. So you start from some point some element G and then you move in some tangent direction and if this tangent direct direction is normalized to one then t is a fine parameter. So then this is the what it was called at the beginning D that um I think it wasn't it was called so D was used for the space-time dimension and for the for the shic for the shic distance we will now call it T and again you just have to normalize this to one and then this becomes a fine parameter that will so we will see how how um the states the mass of the the states how it decays with um with t and the boundaries. So um one when this showics go to the boundary so the boundaries are characterized. So first we are going to define an equivalent well we are going to define it's defined um in the mathematics literature. So this is literature from the I don't know 20s 30 40 some something some things are earlier um so the boundary so two shes are in the same class if they have a finite distance when t goes to infinity so and that defines a point in the boundary so a point in the boundary is like aic as an in asytoic class of of shix so Now for the time being let me forget about the duality group and so make them totally symmetric spaces. Then once one put the duality group they become locally symmetric spaces. But let's first do without the duality group. Then the the boundaries are defined by the parabolic subgroups fixing the point and uh so how is this? Let me just give you the example of uh the parabolic subgroups of of GL4. So we we we move so we're interested in in geodessics that go uh along well let's say let's let's so these are the cartons and let's move along some direction in the carton. The direction in which you move is labeled by this lambda one to lambda four. And one can always order them using the while group. One can always order them in in some way. So we'll order them in this way. Now if we act on some element of CL4 by the the joint action what we get is that each value each sorry each um each component of this element when you evolve it it goes uh it evolves in this way with a difference of lambda J minus lambda I and then now if so if they are all strictly one is strictly larger than the other um so then these two two um two shows. So two sorry two elements are congruent. The only way if they have a finite so the only way if they have a finite distance when t goes to infinity if that the the element is upper triangular so all these all these things where lambda lambda j is where this is positive then all these things have to be zero. So these all these elements are are in the same asytoic class as this some standard element that could be here the identity the identity for example for uh no sorry not the identity because we're sorry we are we have this order so not the identity in this order now for if two of the values are the same then now one has a little bit more freedom this H. So all the elements that are of this form are in the same asytoic class. If three are the same then you have this and actually these are the parabolic subgroups. So the parabolic subgroups of CL4 are the upper triangular matrices. This is the smallest or or the upper triangular with some element in the lower triangular. So it's not very hard to see that um that uh that this define a point uh that the the the boundaries are defined by parabolic subgroups. Now let's for example let's now let's look at the manifold the one can do CL4 or SL4. That's an extra factor of R but let's not care about that. If we now if we look at this space SL4 over SO4 this is the space of matrix. This is a modulized space of matrix in four dimensions. And let's just look at the let's just look at the shodic in this uh in this moduliz space. So we're moving in the cartana and acting on some some element that we call the origin of mod space. This could be the identity. And then when we add in this uh for this uh this kind of shics what one has is the the radius. So this this part is going to infinity. So the radius of the first direction is going to infinity. So this is a boundary point of this form. And then one has to act with the SO4. So it's not a point but is the SO4 action of of this point. The the this corresponds to two radio going to infinity and this correspond to two to three rad going to infinity. and then and the off diagonal pieces of the I can grow off diagonal pieces by putting another putting another group C but it doesn't they they all they are all finite distance with respect to having the off diagonal pieces to zero so one can forget about the off diagonal pieces and and think that all the boundaries correspond to radi going to zero one two or three these are the different >> parabolic subgroups >> there's a fourth one Actually there >> yeah if it's >> there is the 2 plus two parab >> there is the two well here I mean the the >> any partition of force >> yeah yeah that's right I was thinking of uh yeah sorry I was thinking of SL4 here in which the volume is fixed um and then the this is minus the the lambda yeah no and and then yeah any partition so this is up to rotation so any partition of two is equivalent ent as a parabolic is equivalent to this and and then here yeah I was otherwise if uh if we are free to move the if you really do CL4 there's the the R direction which is the volume okay now we introduce the duality group and we're going to demand what um Matilda and and collaborators called us being compactififiable. So it's a um there's an argument that uh that quantum gravity amplitudes are finite if the modul spaces are what what they call compactififiable and this means that a volume of a geodessic ball of radius r grows no faster than what it will grow in aidian space. So in aidian space this bo thish volume will be r to the dimension of the space and they said okay let's call that I don't know material why compactifiability I don't like this terminology >> I mean yeah at best flat and then all the others are finite volume so we had the finite volume example that it's not it's not right now >> yeah so yeah it's not compact but uh and and indeed the finite volume actually here comes comes out immediately by this compactifiability. They are finite volume because in general for G over K the the volume of a shodic ball of radius R grows exponentially with R. So this is some norm of something like the highest root is something that is is non zero. So it grows exponentially with R. So the only way for this to be uh the only way for C over K and divided by a duality group to be uh to grow linearly sorry power law with R is that R cannot uh that R has to be bounded that you cannot um you in in geodessic terms R is bounded and then this and it was shown that then if R is bounded then this has finite volume. So basically the modul spaces if one assumes if one assumes this which they argue about the finess of quantum gravity amplitude so makes sense as a as a conjecture as a swamp plank or landscape conjecture let's say then for symmetric modulate spaces it means that the duality group should be such that this has finite volume and this means that the duality group has to be arithmetic what is called arithmetic which is just uh to be to make it simple like CLN CLK Z if we're looking at the groups can always be embedded in some CLKR and this is like CLK Z but this uh brings a lot of uh mileage because if gamma is arithmetic then the spectrum trans that usually transforms in representations of the duality group now this extends to representations of the group So from now on we can forget about the duality group. We know it's there and it's very important and and there is a latis and this and that. But because this is arithmetic which is a consequence of this requirement then we can talk about representations of the group C. And then this is important because the duality groups are not the same for example in well we won't do eerotic because we will do the split form as I will say but in compactifications of the super symmetric eterotic and the non-super symmetric erotic group the the duality the duality group is not the same the the latis is not the same but it doesn't matter here for for for what we are doing it doesn't matter question >> so does the fact that in this context the compact reliability implies the arithmeticity. Do you see this as support for the original conjecture? Can you have people thought about the extent to which that might be expected or not expected in quantum gravity and therefore this is telling us something about whether the original conjecture is correct or not? Yeah, I think uh yeah, this was surprising uh to us that this is exactly um I mean and this was yeah this was discussed uh before that uh um that the the the that this volume grows exponentially and the only way that uh that this is uh yeah that this can grow power law means uh means that Then this is finite volume. So the the the I mean I think the the non-compactness and finite volume it's I'm not sure it's a conjecture. So you know better what exactly what's conjecture I'm not but uh the fact that model spaces have finite volume um well I'm not sure has a name as a conjecture but >> in all the cases that had to do with this by duality group it was always finite. Yeah. >> Yeah. So yeah, rank one things are always an exception. But yeah, when Yeah. Yeah. And then Yeah, I think that Yeah, that tells you about the the duality group. the the duality group should be such that this these modized spaces have finite volume and that that should be true in general but I think so here it's a I mean yeah here it's very easy to to to determine whether something is a good good duality group or not in general for for general mod spaces I don't think there's such a there's such a crit there such an easy criterium an easy way. Okay. Now so so now once we put the gamma in before the the the boundary points were fixed by some parabolic subgroup and now the relevant thing are the rational parabolics. And maybe one way to see it is uh from the simplest case which is SL2 over SU2 which is like the upper half plane and so if this acts so if if these groups act on by fractional linear transformation the standard here there's only one parabolic subgroup which is the one upper triangular there's nothing else you can do and this parabolic subgroup fixes the point at infinity and then If you do so rotations on this then you get all the all the real line. So the boundary of this is the real line. So be before the gamma right before the duality group is all the real line and the point at infinity. Now once you add once you add the group in what happens is that first this is not the real the whole this is not the real line but is the the the rational points which are dense but uh but still these are these are the rational points and then what happens is that with any SL2C one can go from one rational point to another with an SL2C. So any rational point is equivalent by SL2C to any other and the and so that means that here the boundary is I infinity and um and and that gives you an idea I'm not sure that really uh with that you can really understand in general but why why up to the duality groups then the important piece is the rational part of the of the parabolics of the parabolic subgroup but this will be key in uh the fact that it's rational parabolics. This will be key in showing the the sump plan distance conjecture once you have the duality group in okay and then yeah and then the once you have this then the you can the the you can draw the fundamental cell which is the one the one we all know and it has a single boundary point which is infinity high infinity. So the also something that is interesting is that the only shistics associated to rational subgroups reach the boundaries. The other have energetic motion. So for example this is well known for this case that if the B field is not rational then you keep bouncing bouncing bouncing and you never go to the boundary. Only if B is rational then you can eventually go to the boundary. And that has to do also with I mean that has to do with this rational rational parabolic subgroups. Yeah, you you keep doing SL2C transformations and uh and you stay this way. So if you want to go to the boundary, you have to start from a B field that is rational. Okay. So now to the sland distance conjecture that the clock is so so this is the conjecture and so first we're talking about states that become massless so we have to so states there's a spectrum of states and the spectrum of states is is representation and representation of the group because everything the representation of the um of the duality group extends to representations of the group and if the representation acts on Rn then one can think of the latis as as CN Z and so this is all the possible charges we're not we're not going to assume completeness that all the charges are there but that this the spectrum of states is some invariant subset under this some subset that is some subset that is invariant on the duality group so for example if for OKK Okay, the charges are momentum and winding and then the populated latis could be some sublatis of this like states which have purely momentum or purely winding. These are this is invariant under under SL2Z and um so so again so we are going to assume something assuming that you have all the states could be assuming completeness we are going to assume something that is a little bit weaker than completeness. So, so far this is not an assumption. The spectrum should be uh should be something that is invariant under gamma. We're going to assume something about the the spectrum. And now the mass of the states in some invariant function of the charges and the moduli and this depends on the representation. So let me do the case of OKK because people are familiar. So the masses of momentum and winding states can be computed using this generalized metric that has the metric and the B field inside. So that gives a n / r 2 and then m * r² and then the b field. So it's some function like this and this function is some invariant of the of the group of the duality group. Now this generalized metric you can you can split in vine like here using the vvine for the metric and and the inverse and if we call this thing so this is this is called the momentum. So this is the momenta that depends on the charges and on the modul here are the moduli. So the the the well metric and and field moduli. So this uh this this is the dress momentum or or actually for the the for the nar latis these are called in general these are called momenta. So in general the mass square has to be some invariant function of the group some invariant of the group some function of this momenta and now we assume two things one that the function is analytic which might not always be the case um but we can discuss that and then we assume that one state becomes massless at the boundary. So um so remember the slandista conure is about the tower of states. So we are going to assume that there is at that point of the boundary there is one state becoming massless. So at that point on the boundary one cannot have this constant state this constant term and you start from a from p squ and >> why do you need to uh assume that a state becomes mass boundary? because otherwise this doesn't work. this uh um yeah you you mean this so this is true sorry so one can expand one can expand this assuming it's analytic one can expand this in powers of p but uh to get rid of this constant term you need to assume something otherwise uh um that could be a constant term there. Uh so in general this is so in general what you get this p square and this is the invariant uh this is the group invariant the cmir invariant of of the group um but why not a constant I mean it would be weird to have some constant term in all the >> yeah but you're trying to prove some general the question is what assumptions you have to make in order to >> Yeah. Yeah. So I said >> this is a very important assumption because if you put it there it's the end of the conjecture right >> I I I agree. So I said at the beginning prove that there are mild assumptions. This is one of the m these two are the two of the mild assumptions. There will be another one soon. Yeah maybe you don't like to call it mild. You you think it's strong but I mean just to otherwise I mean what we say can be said. So there will be a tower of states become with who for which this will go to zero that we can prove and then if there's a constant term then they as to this constant the world states as to >> there's no version where you say since you have a compact space and you look at uh things that are analytic by some general analytic function theorem I either the function is constant everywhere or it needs to have a zero for example is that else. >> No, I >> logic like that. >> No, I don't think so. Um, yeah. No, I don't think so. Uh, I mean, but well- compact. It's compact. That's why I don't like the terminology of compactifiable. So uh that that gives you some mileage but you cannot use the the the theorems of compact manifold. So um so I don't think so but I don't know maybe so yeah indeed. So one this is these are two of the assumptions which you can call mild or or strong. I mean it would be weird that the states go to a constant uh and also um it's at every point in modulate space I mean well there are points when this can be when this can be um when this can be important but at all the boundaries of modulate space the masses go to a constant that would be very weird. Okay. So and again to to understand how this is let me again use the example of OKK and in particular O22. So this is the generalized metric has so if I I'm interested in so moving along the cartons because the rest doesn't gives you finite distance and here the cartons there are two so you can you can have only two rad going to infinity. This is these are the inverse radia and the states are momentum and winding states. So in general you can write so here is very easy to see but this is a general formula. So your charges your the space the spectrum is divided into some weights under the h under so under the group. So the the state each state has a weight and then the the the way it it behaves from I mean this term the way it behaves is in this form. So for a given so each one so for example and and this has to be normalized. So we have to have this. So for example, if you take the cartana, if I'm just growing this radius or the way we wrote it, yeah, if I'm just growing this radius, then uh then the well the states that will become massless faster the leading tower are the states that have momentum along this direction. So this is a yeah this is a general formula of the shics just because I mean we know how the shics are. We know that we just have to move along the cartons. The rest will go if I grow some pieces in here, I will go in this direction. And and then the way we choose these lambdas tells you which boundary point you go. You can take R1 to infinity or R2 to infinity or both. Um so if I take just R1 to infinity, then the leading tower will be this. And the the weight of that leading tower we call it omega star. So this is for a given age. So it depends in what direction what what rad you're growing for a given age. Um there is a there is a decay rate that is a um that is the maximum the I mean you you see for a given age you take the weight such that this is this is maximum and that way we call omega star and this is is very nice encoded all this information in a convex hull and this uh this formulation we're going to use and we're going to get mileage of This convex hull is that was worked out by by Jose calron and Valenuela is that this set of alpha define a convex hull. So for example, let's take the three of SU3. So the the tower has this decay with this inner product of the weight and the carton. The inner product of the weights and the carton is given by these bubbles which is called the the alpha hal. So this for for an h this bubble gives me the projection of h along omega and which one is the leading so for a given h like again I'm taking r1 to infinity which one is the the weight such that this is maximum so depends on the h but if h is along any direction here then this will be the the this will be the the weight this was this will be the weight of the leading tower and these alphas then are larger or equal than the distance to the convex hull. So these bubbles which measure the the the decay rate are larger or distance larger or equal than the distance to this convex hull and this convex hull is in nothing else but the wild polytop of the representation. So using this then we get a lot of mileage that these alphas are given by the distances to the to the wild polytope of the of the representation and that's how we got I mean all the a final list of polytopes because now now we're interested is in these distances and given requiring that these distances are what I want them to be then uh then it gives a finite list of of theories is so now the before going into that the proof of the swamp plan distance conjecture. So how do we prove so we assume that there is one state that becomes massless. So how do we prove that there is a whole tower? Well, we have the latis and then the the space the the space on which the representation acts is h is divided into weight spaces and now it's not it's very easy to show because the boundaries are associated to rational parabolic subgroups that these slopes are rational that these weight spaces they um they meet the latis in an infinite number of spo of points. So assuming that one assuming that the function doesn't go to a constant and it's analytic then you have an infinite number of states in this weight space. Now since if we were if we were assuming completeness then this will be the whole story. We're assuming something a little bit milder than completeness which is that for all the vertices in the wild polytope the so there is no state um for all the vertices in the wild polytope then the the the intersection of h of the latis with with this with these spaces is the same as so basically that these spaces are populated um this is a bit milder than complete if but I don't know it maybe it's less intuitive so one can think of one of our assumptions so we call this semico completeness um if you assume completeness of the spectrum then it just goes through because of this parabolic uh this this rational this property of the rational parabolics so and this is true for any boundary point so for any boundary point there will be this infinite number of states becoming light and becoming light in this in this form. Okay. So now to the um prediction. So it's quarter to three. I might skip some things and and go to the result. Let me just explain a little bit what we did and then maybe skip some um some details. So again the question is what modulate spaces satisfy this what we call immersion string conjecture rate. So there they are um particles that decay and decay in this fashion and we are going to do this from the convex hull. Now the convex hull. So when you measure the distance to the facet the the so this the distance of the complex hal is extremised at this point at the vertices or at the middle of the facets and this h the the so what so this extrema should correspond so this this extrema corresponds to a weight. So in the case of uh in the case of SU3 or SL3 this this correspond and I'm moving or and and I'm making one direction in one radius to infinity then this will correspond to the this extrema will correspond to the kuda line tower that has momentum in that direction and so that should be kuda client tower I I call it n plus one because it's simpler to keep where n is the dimension of the facet So a point here corresponds to one. So dimension zero corresponds to a facet one dimension and it's a kooa client tower of one direction going uh going to infinity. Then this middle can this middle will correspond to two directions going to infinity because this line is like r1 is equal to r2 and they're both going to infinity. So we require that all the I mean the different facets according to the dimension have this this distance to the center and the last facet the last co- dimension one facet could be either kaluta client or string it could be a tower of kuta client states if there are only kaluta client states and if there is a string there's a um there is a point in at infinite distance so if the daton is inside our modulate space there will be a point where there is a tensionless thing. So for the last facet one can require this or this but it says that all facets are at the same distance are equal and so they are regular polytopes and they are very simple just so because if you do the the the difference between a facet of dimension n and n plus one this d disappears and there's a relation so the polytopes are very very simple the the you get simplex or orex depending whether you have a string a string um a limit where the where the tension string goes or not but they are very simple these are classified not many is is very nice now what makes things uh what what now for the full polytope so these are the facets of the polytope for the full polytope and this this is where D comes because this was all about distances between a facet a a a point the distance from the difference between this distance and this distance let's say but the overall distance to the center now it's related to the dimension and here we have to put the requirement that the is a natural number and this is what uh really boils down to um a finite a finite list and there's a unique dimension for each for each of these each of these polytops. So the requirement that D is natural. So as a you were saying and natural comes here because it's a dimension of the facet but even natural is not a natural thing that comes out. And the nice thing well this rank polytops agree with this. The nice thing is that the list of all the polytos that appear actually with this are realized by one. So are the wild polytopes of a given group. So all the so in a sense maybe all we need is that the boundary is a locally symmetric modulate space and we don't care about the bulk and we can show this very generally. Um there are these polytopes for um that realize the different boundaries and what's in the middle we don't care. And then yeah we didn't um I mean we we work the other way around. So looking at the polytopes and satisfy this and put this and this is natural and going case by case but we could have done the other way around and then see the final list of polytopes and ask code. So we once once we knew the result and then when we had this result cloud wasn't so cloud chip PT wasn't so powerful but uh then once once once we had this then we asked CHP and then it is yeah they're all realized one given that provided is a natural number they are all real there so they're all the wild polytope of some representation of some league group so um that's why I was saying this might be much more general. Okay. So, effective field theories in the landscape. So, uh one thing is that we assume that there is a so we assume the split form okay K for simplicity. So, split form means this that the real rank is the same as the complex rank. So, we're not doing K + 16 for example K. um we I mean we're we are thinking of it we're doing it now but uh we haven't done it for the for the paper and then we assume that all the vertices are at the same distance and that means that particles are in a single representation of of row. So also another obvious future direction is to is to relax this condition and look at what happens with there are multiple representations and uh well let me know I mean if there's an factor then it's very easy to satisfy this you put this these very regular polytopes at some point from the center and then you you can realize all these groups in any dimension so these these groups once you have the R factor then you can realize this in any in any dimension because you you use I mean this is where you put the the center. So these are a little bit of of a cheat. The nice thing is when these are factors inside like in the exceptional mod in the in the exceptional modulate spaces where the the dilaton is is inside the the group and what we get is this only these polytopes satisfy only this I think there are 11 polytopes satisfy the requirements and each of them is realized by two three or at most five different groups. So these are all the groups and and the representations that you can have in the landscape. This is the I mean that's it. Anything else except for example that I'm going to say but everything else is outside the landscape or does not satisfy this um this uh immersion string conjecture rates one. So there are only 29 theories. So with a four before that we had before it makes it 33. And one interesting thing is that if you look at the maximum dimensions. So for example uh this arises in three dimensions and it's rank three. So you can decompactify this up to six dimensions and you cannot go any further because there are three directions you can decompactify. And one can never we didn't put this 11 that the maximum number is 11 but this comes out this is not an assumption but it comes out that you cannot decompactify up to more than 11 dimensions and then well we see the E87 E6 E5 deceptional theories and there are some some theories that we don't know the string construction for example F44 neither of them we know the string construction But um yeah, let me say so we could in principle get it. Um so yeah, one important thing I'll say quickly is that actually for every for every point in the table there's an infinite family because you can you can you can multiply the highest weight by a number and divide. So there's a symmetry of the system which is multiplying the highest weight by this uh number p and dividing the dividing the the inner product the metric and then you get a bunch of representation for example for C2 you get the seven the 27 the 77 they are related by this so actually if this is possible then all these series is possible and we don't know of some arguments such that this is not allowed h you change the curvatur but uh I mean we don't know of a principle I mean maybe one can put some plank uh plank like it shouldn't be more curve than one over plank square or something like that but yeah I I don't know so this up to that subtlety then the point is whether all these these theories are related and the answer is mostly yes there are slices from each other so slices of the type of orifold that arise in the dimensions or slice of the type of decompactification and yeah let me skip the details but as polytos they are all related so one can all these are slic slices of this but once you put the representations in in uh inside it's not true that they are all related and if we look at the threedimensional the ones that arise sorry not threedimensional ones but the ones that arrive in three space-time dimensions. What we get is that most of them come from the eight but there are very few of them that don't. So this don't come from that and uh so we don't know if string realization and doesn't seem to be uh some string realization if they don't come from this. There are others like for example this f4 with a 52 we can so it comes as a modulate space it can we can construct it as an orifall or a u fold better of this so that it's not hard to to construct and this is we don't know if the string realization of this so we could give you the moduli space realization or the super gravity realization of this but this doesn't tell us that uh it was it's a U-fold so we cannot check I don't know modular invariance for example of that so we we can construct again the sort of the super gravity realization of all these mod spaces but that's not necessarily I mean doesn't necessarily mean that there is a really string theory real a string theory construction >> okay this representation In all of these examples in subra these are the vector multipliers and under this representation of the duality group or >> this is a representation of the so this is a represent so the the group here is a duality group and this is a representation of the of the particles so since we are not assuming any super symmetry um it's a the representation of scalar particles >> for instance the e77 which is like n= 8 to grow in four dimensions. I think the 56 is the vectors. >> So yeah, E77 but we so E77 is not here, right? Because this is the So let me go. The 56 is the is the particle. It's not the vectors. E77 is yeah the 56. >> I think the the scalers are in the 70, right? >> The sorry this is the Yeah. So it's not the scalar. So this is these are the states with momentum winding. Uh and yeah if if you if you want these are the yeah so these are not so the the the 70 are the moduli and these are the states that have momentum or winding in the seven. So if you start from M theory these are the states that have a momentum or M2 or M5 brains wrapped on the parts and the cycles of the Torah. This this is what forms this 56. >> It's just electromagnetic charges. >> The electromagnetic charges. Yeah, exactly. The electromagnetic charges. Yeah. >> Okay. Okay. So, yeah, I was at the conclusions. Uh so, okay. Assuming semicmpeness plus assumptions that some people don't want to call mild and the the rates. So we we show the SL plan distance conjecture and then given that the the the requiring this distances to be this the we have a final list of uh of groups and a final list of polytops and they all have realizations in terms of regroup. So they're all the while polytop of irreducible representations and uh well every solution comes in with this countably infinite theories and um the coming families generated by a seed. These these are things that I say they should we should with this we can give the realization of some exotic modulate spaces but it might not be consistent and there are three families in three dimensions that will come from E8 which are this and I leave with the the introductory the introductory picture of of the landscape. Thanks. >> Right questions. >> This D that appears in alpha the square root of D minus 2 was that supposedly related to the dimension of something originally or >> so this is the dimension of space time but >> it it seemed to not be related to the dimension of spacetime at all. It was it came from the schol it's also the space time dimension in any way or is it >> so it it turns out to be I mean so what you get is that E8 is only possible in dimension in space time dimension three so it cannot be um >> that was gravitational input or that's all pure >> no no that pure I mean we require that the distances are this and then d comes out and d comes out to be what it it is in all examples that we know, >> right? But there was no but your group theoretical construction up here we had no reference to any dimension whatsoever. no no okay and did you did you uh you assumed that d was integer to to call that >> so yeah as I said we did this in two so when we did it we did it in general trying one polytope after the other blah blah blah and yeah assuming that d was integer then once we knew the polytopes uh actually we then we asked CPT okay what are the polytopes that satisfy this simplex or complex plus DV natural a natural number and then it came out uh that that they are all associated they're all the Y polytops of a of a group >> but but just purely group theoretically D could also not be an integer >> based on your input like the assumptions that you made >> yeah yeah yeah I think D cannot be D is not necessarily any integer and then and then it's not going to be I guess my guess is that it's not going to be the well polytop of any league group if D is not an integer but I'm not sure I'm not 100% sure but definitely yeah this D does not need to be an integer once I mean if you do this purely from the group theory >> all right let's take other questions to the break and we reconvene in 15 minutes maybe a little less