 We'll talk about geometrical challenges for the string landscape. First, the sound works, the sound on. OK, good. So good morning, and thank you very much, Johannes, for the very nice introduction, the invitation, and the great pleasure of speaking at this more mathematical and physical conference. So I am in a slightly difficult situation. Probably what I'm going to say is going to be much more critical for my physics colleagues who I've seen here. And the majority of my students are going to be too physical and too critical. But nevertheless, I think it may be useful to have one talk where one might be thinking of what physics implications are, and what elements of physics. And so let's use this more on occasion, like some discussion and questions. Please interrupt me with questions. I'm not supposed to be very formal. I'm not going to spend the short original result, like you often do, or you can use it to do something. And in part, I won't take much of it. All right, so the title is geometrical challenges for the string landscape in the 2000s, I guess. And so the plan is I will very briefly trust with you the landscape of string landscape as it was accepted for the last 20 years. And then the focus on problems, which I can discover in the aftermath of the city and other swamp lands that I eat recently, and maybe even physicians have heard about the swamp land thing being in physics recently. And so I will focus on things which are close to my heart. I think I'm important, which are the people of both problems of virtuality. The tetral constraint of a large volume scenario and the tetral conjecture, which is made of a more mathematical of these issues, which could, if true, which I don't think is actually going to form, could be a short stop for the whole length of it. OK, so let's start. OK, thank you. Now it works. OK, so let's recall. So we talk with string theory. And for the purpose of this talk, it's going to be mostly the typography side, the tangy typography for which we will look on solutions. The field theory, I mean, the coefficient of all of these problems, we'll know what the field theory is, the classical field theory. And we are going to look at solutions on a base, which is in Kowski's case, the time is x. And x is going to be a collabial. More precisely, it's going to be a collabial oriental. Collabial, modular sum, discrete final k. It's so-called collabial oriental, because together with this evolution in space, we have the orientation reversal of the string. But this is not completely too important for the purpose of the whole length of typography. And the visualization is, I mean, what kind of typography can have, it's a power tp by v2. We will use such a base, which is called kilo or newly, by some, it has four conical similarities, which are called the different oriental planes. And depending on what is going to be done in this epilogue, and then depending on the action of this group, these similarities may have different co-dimensions. And the point in the picture is full co-dimension, or like this co-dimension complex one. That's exactly what I'm talking about. Okay, so this space is basically the epilogue after this whole division. And then of course, on this space, we are imagining that there is a field theory within the supergravity. And there are metric fields in particular and undersymmetric tensile fields. And so the full dimensional effective future, then it's supposed to describe our real world based on the geometry obtained a certain field content, including in particular, not only, but in particular, the module line of the Kulabial Act, which is now not pure mass in the center. There's really a dynamical metric field in 10 dimensions in spring theory or in supergravity. And then in 4D, the dimensions of the Kulabial really become dynamical fields, which we observe in our real world, which we think of as strictly massless scale of fields, and which are supposed to be a problem. But let's just a moment, we'll come to that at the moment, let's write down the four dimensional Lagrangian governing theory. And the key point of this is the connected terms of the module line. And they have the form of a scalar metric and modular space for the complex structure part, on which I think all that, and the k-node part, which I think that's key. And this t is not quite a mathematical piece, it's an exercise version and the multiplication and its origin in certain integrals of p-form fields, like in the D here, in the, it is really now also the complex versus truly physical. And yeah, so we can go on. And then with this telemetric, the telepotential, which you can do down in a very well familiar kind of way. I mean, the structure of the life, which is a lot of integrals of, either the Kulabial, or the Kulabial, and then of course it can be rewritten in this more explicit form, using the period vector, which I will denote like this. So one period from the community, then there are z and five, which are actually, I'm wrong here by one, never mind, it's not my fault. So basically the key point is that there are only n or h to one independent complex possible like that. And then there are complicated periods, which of course you know much better than I know. And in addition to this, there's a similar telepotential for the telemoduline, t, I'm specific, I mean, depending on which symmetry, which is a string theory, let me focus on one example here without much detail. I take the case where it's defined in the way I wrote down here. It's the large volume limit. T are the volume of two-cycle moduline. And the kappa-alpha-beta-gamma are the intersection number of the corresponding, four-cycle. So this is the expression of the keys and then the keys are related to the tau, which are the volumes of the four-cycle moduline. And for a reason of super symmetry in this particular model, I'm considering the good variable of tau, which are the real part of t plus t bar and t is the real complex variable that you have. It's slightly indirect, but in the end it's not important for what I'm going to say. It's not important what the delta is in principle, well-defined function. There are closer corrections. I mean, you can, in the law of course, more than neurosymmetry, these key keys are related to here. I'm missing what is called incident corrections, which are not important in the large volume limit. So this was very, very introductory. I'm not sure maybe if you think too much so, but I hope, I mean, it's really roughly something that doesn't make sense. Okay, so now things become, I mean, this is not too soft to know better than me, but now things become more physical, namely in the super gravity, in the super gravity, they are also, as I already mentioned, they are form-cured. And in particular, there are two very important form-cured purposes with three form-cured tracks, X3 and H3. And it's common or it's useful to combine them into a complex, three-form, which I wrote down here. And the coefficient itself is also a few, in fact, a combination of the inverse string coupling and the zero-form here. So this is maybe more detailed than in the moment, not sufficient time to accept. It's not too crucial. There's a complex field, dynamical, and these two real three-form fields combined into these two. Now, what is absolutely important is that this field in the equilibrium can take a vacuum expectation value. A vacuum expectation value in the sense that even the vacuum solution, the unexcited, calm world has a certain value of the field and that value is called background-collectionist quantum. Quantum is very explicitly here by this X3 and H3 being part of the integral form-cured. So the reason why this quantum is so important for you is that the world has a quantum mechanic that will be directed back to the quantum theory, but that's not crucial. So we have a choice. We have a choice of taking a quantum for those fields, picking a vector in this huge, in general, very large level. And once it's done, the first key point is that now the modular, which we have to take a quite a potential. So if we see, if not anymore, it is described by these specific terms, which is to do with telemetrics, but this is also a state of potential, which tells you where the minima and where the real world sits, and this is even here. In a very compact way, it can be written as this is the inverse telemetric I had before. So far, sorry, it applies so far only to the complex structure modular, which is kind of natural because both govern the three cycles and these are three-form plucked with the outgoing and pretty much even how to derive it makes sense. It's not natural, but it depends. And then we have here the covariant derivative of the two potentials. Two potentials is given here, often known as super far for written two potential. And it's the integral of this complex pipe, the three-form plucked with omega, and the derivative, the covariant derivative of covariant is what depends on the scalar potential given here. So, it's a relatively simple natural expression. And it's going to be very, very explicit because of course the W, which I told you this integral can then be translated to sigma being some vector matrix between these two vectors. The one vector is built, the one vector which I will call the flux vector, it takes two flux vectors, F and H, so these are now the integer number vectors for the two lettuces of the integral three-form normality I had before, as is still the complex field, pi is a pure vector. All right, so now comes maybe a key point to think about the potential. You can now take two perspectives. You can think, okay, I just have potential, I'm going to take the world to the minimum and then the conditions around the minimum level. And it's equivalent to say that this will think it's symmetric because you are on a color yellow and principle the feature will be broken by the fluxes but it will be able to restore the next minimum from a particular point where the complex structure model is given that these flux vectors are chose and the conditions are here. Those are probably everybody who has seen some simple symmetry they're carrying the root of W being zero with the I, I labeled the complex structure model, I and S. S is this one extra field, the string cut. Okay, and this generically stabilizes all this model line that I and S and through the n plus one operations for n plus one variable and now they're as many vector as their choices of this vector as an H. Now, F and H are both vectors of the size of the three sides. Stop, I mean, stop, anything is not true. So now unfortunately, there's even more physics that has to be taken. I mean, I want to take a picture, pay where the physics comes from and then formulate the mathematical requirement following the list. This tend to supergravity is a complicated animal and has more of these forms as another high-form field strength. And this high-form field strength is sourced by the flux, the flux meaning the three-form flux with the stress and it's also sourced by the old planes and also brains which I did not discuss too much, which I already mentioned. The single of corners will be down to then sources for the size. And so the equation which is basically just like Maxwell equation is that D star at five if the sources and the sources are in this combination of the three-form field strength and some local contribution like for J's law force localized. And so there are two pieces which have to add up to zero because this integral of D of something of compact space. Everybody knows why this is called a pet code? For the pet code because if you do physics with this you basically have some things you source a diagram. You have a diagram with sources of the field with one end. It looks like that. It looks like that. So this is a particular equation that everybody knows. Anyway, the stupid word. So this however now is extremely important because it means that you cannot switch on this three-form flux at liberal. You're constrained by this equation. You're not constrained as this product of integral of the product which number has to be zero but it has to be exactly compensated what you have in this localized source and these localized sources are based on geometry of X. And now it's important that really the geometry of X not of the Calabria. Because if you work it out in detail it turns out that this can be written as one fourth of the number of four fourths or three-play. Those are those singular points which are full full dimension. So really the singular point in the Calabria. And then the other number of the fourth or seventh then these are those which are surfaces with full dimension complex one in the Calabria. And then there's also contributions from the given range which come together with the old play and this should be read from the Calabria which I basically should not discuss here. I mean, I'm happy to discuss the Calabria time all the time though. Okay, so there's anyway, the key point is that given the Calabria you have a number here and then the fluxes are constrained by that number. And there is, I mean, what I'm going to say next the most of his models and the problems on them are in this so-called type computer gravity. But actually the better setting to discuss is the more mathematical, nice setting, it's f-series. And then f-series, one does the following. One looks actually at the Calabria of four fourths which is an elliptic vibration, which is two vibrations over some complex space in three. And in this context, this torus vibration actually, I mean, the way the torus literally vary the complex structure torus varies over the base, describes the profile of this field S which I have here, the string coupling in it, which then is not constant. We have a meter constant but then f-series constant varies. And in that context, I mean, many things are much more complicated but some things are more elegant. For example, this localized contribution that's all a number of Calabria of four fourths that you have for a single clean quantity which is not exactly the same. But still the same thing happens, it limits the flow. Okay, so now we are at the key point here. So the finite net of this available end because it has to compensate this Q which was a feature of the geometry. It meets with the finite net of the density. And it's pay payments, I mean, it started a long time ago but then it's back to my point that the key papers that was in the density and let me mention a few more things on the focus on this very important issue of the tightness or sorry, Thomas, I know I'm boring you all the thought but then we're gonna need two or three slides. Interesting. All right, so this gives you this huge but finite length. No question, everybody knows. So now we only discussed construction modernization and now I have to turn the kilomodernize because the ugly and really physics like part of the talk but I have to do that because it somehow very difficult as my work and I think it is the point where at the moment the decision is about to be made for the landscape actually it's a form we love it and I think about it exists as well as it's described in the world and the issue is really the kilomodernize, in my opinion. Well, it was completely cool. People who believe in technical construction maybe think it's also connected to the universe. Anyway, let's see. So kilomodernize are also important to me and see the story is far less nice, I have to admit. As the big progress came by now almost 20, no. Well, it was like 20 years ago by KKLT, famous scenario and the first step of KKLT, cultural colors, limited creativity. So we assume for the moment that construction modernized by the mechanism as I've explained before that just indicate about their stabilize, their fix over the beyond can only be formed in the kilomodernize informations. And so here's my cartoon of the KKLT now there are various cycles there in particular I brought four cycles here and these four cycles can be read by something that's called a Euclidean B3 brain, read them a four cycle, an incidental correct. The brain is dashed because it's really not there. It's something like a quantum effect that the brain occasionally can nucleate off a vacuum and it appear again. Typical quantum effect associated with the brain cycle and leads to a correction in simple tension. We should remember what flux induced and this is the formula at the constant now and plus exponential exponential in kilomodernize T, T governs the site of this foresight. I explained that. And again, there's a formula for scale of potential which now it's well it's actually now the more general formula. Previously I gave you a simplified formula which gives you a picture of the kilomodernize and now it's very abbreviated and has a minimum and can show with this answer that it has an anti-discipline here. So basically now at this point, the following thing that you come to such a model that are gone, the kilomodernize stabilized people use instant and effective a certain point and the complication is now even though I thought it was, you know, I was facing on myself I'm gonna be out of time in Copsky but now it's actually gonna be out time. Let's just look. Okay, so that's not good enough because our world is not under the center. Our world is very precisely in Copsky they didn't look very carefully it's a little bit different. So step two of this famous KKT proposal is that you need to consider the case which is very common that the Kola Biao has a conical singularity that was thankfully yesterday already blamed and discussed and many have been like very well I'm going to explain what conical singularity is and if however together with conical singularity there is flux then on the various cycles of the Kola Biao then it turns out that you're actually not a singularity but slightly away from it and the deformed side, not to explain that what the deformed side is and so the three cycles take the finite size and then actually something else happened which is now very, very physical which is both working. So my picture of the Kola Biao is now this here goes into this so where you had a singularity there now emerges some usually drawn like this and called the throat, it's and this picture is I mean trying to be faithful in the sense that it shows you what the geometry is a wave that was the Kola Biao geometry but this is the metric we started with and including this flux in this particular conical situation the metric changes like that in other words, you get a so-called warp backbone depending on why being the Kola Biao the point of the Kola Biao multiplying the four these days so it's actually now not a slow death like in a complicated that's not important for us crucially the Kola Biao itself the compact geometry is forced by a tree factor both conformal Kola Biao now and in such a way that this region is extremely over-enhanced and in fact this region part of the region that the singularity was used to be it's now it's like a huge potential sink in this Kola Biao we have to think of the Kola Biao not with a real face so that region where gravity I mean actual gravity because there's an actual dynamic metric here pulls everything to both the point and at this point you place a metastable well metastable in the context until you show it so you add an ingredient subject to all consistency requirements of things you can plot something through by hand it's something that you print it all out and it will print that thing which will break super symmetry break a super symmetry into this important vacuum energy uplifting in what the word technical terms you will come back into the future technical terms I just wrote them here I mean I hope you know Klaven of Krasnowski the name of this strange geometry which is by the way I mean this I mean of course you know that the the form 24 is one of the Kola Biao metric which I know explicitly and moreover even the formed version with the whole thing and everything it's fully not the metric here is the famous solution really and this is the metric that's all for the definition and this is called the KPD kashu pi of nalini so you're going from this situation you access these three under these three potential and you are adding it you can't avoid it having being this slope and so you kind of you contrive everything I mean I have to use a negative word contrive here to have a little minimum here somewhere which can only be metastable where you sit and you expect to do the sitter, metastable, the fitter, long-lived let's say the fitter described our way of doing it and that's with this new variance one has to be honest the only thing we have achieved and that's how we describe the result and think it's really very intricate and detailed and painful coming to the point which may or may not care but if you do care I mean if it strings you, this could be our business it's really relevant for the real world I mean it can be in many other ways but this is the best way we understand and as I already said, I'm an architecture expert so in fact I I miss saying that I wrote it down here for this to work with minimum here you actually need the W0 being in a small number which is possible but I think you need to try to start it appropriately then um you have it's under the three uplift follows from the 10D effective field theory I mean there's no string of 40-degree derivation it's a horrible nightmare to actually give this I mean this, this is not beautiful math this is something that we really need to be critical of uh and now come more recent development this and some important variant like large volume scenarios should be mentioned uh have remained the main evidence for uh stringing the citrate has been imposed based on this in the citrate it's impossible to matter of principle the possibility that it is so unlikely the capability to work and only there are good reasons why why strings here are in principle the four types of things and so on so and subsequently because of this kind of counter movement people have started very carefully to study the construction of the outline KKD and LDS um have been in terms of subject to intense scrutiny with varying success some things it has passed 30 tests and other tests it's about tail and so I would focus on what I think is most critical so for example one critical thing is that as it turns out you can study parametrically what I mean this short region that you have that you need to break your symmetry has to have a certain size a significant depth because it's uplift I suppose you have to be very small not to destroy your reconstruction then it has a certain width now this which tends to be bigger than the size of the collabial we are gluing some some some strange work within the collabial which is in principle a smaller size that's a parametric clash here and I uh let me draw redraw it here so you're looking at something very weird you're doing a big stroke you can be able to small somehow swapping the function h has to change the design way from here to here it turns out you can show this is our I mean this among other our work that with h of y goes to zero metric becomes undefined and it actually turns out that this scalar potential for the scalar modular k of e at the bar is not calculable our metric control is lost because we are in a strongly curved regime in a singular way so uh the thing becomes internally inconsistent and not in a not in a actually deadly way I mean we cannot show that it cannot work but uh certainly it cannot work in a regime when you are sure that all the physics corrections which are all over the place here are not important and then the beautiful marsh is still there but it's not the regime there that uh we call it a singular bulk problem and that people have written counterpapers saying that things you you feel it if you look carefully if you think about what you do after all it's all somehow it's like don't think not to think uh I have you one slide where I go with you to the parametric analysis of this but I think I should I think I mean I have to do something um to possibly better shape is a so-called large volume scenario we are seeing you as you see I have to I mean people now start adding at the side I mean you have to add at the side it turns out I mean it's natural that we have more than one kilometer what I described for this and the Calabriao with many things of plastic moduli for the fluxes and one kilometer now I go two kilometers now the calamatic which I explained to you how to construct from the mass input explicitly in certain Calabriaos take its form D and S standing for big and small cycle this is an important higher curvature correction which at least in principle I would mention of this and uh then you again get get such a two potential and the interplay of this one can now show what we get an ADS minimum similar to KTLT but with the following key different we now have two four cycles in the Calabriao and they get stabilized at this ADS minimum such that one of them is exponentially much smaller than the other I mean this has to be big for for us to trust the classical geometric description but this is much bigger and uh explicitly with this volume it's exponent of one of the spring coupling this can be tuned with small values using the enormous amount of flux showing the discrete numbers but it turns out that this volume here that we get here is exponentially large but may as a matter of fact still not be large enough with sufficient quantitative control very surprising important paper of Wienhans in fact there are various corrections here I mean the complicated story and the corrections basically come or they're related to the fact that the action contains not only the actual Hilbert term but in principle contains many hierarchical terms so these are of course not irrelevant and not obviously zero if you have a Calabriao you know for the Calabriao you know you know ancient legend you know these terms in the case of Neymar Salt that's all this beautiful story all based on this term but we go beyond this there's other stuff which cannot avoid which seems to be big and so problems arise yes control means something which will certainly very much of objective control means that I have in my equation of motion or in my calculation for the point which I stabilize my moduli higher order terms so I cannot really calculate because I have extra terms in my equation of motion which I neglect and I neglect them on the basis on me stabilizing since in a regime where these terms are small and the regime being that regime it's not a mathematical limiting decision I have no mathematical limiting decision because I'm in a finite I mean I'm in a theory which is string theory is no parameter all I can do is choose a flux by choosing the flux I'm choosing this quick number that in a very large set which is finite and in this usually this choice I make this G string which is stabilized by the ordinary equation it's more it's not that this is done but it should be possible then the volume becomes large then in this large volume which I now treat the parameter I have correction which is sub-reading sub-sub-reading and so on and control mean that I am in a regime when these corrections are numerically small when this ends just numerical but it's not really and it's not really trustworthy numerical like I have like I have a full series I can sum all terms and look at the first and I can feel the error I have the first term and I have the second term being which I don't know the coefficient but I know the expansion parameter is one time okay so now that is the situation unfortunately but surely I mean this is the large cycle this is a large cycle yes so surely it's exponentially large that will be some number of it's being changed right it will be many string lengths but I will argue so what is the when you say it doesn't I mean I would be worried about the small volume like that I mean what you say you're not large enough for control if the large one is not large enough for control yeah so there are what what's the other parameter that makes I that was exactly what I was about to explain that so what happens is the following so only in words I mean that we are running out of time so only in words and I apologize for each of this it's a series of several papers right I mean the volume is very large the volume is large the ADS minimum the minimum we are stabilizing it's it's very shallow it's back to one of volume few so if you want to uplift a shallow minimum and you don't want to destroy the minimum you uplift it by little bit uplifting a little bit means the throat which is clever as far as I think the only thing has to be the ideal that requires that you have a lot of flux in your throat because number and the flux contributing from the throat being has to be large now but remember to the whole thing to their curvature direction their curvature direction with higher terms which could be behaved like and throat or volume to the third so now you have things look and mainly you think okay things are perfect because the volume exponentially large and it's largely large in the throat now and so you can be sure that for very large and throat these corrections which come from the high curvature term which are of type and throat of a volume to throat will actually be small because this is large and this exponentially large in end everything is fine but the problem is then then you work out this formula with all the small patients I mean all the corporations and pies into only load and then it turns out that again go in the regime where before this beautiful logic this is small this and throat it would be large but you cannot put a lot of the end and and it's limited by the available Calabria or Yentic or Pionic this Q kills you this Q prevents you from going into the regime where you think you are safe and focusing on the most optimistic regime basically where all the flux number that we have it will drop a few up in the throat and also defining this control pyramid I call it control pyramid it's kind of it is an expression for the relatively important sub-leading term in the series which should drop it's like whatever excellence function feels very and not excellent because it's small because you want to consider only for the excellence that's what I was trying to do so it then there's also the fact that the tip of the throat I mean the reason that the antebraid is has to be controlled the tip of gravity and so then there has been a series of papers between us and Wienheim and we cast this a lot and so the conclusion for us is that there is something that we call the large volume scenario primitive hetero from today which gives an expression of how large your end has to be such that you can be sure sure at the level of numeric that you probably need some more if you put in all the detail and the pipe and the various coefficient and there are some some couple from the intersection numbers of the cycles and excite which has to do with the Euler kind of Euler number to be out to let me not go to go to go there I mean we I think we fairly honestly put in the numbers try to get the mystic to get here we need an attack full of order five hundred to get some I mean I I chose a quantity a suppression quantity type a series it's good enough for us if you know it goes like one plus one fifths plus one hundred fifths I sound must sound horrible but I'm not efficient but there's some some people okay I'm okay so if you choose 10 here things get worse but not too much and now you go back to to mathematics or to apply mathematics and ask what people have found in terms of completely calculating this Q in the Calabria or Yankee Calabria or Yankee Calabria that you have and you get numbers like this so for the pure Calabria entity I'd explain it's 225 and actually borderline push more so it's not that you can say oh all is lost and what's not that you can say you're safe kind of you know borderline maybe okay or not okay depending on how you feel that day and then things become much better if you allow yourself more complicated geometries I mean the seven brains contribute if they start to do crazy things in the Calabria you can choose a geometry you can go up to 3000 here if you do f-series this number is known to go up to 75 so you're perfectly good shape but then these are not this these are not the model they are all but as that before works perfectly there are now other issues which in principle make me lose control because here for example the string coupling cannot be small at all which is my main small parameter and here the string coupling varies crazily because the seven brain sources there are some I mean so they're not going to the profile which I did not include in the calculation which you cannot include in the calculation because you don't know the Calabria metric so it's it becomes a nightmare and so yeah clean mathematical question what is the maximum Q in a pure Calabria for which one the most not not people have just thought that more physics question how to how to get that to control if you better that so you know that that Q could be done okay head for thought so now comes to now comes to the better the best part okay the best part I mean yeah the best part for you because maybe this is the part where I really know it seems to be concurrent so we are driven the polling situation we have this Calabria in the certain flux in the box people are from the collection of the lot a certain flux in the throat which I just discussed we need to be very large have a very large open source these two numbers and up add up something which I call n and this n has to be smaller than Q max Q max for this yeah number of three planes I think this goes for a or a number of 70 yeah yeah yeah yeah I'm sorry just this question of what is the maximum Q for Calabria the question is you have Calabria with holomorphic involution Calabria has no Q has no Q as you know right if Calabria has no Q no no I know I know I know just trying to yeah some idea okay some idea what the Q is so you have a Calabria with a holomorphic involution and you look for roughly speaking the homology class of the very roughly speaking the homology class I have to call you somewhere right here so so so the fixed point the simplest case the guys are two and put a mention four six so cross this out and make it one fourth here then this is the number looking on Calabria okay cross this out and make the simplest case these are these are related to the space so then we can remove this and have this number of three and this is one fourth and so the holomorphic involution is these are the fixed points which locally look like all three z of the Calabria will be minus that and this is the locals of where the reflection is that the minus that and the other so that one minus that one and the three of the Calabria are untouched there are two specific types of singularity one of them is is instead of loci it's counted here the other is the surface this is the other number that we found here this number is essentially it's 252 that's all right all right I don't know I need to have five more minutes or something right oh well all right sorry no I have to go back go back here so what we actually want is I mean let alone all the details that I just discussed the horrible details we really want n-bar to be small we want all the flux that we can afford to use for the throat for the energy breaking and you want the Calabria will have almost no n-bar that would make us more happy than the other way around and now the question is will bulk flux with a small head full n be able to stabilize all construction model like the head full construction in tech claims exactly the opposite the people have people claim exactly this work one is not what you will get namely the claim is that if some flux vector stabilize a large number n of construction model right then this n flux induced by the by the slack will be larger than alpha sometimes over one and so the more more do they want to stabilize the more the tech for induced by the three form flux and now this contextual comes in barrier there's a refined version where people claim they know the number here there is versions that you may or may not require in this context that the Calabria will stabilize at a smooth for the clear I mean maybe it will not collection not hold if you allow to be able to stabilize the feeling of for example where some 24 points will stabilize actually on which however it's not demand and then there's a variance or whether or when not to require that this n of the module you stabilize is really the maximum n I mean it's actually one or you just you want to do this is for strong and deeper so in fact they are eight different contextual but I think I think I do and people are not and some of them the strongest one is to take all the strongest of the regular route out you have some counter examples already and it's of course also unclear what it means at large end I mean the counter examples for Calabria also should model that so it's known that Calabria was one or two such a model that increased on flux you can actually stabilize this one or two such a model that the claim is that's a very large number for the Calabria that really make up with with the typical Calabria for the hundred from the structure of the line this will not be so what are the odds so the arguments in fact in the court of the convection in my opinion already we so one argument is based on case three times case three down but there is a particularly simple factor of the theory effect other examples then there is there are example analysis but even even proofs in fact in fact by Thomas again at large complex structure in the extreme large amount such a limit where however again the structure of the period becomes particularly simple so all this rests on simple form of period but my my personal argument again this connection why I'm skeptical is that really the main argument why I can stabilize or mobilize relies crucially on the complicated form of period namely you have this expression for the continue breaking after I'm at basically the condition for the minimum and there are any equations for invariable and with generic functions in periods I you know horrible expressions with periods of exponentials so that's actually we expect only this resolution in other words for modular other narrative stabilize so maybe if I'm clear why you would expect I do now I'm still personally worried for the following reason this technical connection could still be true I also would say in in in period for the following reason it could be that you generically have these these solutions but actually there are no solutions in the physical domain of that in other words the flux potential for the commercial to model life would be such that you're driven to a runaway in multiplication or to singularities of a type which are too bad to be physically controlled in the actual model is destroyed because you're driven either to extremely large in the structure or to singularities that I cannot rule out and um so an interesting final point is uh that is that work and actually in chance it's maybe not the not the only but any important part of the struggle for organically can be mathematically more precise because it can give can be given a hot static formulation so this statement of E w equal to zero which was e field periods approach to minimize the potential and the precise statement in the sense that this uh flux g g3 of course of the problem uh that's the fact of the flux actually uh lips only in h to one plus eight zero three there's no no component of the the other four and together with of course h3 and h3 being a integral normality and also this and flux with this integral product this integral the product of h3 has to be a big number so there is a that set of conditions which together will define for you this more more physics way of saying what the minimum is purely and so then you can start talking about what you're under the table for the really flux letters so all flux choice is satisfying the above conditions we can denote them like this so this is these are the points which which I said at the before satisfy these three statements here and so then we can say and and this is of course true for a given value of this activiton and and then to the definition would be to the do you know what in this case of the modular of the label are those values of s and z that is that that I from that one to that end this is hyperbolic plane this is the modular state of the label such that the rank of this flux that is it's larger than zero in other words all points at which you can be stabilized by a by a lecture then the number of stabilized motor line at such point it's a core dimension of this top money for of the motorized space and to the proposal then one of the forms of the proposal which one can state which which one of the paper is that sorry I should mention that of course there are of course there are you know I'm very related to your site on the London by so much spring again before so the proposal would be that precise step of complexion that you take this core dimension i.e. the number of stabilized motor line divide by the the step fold of the corresponding flux and then you maximize over all flux and you say is that you can never get to large and one while this is the a film will take you from a fine point it's now what exactly the core dimension for the final for them proposal there by your funders is that who takes the risky for dimension and this means that actually be stabilized by high potential terms not called so you really it's kind of a compromise suggestion because it means that work in lecture can be true but not deadly for physics because many modern lie remain unstable life in the sense that they remain martial skill but they still don't change what to provide in the sense the girl is not running away in that direction so but that maybe I don't know this is the technical I wanted to advertise your hundreds recent news in some light work on this maybe you can if questions come apart come up to this slide I would have to refer to your honest to the end I think I will not be able to defend rather than I guess which brings me to the summary so three sentences so some of the most pressing issues in establishing a realistic string landscape has to do which higher dimension operator of the text that what you who really study in with the real contact to see the only leading order and that they are very well the time directions which are really depending on the key to it that the metrics beyond just being which so that of course you don't want to deal with it you really want to go in situation when it comes on the reporting or on the title is that of these that you might be to remind first this is possible by then one needs to go to geometries with a large negative conceptual too which also is a problem and probably fundamental limit and a related issue a critical issue can one stabilize many motorized by a flux with a limited head to like a flux which does not require which is also an open interesting question all right thank you very much so are there any question you sure is there no question or comment we're ready for okay thanks yeah so this is a very naive question so so looked at from far away as a mathematician observing the string landscape and the swan planned over many many years in some sense it looks like because it's difficult to know what all Columbia manifolds are what all Columbia module I look like we're trying to replace the physical questions of finding a unique and application space with finding properties maybe hodge theoretic properties of Columbia medicals in general and and drawing conclusions then using using hodge theoretic methods and this is just big picture okay if you flip that around though I'm just wondering can we modify our mathematical expectations or what Columbia geometries could exist based on optimistic physical scenarios hodge theoretically I mean I way back when this has always been an issue of of could there be Columbia modular spaces of one dimension that are of genus bigger than zero I don't know but is this the kind of question that maybe we could start turning around into a the feedback into the I'm not exactly sure that I fully understand that maybe maybe that like put it that way so I think the generic expectation that there will be a large landscape and many solutions and relevant to the real world it's still there but it is attacked by skeptics which are who I have higher expectations justified on the basis of us not being able to produce a single example so if you have a Columbia we therefore some reason I mean and okay and we know that that's paper like the modular like in a Columbia in a large Columbia with many cycles is the computationally hard problem it is a heart in the sense that we cannot hope to solve it that's a good point right so if you're maybe you're saying that let us look for the specifics model what's probably going to be out yeah for some reason I'm able to specifically produce that flux number stabilize the flux find W zero maybe not find maybe in the future even find the metric I don't know but in a sit in a so if you if you find models which avoid a problem which I maybe haven't emphasized them now namely the problem of explicitly stabilize the model right if you be the way I mean I didn't emphasize them now this looks innocent but if you have 150 z it's completely undoable I mean with integral conditions if you find it because of but I can to do well in special cases if you find it going to be out but this is doable things can suddenly look much much better is that the question well not really so so I think it's not less a question than a comment I feel like the problem of of finding Columbia compactification spaces was specified properties has motivated the field tremendously in the early days but my own sense is that we're only at the tip of the iceberg that that we don't really have a sense that any one construction of what the amount of folds is the construction and so the great virtue of the techniques coming out of hodge theory is that they provide absolute universal constraints on what could exist that's also interesting you mean you could prove for example that we are not the end of the don't discuss because because there's two of the pictures somewhere but because because it's a limit there are three yeah so so I I you know I'm not sure this would this approach necessarily I mean I would say from from my perspective the the the transmission of questions of modular stabilization into hodge theoretic questions was a good move because we simply don't know about the compact Columbia spaces and their modular spaces we don't know enough we have lots of examples millions and millions of examples but we have no idea whether that's all right so but what I'm wondering is whether the field is mature enough to have the mathematics the geometry of those kind of foundational questions now informed by predictions from physics so I'm what I'm hoping is that what will come out of this is okay physics says for some kind of consistency there's a very strong hodge theoretic constraint possibly having to do a certain mix of the infrastructures and then we can turn that around into a question of construction of bloody else so that this this is this is more of just a general comment of where I I hope maybe we'll go the microphone well I think that's generally true but it's also very difficult because the the physics trajectories are not always formulated in a way which nicely translates into mathematics so one has to be very careful to pick the right point of view on these convectors and then translate into mathematics I will show you one conductor which came from physics this translated into mathematics and turned into mathematical theory but some of the other swamps and from lectures they're kind of quite a few now at this moment but it's not so easy necessarily to translate them into say hodge theory but but but I mean my point of view that's a very fruitful avenue because then at the end of the day at least you can prove it in this setting and then of course in physics you're not certain that these are the only options so they could be much more beyond that mentioned kind of other correction and so but at least within these hodge theory settings you should be able to prove this in the mentioned like the that book and one what I'm optimistically hoping for is input from physics that will constrain the possible proposals for the possible glabia geometries and the more possibilities and we are the depth of our ignorance when it comes to that is pretty much the same as well as 20 years we got to go all right I'll see you in yeah 1045