 this workshop. What I will be discussing today is based on these two recent papers that I did in collaboration with Jose Calderón, Matilda Delgado, who are also PhD student at IFT Madrid, and our advisor Angel Leurang. So let me start with some introduction about the things that I will be telling you about and why it is interesting to look at them. So suppose you have a theory that is not sitting at a minimum of the potential but rather it is moving on the slope. This is something very general that can happen, for example, if you break supersymmetry and then you generate a potential of an awakened. We refer to such situations as dynamical tadpoles since these are tadpoles for dynamical fields, and so they are a different thing than topological tadpoles such as Ramón-Ramón tadpoles, which are for non-dynamical fields and in post-consistency conditions on your theory. As opposed to that, dynamical tadpoles are not inconsistencies by themselves, but you really have to take them into account. And in fact, it was recently shown that ignoring these kind of tadpoles can lead to some violation of swampland conjectures, and this was in fact one of the original motivations for this work. The typical situation we will be dealing with is when you have a theory that has no tadpole and you have a nice and smooth solution for your equations of motion. Then you add some ingredients in the construction in such a way that a dynamical tadpole appear. The presence of this term in the potential has the effect of destabilizing your solution. And since obviously the equation of motions are modified, and then in order to solve the tadpole you allow your solution to run in such a way that it acquires some dependence on spacetime coordinates. And this is the general case, but in fact we will always be considering space dependence for this talk. So we have these running solutions that are sourced by dynamical tadpoles, and they have some interesting features. In particular, they often contain metric singularities, or they involve strong coupling regimes, and this obviously makes their interpretation difficult. What we can do, and what is also the plan for this talk, is to investigate these solutions further by looking at many examples that we have from string theory, and see if we can identify some general properties or some universal pattern that is based on this example that we studied. Another question that I will address during the talk is how swampland conjectures manifest in these kind of setups, and in particular I will focus on the co-bornness conjecture since it seems to play a key role in the discussion. Then in the second part of the talk I will also try to connect with the distance conjecture. So let me now introduce a bit this co-bornness conjecture and what it is about. This can be seen as a particularization of this statement that in quantum gravity you cannot have global symmetries, and this is in a sense the oldest among the swampland conjectures, and we really have a lot of evidence for it. We know it is true in any string theory compactification, and more recently we also have an argument for it that comes from holography and ADS-CFT correspondence. So we expect that there are no global symmetries in quantum gravity, and what McNamara and Vafa proposed was to look at a particular kind of symmetries that we can construct in interiors of gravity, and the idea is that in gravity we are allowed to have topology changing transitions, and since topology changing transitions can be interpreted as co-bornisms, any topological charge that is associated to a topological symmetry must also be a co-bornism variant. So if you have a topological charge that is associated to your original manifold m, and there is a topology changing transition that takes you to a different manifold n, this means there is a co-bornness between these two manifolds, we require that this topological charge is a co-bornism variant so that it doesn't change in the transition, and this means that that a natural global charge to look at is the co-bornness group, which I call omega. So for manifolds of dimension k, you say that there is a co-bornness between any two of them, if there this joint union is the boundary of another manifold with one dimension more, and this is the usual geometric notion that you have in mind when you talk about co-bornness, but in fact there is also another way to look at it, which is equivalent. Suppose you have a theory of quantum gravity that has these space-time dimensions, and then you are willing to define its co-bornness group as the set of in equivalent configurations of the theory, where any two configurations that can be connected by a domain wall are identified, and this makes sense because if your theory in the dimensions is arising as the compactification of a higher dimensional theory on a compactification space of dimension k, which is by the way what is always happening in the context of string theory compactifications, then these two definitions are just the same because any non-trivial co-bornness between two compactification spaces will correspond to a domain wall if you look at it from the perspective of the lower dimensional theory. So to recap, in a semi-classical theory of gravity, you can have topology change in transitions, and there is a natural global charge which is given by the co-bornness group, but since we expect there are no global symmetries in quantum gravity, what the co-bornness conjecture is telling us is that at the fundamental level this co-bornness group actually has to vanish, so all co-bornness classes must be trivial. What does it mean? If your manifold m is carrying some global charge, then you really want this charge to disappear, and there must be a transition in the theory that allows you to shrink this manifold to a point, or in other words this manifold is co-born to the empty manifold, so it's the boundary of something else. And equivalently from the perspective of the lower dimensional theory, this means that there must exist a boundary on which your theory cannot own, and we call this a wall of nothing since it is an end of the world wall with nothing behind. So, suppose you cannot do this in your low energy description with some smooth cuborgism, but maybe you can do it with some singular configuration that is not part of your low energy effective theory, but is a valid configuration in the full quantum theory of gravity. And the idea is that on top of what you have in the IR, there must exist certain defects that really must be there in the theory. And in other words, an interesting implication of this conjecture is that you can use it to predict the existence of defects. And the idea is that whenever you find that the co-bornness group of your theory is non-trivial, then something you can do is to include some additional ingredients to produce new co-bornness in such a way that otherwise non-trivial co-bornness classes are killed. As a very simple example, just take a two-sphere, which is the boundary of a three-sphere, so this is trivial in co-bornness. However, if you put some U1 magnetic flux, this is non-trivial because the flux cannot just disappear, so the sphere cannot shrink. And the way to solve this is to include the magnetic monopole as a localized source that is able to absorb this flux in such a way that the sphere can shrink. Now, obviously, there are many co-bornness groups you can think of, and which one you have to look at depends on the theory that you are considering. For instance, if your theory contains gauge fields, then you would like your gauge field to extend to the co-bornness in a smooth way. And you can use this to predict the existence of D-brains in type 2 string theories just because there are Raman-Raman gauge fields that can be turned on. Formally, given a P4μ1 gauge field, the Bianchi identity for the field strength implies that the total magnetic flux over any orientable manifold is a co-bornness invariant, and this is what is labeled in the co-bornness classes. This also corresponds to the existence of d-minus p-minus 2 for symmetry, which is given by shifting the magnetic dual potential. And the way to kill it is to introduce d-minus p-minus 3-deffences, which are magnetic monopoles, again. Now, suppose that you turn off the gauge fields and you want to use only the fact that string theory contains spinors, then you have to look at spin co-bornness, so the co-bornness group for manifolds for which you have a definite choice of a spin structure. Let me just mention some of the examples. For zero-dimensional manifolds, the spin co-bornness means set, and this is generated by a positively oriented point. To kill this class, for example, in M-theory, you have a boundary, and such an object is, in fact, very familiar since it is given by the Aurawe-Wittham wall. Similarly, for type 2a, you have an o8 as a boundary, which is the Aurawe-Wittham wall wrapped on the M-theory circle. Another example is for dimension 1 manifolds. In this case, the spin co-bornness is z2, and the way to kill it in, for example, circle compactification of type 2b is to introduce to the orientation of seven planes. Another yet possibility is when your theory makes sense on non-orientable manifolds, and it also has fermions. In this case, the co-bornness group you must consider is even more general. Now, in all these examples, all these objects were very explicit and very familiar, but there can also be other setups where you find a non-trivial co-bornness group, and there is no known object that kills it for you. And then you have to rely on the conjecture to claim that such an object indeed exists. This, for example, was used recently to argue for the existence of bifolds, which are a generalization of orbifolds and orientifolds, and to put constraints on the rank of gauge groups in dimension greater than six. The fact that these co-bornness defects are not always known is also true in our setup. Sometimes we will be able to identify them, and they are very familiar objects, such as orientifold planes or d-brains, but there were also some cases in which it was not obvious how to characterize them, and we had to rely on the conjecture to claim that such objects indeed existed. Let us now go back to our case with dynamical tadpoles. There are basically two tadpole lessons that we learned by looking at examples. The first lesson is what we call finite distance. We observe that in the presence of a dynamical tadpole, which is controlled by an order parameter t, the space-time-dependent solution of the equation of motion can only extend up to a finite space-time distance, which I call delta, and this delta scales inversely with the strength of the tadpole. The larger the tadpole, the smaller the distance. This scaling obeys power law, where the exponent, we determined it case by case, and it was always true in Minkowski-like examples and one in ADS-like ones. The second lesson is what we call dynamical cobaltism. The proposal is that there must exist a physical mechanism that is able to cut off for use space-time dimensions at this distance scale delta of the finite distance lesson, and this corresponds to the cobaltism effect of the original theory with the tadpole. The reason why I call it a cobaltism effect is precisely because this is the effect that would be predicted by the cobaltism conjecture. These are the two lessons that I wanted to show you, and we are ready to see them at work in some examples. The examples that I would like to discuss include the conifold theory, then some toroidal compactifications with three-form flux, also magnetized models. We have non-super symmetric TND strings, in particular the USP32 theory. Finally, we will see a model building application where we are able to use these ideas to get a semi-realistic MSSM-like particle physics model. The first example I want to discuss, and which is a prototype for also the other ones, is based on the near-horizon limit of n regular D3 brains at a conifold singularity, which is famously ADS-5 times T11, with 10 units of Ramon-Ramon 5-form flux on the T11. This is the Klebano-Wittham solution, which is the first picture on the left. We can consider adding m units of Ramon-Ramon 3-form flux on the S3 that is inside the T11, but this modification is dramatic, and the resulting solution is the Klebano-Zitlin solution, which is the picture in the middle. The reason I say dramatic is because of the nate singularity that this solution develops at some finite value of the radial coordinate. What is happening is that in the back-reactive metric, the T11 shrinks to zero at some finite value of the radial coordinate, but this will not be a problem because the 5-form flux also disappears by then. However, the S3 is supporting the 3-form flux. This is also shrinking, but the flux is constant. The flux density to infinity, the back-reaction is also infinite, and this is why you get a singularity. The smoothing out of this singularity was studied by Klebano and Strassler, and it corresponds to the third picture on the right. The idea is that you can give this S3 a finite size in such a way that the solution terminates in a smooth way. Space time is capped off at this point in the sense that after that you have no meaning for the radial direction. Until now, this was just a review, so let me now to reinterpret the whole thing in our language based on TATPOL and running solutions. That there is a TATPOL for the dilaton can be seen very explicitly from the 5D effective filtering point of view if you look at the 5D action. Introducing M units of flux generates a potential for the dilaton, which is this blue term, and which, as you see, is of a runaway kind and has no meaning. In the dilaton, a question of motion, there is a source term, and the only way to balance it and keeping the dilaton constant is to introduce some space-time dependence for some other scalar, which is the NSNS section that comes from integrating the NSNS2 form over the two spheres inside the T11. In this way, the TATPOL is sold and the specific space-time dependence that you get is this logarithmic running in the radial direction. This is also equivalent to introduce NSNS3 form flux, which has two legs along the S2 and one other leg in the radial direction in space-time, so that the combination of the three form fluxes is imaginary self-dual. Then we can compute the distance to the singularity and compare it with the TATPOL, and what we get is in agreement with the finite distance lesson with a scaling coefficient equal to 1. In this language, the Klebanos-Trasler solution is providing you with the coborgis effect you need for our dynamical coborgis lesson. The next example is also based on three form fluxes, but now in the context of toroidal compactifications. So consider that to be on a five torus with Ramon-Ramon three form flux along the direction x1, x2, and x3 inside the T5. This introduces a TATPOL for the dilaton, and in fact the equation of motion that you get is very similar to the one that we had before. The way to solve it is also very similar, since you can introduce some NSNS3 form, which has two legs along the compact direction y1 and y2, and another leg in one of the space-time directions, which I call y. Since the combination of these fluxes is an imaginary self-dual, it is easy to study the back reaction on the metric. It is convenient that we put this y-coordinate together with the coordinate of the torus, so to get complex coordinates z1, z2, and z3. The word factor obeys a Laplace equation that is sourced by the f3 flux, and in particular it develops two singularities at some finite values of the y-coordinate. Also if you compute the distance between these two singularities, you can check that this distance is finite, and it obeys the scaling law of our finite distance lesson with scaling coefficient equal to 2. The last step is that we have to identify the co-bordance defect that is able to remove these singularities and make this direction compact. This co-bordance defect of 2B on T5 is provided by orienting for 3 planes, which will appear as sourced in the Laplace equation, and in the end what we get is a compactification on T6. This is of course familiar, and what is new is that the 4D model is obtained in two steps. You start from a 5D model, and then you end up in 4D. We like to call this a spontaneous compactification because it is triggered by the tadpole that was present in the original 5D theory. The way to solve the tadpole, as we saw, is to introduce dependence on a special coordinate, and what we discover is that this coordinate along which you run is forced to be compact. Let me move to the next example. We consider type 2B, compactified on T4, where the orienting for action is flipping the coordinate of the first T2. There are four orienting for 07s that are localized at the fixed points of the first T2, and also 16 D7 brains, and of course they are orienting for images. Something we can do is to introduce M units of D7 world volume magnetic 3-fold flux along the second torus, and this breaks supersymmetry and generates a tadpole for the dilaton and the color modulus of the T2. One way to see this is that the charge of these objects cancel, but there is some unbalanced extra tension. Now the interesting part comes that is we have to solve the tadpole. One possibility that we have is to introduce magnetization along two of the six non-compact coordinates, which equal Z3, in such a way that the combination of fluxes you get is supersymmetric. Then the discussion of the black reaction can be carried out in a way similar to our previous example, if you consider the lift to F theory. In the F theory picture, our volume magnetic flux corresponds to G4 flux. Then you can look at the warp metric, and again you find that the warp factor is singular, and that the singularities are separated by a finite distance, that is by the way inversely related to the tadpole, and the way in which these singularities can be removed is again by making these two directions compact, and the cobalt means the effect that you get are some extra orientival 07 planes and D7 brains. The next example that I want to show you is quite interesting, since it has to do with non supersymmetric 10D strings, and in fact as soon as you try to formulate these non supersymmetric string theories, tadpole will appear because you have no protection by supersymmetry. Let's focus on the 10D non supersymmetric USP32 theory that was constructed by Sugimoto. This can be obtained as an orientival of type 2B, which is very similar to type 1, and the difference is that you have positively charged orientival 09 instead of negatively charged 1, so this is why you have the SP projection. Cancellation of Ramon-Ramon tadpoles requires that you also introduce 32 antidenides, and the resulting theory is anomaly free. There is no space-time supersymmetry, and there is a dilaton tadpole, which is this blue term here that appears as a cosmological constant there in the 10D action. This implies that the theory does not admit a maximally symmetric vacuum, and solutions with lower supersymmetry must be looked for. Actually there is a solution with 9D Poincaré invariance, which was originally found by Dudas and Murat, and this is a running solution where both the metric and the dilaton depend on one of the special coordinates, which I called y. There are two singularities, one is at zero and one at infinity, but despite appearances these are separated by a finite distance, something that was already noted in the original paper by Dudas and Murat, and it is also easy to check that this distance obeys our scaling law with exponent n equal to 2. Now a legitimate question to ask is which is the cowardice, the effect that is able to remove these singularities? The answer is that we do not know, so in a sense we stop at the same point where Dudas and Murat did in their paper. What we can say is that such an object should exist by the co-boarding's conjecture, and in fact it was recently proposed that such an object is given by AD8. However, there is also another way to solve the tackle, which is to introduce magnetization, and I will be sketchy about it because this is not new, we already saw it in the other examples. I just want to mention it because it is quite nice that you start with a non-super symmetric theory and you are able to get a supersymmetric compactification of it. The idea is that you introduce some word-volume magnetic fields along three different complex planes, and this triggers a compactification of an orientation of T6. The last example is now to have maybe a bit of fun and see if the standard model is the outcome of one of these spontaneous compactifications that I have been telling you about. The idea is that you start from a 6D theory with a dynamical tackle and you solve it by a running solution, which makes some dimensions compact so that in the end you have a 4D model and the spectrum is an MSSM-like. So you can start from type 1, compactified on an orienting fold of T4, with 32 magnetized D9 brains and some D5 brains that are obtained as induced charges. You also have some orienting fold O9 planes and O5 planes. You can find the details in the table. There are different stacks of brains and these numbers N and M correspond to the rapid number and the magnetic flux units for each of the two T2. Just for completeness, let me just mention that this system can be tidalized to a system of intersecting D7 brains. Okay, in this setup, Ramon-Ramon tuples, you can check that they are cancelled, but they are uncanceled NSNS tuples. One way by now familiar to solve them is that you can introduce magnetization along two of the 6D space-time dimensions. This triggers a spontaneous compactification to 4D, where the co-ordinate defects correspond to some extra orienting fold O5s and also D5s that are needed to absorb the orienting fold charge. And in this way, Ramon-Ramon tuples are still cancelled. So in the final model, you have some extra stacks of brains, which are label B and C. And this 4D model can be made supersymmetric if you choose appropriately the areas of the T2s. And this guarantees that in the end, you have no tuples that you have cancelled the tuples. Now, if you look at the spectrum of this theory, this is a three-family MSSM-like spectrum, which in fact was already built by Marchesano and Schu. What is new is our interpretation of a 6D model that spontaneously compactifies to 4D. An interesting observation is maybe that in the original 6D theory, you knew nothing about the standard model. And all the MSSM-matter, the X-chiral multiplex, and the electro-wicked gauge sectors all arise from the co-ordinates' brains. So in this sense, the spontaneous compactification from 6 to 4D that was triggered by dynamical co-ordinates implies not only that some dimensions are removed, they are made compact, but also that there are some new degrees of freedom that appear. Okay, this is not yet the end of the story since in all the cases that we considered so far, the relevant co-ordinates was a wall of nothing. But in principle, domain walls that interpolate between different theories are also possible. So one may ask what is the difference and how to discriminate between these two different kinds of co-ordinates. Suppose you have a dynamical tuple in your theory, and then you find a running solution that solves this tuple. Then you can look at the scalars in your theory, and some of them we run, and you can compute the distance that they traverse in field space. Now our proposal is that if the scalars run into a wall that is at infinite distance in field space, then this must be a wall of nothing. But if the wall is at finite distance in field space, then this must be a domain wall that takes you to a different theory. What is interesting about this proposal is that in some sense it allows you to relate the co-board this conjecture and in particular walls of nothing with the distance conjecture. So the idea of the distance conjecture is that as you go along geodesic paths of infinite distance in field space, you will get an infinite tower of states that becomes exponentially light. If walls of nothing appear at infinite field space distances, we would expect from the distance conjecture that there are some light states coming down, and then we can also play a bit with the scalings that we have to relate, for example the mass of the sdc tower with geodesic quantities as the spacetime distance to the wall or the curvature scalar, as we will see in the examples. These are again examples from string theory, which include a massive type 2a, m-theory compactified on a Calabi-Yau threefold with g4 flux, and as a final example a circle compactification of 4d n equal 1 effective field theories with some axiom flux. So let us start with a massive type 2a, so type 2a with an additional Ramon-Ramon 0 form flux, which is the Romance mass parameter m. This theory is supersymmetric, but it has a dilaton tuple that is controlled by m, and it admits a one of BTS solution with 90 Poincare invariance, where both the metric and the dilaton depend on one special coordinate, which is x9. This metric becomes singular at some finite value of the x9 coordinate, which corresponds to the vanishing of the wall factor. And it is easy to check that the distance in spacetime to the singularity is finite, and it also base our scaling relation with coefficient n equal to 2. The singularity is smoothed out by the non-cogordin's defect of type 2a, so this is an orienting for 08 plane with possibly some additional d8 grains that are needed to absorb the flux, and this corresponds to a wall of nothing. In fact, this matches the type 1 prime setup and singularities that were discussed, for example, many years ago by Polchinsky and Wittner. There is a single scalar in the theory, which is the dilaton, and this runs off to infinity at the singularity, which is exactly what we would expect to happen at a wall of nothing. We can also compute the distance in spacetime and the curvature scalar in terms of the distance in field space, and then relate them to the mass of the light states, and in both cases we get some power loss scaling. However, this is not the most general solution that you can have, since you can also put d8 grains along x9, and if each of them will serve as a domain wall that is interpolating between different type 2a theories corresponding to different values of the romance mass parameter. The form of the solution is the same we had before, except that now this function z is piecewise linear with different slopes between each pair of d8. This means in particular that the dilaton stays finite across the domain walls, and this is in agreement with our general picture. The next example is interesting because it allows us to address the question of what would happen if you are at finite distance in field space, but you hit a singularity. So what do you get in this case? Do you get a wall of nothing or do you get an interpolating wall? So let's see. The setup that we consider is M theory compactified on a Calabi out threefold in the presence of G4 flux that is closely related to Witten's strongly coupled atherotic. The 5D effective action contains some scalars which are V, which is the hypermultiplex scalar and corresponds to the breathing mode of the of the Calabi out. And then you also have the color moduli which are vector multiplex scalars and these are the BI's. There are also axioms that come from integrating the 4 4 over a basis of four cycles which are these A's and also they are dual 4 4 potentials lambda. This G4 flux acts as a source and generates a tuple for the overall volume and color module which is this blue term. And this action also contains also contains a boundary term to include four dimensional localized sources and this correspond to M5 brains that are wrapped onto cycles and this serve as domain wall in the solution. This is a running solution that sold the tuple and both the metric and the color moduli are allowed to vary along one of the 5D coordinates that is the coordinate transfers to the M5 brain walls. Everything is controlled by these one dimensional harmonic functions H which are linear in the in the direction Y. So the picture is very similar to the one that we had before and the only difference is that now we have a bunch of piecewise linear functions which are this H which control the devolution of the Calabi Yaw along the 5th dimension. Okay, in particular if the world Calabi Yaw goes to zero size what happens is that you eat our other weekend wall. On the other hand if you have an M5 wrapped on a two cycle then you have an interpolating domain wall between different values of the G4 flux which jumps as you as you cross the domain wall. Now something we can ask is what happens if the evolution of the Calabi Yaw is such that it suffers a flop that is a topology change in transition which in this case can occur dynamically. A typical flop is for example when you have a positive two cycle that goes to zero and then it becomes negative. This was already studied by Green, Schalm and Schiu and they checked that the solution is continuous across the flop which for simplicity can be located at Y equals zero. One can choose a more convenient basis for the color module in which the flop curve corresponds to W minus U. Then what you have to do is to match the solution on both sides of the flop in such a way that all these harmonic function H are continuous and in particular this is true for the harmonic function that corresponds to the to the flop curve. What is interesting for us is that these functions are finite at the flop point and since the color moduli depends on them also the color moduli are finite so despite the singularity in modular space the solution remains at finite distance and this answers our original question about what happens if you have a singularity which is at finite distance so you do not get a wall of nothing you will get interpolating walls and in fact you can also check that the G4 flux is discontinuous across the flop and so the flop works as it was an M5 brain interpolating cobaltism. What this points to is that walls of nothing really have to do with infinite distances in field space. The last example that I want to discuss is based on a circle compactification of four-dimensional N equal one theories with axiom flux. For simplicity we take a single axiom-saxiom pair so A is your axiom with unit periodicity, S is its it's its axiomic partner which we canonically normalize and call it phi. Then we take we take an ansatz and compactify to 3D. Regarding the axiom as a zero form gauge field we can introduce two units of flux for its field strength on the on the circle and we also allow for a general section profile to account for the back reaction. By looking at the 3D action we see that there is a dynamical tackle for a linear combination of the radion and this axiom which is precisely induced by the presence of the axiom flux. This tackle can be solved by a running solution where you can keep the axiom constant but you allow the other scalars to acquire a non-trivial profile with dependence on one of the 3D coordinates which I call R to suggest it is a radial coordinate and there is for this I will explain it in a second. Observe that as R goes to zero the radion blows to infinity and this means that your compactification circle is shrinking to zero and then the metric becomes singular. The axiom also goes to infinity in this limit so according to our general picture the singularity should correspond to a wall of nothing that is capping off your spacetime so that you have no region with negative so it is in fact a radial coordinate. Keeping the leading order as R goes to zero you can compute how the spacetime distance and the scalar curvature scale with the field space distance and then you can get the relationship between the mass of the light states and these geometric quantities and as we expect we get some power law scalings. Something interesting that I want to point out is that this setup matches precisely with the EFT string solutions that were recently studied by these authors. They were looking at BPS axionic strings with co-units of axiom charge and which back react on a complex scalar and there is in fact a clear dictionary between our running solution in the S1 compactification and their EFT strings namely the angle around the string maps to our compactification circle then the string monodromy for the axiom map to the axiom flux around the maps to the axiom flux on the on the S1. The radial coordinate in the complex plane around the string sorry was lost so the radial coordinate in the complex plane around the string maps to the direction of the running so the running along the coordinate are and this is semi infinite because you have a wall of nothing at the singularity. Then the the string back reaction on the section which is realized as the string argiflow maps to the running of the section that for us is sourced by the dynamical tadpole induced by the axiom flux and then the the scalars that run off to infinity as one reaches the string core map to the scalars that are running off to infinity as you eat the the wall of nothing in particular the fact that this wall of nothing cannot be described within the effective theory exactly correspond to the fact that in the in the EFT string setup the EFT string is regarded as as an UV given the effect the the upshot of of the analysis in this landslide paper was the distance the distance of axionic string conjecture which states that infinite every infinite field distance limit of 4d and equal one EFT consistent with quantum gravity can be realized as an argiflow UV endpoint of an EFT string using the dictionary that I showed you before this can be translated in our language to the statement that every infinite field distance limit in this setup can be realized as they're running into a wall of nothing abortness in a circle compactification of the theory with some with some axiom flux then we can make a wild jump and suggest that this may even be more more general so that every infinite field distance limit of an EFT consistent with quantum gravity can be realized as they're running into a wall of nothing co-ordinates in the in the theory or in a compactification of the theory so I was maybe faster than I thought but I I I can conclude this point so let me recap what what we did what to study the the properties of space dependent solutions in theories with dynamical tadpoles what we learned was that this solution must have a finite extent which is inversely related to the tadpole and on top of that there is a physical mechanism by which spacetime is cut off and this is provided by the co-ordinates the effect of the co-ordinates this of the co-ordinates conjecture and one natural question that one can still ask is is about time dependence and this would imply a singularity in time so it will certainly be interesting to look at that also from a phenomenological or cosmological perspective and then we we try to connect with the the distance the distance conjecture and what we saw is that that infinity in field space is is really connected with with walls of nothing and then one one question that that will be interesting to explore is whether there are other one-plan conjectures that enter this this tadpole game so we we hope to address this this question in the in the future and with this I I think I can conclude and I I thank you for for your attention okay so a bit earlier I don't know if there are people who who make this group photo present now maybe okay maybe not okay let's let's just proceed with the discussions and so are there any questions yes I have a question so very nice talk general bro so I have a question about the couple questions one is the collavia three-fold case you were talking about the flows in the extra dimension extra face direction in the context of flop did you also study the four cycles shrinking there well we we look at it and we considered as a variety modulus just the volume of the world calabi out and if one does this naively then and then if you study the so if you study the situation in which the world calabi out shrinks if one does the computation naively you will get you will get that the the distance in space time is finite and and it obeys our finite distance lesson then one one may may not be so happy with that because if the world calabi out shrinks you are losing control of of the theory so this is something I guess I could be a criticism and just want to make sure maybe I maybe just I'm not sure I completely understood what you said so let me ask my question again so suppose you have a collavia three-fold with a let's say just for the concrete that's considered a p2 inside the collavia so it's a four cycle and you can change the moduli of the collavia the compact collavia such that this p2 shrinks to a point that's a finite distance in the collavia so this can happen a finite distance so according to your conjecture if I understand correctly if you vary this collavia over a line interval where this four cycle shrinks then you expect that there should be another side because otherwise you're getting a somewhere with finite distance and you're not getting it so you saw it shouldn't have been a infinite distance I'm just asking you if that's the case and if so I'm not sure what the other side is I'm just trying to understand whether you're making such a prediction and if so can you check it because it should be relatively straightforward to check if it's true or false okay I got it I got it now well then yes our proposal is that that should correspond to an interpolating wall since you are at finite distance and then I do not know how to answer right now but I since the the only case we considered where you have a singularity at finite distance was the flop case and in that case it was clear that you had two different theories at the flop with the flop serving as a domain word but in the case that you that you are mentioning well I do not have a a definite answer now right these are going to be challenge challenges for the conjecture because there are these things like terminal singularities or other ones which stops and you don't know the other side in the collavia jar so it suggests that there's no other side but they are finite distance so I think it'd be interesting to study them and if indeed your conjecture is true then at least in the category of supersymmetric compactification there must be an easy proof of it so in other words you should be able to say why if for example if you take a collavia and you fiber it over some geometry and let's say get a g2 out of it then it must be an interplay between condition of having a supersymmetric g2 let's say and and what is the other side or not or it stops and so on so from the viewpoint of a fiber geometry so there should be a I would say at least in the supersymmetric category there could be a proof for counter example to the conjecture would be interesting to study so that was one question the other one is so you were trying to connect the distance conjecture also to these to these cobaltism walls so the distance conjecture would be is usually formulated in a situation where things are static so when things are varying dramatically then that's going to be not correct for example when in the string example you mentioned where you have for example in an axiom or so forth or similar to what happens in f theory where you have elliptic vibration then at infinite distance in in in tau space the circular strength and well of course if you use as a circle compactification you would think that it's it stops because because on how you look at it but it's the end of a cigar or what but there's no masses there's no tower of masses modes there so that even the naïve the distance conjecture says there's a tower of states in the actual colabia metric where you write the elliptic vibration with the correct metric there's nothing singular happening there it's just a coordinate singularity so there is no evidence of the tower of states so how are you how are you mentioning how are you connecting the distance conjecture with these because in the generic cases i'm familiar with things were very too dramatically to be adiabatic so you don't have a you don't have a situation where things are static enough to apply the distance conjecture so perhaps i don't know how you think about it in those cases okay well in the last example that i showed the the one with the s1 compactification that that one seems seems fine to me since it is it is equivalent to the formulation by lans et al where they study strings these eft strings as a way to attain infinity when you approach the string core so in in that case i think at least to me it seems the connection is clear so so what do you mean by clear that that's not clear to me so do you have a tower there you're claiming there's a tower of states there for example let me just make sure i understand take the take let's take an eight-dimensional version of it so you have like fd area elliptic three elliptic k3 so you have a p1 and there are some points where elliptic vibration strings at some point so elliptic vibration string means tower was to infinity and therefore it's a bit of a distance in tau space but there is no tower of states near the d7 ring so the fd area description there's a d7 ring and there's no apiary tower so what do you mean by that tower i'm not i'm missing it i'm not quite understanding okay well it is true that we are we are not able to identify i would say any concrete tower in the in the in the in the examples we looked at what we the the reason why we i mean the the way in which we were trying to connect with the distance conjecture was just to say okay whenever we we we observe that whenever we run into a wall of nothing there are scalars that are running off to infinity if you look at the distance travels traveled in in field space so we are not claiming that we are able to identify any any any tower indeed so thank you other other questions or comments well i have myself a kind of general question so i guess in for this kind of full-fledged quantum gravity cobaltism theory it's not mathematically well defined but i wonder if there's some sort of intermediate version which is between geometry usual geometric cobaltism and this quantum gravity cobaltism which can be mathematically well defined and well indeed in the kind of examples that we that we studied which we propose as some sort of dynamical realization of this cobaltism conjecture where you really identify some mechanism that is able to cut off space time so you find these walls of nothing well i i don't know if i can formulate it in some more mathematical rigorous way but it is also true that if if you look at it from the perspective of compactification spaces then you can you can look at the at the cobaltism groups in the in the usual geometric sense so in that case that is mathematically well defined but for this kind of this type of kind of solution so you say you would you would consider kind of geometric manifolds with particular kind of solutions of some fields with some singularities that that how would you propose it to define mathematically i as i as i told you at the beginning you you you can look at these cobaltism from a more physically motivated perspective which is you look at the effective theory and you see if the theory can be connected to another one via a domain world or if it ends to nothing and this is the perspective that we took in these examples or you can look at compactification spaces and look at the more geometric mathematical thing so i guess from a mathematical point of view these walls you kind of include them by hand just possibly that there are some defects yeah the reason in which we identify them is uh in most cases these objects are already known so we we already know what they are and in that case it is easy when they are for example d-branes of or or antifold planes and and that is easy there was for that there was the the usp32 case which was more subtle because in that case it was not obvious which the defect was and in fact in that case we we were not able to identify it okay maybe i have a question related to what you just were discussing so in some cases as you said we don't know how to identify these domain walls and uh in the paper we wrote with Jake uh when we had the examples that we didn't know how to construct and uh simple example of this type to be in 10 dimensions and we expect that to have a boundary but naively that sounds like a very difficult thing to do given the chirality of the theory and all that there's no no-go theorem so it's in principle possible but there will be have to be very strange boundary conditions mixing up the the the the the self-duality of some of the form around one field which has some kind of a chirality in it mixing up with the gravity and all that in some form so there will be some very bizarre boundary conditions on these domain walls now coming to your conjecture in that context if you imagine i do not know if your conjecture also has the inverse that is if if you assume that what we said that there is no the cobaltism is trivial that means there is a domain wall and therefore according to your conjecture you would think that okay there must be some field that infinite distance which gives you that domain wall in type to be but the only field that you have in that context i think i can think of is just a coupling constant type to be and in that case how going to infinity the theory becomes weakly coupled in the weakly cobalt theory there is no such domain while we are aware of because i think we already know the domain wall so somehow this would be against the fact that we we thought we understood the perturbative objects rather well in this theory so i would expect that there should be something non perturbative maybe a tie goes to high or something which we are not familiar with that would do the job but how would your conjecture fit with the with the triviality of the cobaltism for type to be i i would say let me let me let me think for a second because this is something we we discussed that at some point what we are saying is that uh yes when whenever you have a wall of nothing there are if you interpret this as whenever you have a wall of nothing then you have a scalar standing off to infinity then i i i agree with with with this criticism so i'm not sure i have a definite answer right now okay thank you other questions comments if not let us thank general reggae for talk for the talk and uh discussion so i don't know if the cts people are present here who who would want to take a picture yeah we are supposed to take a look now okay so yeah if you if you don't mind you can turn turn on your cameras and we'll okay someone is missing