 So I would like to talk about some recent work done at Daisy in collaboration with Markus Fierigl, Fabian Rühle and Julian Schweitzer, which is the work I talk about is contained in these two papers. One came out a little more than a week ago, is this one, the other is essentially finished and will be out I think in a few days. So let me start with some very simple questions, everybody knows. I think it's well known that fermions and bosons play very different roles in the standard model. Now quarks and leptons come in three copies of complete gut representations, whereas gauge bosons and hex bosons form single, incomplete, sometimes called split multiplets. Now in the following in this talk I would like to argue that there may be a simple reason behind that which explains this difference, namely a connection between gut symmetry breaking and supersymmetry breaking. Now so the result will then be that the scalar quarks and scalar leptons, the partners, Susie partners of the quarks and leptons, they are very heavy because they belong to complete gut multiplets and because of that they also come in several copies, three not being distinguished. Whereas a supersymmetry breaking should be small for gauge and hex fields because they form incomplete gut multiplets. So that is the argument is there should be kind of a double split. Either you have, if you look at the multiplets you have a split in supersymmetry, a split supersymmetry multiplet,t but then you have a complete gut multiplet. Or the other way around you have a split gut multiplet and then small Susie splitting. That is the picture of split supersymmetry as suggested by these people. And in fact I should say to us this outcome was a little bit of a surprise. We were not planning to work on that, rather we were thinking about constructing a detailed model of inflation in from extra dimensions. So in order to explain what is behind this statement, let me start with a few technical points concerning compactification of supergravity on obi-folds with Wilson lines and flux. So we consider 60 supergravity with a U1 gauge field to start with. It will become a little complicated, more complicated later. And we look at the compactification on the toes. So then as a standard metric as you know is this, you have the four-dimensional metric for essentially the Minkowski space, you have the internal metric of the compact space, rescaled by the radion field, that's a six-dimensional metric. And here you see the lattice corresponding to the torus, the fundamental domain. And then as you know on the torus you can have two Wilson lines, the one going in this direction, one going in this direction. And you can write them as a line integral over the gauge field. Now what will be important in the following discussion is the following. It's not really important, but I think helpful is the following. Let's now go to the obi-fold. So that means we take this thing and take half of it, which is the fundamental domain. So here you see this fundamental domain, which now has four fixed points. And as you know, in order to get the obi-fold you fold this thing around this line and you get the well-known pillow with these four corners. So topologically, as you know, the obi-fold is a sphere with four points taken out. Now therefore, on the obi-fold you have canonical, I would say, one cycles, which in fact just encircle these fixed points. So you have one here, one here, one here, one here. And here you see the projections of the Wilson lines on the torus to the obi-fold. So this one Wilson line now is this, the other one is this. And you can decompose it in this way. You can write, say, this Wilson line as the sum of Wilson lines around these fixed points, T1 in this way, T2 in this way. Now as you also know, once you are on the obi-fold, you no longer have continuous Wilson lines, but they are discrete. And if you look at this line integral, the exponent of that, then it's quantized and you have essentially a parity. That means these factors can only take the values plus minus one. This is well known. Now let's introduce, in addition, a flux. So we have, say, a gauge field, some are characterizing the Wilson lines and we have now a flux, which is constant on this obi-fold. Then we know the flux on the obi-fold is quantized also in this way. And the total background field is now the field related to the flux. And in addition, it's the obi-fold field. The obi-fold field produces no flux. So if you have, say, you can have fractional flux at different fixed points, but they add up to total of zero. And then, as you know, what is important for the following discussion is the flux generates chiral zero modes due to the index theorem. So in our case, that means if we have, say, with a minus sign here, if n is positive, we have n flux quanta, then we have n left-handed fermions. So we have chiral, so with flux, we have just a number of chiral zero modes. Okay. Now let's look in more detail at 60 supergravity. The Lagrangian or the action for that is well-known. You have, of course, a metric. You have a dilaton field. You have the field strength of the anti-symmetric tensor field B. And we have now the field strength of a U1 gauge field. And then one includes in the field strength here, transimons terms for the spin connection, local Lorentz symmetry, and for the U1 gauge field, which is this one. In our discussion, we will ignore gravitational anomalies. So I can comment on that if you want. So we will, in the following, ignore this term. Then if you now consider also flux on this orbit fold, it turns out you have to redefine your tensor field in such a way, so that you can do the usual proper Kalutza Klein expansion. And so you have then, once you do this redefinition, you can write the derivative of this tensor field as, say, the two form on your compact space times the derivative of a scalar. Little b is now a real scalar plus another three form. And this derivative of this tensor field is, we are interested in the 4D action, then this is, this field will not depend on the coordinates of the compact space. Having this, you can, in a straightforward way, derive, write the field strength of this anti-zometric tensor field in this form. So we have the scalar here, which comes together with this fluctuation, and you see it's proportional here to the flux which you have, and then the rest. Now, there is a, if you are familiar with this, there is a standard way of how to manipulate the thing. If you want to derive the 4D effective action, you dualize this field strength, that means you, of this thing here now, so you trade this for another real scalar C, and then you also, instead of radion and dilaton, introduce two other scalar fields called T and S. That's a standard procedure. And now we are ready to discuss anomaly cancellation. A crucial problem in this business is the anomaly cancellation for, which can be done for, in fact, by adding a green Schwarzstone. Now, this is well known for the case of the obi fold. You know, once you have an obi fold with, which generates, say, zero mode, then you know that the anomaly cancellation for that, that means, can be done just by this green Schwarzstone. But, and that cancels the anomaly on the one hand, the bulk anomaly, and then the fixed point anomalies, which you have in this obi fold. So that is all fine. However, what created a puzzle for us and what, to my knowledge, had not been discussed before in the literature is the following. If you now introduce the flux, you get, in addition, zero modes due to the index theorem. So you have the bulk anomaly, you have the four-dimensional anomaly associated with these zero modes, and you have fixed point anomalies on the obi fold. And the question is, what do you do about these anomalies? Well, Andy turns out that if you work this out properly, then in the end, you don't have to worry about that, because this Simon's term automatically takes care also of the new zero modes. The way that works is you insert here another full gauge field. That means the background field plus the fluctuations. And then, due to the quantization of the flux, you automatically get the contributions, which also cancels the anomaly due to the zero modes. So, if you work this out, then finally, your full action has the following form. You have the field strength for your U1 gauge field. You have a classical, this is a flux, you have a classical contribution to the energy density. You have the B field, this real scalar together with the vector field. You have this additional field scalar here, again, together with the vector field. And you have a linear combination of these two real scalars coupled to this Schrodinger-Siemann's form. So, the physics of that is that one linear combination of the two real scalars, which are contained in the anti-symmetric tensor field, combine to make the vector massive. So, that's a Stuckeberg mechanism, a version of the Stuckeberg mechanism. But as the other linear combination remains a massless field, coupled in the standard way to the gauge field. So, that you can easily write like this. You find that sort of the end of the story. You find the field strength here of your U1, this classical contribution to the energy density, the mass term for this vector field. And here's a coupling of this, the kinetic term for this one massless axion. And here it's coupling to the U1 gauge field. And if you look at the formulae which you have for this vector boson, for this kinetic term, for this coupling, then you see here always a factor N, which denotes the number of flux quanta. That means you can now construct various limits. For instance, one is where you put flux, you put the flux to zero, then you just get the usual Stuckeberg mechanism and you get the mass for the vector field just from the normal term. If there is also the flux, you can have just the flux without even a fermion, then just classically you generate a vector boson mass. Or you can have the linear combination of both. Then what enters is, in these expressions, is a flux. It's the term from the normally and that gives you the mass for the vector boson and also for the couplings. And what is interesting is you can now construct wave functions also for the bulk fields which you have on this orbital. In fact, for that, in fact, for this whole work, what is very important is a paper written a number years ago by Bakas, who had some wave functions on the torus, there were extensions on the torus, also on the orbital. And then you can get explicit expressions for these wave functions. First of all, let me say what the spectrum is. The mass spectrum, what is well known since this work of Bakas is that if you go away from the orbital and you introduce a background flux, the spectrum changes drastically. You don't have the usual thing with Wilson lines. Rather, the spectrum is now independent of the Wilson lines. And instead, you have a spectrum for the scalars. It starts with zero here. So 4 pi n over the volume, the two volume would be the mass squared of the scalars. And then for the fermions, if you take here n equal to zero, then you see you get zero mode and zero mode and also you get a heavy one. So the 60-wire fermion splits into two components, one of which is massless and the other one, massless. It combines in the Kaluta client tower. You get nice wave functions and just watch here. If you have the full dots, it means that the wave function at these fixed points is non-zero. So this is a situation with flux, no Wilson lines. Now you can introduce Wilson lines and then you see that this pattern changes. You now get at two fixed points, you now get vanishing values of the wave functions that's indicated by this empty circle. And on some points, say here and here, the full points, you get non-vanishing wave functions. And then as you change the pattern of the Wilson lines, that changes where the wave functions are zero or not. But if you take, say, these canonical cycles, you can immediately, and assigning fractional flux to them, you can immediately understand where the wave functions are zero or not and how qualitatively such a pattern of wave functions should look like. So this is the kind of technical introduction. Let me now come to the main point. And so this way of how you get zero modes, on the one hand from the singularities of the orbit fold, on the other hand from the flux, if you are familiar with, say, guts in higher dimensions, as discussed by, say, Olnomura and a number of people in the past, you immediately realize that there may be an interesting possibility. And I will illustrate that with one, say, SO10 gut in six dimensions, which a couple of years was discussed by Asaka, Kuvir, myself, and also Olnomura. So there you have, as bulk fields, 45-plates, you have a 16-plates, a 16-bar, and 10-plates, the standard SO10 representations in the bulk. And we will now, and the nice thing or the way the symmetry is broken in this model is that you have a fixed point with a pillow with four corners. And in fact, you can break at fixed points by adding Wilson lines as a symmetry in the following way. In the bulk, you have a symmetry SO10, which is unbroken at one fixed point. And another fixed point by adding a Wilson line, you break it to the pati salam group. It's the other fixed point to Georgie Glashow's standard SU5. And then here, automatically, it's broken to flipped SU5. And then you get the standard model out as the standard model gauge group as an intersection of these four gut groups, which you have at these different fixed points. So this is the picture. And now let's extend SO10 to SO10 cross U1. Let's add an anomalous U1. Typically, this is what you would expect if you do add out extreme compactification. So if you embed this in E8, you have the minimum group would be, say, SO10 times U1 cubed. And it's anomalous under them, and you will have a more complicated sector. So in some sense, we are aiming at constructing a subsector here. Now, at these different fixed points, you now have the standard decompositions of the echelon representation, the 10 plate, the 16 plate. And what you do at these fixed points is that you always project out certain parts. So, for instance, at the standard SU5 fixed point, you keep for one 10, you keep only the five. From another 10, you keep the five star. At the party's alarm fixed point, you just keep the SU2 fields. You throw away the color fields. And so this is the way you realize the doublet triplet splitting. So from these projections, you break the gut symmetry and you reduce the complete SO10 representations to split multiplets as far as zero modes are concerned. So for the Higgs, which comes from these 10 plates, you are left in the end with two Higgs doublets. And from this 16 plates, you are left with fields which can break B minus A. This is the old picture. And now what is known is if you have just this, then you have the usual picture of gut breaking by all before boundary conditions and you get a split Higgs sector, the doublet triplet splitting. And you can cancel the anomalies by means of a green Schwartz term. But now what is interesting, if we have the U1 and we add now flux and we treat this U1 as the chiral symmetry, that means we assign, say, U1 charge, say, just to the 16 plate and not to the other fields, then we generate, in addition to these split multiplets, you generate copies of complete gut multiplets. So for instance, if I take the 16 plate here, give it a charge Q and add now the quantized flux. Then depending on the number of flux quanta, which I have on my orbit fold, I will generate a couple of 16 plates as zero modes. So these will be my fermions. And you see the point is now if a field transforms non-trivially under this anomalous U1, then you will get on the one hand a multiplicity. And on the other hand, you will get a complete gut representation. So that always comes together. So the fact, in that sense, if you look at the forks and leptons, the fact that we have three copies of them is tied to the fact that they are complete gut multiplets. Whereas if you look at Higgs and Gage fields, they have no, they don't feel this anomalous U1. Therefore, you don't have multiplicities. You don't have copies. And you don't have complete gut multiplets. They come as split multiplets. So that is somehow the observation, which is related to this the way the symmetry is realized. And then the other thing is now, of course, what is crucial is due to this work of Bacchus, that this way of having gut multiplets and split multiplets on the one hand and the other hand is directly related to the way you have supersymmetry breaking. But before I come to that, let me first discuss what you can now learn in such a model about Yuccava couplings, because there are some implications for the flavor structure. As I said at these fixed points, these multiplets are projected. And now you can have, for instance, at one fixed point, you have the coupling 16, 16, 10, 16, 16, 10 to another 10. And you have always for each bulk field, you have one coupling. This is a coupling which is given here. If you go to another fixed point where the symmetry is broken, you can have couplings which are compatible with the unbroken group. So here is your five, so you have one coupling more here, but his alarm, you again have a coupling less and flip less your five. And now, if you now have the flux for the zero modes, is you, the single coupling turns into a matrix simply because the 16 plaid say becomes three copies of zero modes. And these three copies of zero modes, then for them, you have different products of wave functions at the corresponding fixed points. And in this way, the complete super potential for all these zero modes look now like this. You have the couplings, which you can have for the bulk fields. And they always come together with a matrix, which just depends on the fixed point. So at the fixed point, you get matrices which are determined by the flux quantum numbers and also the geometry that fixes these matrices. And then you have, in addition, these couplings which come from this. Now, to work that out in detail is something which we are looking at at the moment. But now, let me come to the main point. As I, yeah, as I try to explain this idea of assigning an anomalous U1 in the bulk, so to start from the group of SO10 cross U1, gives you copies for complete gut multiplets and just single fields for split multiplets, for split gut multiplets. Now, the flux also introduces supersymmetry breaking. That's due to this work of Bacchus. So you will have a universal scalar mass plus calypsoclin excitations. So the lightest scalar quarks and leptons will have a universal mass, which is 4 by n, where n is the number of flux quanta. In our case, n is 3. So it will be this in units of the volume of the compact dimensions. Now, I should say the volume of the compact dimensions, here's the dynamical quantity, because it depends on the radion field. So it has to be determined from stabilisation. And of course, this is something, well, work on that you can find in the literature. We are trying to understand, looking at that now also. And of course, what you then get from that has to be consistent for this mass to be consistent with gauge coupling unification and this proto-dicates. That puts some constraints. But typically, this mass, this scalar mass will be of order, the gut scale, say, 10 to the 15 GeV. Now, so that means at three level, you start from a spectrum where only the scalar quarks and masses, because they are in a complete gut multiplet, are massive. And they all have masses of the order of the gut scale. The rest, at three level, is muscles. Now, so that means, gravity, no, gauge, no, six, no, six both, so they are all muscles. Then, the flux introduces an energy density, which corresponds to the term breaking. Therefore, you generate a gravitino mass. I think that's unavoidable. So, typically, if this is, say, 10 to the 15 GeV, the gravitino mass will be around, say, 10 to the 12 GeV. Then it depends what happens. Now, there's a normal mediation, quantum corrections and all that. So, in the standard picture, one would say, I have a normal mediation. And then, if this is 10 to the 12 GeV, I will generate gaugeino masses, say, between 10 to the 9, between, say, 10 to the 10, 10 to the 9 GeV, roughly, also very heavy. The biggest protection you have for the hexynos and the chargenos, they can be even lighter. What their mass is depends on the way you treat the quantum corrections for that, but that remains to be seen. So, in any case, qualitatively, the spectrum will be that you have really a universal squawk and slepton mass of roughly the gut scale, maybe a little bit smaller, much bigger than the gravitino mass, much bigger than the gaugeino and hexino masses. Now, what comes out in detail, I think we are learning now about the topic because, for us, this is rather new. It depends on how you treat the quantum corrections in detail. And you may have the typical split supersymmetry mechanism with everybody in the TV range, or you may have spread supersymmetry where the lightest are the hexynos, the gaugeinos are a little bit heavier, or if all that becomes too heavy, you may just have the standard mole and an axion. But that's the question of the quantum corrections and, of course, you have now all the problems well known from split supersymmetry of the fine tuning, how do you understand the Fermi scale and all that. I have nothing to say about that. We are also just looking now again at these problems. But at least, I hope to make the main point clear, namely that the fact whether you have complete gut multiplets or not is related to the multiplicity and to the size of supersymmetry breaking. And if you take such a picture, then you essentially qualitatively get out the picture of split supersymmetry. And, of course, it will be very nice if one would find some evidence for that at GLC, which could be light hexynos. Okay, so we have time for a couple of questions. Yes, I understood that if you take into account the bound on the hex mass, I mean the value of the hex mass, that rules out the squares and the sleptons at the gut scale. Have you considered that? That all this has to be consistent with 125 GB heaps? Well, of course, I mean, I'm aware of that, but we have not yet that worked out. I mean, of course, once you have this picture, there's fine tuning for the Higgs, the well-known fine tuning of split supersymmetry for the Higgs, as well as radiative symmetry breaking. Yeah, but on top of that, in principle, you see that there's this plot and this paper by Yudich et al, where you see the line of the Higgs mass and then see how the split suzie fits, and it brings the value of the split very, very, very slow. Yeah, at that point, the question is what split suzie means. I think what split suzie means in that paper is not what it is here, because that's a special assumption, I think, on the masses of the other particles. Well, we can discuss. Yeah, we can discuss. Just a question. Typically in these dimensional models, when you compute the mass of scalars, very often they become techonic. Yours are positive definite all the time? I think so. I don't see what... You mean the scalar mass is here? Yes. I don't see what makes them techonic here. The term can make it... Well, no, but that is... The term can work in the nice and the wrong direction, positive and definite. Well, I think this is, as far as I understood and talking to some people, this is sort of established that for... I mean, this is essentially the mass spectrum worked out by Bacchus. So then, Bacchus, I think, didn't even ask the question, what kind of supersymmetry rate do you have? I mean, he just calculated the mass spectrum for which you need, essentially, that once you have this flux, I mean, the wave functions behave like harmonic oscillator wave functions. Then you get this kind of spectrum that was generalized to the case of the obi fold, and it was also figured out that I think the vacuum energy density, which you generate by the flux, corresponds to a deterrent breaking. As you know, later on we did a more detailed computation with Kremlade set out, and I think you get both sides in the spectrum, positive and negative, but I don't know... Well, I mean, I would be... In the presence of the orifice, it's true that the orifice can project out some of the negative states. Okay, I mean, this is something we have to... I mean, we look at your paper in connection with these wave functions, and in fact, we learned a lot from that. What that means for the masses of the scalar particles, I don't know, but I would be glad to learn more about that. All right, one more quick question. Okay, I don't know, it's a question or comment. If you think about the stabilization of radius or the electron to cancel, then I think, generally, you can do this, but the generically f-time of these things could be easily generated, because you have supersimilar breaking, and you have radius t, f. And if the f-time of these things are generated, and even it's this size of gut scale or complication scale, and if that's the case, Gagino will also get the mass. Of course, you can kill that by going to no-scale type scale or whatever you can do it, but you may need additional things to get actually this. Once you think about all these dynamics of this... Yes, I fully agree. I mean, what happens now, whether there are other sources of supersimilar breaking, one has to take into account, and what that does to the rest of the spectrum remains to be seen. Yeah, I agree. Great, so let's thank the speaker again.