 Very good. Well, again, I'm sorry I can't be there in person, Mike. As I mentioned, this was a difficult week, but it's nice to see you virtually. And so everybody can hear me okay. Can you see my pointer on the screen? Great. Okay, so I want to talk today about some questions about string vacuum and the standard model. But let me start first with a little brief prologue. So I met Mike at Caltech when I was visiting as a prospective student, I think in 1988, spring of 1988. So quite some time back. And I have a very strong memory of that because Mike had just bought a new Mazda RX7, if I remember correctly. And we met, we started talking about physics, and Mike took me for a drive around in Southern California in his RX7. And I thought, wow, this is really cool, this world of string theorists. And I wanted to go to grad school to learn about string theory. And Mike, I've always been very inspired by Mike's adventures in the world of string theory. And at that time, as I was starting as a graduate student, you know, there was like, you know, Ed Whitten, and John Schwartz, the gods of the field who were kind of working at a, you know, very, you know, far away level. But Mike was just a few years older than I was. And he was, you know, a real person doing this stuff. So that was very inspiring to me. Anyway, I remember fondly that that visit and tooling around in your car, Mike. Back then, decades ago, I think this picture is not quite that from that era, but it's somewhere between here and there. So anyway, you know, since then, Mike has pioneered a lot of interesting directions. And a lot of his work has inspired a lot of my own efforts over the years. I just want to bring up a particular, you know, a paper that Mike wrote almost 20 years ago, which was part of his pioneering work also with others with Ashok and Denif on using statistics, ideas of, you know, systematically and statistically thinking about the enormous landscape of string vacuums. Mike was really, you know, a pioneer in this, in this direction. And I just want to call it to your attention, a couple of aspects of this paper. First, Mike, at that point already was affiliated with the IHES. And although his US institutional affiliation has bounced around a couple of few times over the years, his affiliation with the IHES there is really the most stable of those, of those things. So it's great that you all are hosting this event. And it's great to virtually be there where he's been having a lot of his intellectual life over the decades. And in particular, this paper starts by talking about classifying string vacua. And I don't know if you can read that on the screen there, but, you know, studying the ensemble of string vacua and ending using these ideas, we outline an approach to estimating the number of vacua of string M theory, which can realize the standard model. So, you know, this, this idea is something I've been, you know, fascinated by for quite some time. And today's talk is really building almost 20 years later on what Mike was doing, and still trying to move towards finding a systematic approach to estimating the number of vacua of string M theory. And now I would add F theory, which can realize the standard model. So that's kind of the framing of today's talk. And it's great to be able to do this here at the IHES with Mike there for this, for this nice event. So please, as I talk, feel free to stop me with questions, particularly in this, this virtual format, you know, it'll enable me to connect more directly with what's going on there, people ask questions. And if I don't quite hear your question, just just shout out. So I hear you. So I'm going to start by talking sort of generally about the F theory approach to understanding the landscape of string theory solutions. Just give a general picture of this and how this fits into the program that Mike, you know, initiated those, those many years ago. And which has been, you know, proceeding steadily over the years, it's just, it's a hard problem. So, you know, you, you, you start working on something like this, and it'll be a while before we're at the end of the path. But, you know, Mike's work here really framed a way of thinking about this that I think has informed people over the over the time sense. So I want to fit this F theory story into that context. And then I have some more detailed analyses of two new constructions of the standard model using F theory. And so I want to present those and think about them in the context of this global picture of the landscape. So the first part of the talk is but more of a big picture of what's going on. And then I get into some more technical things. And I probably have more technical slides than I'll have time to go through. And I know the audience is broad. So I'll try to focus on the important big picture issues. And again, if questions come up, I'm happy to delve into those and end up spending a little less time on some of the technical stuff, which, which may be more of a more appreciated by best specialists in those particular areas. So, okay, so without further ado, F theory is a very powerful approach to thinking about string theory vacua. It's a non-deterpretive formulation of type two B string theory. And, you know, back again, just going back to Mike's abstract here, an approach to estimating the number of vacua of string M theory, which can realize the standard model in order to really implement that program. In my mind, you know, since starting to think about this, you know, 20 years ago, the key thing that we need is a really good global picture of the set of possible string vacua. And for many years, people had different constructions. And, you know, you had this Calabi or that Calabi, but we didn't really have an overarching picture of how they all fit together into a big framework. And to me, F theory, the big one of the biggest values of F theory is that although it is difficult to compute things in specific solutions, because of the non-deterpretive nature of the theory and the fact that the string coupling gets strong somewhere in the space of the compactification, somehow the power of holomorphy gives us a very powerful global picture of the set of string vacua that fit together into the largest set of string vacua we know of, which incorporates through dualities many of the other formulations like heterotic vacua are really sort of a corner of F theory in a particular duality frame, at least the heterotic vacua that people understand. So F theory really gives us this very powerful global framework in which we can really sensibly start to talk about what is natural, how might we formulate statistics more precisely, all those kinds of questions. So basically, what is F theory, it's a dictionary that translates between geometry and physics. So the idea is we take type two B string theory and we compactify it not on a collab-y-ow with no curvature but rather on a general compact-calor manifold like PN or some blown up version of that, there's really a huge, as I'll say later, there's a huge class of compact-calor spaces on which we can compactify the theory. And we then have to do a non-deterpretive compactification. And if we're going down to six dimensions then the space B is a complex surface, a complex two-dimensional space. And if we're going to 4D, we need a three-fold B. And the axiodilaton of type two B string theory then encodes a fiber which can be interpreted as a kind of auxiliary elliptic curve for each point in the compact space B. And so we have a Weierstrass model, which y squared equals x cubed plus fx plus g, which is a simple way of formulating an equation for an elliptic curve. But in this case, f and g are functions or more technically sections of line bundles over the base, either B2 or B3, that encode an auxiliary geometry, an elliptic fiber over each point. And this fiber, this elliptic curve, basically a copy of T2 at each point in the base space B, degenerates at certain loci. It needs to degenerate at certain loci if the space is not collabiao because we want to find supersymmetric solutions and supersymmetrics and solutions that solve the gravitational equations. So to solve the gravity equations, we need sources for the axiodilaton in the type two B language. These are seven brains, D7 brains and more general PQ7 brains. These seven brains source the axiodilaton, and in the geometric picture, they are loci where the elliptic fiber or the extra auxiliary torus shrinks. And in the beautiful work of Kodaira Nauron in the mathematical literature, these kinds of singularities in elliptic vibrations have been classified in terms of Dinkin diagrams. And the beautiful insight of F theory is that those Dinkin diagrams that have been associated in the mathematics framework with singularities of elliptic vibrations precisely and naturally correspond to gauge groups, not just perturbative gauge groups like SUN, but also non-perturbative gauge groups like the exceptional groups E6, E7 and E8. And that geometry can be interpreted in several ways, one natural ways in terms of an M theory limit, where we have brains wrapping these cycles which have shrunk to points to form these singularities. And so we have an emergent gauge group from the co-dimension one singularities in the elliptic vibration. And at co-dimension two singularities like these green loci here, for instance, where seven brain loci cross, we get matter representations. So F theory is basically a dictionary that allows us to take a Weierstrass model, an elliptic vibration, which is a very mathematical object, but which we can think of in type 2b as just an axiodilaton over some base space b, which we're compactifying on, which encodes singularities that give us a non-perturbative gauge group and matter content. So this powerful framework has allowed us to get a very good handle, as I said earlier, on these large classes of string vacua. So a lot of the work on F theory has been built on F theory as a limit of M theory. You take M theory in one dimension lower. So for instance, if you want four dimensions, you take M theory on a collabia fourfold. I should add that for supersymmetry, this total space, the total space of the elliptic vibration is a collabia. So we are still thinking about collabia manifolds, but the geometry of the string compactification is not collabia. The collabia will only emerge when we add this auxiliary torus. So a lot of the work on F theory has started in one lower dimension. For instance, by going to three dimensions and thinking about M theory, which is an 11 dimensional theory, compactified an elliptic collabia fourfold, and then taking a limit where one of the directions decompactifies, you can define that as a non-perturbative type 2b theory. The general philosophy of this talk is that I want to take the type 2b description seriously. That is, from the geometric point of view, we have a singular collabia, and while the M theory picture involves doing what's called a resolution of that where we make it into a smooth space and then do the physics on the smooth space, that is essentially just a workaround for the fact that we don't really know how to think about the singular geometries. And one of the things I'll talk about today is the fact that some of the physics, certainly the physics is independent of how you do the resolution, but it even seems that there are parts of the mathematics which can be extracted, which are independent of how you resolve the singularities. So I want to focus on that resolution independent structure. The physics, as I say, has to be independent of the resolution, and we should be able to develop a systematic way in the non-perturbative type 2b framework of thinking about how the resolution independent physics is encoded in what the apparently singular geometry there. There's some recent other work paper with Sheldon Katz that came out recently where we work on, again, an extension of some other work of Mike's that he did with Schnell and my former student, Daniel Park, which was essentially pushing in this direction. How do we understand that the physics of type 2b from directly without having to go through M theory? So in that work of Douglas Park and Schnell, they understood in eight dimensions how to get the Avillian gauge groups coming out of F theory, and this recent work I just mentioned obliquely here with Sheldon is basically taking that other aspect of Mike's work and generalizing it to higher dimensions. The focus of what I'm going to do today is on the structure of the intersection theory on these singular elliptic collabials, which enables us to understand chiral matter and gives us essentially a toolkit for understanding these different constructions of the standard model that I'll be talking about. Again, just stop me if there are any questions. So as I've said, F theory is very powerful because it gives us a global picture of a large part of the string landscape. And one of the big puzzles over the years has been there are lots of collabials. Can we even show that the number of collabials is finite and get a handle on that? Again, the finiteness of the landscape is a problem that Mike has had many contributions to over the years. He and I wrote a paper actually on one aspect of that. But we still don't know whether the number of collabial three-folds or four-folds are finite. On the other hand, we know that there are a finite number of elliptic collabial three-folds and four-folds that were shown by Mark Gross based on some work of Antonella Grasi for three-folds. And for four-folds more recently, there's been some nice work also showing this in the mathematical literature. So we know that there are a finite number of elliptic collabial three-folds and four-folds. And in recent years, we have come up with more and more evidence and stronger analytic arguments that in fact, almost all collabial three-folds and four-folds are in fact elliptic. This was found by this group from Virginia Tech. Anderson, Gao Gray, and Li have found strong evidence for this among the complete intersection collabial manifolds. And with my former student, Yu-Chan Huang, we looked at the Croitzer-Scarca database of 400 million different toric constructions. And of the Croitzer, so I think Anderson, Gao Gray, and Li found numbers like 99.9 percent for four-folds and 99 percent for three-folds. Looking at the Croitzer-Scarca database of the 400 million toric constructions there, all but about 30,000 admit what I would call an obvious elliptic or genus one vibration in one in some phase. So here's a picture of the hodge numbers of the collabial three-folds in the Croitzer-Scarca database. And the red points are the hodge numbers associated with cases where we can't just immediately by looking at the toric construction see an elliptic vibration of that geometry. And I suspect that many of those remaining 30,000 will still have elliptic vibrations. It would just require some some more sophisticated analysis to show that. So basically the point I want to emphasize here is that the set of elliptic collabial three-folds is bounded, it is finite, and it is very well described. And the same is true for for elliptic collabial four-folds. The classification is less complete. There's a picture which looks a bit like this, which is only a part of what you get even torically. And I'll come back to the question of how many toric even bases there are that support elliptic collabial four-folds. But the point I really want to emphasize in the context of this vision of Mike's from a couple decades ago of systematically doing statistics on the string landscape, the F theory landscape of elliptic collabial four-folds is under better, even though it's complicated, we don't have full control over it, we have a better sense of what the global picture of that landscape looks like. And in particular, how all these different vacuums are connected. I should emphasize that this picture here is some 40, 400 million different elliptic collabial three-folds. Those are all provably connected through certain kinds of transitions through higzing and tensionless string transitions in the 60 theory from the physics point of view and through geometric transitions on the geometric side. And the same thing is essentially true for four-folds, although the things aren't proven quite to the same degree of precision. So we have a big landscape and we have some control over it. So let's look in that landscape to see what we can find for physics that looks like the standard model. That's really the top-down goal of this talk and some part of my research program and part of Mike's vision of how this might lead us to connect string physics with observable physics. Any questions? Okay, so what do we learn by looking at this big landscape of F theory models? So one of the things we learned, which might seem a little surprising from other points of view, is that virtually all of these elliptic collabial three-folds and four-folds have a structure which I would refer to as a rigid or geometrically non-higgsable gauge group, which is basically that everywhere in the modular space for most of these bases, we are forced to have certain kinds of gauge groups, at least geometrically. From the type 2B point of view, there are certain loci corresponding to so-called divisors in the base geometry where that codimension 1 locus for a 7 brain has a negative normal bundle, which essentially forces 7 brains to live at those loci and support gauge groups. So this is true for virtually all three-folds and two-folds, complex, scalar, two-folds and three-folds that support elliptic vibrations. And these rigid gauge factors that are forced on us, if we take say four dimensions, very similar in 4D and 60, but my goal here is to get to 4D, so we'll focus on that. These rigid gauge factors include certain gauge groups like the exceptional groups, E6, E7 and E8 and F4. It also includes some of the classical groups, SO8, SO7 and SU2 and SU3, but it does not include all of them. It does not include in particular SU5, which we often use as a gut group in physics. So you might ask, why can you get SU2 and SU3 but not SU5? And the answer is that there are actually non-perturbative realizations of SU2 and SU3 through some of the codyrosingularity types, which are basically forced on you by vanishing of F and G to some orders in the base. And these non-perturbative realizations are possible for SU2 and SU3, but not for SU5. In addition to single gauge factors, there are also products of two gauge factors, and in 4D there are only five different products that can emerge with G2 and SO7 and SU3 times SU2 or products of SU2 and SU3. So a typical F theory geometry, if we go to this big global picture of all these different possible things, a typical global geometry will have a whole bunch of these gauge groups forced on you and they will appear in clusters which are connected to each other through intersections of the seven brains where matter lives, and those clusters will then be separated. So you will have multiple, perhaps a dozen clusters in a given geometry, each of which supports a particular gauge group. And those gauge groups can only talk to each other through gravitational interactions and scalars. So it ties in very naturally to the notion of dark matter that we have one sector where our SU3 cross SU2 cross U1 lives, and then there are other hidden sectors which only communicate gravitationally or through massive scalar interactions. So this fits pretty well with the fact that we're having a hard time finding, for example, weakly charged dark matter. If much of the dark matter is hidden in these extra sectors associated with geometrically forced gauge factors, it would fit very naturally with what we've seen. So as I say, these rigid gauge groups are ubiquitous throughout the landscape. If we look at the better understood case of 60 supergravity theories controlled by elliptic Calabi-A 3-folds, if we again go back to the creature scarca picture, of the 61,000 toric bases that support elliptic Calabi-A's, only the generalized delpetsos, so only about dozen different bases, those are the orange points in this diagram, do not support some kind of rigid gauge group. The typical gauge group in 6D is something like E8 to the fifth times F4 to the sixth times 10 factors of G2 cross SU2. So there's a big gauge group with these different factors which are disconnected and in 4D the story is very similar. For 6D we've done a systematic classification of all of the toric bases that support elliptic Calabi-A 3-folds and there's about 61,000 of them. For 4D constructions there are far more and you can't systematically enumerate them. We've done some Monte Carlo studies and a good estimate is that there are something like 10 to the 3,000 base geometries. So this is not even including the exponential statistics of fluxes that Mike and Frederick Deneff and others studied back in the early days of the large landscape discussions, but these are actually distinct geometries, something like 10 to the 3,000, even just toric geometries that would be 3-fold bases supporting elliptic Calabi-A 4-folds. And there's a little bit of a question in that counting which has to do with whether you count different phases of the same or different flop phases differently and that's I think a key question which I might come back to if I have a few minutes later for dealing with these questions that Mike and others have studied since the paper I mentioned of bikes in trying to understand how to really be systematic about the numbers here. But anyway in 4D of those 10 to the 3,000 only about 4,000 are so-called weak fano bases which lack these rigid gauge factors. So almost everywhere in the landscape we've got a bunch of these rigid gauge factors and so it's a natural way to try to understand what's natural in strength theory and what's not. We don't know how to do precise statistics but here we're talking about something like 4,000 out of 10 to the 3,000 which would be branches of the modular space where we don't have a forced gauge group. It makes it seem very compelling that the place to look for a realization of the standard model or our observable physics is through these rigid gauge groups that are forced on you from the geometry. That is if of course what we're doing in these supersymmetric theories has any relevance to non-supersymmetric physics. I believe that it does in part because the story is so similar for the 6D theories with eight supercharges to the 4D theories with four supercharges in terms of these rigid gauge groups that are forced on you that I suspect when we really understand non-supersymmetric vacuum of strength theory in an equally systematic way we will also have these kinds of rigid gauge groups forced on us there as well. Okay so what do we do from there? We have this big picture of what the strain landscape looks like from the f theory point of view. How can we realize the standard model in that context? So there are a lot of different choices for how to proceed. I've emphasized this idea that there are these geometrically non-higgsable or rigid gauge groups in the majority of the different geometries. We can also try to directly tune the gauge group. Remember we have this virus truss model and if the virus truss model is such that f and g vanish to various orders we get what are called what we would call tuned gauge groups where we choose we basically fine tune a bunch of moduli to get some specific gauge factor. So we can either try to realize the standard model by tuning it through either through tuning a unified group like an su5 or by directly tuning the standard model or we could go to a rigid or non-higgsable gauge group and try to realize the standard model as a subgroup of that. Again I emphasize the geometric and the non-higgsable because of course a standard model realized in that way will still be higgsable through the standard Higgs mechanism along with supersymmetry breaking. The non-higgsability here really is a geometric non-higgsability related to the rigidity of the geometric construction. So we can imagine using a rigid or non-higgsable gauge group to directly tune the standard model or we could get the standard model from a gut but where the gut is realized through one of these rigid gauge groups which would then have to be something like e6 or e7 or e8 instead of su5. So the first construction I will talk about of the specific constructions today is a class of models where we directly tune the standard model gauge group without a unified group in the context of f-theory. So that'll be construction one that I'll get into in more technical details of in a few minutes. The biggest body of work starting with the work of Beasley-Hackman-Vauffa and Denuggy-Wineholt quite some years back was focused on tuning a gut group like su5 that is not having a rigid gut group because you can't have a directly a rigid su5 but having a tuned su5 and then breaking that through fluxes down to a standard model gauge group. But from this top down point of view however as I've been emphasizing these tuned models in several senses are rather rare in the landscape. On the one hand they require tuning many moduli if you have a given geometry you have to tune many of the complex structure moduli to get these gauge factors and they're rare in another sense which is that most of these bases are kind of chock full of these rigid gauge factors and that doesn't leave much room for tuning further gauge factors. So there has not been a systematic study of this but my general sense is that it will be very hard to find typical Calabi-Yau three-folds or four-folds with large hodge numbers where there's enough room to tune either an su5 gut group or even a directly tuned standard model gauge group. Again that's an interesting project for further work but I think that's it's certainly easier to imagine getting things with the rigid gauge factors. We can get the su3 and su2 as a rigid gauge factor as I've mentioned before. Sorry was there a question? But it's difficult to integrate the u1 factors. So a natural approach just to try to take one of these rigid gut groups. Sorry is there a question or am I getting a little echo? You are beginning to go I think. Okay great I'll ignore that then if there is a question. So the second construction I'm going to talk about today is one where we take a ubiquitous rigid gauge group in particular e7 or e6 and we use fluxes to break that to get the standard model. So this is kind of the broad context the f-theory picture which gives us this better handle on the global structure of the string landscape. And what I'm going to do now is dig into a little bit more in detail how we actually get these different kinds of standard models. So this is a good a good point to stop for any questions at this stage of the big picture. Any questions? Okay merely we roll along. So let me now turn to a slightly more technical aspect of the talk. You have a little interlude on a mathematical piece of structure which gives us a powerful set of tools that will enable the two constructions I just talked about. So this is some work that came out with Patrick Jefferson who's a postdoc at MIT. Patrick is fantastic and I think we'll be looking for jobs in the fall so I urge you to consider him and Andrew Turner who's a former student who's now a postdoc at UPenn also an excellent physicist. So with these guys we recently put out a paper where we systematically looked at what seems like a technical issue the middle co-amology on elliptic collabia fourfolds and basically the upshot of this is that we have a way of describing this middle co-amology and in particular the middle the intersection on the middle co-amology the intersection form on that middle co-amology which streamlines and simplifies understanding how chiral matter works and also how fluxes work in four-dimensional f-theory models. So this is a subject which has been studied very richly and I'll give some references in a few minutes but in general the tools were applied in a sort of case by case fashion and I think what I'm about to describe gives us a nice unifying framework for doing a lot of these calculations. So in order to talk about this we need to say a little bit more about the topology of elliptic collabia fourfolds. So as I mentioned divisors are co-dimensional one algebraic three folds living in a four fold for example the seven-brain loci are two-dimensional divisors in the three-dimensional complex base and there's a relationship so basically the number of divisors the dimension of the space of divisors is is given by the hodge number h11 of a complex variety and the h11 the hodge numbers of the h11 hodge number of the elliptic collabia three or four fold by a nice result of sheodotatin wasier is given by the h11 of the base plus the rank of the gauge group plus one in physics language. So basically what this means is that the divisors in the collabia three fold or four fold can be broken up into and I'll use this as an indexing structure there's a zero section I don't think I mentioned this but the elliptic vibration structure requires not only a torus over every point but also a section of that elliptic vibration so there's a a section called the zero section which is a divisor in the total space there are divisors which I'll index with an alpha index which are pullbacks of divisors on the base that corresponds to this h11 part of h1 on the base part of h11 of x there are carton generators and these are the divisors that are associated with the resolution of the singularity so when we go back to this picture of the collabia as an elliptic vibration when we resolve the singularity that gives us say an su3 gauge group there's an a2 dink and diagram there and those curves there fibred over this divisor in the base are what give us the carton divisors of the gauge theory in the collabia construction so those I'll index by a di and then if we have multiple u1 factors relevant for example for the story of Douglas Park Chanel that I mentioned earlier or other things where we have multiple u1s we'll index those by an a although nothing I'm going to do in my talk today involves extra u1 factors so there are little sorry that's not true some of what I'll say will involve extra u1s but not too much so collectively we use a capital I index and denote these and I'll just note that the non-Avillian gauge factors of the model are supported on a divisor which is a sum slightly unfortunate notation here the first sum is a sigma the second sum is a coefficient sigma alpha d alpha so two different meanings for that sigma alpha um okay so that's some notation so in general the collabia fourfold will of course have other hodge numbers h3 1 is the number of complex structure moduli generally h2 1 is small h2 2 which will be interested in is related by an identity to the other hodge numbers as is the Euler character so we're interested in h2 2 the the 2 2 columnology of this collabia fourfold and to understand fluxes and chiral matter in f there we're we're interested in a particular class of vertical cohomology of cohomology classes characterized vertical cohomology which are basically generated by the intersection forms of divisors in so at least in the Poincare dual picture so we want to look at if we take two divisors and intersect them in a fourfold the divisor is a threefold so the intersection of two divisors is a surface so we have a set of surfaces which i'll denote si j which are in the vertical homology and these si j's for different i and j indices are not all linearly independent there are homology relations uh that give us linear dependencies so there's a subset of those but taken together those si j's generate the vertical homology and this vertical h2 2 lower is the vertical cohomology which is point grade dual to that so the vertical homology and cohomology are relevant to us because if we want to understand chiral matter or fluxes in f theory uh this where these things live so fluxes in the h2 2 vertical part of the the four four cohomology of the fourfold generate chiral matter um so in general the h4 of a fourfold has an orthogonal decomposition into three parts there's the vertical part which i've just been describing there's the so-called horizontal part which has to do with deformations of the complex structure and then as was gradually understood in part driven by the physics of this there's also what's called remainder cohomology h2 2 ram um and then there's the unimodular intersection pairing on this whole thing so what we're focused on today is the is the intersection pairing on h2 2 vert that is if we take two uh surfaces si j what's their intersection form or in the point grade dual picture we take two vertical to two forms what is their intersection so this is important to understand chiral matter um so flux in a 4df theory model if we think about it in the m theory picture it's given by an element of h4 of xz and it was shown by witnesses shifted by something which might be half integral depending on the second churn class we won't worry too much about that if we impose supersymmetry the flux has to satisfy certain conditions it needs to live in h2 2 and it needs to satisfy a primitivity condition with respect to the caler form j there's a tadpole condition which says that the integral of g wedge g uh contributes to a m2 brain tadpole um which the total of that tadpole has to match the Euler character over 24 and so we want to find the fluxes that satisfy these conditions and you know these are the fluxes that that uh mike and and fredrick mike and and denf and and ashok and others uh quantify when thinking about counting vacua of strength theory um here we're doing it in f theory and we want to satisfy conditions of point korean variance to give us a good four-dimensional super super gravity background that requires that the g flux integrated over certain surfaces s0 alpha and s alpha beta again where alpha is a base index have to vanish and if we want to preserve the original gauge group we need to have g integrated over si alpha where again i is a carton to vanish so when we want to break e7 by fluxes in the second construction i'll talk about today we will take that to be non-zero uh so basically as i said earlier chiral matter is determined by fluxes through vertical cycles so in a given representation the chiral matter index in that representation is given by integrating the flux over a so-called matter surface s r so the problem of understanding gauge symmetry breaking uh through this equation and chiral matter through this equation boils down to understanding the intersection form on the middle co-amology on the vertical middle co-amology of the collabia fourfold okay so this is what we've been studying um again previous work on chiral matter tended to use explicit resolutions but basically what we do what we have done in this in this work with patrick and andrew is to think of the fourfold intersection numbers m i j k l which in principle and in fact depend upon the resolution that you choose of the singular elliptic collabia four and organize them as a matrix where the indices of the matrices are pairs of indices so for instance m i j k l which is the intersection between s i j and s k l can be thought of as a matrix in this space of two two vertical surfaces we then have an equation for the fluxes theta i j coming through the surface s i j integrating g through s i j which can be written as this matrix times a vector where this vector phi k l basically encodes the flux through point-carre-dual surfaces so the idea is we're reformulating this whole story in terms of linear algebra and a matrix so that we can manipulate things in a fairly simple and straightforward way if we remove the null space associated with trivial homology elements then this matrix m which again depends on the resolution um at least the entries depend on the resolution removing the null space gives us a non-degenerate intersection form which i'll call m red or m reduced and the observation leading to a conjecture is that this m red is resolution independent in every case we've looked at um up to a basis choice so that if you take two different resolutions and you compute the m red they are equivalent under an sl and z transformation and we've seen this again in large classes of examples and we can show it with a general argument where we only really need to make one assumption about uh basically where something lives in the in the root lattice in the weight lattice rather of the of the group um so the general form of m red is roughly the following um for if we just have a simple non-ability engaged group g and we look on a certain basis we can write this m red in terms of intersections on the base where k here is the canonical class of the base the d alphas of the divisors associated with um divisors on the base and then there's uh this is the in the kappa here is the inverse carton matrix of the gauge group sigma again is the divisor carrying the gauge factor and then that gets dotted into the v's and after we do a non-integral change of basis again these are all the same with these asterisks being undetermined they're all related to each other for different resolutions under linear transformations or under a non-integral chase change of basis we can diagonalize that last part and this then encodes the physics of chiral matter and flexes so in all the different constructions that we look at this is essentially the form of this reduced matrix and having this resolution independent form allows us to very readily do calculations with just understanding the geometry of the base and the gauge group and a single computation of this this m fizz part so that we can compute things like the chiral matter spectrum and the flux breaking of different things so you know for an example one that has been studied extensively in the literature uh from different points of view from this reduced matrix m red we can write that the uh theta three three flux one of the components of flux is given by this intersection product times a linear combination of the of the flux as phi relating that in various ways either through matter surfaces or connecting to 3d churn and simons couplings uh to the chiral multiplicity we get that this flux term controls the multiplicities of the five and ten representations of su five and so we can just read off a base independent formula for the chiral matter multiplicity in an su five theory in terms of the geometry of the objects on the base um you know so for example if the base was p three then the multiplicity of matter is uh four times five n minus 24 times m where m is an arbitrary integer and the n is the uh choice of divisor oops sorry on which the um on which the group is resting so this gives us a very general way of constructing um f theory constructions with chosen gauge groups and analyzing them in a very general geometric way so let me now just fairly briefly describe the two different constructions that I want to tell you about one of them is a universal tune standard model structure uh with again Patrick and Andrew and also my former student Nikhil Raghuram um let me just say a few words about what I mean by universal engineering so um in f theory there's a notion of generosity for the representations of matter associated with given gauge group so in particular if we take some gauge group like su five su n su three cross su two cross u one and we construct an f theory model with that gauge group there are various different kinds of representations we can get depending on what kinds of co-dimension two singularities we have those different constructions will have different dimensionality in the modular space and so what we would call the generic matter is the matter which occurs on the branch of the geometric modular space with the largest dimension and it turns out this notion of generosity matches with the simplest singularity types in f theory you really have to stand on your head to get more exotic kinds of matter in f theory you have to get very complicated singularities which which go outside some of the more simple understood classification of singularities so for example for su n the generic matter is the fundamental the two index and the symmetric and the adjoint interestingly if we take su three cross su two cross u one is based on the the algebra of the standard model the matter of the standard model is not even close to being generic in particular that generic matter doesn't have anything which is charged under all three gauge factors on the other hand if we take the standard model gauge group to be su three cross su two cross u one mod to six then the standard model matter plus a few exotics are in fact generic so for a given gauge group generic matter is in some sense typical and that you live in the largest branch of the modularized space anything else is fine tuned so i'm going to focus on generic matter is some subtleties in that story but i'm going to skip that so with that notion we can talk about sort of a universal g model in f theory which is a class of virus drops models where we do the minimal amount of tuning needed to get that gauge group g and so in general that will give us generic matter and some high dimensional class of virus drops models with the chosen g so for example the famous tape models or the morrison park model for you one are examples of universal models for a particular gauge group and with raga rum and turner we identified based on unhanging a somewhat exotic model that raga rum had written down for a u one theory with charges one through four we managed to identify the universal virus drops model giving us the standard model gauge group by which i mean three two one mod z six so we have this very complicated looking virus drops model if you've studied f theory and you're familiar with like the tape models or the morrison park model this is sort of another example of that those parameters are choices of polynomials or sections of line bundles on the base and so this is a very high dimensional parameters we've got a high dimensional number of parameters parameterizing this universal virus drops model and we can check that it has the correct number of parameters needed by anomaly cancellation so that we have all the uncharged scalars you might expect a special case of these is the so-called f 11 standard model constructions which have recently been used extensively by the pen group in their constructions okay so for these models we have three families of anomaly free generic matter there's the mssm matter and then there are two families of exotics and in a hopefully soon to appear paper with Patrick Jefferson and Andrew Turner we take this general virus drops model and analyze chiral matter in that model using this resolution invariant structure i described and we fact and we find that you can get all different all three families are possible there are no constraints on what families are allowed beyond anomaly cancellation and we can write down a closed form formula for the chiral multiplicity so for example again on p3 here's a formula for the chiral multiplicity of the 32116 matter it's given by you know here n2 and n3 encode the divisors that support the su2 and su3 factors and fives of flux so this gives us a very general construction of a huge class of standard models starting to get to the point that we can sensibly ask questions like those mike was asking in his 2003 paper how can we do statistics now really having a global picture of what's the set of models where can we tune this and what kinds of fluxes can we turn on and start to analyze those models in more detail i'll just mention that a year or two ago there was a paper by the pen group aesthetic chalerson win and with other papers with liu dn and long um looking at the so-called f11 model they had a paper where they had a quadrillion standard models where they had a quadrillion constructions that led to a standard model like thing that's a very special case of this broader class of constructions we've got here so very large class of constructions okay but as i said earlier this is great but it's tuned so if we really want to find something typical or generic we should break one of the non-higgs of geometrically non-higgsable exceptional groups so in a recent paper with uh kouhi shingyan li student at mit and another paper forthcoming shortly um we have outlined a construction where we take e seven as a rigid or geometrically non-higgsable group and break it down to the standard model so again remember we were thinking about through this kind of unifying resolution independent picture the flux as being this matrix times the flux phi and when we have non-zero theta i alphas that will break some of the carton generators breaking the gauge group down so one simple thing to do is just take an e seven and choose fluxes that break the i equals three four five and six and that gives us an s u three cross s u two furthermore generally there's a residual u one factor and you can get different numbers of u one factors depending on the fluxes but we can do this to preserve one u one factor typically if we do this we get some exotics so we can either do this and have exotics and say look we've got the standard model plus exotics or we can go one step further and say let's choose the fluxes so that the u one is essentially the u one hyper charge and then we don't have exotics on the other hand so this is actually approvably a unique up to automorphism this embedding of s u three cross s u two if we do choose a u one that corresponds to hypercharge then somewhat you know everything comes together and you get another non you get another unbroken non abelian factor so you actually have an s u five so to get the standard model without exotics it seems that we want to first do a flux breaking down to s u five and then use a more exotic mechanism called um hyper charge flux breaking which requires a slightly more complicated geometry to break the s u five down and this is actually the mechanism the breaking of s u five ends up being the mechanism that was used in this earlier work I mentioned on the tuned s u five models so the idea is we first use vertical flux which is what I've been describing it most explicitly to break e seven down to s u five and then we use what's called remainder flux breaking that is we're now using co homology in the remainder part I told you about the vertical plus remainder plus horizontal h four decomposition we're now using remainder flux which is a subtle beast it's been studied by various groups brown color initially volando probably most extensively you need to find a divisor sigma in the base which contains a curve which is non trivial in that divisor but which is homologically trivial in b and you can't do that if your base is toric but you can't do it with a toric sigma if your base is toric but you can do it on a non-toric base so you have to go to a slightly more exotic construction but you can do this and then you get a theory with only the standard model matter so just a very simple example I got only two I'm going to stop in two minutes so let me just say a couple more things one simple example is if we just look at the s u five part of the construction we take the base b to be a certain variety of p one bundle over here it's a brook surface f one and a segmented be a certain section of that we look at a particular geometry we have to solve the primitivity condition we have to solve some constraints in the caler cone which tells us that the fluxes we turn on have to be have basically opposite signs small integer fluxes and we can then compute the chiral index which is seven times one integer plus four times another integer and then we know by the tadpole condition that the number of possible directions in h two two is much bigger than chi over 24 which means that typically in any given direction the flux will either be zero or some small non-zero value so there's kind of a minimal solution to the equations that allow us to get a good construction which is that one of the one of these numbers in is one and one is minus one and when we turn those numbers on we exactly get three generations so this is just one example but the point is that the analysis of these models gives us constraints on the chiral matter multiplicities which are naturally linear combinations with small integer coefficients of integers and that can very naturally lead to three as a somewhat minimal solution. I have some details on a more exotic construction where we really use the speed rem curve I won't go through that. Let me just summarize the features of this construction. So this is the ubiquitous construction it's possible on typical bases based on these Monte Carlo's it looks like something like 20 percent of the base three folds have a rigid e seven it allows us to break e seven as a gut even though e seven doesn't have its own chiral matter we get chiral matter after the braking there's a little bit of technical details I didn't have time to go into there. We don't get chiral exotics if we go through this braking pattern with an intermediate su five a similar construction is possible for e six and it's more complicated. Okay so let me conclude and come back to the overall questions we've got this approach to understanding resolution independent intersection forms on h2 to vert which sounds pretty technical but it allows us to construct a variety of new models including these universal gsm models where we can compute the chiral matter and these models where we break e seven down to su five down to the standard model. So now we have a global picture of the f theory landscape and a number of very different ways where we can get the standard model. So I want to come back to Mike's question from the paper that I put the abstract up at the beginning. Can we now take this and formulate an intelligent and informed statistical framework for really considering in detail the relative merits of these different constructions and identify what is natural as beyond the standard model physics assuming say that there's supersymmetry we're doing everything here with a supersymmetric extension of the standard model. But this basically I feel is trying to make concrete vision of Mike's of setting up a framework where we can really think about what is typical and what's not and one thing I want to emphasize is the fractions here are enormous so even if you don't have a perfect handle on your statistics if you have two constructions and one of them can happen in 10 to the 3000 ways and one of them can happen in 4000 ways with no exponent that's an enormous ratio that is very hard without some physical principle to imagine evading the conclusion that the by far more prevalent system will dominate it's just like you know the gas in the room there is very unlikely to end up all in a corner suddenly and so we just do statistical physics based on typical notions of average things and I think we're really getting to the point where we can start to ask these questions more intelligently so with that I want to say happy birthday happy 60th birthday Mike and since I think I'm the last speaker I'd also like to thank you organizers for putting together this celebration for Mike's birthday again I'm really sorry I'm not there in person but happy birthday Mike and thanks for giving us all interesting things to think about we have any questions yeah yeah thanks Wadi that that was really excellent talk it's amazing how far you've come with this question I look forward to like you say try to come up with some interesting you know quasi real world statements from the right from this stuff just one check of this claim that this is a you know a very large fraction of what could be out there is it is it known how that original or the had erotic manifold under the time under time now manifold that type of construction can be realized enough theory good that's a good question so sorry I had a fun ringing for part of that the heterotic let me say a little bit about heterotic actually first where I can say something intelligent and then I'll just punt on the rest of it um when I say that the heterotic vacua are mapped to a small subset of the f theory vacua basically had if we take a general heterotic vacua on a collabia threefold as I've argued most collabia threefolds are elliptic so it's the only the elliptic ones where we have a clear f theory dual but because it seems that most of them are elliptic I would say that the lion's share of heterotic vacua are on elliptic collabia threefolds which can then be mapped to f theory models on basically p1 bundles over that same if we take a heterotic model with a given threefold we can we can map that to a p1 bundle base f theory model the more general and heterotic like models with like Tian Yao that you mentioned in particular things with some these are things with some h flux right and they're they're non they're non collabia but yeah that's right it's just to do a model and do a language that's right has a non-trivial vector right um that's a good question I think there's been a little bit of work on some I mean I think um Dave and collaborators have done a little bit of work matching some of these as I that was more on the non geometric uh framework I don't it's a great question and I don't really know how to fit those in in this context um I'm optimistic that there will be a way of doing that but I haven't thought heavily about that it's great question thanks let's talk about that next time we're in the same place at the same time another thing is how to fit g2 into it actually I mean there's been some work on matching g2 to f theory and actually just briefly you know finding a way of really understanding how to go back and forth between g2 constructions and f theory I think would be a huge uh progress because we don't really understand nonability in gauge groups very well in g2 and it would be great to see these same kinds of rigid gauge groups popping up I know Sakura Shevronemakey and Andy Brown have done some preliminary work in this direction and hint at seeing the same kinds of of rigid gauge groups in g2 but I think that would be that would be a an important thing to really nail down um to understand whether we're really all looking at the same thing or whether there really are differences between these different approaches yeah so I'm not in let me let me draw back a second in six dimensions I have a great level of confidence in f theory because we have a close matching between what we can get from f theory and what's allowed from low energy physics modulo some funny swamp land type models that we don't know if you can get in string theory but basically f theory can get you almost everything that you think you should be able to get in terms of n equals one 60 supergravity theories um it's not as obvious in four dimensions that it covers the space as completely but it does so well in six dimensions that I have a hard time believing that we don't we won't be able to extend or or connect the theory with these other things like g2 does that address your question there yeah thanks great yes several very many questions sure can you get something exotic like nine gauge groups like sorry what gauge groups like a nine the problem with something like that is okay so this is actually tied into some difficult questions in f theory I think e9 gauge group will not work because that would entail encoding a singularity that goes beyond the kodara table and it would be in some sense that infinite dimension in modular space now that's kind of related to a problem that we don't have a clear answer to which is that sometimes that so that would give you what's called a four six singularity which goes outside the kodara classification we could try to look at codimension two singularities that are a four six type and it seems that there are some exotic types of matter that may be realizable at these four six points that we don't really understand geometrically from the f theory point of view but in some cases there may be a heterotic dual which does make sense and this is an open problem actually um is understanding what happens at codimension two and higher four six loci you know another aspect of this is there can be super conformal field theories coupled to the gravity theory at those loci and there are a lot of open interesting questions in that regard I don't think we'll get an e9 gauge group but I think doing the some similar things that codimension two can give us exotic matter and is at the boundary of our understanding and and there's lots of I mean let me even say one more thing about that I gave you a construction for e7 which generalizes to e6 e8 is more prevalent in the landscape of these rigid gauge groups so of course what I would really like to do is break e8 there was one paper by by Wong and Tian that made some progress on this the problem is that the e8s have these only have uh extra singularities like matter at codimension two loci that are four six and we really don't understand how to treat those these are some kind of non-perturbative physics coupled to gravity in some cases they're super conformal field theories but we really don't understand them well enough to to understand what that physics looks like or whether we can sensibly build a standard model based on that so that's kind of at the frontier of the theory research right now yeah you said you had multiple questions yes yes so in the law of classification tools which you used were more or less classical algebraic geometry so classical topology so like in the invariance like cohomology large numbers intersection theory but do you think that more refined invariance of let's say four manifolds if in case of the six dimensional configurations would play a role of and may actually reduce that in some way yes and actually in particular the thing I talked about in terms of the resolution independent intersection for my middle cohomology that sounds like a classical algebraic geometric object but the point is that I'm arguing that this should be a sensible construction even for singular varieties so for a singular collabia fourfold there should be a singular version of intersection theory that gives us this and I know mathematicians have made some progress on intersection theory and hodge theory for singular manifolds but at least as far as I understand it it's not a completely understood topic and I don't know for example how to mathematically compute this intersection form on middle cohomology that we are identifying from physics as a resolution independent object directly from some kind of I don't know I think it was like a rescue McPherson intersection theory on singular varieties you know I don't know how to connect those dots and I don't know if the math I don't think the mathematicians have that more refined version of cohomology intersection theory working at a level that we could use the outside so very much I do think that the development of the mathematics particularly for singular manifolds and this is also relevant in the g2 case that you have to go to singular g2 manifolds to get non-ability engaged groups so one place where I think a lot of exciting development may be relevant in the mathematics side is this kind of more refined intersection theory and other structures for singular varieties no but that's going kind of in the opposite direction you're talking about more and more singular spaces so that you're trying to move away from geometry I'm asking what if you're I think it's smooth but like exotic smooth structures which are yeah sorry sorry I misunderstood I don't yeah I suspect that when we get to I think there've been a few papers on exotic smooth structures it may particularly be that when we go to non-super symmetric theories we have to understand those things a bit better with supersymmetry we're in kind of in this nice algebraic geometric framework where everything is analytic and we don't have to worry so much about these kinds of exotic things I think but that's a great question I wish I understood non-super symmetric theories better okay thank you so much