 All right, so welcome to this morning's session. Our first is second lecture of Schlomo, so we'll continue with his course. Yesterday, I outlined a very general program of how to engineer four-dimensional theories starting from six dimensions, and I was making certain claims about being able to understand certain properties of these four-dimensional theories in this setup. So today, and we will start adding details and trying to understand what actually we can do and what we cannot do. So I remind you that the basic thing that we want to do, we want to engineer 40 CFTs, place some CFTs in the IR by starting from a flow in six-dimensional, with six-dimensional theory, let us call it CFTUV, which we'll call T6D, and this will be some T4D. So we flow from higher-dimensional theory to lower-dimensions, and we flow by turning on a relevant deformation, which is geometric. We basically put our six-dimensional theory on some geometry, and what we also want to do is not just understand some properties of the fixed point, which might be strongly coupled. We want to find a dictionary between such compactifications starting in six dimensions, and hopefully some weakly-coupled four-dimensional CFTs which flow to the same fixed point. And the main idea that I was telling you about why this program might be useful was this factorization property, that if you start from some six-dimensional theory, T6D, and you try to understand what is the theory you get in four dimensions if you compactify on some surface C, that you can decompose into a sum of two surfaces, geometrically, like you just literally break the single surface that you have, say something like that, into two surfaces that you glue together. So this is C, this is C1 and C2, and also we can have some fluxes, and we split the fluxes such that some of the fluxes is equal to our original flux. Then the claim is that in certain situations, this, to understand this problem, we first can understand each ingredient, the compactification on each ingredient, and then combine these ingredients directly in 4D. So this was the statement. So we can understand first the first piece, then the second piece, and then combine them, and the combination is some filtering operation. So this is what we are going to do today. We are going to understand what goes into this statement and how to derive it or argue for it in more detail. So there are many ingredients to the story. The story is rather complicated and rather non-linear. So I will try to present some linear story with some perceived logic to it. But as usually is in physics, the way things work and the way things were derived, they don't follow a linear logic. So please stop me with questions if you don't understand the linear logic I'm going to try to sell to you. So there are different parts of the story. So one part of the story that I will be focusing on eventually is the four-dimensional physics. And remember that we had the phrase several questions about the four-dimensional physics, and the main question I will try to understand in these lectures is this phenomenon of emergent symmetries. The phenomenon when the symmetry in the IA is bigger than the symmetry in the UV in this type of a setup. So this will be my main focus. However, we are engineering this four-dimensional theory starting from six-dimension, so we will need to understand something about six-dimensional physics. And a very useful step on our way down from six-dimensions to four-dimensions will be actually five-dimensional physics. So in order to thoroughly understand this program, you need to understand each one of these pieces separately and together and the way they interact with each other. So again, the way I will present the things, I will give you very little details about 6D. There is very little that we need to know about 6D in order to work on this type of a program. We need to understand a little bit more about 5D, actually, and I will talk a little bit more about 5D. And then the main focus with most, let's say, technical details will be about 4D. So about 6D and 5D, I will just give you the bare essential details and the bare essentials that you absolutely need to start thinking about this program. Okay, so that will be our plan. So first I will be setting up the general understandings that you need to have in order to start with this program. So first I will talk about some generalities, some general setups, general ways to, general questions, technical questions you need to understand before understanding this physically interesting question. And then eventually, I hope, in the second lecture today and tomorrow, we will discuss in detail a very particular example of all this program. So before starting, let me also tell you what the moment you understand that you need to, or you conjecture that in order to understand certain interesting properties of these four-dimensional theories, you are interested in this type of factorization program, then you know what are the basic pieces of the program that you need to understand. If you have a very generic Riemann surface and you conjecture that anything, combatification on any Riemann surface, on this generic Riemann surface can be understood by understanding the pieces, you can decompose any Riemann surface into small pieces, which will look typically like that. You will have just a two cylinder with two boundaries which have some, has some flux. And again, we will give some details. So this is one kind of Lego piece that you need to understand in order to build arbitrary theory. Another Lego piece that you need to understand is a combatification on three punctured spheres and say with zero flux, so you can have some flux. And then from these two Lego pieces, by combining them together geometrically and in field theory, you will be able to construct any combatification that you want. So I will talk in these lectures mainly or almost only about this piece. It turns out to be much easier to understand about these pieces. We have very rudimentary understanding. We understand them in some particular cases, but we don't have a generic understanding. And if I will have time tomorrow, I will also talk a little bit about that. So this is where we are going. So setting up. So first thing you need to know is some things about the six dimensional theories. And you don't need to know much about them. Okay, what, thing number one you need to know is that these theories exist, okay? And these theories exist. There are many attempts to classify these theories. These theories are highly non-trivial. So if you try to write Lagrangian in six dimensions, you cannot write anything which is interacting. So the only things you can write are free Lagrangian. So these theories in six dimensions are intrinsically strongly coupled. So there are very sophisticated techniques to describe them in string theory, M-theory, F-theory, and so on. We will not get into this discussion. There is a very nice review that I want to mention. If you're interested in details, which was written recently by Heckman and Rudeius, which is very pedagogic and I encourage you to read it. But the only thing that we will need to know is that these theories exist. They have some supersymmetry, the minimal supersymmetry that you have in six dimensions is called the minimal supersymmetry. Super conformal algebra is called the 1,0. Algebra and it has eight real supercharges and there are many theories which you can build. Some of the theories you know, some of the theories were mentioned before, say by Leonardo. So the theory that everybody should know is the so-called 2,0 theory. Okay, this theory is a theory which lives on M5 brains. It's a very nice theory in particular in this 1,0 language. This theory, so as you know, as in the lower dimensions, if you have a more supersymmetric theory, you can think about it in terms of lower supersymmetry. So here you have something which has more supersymmetry and you can think about it in 1,0 supersymmetric language. This particular theory, the 2,0 theory, has a global symmetry, okay? So one thing that will be important for us in this program in understanding emergent symmetries in four dimensions will be to understand the symmetries in six dimensions. So one thing you know about this theory that the global symmetry of this theory is SU2, for example. There are many, so in terms of 1,0, it's not an r-symmetry. So you have an r-symmetry and you have a global symmetry which is SU2. This all of it fits into SU5 r-symmetry of the 2,0, like the r-symmetry and this global symmetry. So we will be talking always in terms of minimal supersymmetry in these lectures. Then you have many other theories that you might heard about or might not heard about. So for example, you have theories which sometimes are called minimal CFTs, which have no symmetry. So one way to describe these theories and theories, one way people study theories in six dimensions is very analogous to what people do with n equal two theories in four dimensions. So the amount of supersymmetry is the same. So what you can do in four dimensions is study Coulomb branches of theories. Something Leonardo has mentioned yesterday. And you can do the same thing in six dimensions. You can go on some branches of vacuum in six dimensions. You can give some scalars vacuum expectation values. And the analogous branches in six dimensions are called tensor branches. And on these tensor branches, some of these six dimensional theories have an effective description in terms of a gauge theory. It is an infrared free gauge theory. Like when you go on Coulomb branches in four dimensions, you have physics which is infrared free. So these type of theories, for example, correspond to tensor branch effective description based on some groups, some gauge groups, pure gauge groups with no matter. So there is no matter. So there is no symmetry, no continuous symmetry. And you can argue that, people can argue that these theories, on these gauge theories on these tensor branches, once you switch off the vacuum expectation value, they get to describe some strongly clapped couple of CFTs. So there is some list of groups that you can have. There is some finite classification of these theories. So again, only thing I want you to know is that these theories have no symmetry at all. Then there are other theories which will be interesting to us. For example, you can take these M5 brains, these six dimensional objects and put them to probe some singularities in string theory. Again, details don't matter, but the point is that you get some 6D theories which have some properties which we know. For example, you can take M5 brains on AD type of singularity, and let's take some element, some algebra in this AD classification, and then you get some six dimensional theory for each choice of G, which has some global symmetry. And typically this symmetry is some subgroup of this G times G. You have two copies of G, the global symmetry is two copies of G. Sometimes it is bigger than this, but typically it is at least that. And one very particular example that will be important for us will be the theory that one obtains when one takes one M5 brain on D4 singularity. This theory is sometimes called an E string theory. And the global symmetry that this theory has is E8. This is some theory you can get in six dimension. It has an E8 symmetry. Following this logic, you should have gotten a D4 times D4 symmetry, but the symmetry in this case is bigger. It contains D4 times D4. It's actually an E8. And this will be an example we will discuss later on in detail of this program. And finally, I want to tell you another thing about six dimensional theories. So there is a variety of theories. There is some global symmetries. There is a wide variety of symmetries. And again, these theories are strongly coupled. We know very little about them. So we know about their symmetries. We know some other things. For example, what we can know is not just what are the symmetries, but also what are the Hooft anomalies. And Hooft anomalies you can package nicely into something which is called anomaly polynomial. The fancy way of saying that you know the Hooft anomalies of these symmetries. Yes, there was a question. So you have a space which is singular. You have some singularity. You have some part of the space in which you act with some discrete group. And then it has some singularity. And on top of that singularity, you put the brain. So it will not be important for us how you engineer these theories. I will not get to it again. Really encourage you to read this beautiful review if you want the details. The only thing which is interesting for us is that these theories exist. And you can understand their symmetries and can understand their anomalies. How you do that, it's rather complicated. It's a very intricate business how you do that. All of these things are done basically now. They are working now or in the last five years or so on classifying these theories, computing these anomalies. So it's rather intricate non-linear business with a lot of different consistency checks. Definitely, I'm not an expert on this thing. And I think it will be very hard to present things coherently. And this is a very coherent review. And that's why I'm really encouraging you to read it. So these things I want you to know about six dimensions. Now another nice thing that we will discuss is 5D physics. 5D will be very important for us. And it will be important for us because we want to do this, to study this factorization. And the way the 5D physics enters the story is the following. So again, we want to understand the compactification of some 6D theory on an arbitrary Riemann surface. So one thing you can do is take this Riemann surface and represent it like that. Like a very, very, very long tube with some pieces of the surface at the ends of this tube. Now if you focus on this long cylinder on the next two different pieces of the Riemann surface, it looks as if you took your 6D theory and compactified it on a circle. So this is the sixth dimension, if you wish. This is the fifth dimension. Eventually, we're going to compactify both the sixth dimension and the fifth dimension. But we can ask ourselves, can we understand these long cylinders? We have an intermediate step where we first go to five dimensions. Now the interesting, again, fact about this thing is that when you take the six-dimensional theories that you know and you compactify them on a circle, again you can ask these questions what type of an effective theory you get in five dimensions when you go to energies far below the scale set by the radius of the circle. In some cases we know what these theories are. In generic cases for generic six-dimensional theories is actually an open problem to figure out what these five-dimensional theories are. In some cases we know. So let me list some examples. So for example, if you take 2,0 on a circle, the claim is that what you get in five dimensions is a maximally supersymmetric Young Mills theory in five dimensions. It's some gauge theory based on the same algebra that you had here. And it has a maximal supersymmetry. Now, as in six dimensions, five-dimensional gauge theories are infrared free. So they seem to be boring theories. So what happens in this particular case, the claim is that this theory, this maximally supersymmetric Young Mills theory is not UV-completed by a five-dimensional SCFT, but the only UV-completion it has is the six-dimensional 2,0 theory. So again, this comes almost trivially from what I have told you. There is a 2,0 theory in six dimensions. You deform it in some way and you get an effective theory which is this maximally supersymmetric Young Mills theory. So by definition, the six-dimensional 2,0 theory is a UV-completion of this theory. And the claim is that there is no... You cannot find, you know, a really five-dimensional theory, five-dimensional SCFT, which UV-complets this theory. So in this case, we know what happens. In some of these cases that I listed here, we also know what happens. So there is a long list of theories here. Not very long, actually. It's very rather short. So there are some gauge theories that you can write. And for some of them, we know what happens in five dimensions, for example, for SU3 and SO8. And it's actually rather non-trivial, but for others, we just don't know. So the idea is that you can list all the possible gauge theories in five dimensions. Again, that's something people do. And then try to understand, how much these five-dimensional theories to any six-dimensional starting point. And in some cases, we know what the answer is, or we conjecture some answers. In other cases, we don't even know what the answer is, and the effective theory in five dimensions might be strongly coupled. So one ingredient in order to go through with this program, with this factorization program that I'm trying to tell you about, is to know what the five-dimensional gauge theory is. So for many cases where we don't know this, what I'm going to tell you about will be in a sense useless. And why is it the case? You will see soon. So for other cases for this type of theories, for example, we also know what the five-dimensional theory is. And in particular, I want to tell you about the E string. So in the case of the E string, the conjecture is that if you just take this E string theory on a circle, you just compactify it on a circle, what you get in the five-dimensions is a five-dimensional CFT, which is called sometimes Cyberg's E8 theory. If you take the strict limit of the radius going to zero of the circle, you don't do anything else. You get some strongly coupled theory, which has an E8 symmetry. And this theory has some relevant deformation, some mass deformation, which takes it to be a gauge theory. So this, after some deformation, this theory actually becomes NSU2 gauge theory plus seven flavors, five dimensions. And this particular gauge theory is conjectured to be UV-completed in five dimensions by this strongly coupled CFT. Another important thing I want to tell you is that you have choices. You always have choices upon compactification. So here, when I took this E string theory on a circle, I did nothing. I just put the theory on a circle and went to low energy. But what you can do on a circle, since these theories have global symmetries, in particular, this E string theory has an E8 global symmetry, you can turn on holonomies in these global symmetries. So the statement is that if you take the theory on a circle and turn on the holonomy, and we will soon discuss these holonomies in a little bit more detail, such that E8 is broken to a subgroup, which is SO14 times U1. The effective theory you get in five dimensions is a very similar theory to that, but it's slightly different. It's an SU2 gauge theory plus eight flavors. And this particular theory, it is conjectured again, doesn't have a UV completion in five dimensions. This theory has only UV completion in six dimensions by the E string theory. This is a very confusing set of statements. So you need to digest them. So taking this E string theory with some holonomy, and holonomy is an integral over the gauge field around the circle, so it's a mass parameter. So with this additional deformation, you get a different theory. Now things become even more intricate. Another thing I want you to know about five dimensions is that when you make different choices of holonomies, sometimes you end up in five dimensions with different theories. So different choices of holonomies to different 5D theories. And let me give you an example of when this happens. So for example, if you take this type of theories, these M5 brains probing some singularity, they are sometimes called conformal method theories. If you hear this name. And you take this theory which corresponds, say, to one M5 brain on Dn plus three singularity. This is called sometimes the minimal D type conformal method, SCFT. And you turn on different holonomies in the global symmetry of this theory, which is actually the G6D. In this case, it's D to N plus six. Again, it contains this group, but it is a bigger group. What you get, you can get three different descriptions. So description number one that you can get is in terms of USP to N, gauge theory plus matter, plus some number of hypers. It's not important what it is. Turning on different types of holonomies, breaking the symmetry in a different way, you can get an SUN gauge theory plus matter. And doing it in yet another way, the gauge theory that you get in five dimensions, the effective theory you get in five dimensions is much more complicated. It's a quiver theory. It has the following structure. It is a sequence of SU2 gauge groups, ending with a flavor group. So I remind you the notations that Leonardo introduced yesterday. So circles stand for gauge groups, and the number is corresponds to... So when we write circle with N, we mean that we have an SUN gauge group. And when we have a square with a number inside, for example, here is a four, we mean a flavor symmetry of SU4. And the lines are bifundamentals charged under these symmetries. So when you do some other type of a holonomy, and the number of groups here is equal to N, you get this type of a theory. So all these theories look very, very different in five dimensions. They have actually different symmetries. When you study what are the symmetries of these theories, they are very different. And the reason the symmetries are different, because to arrive at them, you have done something different. You've broken the 6D symmetry in different ways. You have turned on different types of holonomies around the circle. So you get these types of theories, which sometimes are called 5D dual to each other, I don't know, 5D or 6D dual to each other. And the way to think about this duality is not in the usual way you think about the duality. What you can say about these theories is not that they flow to the same theory in the infrared. They are different. But what is true about them is that they started their life from the same theory. They are UV dual. So you start from some same as CFT in six dimensions, you do something different, and you get different descriptions. So this will be important to us. That you can, by turning on suitably holonomies, you can get, in many cases, gauge theory descriptions, and turning on different holonomies, you might end up with different gauge theories. And you can see that when n equals 1, so when the singularity is d4, and 1 and 5 brain on d4 singularity is what is called the E string, all these three descriptions become the same. So all the groups become SU2, and also the matter I did not write here becomes the same. So this is very, very fast. I just dropped on you some facts, and I will not talk about them anymore. This is all you need to know about these very generic features of six-dimensional and five-dimensional physics. And now we will discuss a little bit more details, and I will give you a little bit more explanations of what I'm going to say. This is an extra question, so the question is... Repeat, please. If somebody gives us these theories, how can we know that they come from six-dimensional, from the same 6D theory, and the answer is that it is a very, very complicated thing to do. Again, this is a topic of research which is happening now. In certain cases there are some string dualities that you can employ. You can do some string dualities, go to five-dimensions in these different ways, and understand that you get one of these theories or get the other theory. If you ignore string theory completely, ignore this higher-dimensional physics completely, it's very, very hard to understand that they come from the same theory. And some of these statements are conjectural. There is very little evidence for them. In a sense, what I'm going to tell you about four-dimensional physics is going to be a check of these statements. Again, as I said, the logic here, I present it in a linear way, from six-dimensions to five-to-forty, but actually it's a big set of consistency checks. So there are a lot of claims that you make, and then if all these claims are correct, they should be consistent. If one of these claims, one of the checks that you perform fails, something is wrong, and you don't know always what is wrong. Maybe your starting point is even wrong. Like some of the things you said in the beginning is wrong. So this is an ongoing research. It's not a closed chapter in physics, of which you become this one. So UV completion of these theories of all of them is the same 6-D SCFT. The only UV completion that they have is in 6-D. You should think of them as 6-D theories on a circle with certain deformations. Okay? Yes. So the logic is the other way around. You start from 6-D. So this is the, you know, papers are being written right now about exactly the question you're asking. So say I give you a 5-D theory. What you can say about it? There are three options. Option number one, there is some CFT in 5-D which UV completes this theory. Okay? So there is some CFT that if you deform it, you get some gauge theory description. This is option number one. Option number two, that there is some 6-D, higher dimensional theory which UV completes it. And option number three, that maybe this theory is inconsistent. And what are exactly the criteria, you know, given a theory where does it fall, it's ongoing research. I'm not qualified to say much more than I'm saying now. But there are certain techniques. Again, some of them are just brain constructions. You just have some brain construction which engineers a theory in 6-D, then you compactify it on a circle, use some string dualities, you get some 5-dimensional theory which is described by a gauge theory. This is the simplest thing you can do. Other things is literally there is a finite, not finite, but simple list of gauge theories you can write in 5-dimensions. There is some list of theories you can write in 6-dimensions and you can basically try to match them using some simple parameters like matching the modular spaces and so on. So the evidence is very... There is not... there are not many complicated computations that are done at the moment. So we will be in a sense users of these... of these results we will be making quality assurance that these claims are correct. But by the logic we will be performing. Okay. So with these general statements about 6-D and 5-D theories let me tell you why 5-D theories are actually useful. So till now I didn't tell you why they are useful. I just told you that you can think about these 3-man surfaces in a certain way and then this 5-D 5-D description might be relevant. So why is it relevant? There are two ways in which 5-D description is relevant. So one is to understand what is the flux. What the flux is. What does the flux do? Okay. So again I mean flux what we are doing is taking this 3-man surface we have some global symmetry say this global symmetry is A1 and we just turn on some gauge field configurations such that if we integrate the field strength over the Riemann surface that we are discussing we get some non-zero answer. So this is the flux supported on this Riemann surface. So let us try to understand what this flux means if we take this Riemann surface to be a very very long tube and we will cut it at the end soon. So think of it as very very very long finite but finite tube on which we turn on some flux. So what is this flux in 5 dimensions? So let us engineer this flux explicitly in a toy example. So this is the 6th dimension this is the 5th dimension so this is X6 and this is X5. So let us turn on the gauge field for this U1 global symmetry which has the following profile it's just some parameter M times a hyperbolic tangent of some other parameter lambda times X5. So this is some gauge field compute the field strength F5 6 which is just delta 5 of this A6 and you get that this is equal to M times lambda over cos2 lambda X5. It's a very simple thing so what you can see immediately that if you integrate these field strengths over this Riemann surface you get some non-zero answer. So this type of gauge field configuration produces flux supported on this Riemann surface. Now look what is the profile of this of these quantities the A6 and F. So A6 is hyperbolic tangent so it just runs something like that from minus infinity to plus infinity and interpolates between minus M to plus M. Okay? The flux is localized around zero. So this is the flux and this is this gauge field A. Okay? How should we think about this gauge field A? So what is this gauge field A? You can compute the integral of this gauge field A6 around the circle around the six-dimensional circle. So you just compute the integral of this A6 around this circle. This is what is how the holonomy is defined and this is trivially computed from this expression. It's just M times the radius of the radius of this circle times hyperbolic tangent of lambda times X5. So what you get here is that you have a holonomy okay? You have some holonomy, value of which depends on your position along the fifth direction. Okay? So a holonomy which changes in space can engineer for you flux supported on this Riemann surface. And you can engineer the parameters that take limits on this lambda that really like most of this cylinder has holonomy minus M. On this side you have holonomy M and there is some, you know, sharp transition somewhere on the Riemann surface. Okay? So flux in five dimensions flux, the six-dimensional flux on this surface is equal to domain wall in five dimensions. Domain wall interpolating different values of these holonomies. But what did we say about different holonomies? When we compactify theories, six-dimensional theories on a circle with some holonomies sometimes we get a five-dimensional gauge theory description, right? And here the holonomies are different. So here they don't look very different, it's just M and minus M, but more generically we can engineer which are much more different. And the important thing is that when you look far away from where this gauge field is changing this looks like 5D theory 5D, T5D, A some 5D theory which corresponds to holonomy minus M and here it looks like T5D B, something which corresponds to compactification with holonomy M. And sometimes these domain walls are called duality domain walls in 5D. They interpolate between two different 5D theories which you'll be completed by the same six-dimensional theory. So one thing we learn immediately that there is a very, very nice way to understand flux in five dimensions, these are just these domain walls. So if you want to understand compactifications from six-dimensional theories which are very complicated we don't know much about them. You can first go to five dimensions and at least the flux you can understand in these five-dimensional gauge theory descriptions that you might have in terms of these domain walls. No, it is. I just, you know it will be the minus of that. Okay. Any questions about this? Yes. So at the moment I ignore boundaries. Boundaries are... So the question was what happens with the boundaries? So this is a toy exam. So the question is what how should I think about the boundaries? So you should, the only thing I will say you will properly define it. Whatever you need to define you will do. Okay. I'm just telling you some, you know a toy understanding of the story. Okay. So the details are usually rather complicated. You know, I didn't tell you what the 6D theory is. I just told you some symmetry and so on. And as you will see in the particular example of the E string that I will discuss I will not actually derive. I will tell you what the domain wall is. It will be some rather, there will be some rather simple description of the domain wall. But the checks that one can perform about is there is no derivation of it. There is no simple derivation of these domain walls. It's possible checks that you can perform but we will get there. So at the moment, actually it's a very tricky and important thing what is happening at the boundary. I will completely ignore it. It's just too tricky. For example, let me say one thing you know, the types of flux you can turn on will depend on what types of boundary conditions you put in at the boundaries. So things are correlated and really tricky. Okay, so this is one thing why one reason why five dimension is useful. Another reason why five dimension is useful is exactly that. We can try and understand what the punctures are. So for this factorization program we break our theories into pieces which have punctures and then we geometrically glue the pieces along the punctures. So what are these punctures? Okay. So we can learn a little bit about this again by studying five dimensions. So again we think about the puncture. We say we have some very general ribbon surface, something very generic and then it has a puncture. It has a hole in it and then we kind of zoom on the region of the puncture. And again we just think of it as a very, very, very long cylinder and on this cylinder we can understand the puncture in terms of five-dimensional effective theory. And in cases where the five-dimensional effective theory has a gauge theory description, again it doesn't happen always, so we don't always know what is this description. In those cases, since these are just fields which live in this five-dimensional space we can understand what type of boundary conditions we can put. We just need to specify boundary conditions for 5D fields. And since we will want to preserve some amount of supersymmetry in all these business, we want to understand what are the supersymmetric boundary conditions that you can put at these boundaries. And there are a variety of choices. There are a huge variety of choices. In some cases for some of these compactifications they were classified and understood, but generically again, it's an open problem classifying all the possible boundary conditions. However, there is one type of boundary condition which is kind of universal. It exists for all theories which have this gauge theory description and it's sometimes called the maximal boundary condition. It's very simple, so you have gauge theory description in these five-dimensions and what you do, you just take Dirichlet boundary conditions for the gauge field. And you have other matter fields and you need to specify boundary conditions for those matter fields. What are those matter fields? Depends on specific theory you are discussing, but basically what you do, you put Dirichlet boundary conditions for gauge fields and then whatever supersymmetry tells you, you need to do for other fields, you do for other fields. And what happens when you put this boundary condition, the Dirichlet boundary condition for the gauge field you freeze it at the boundary, so what you acquire is that you have this large gauge transformations which become global symmetry. So your theory, because you have this puncture for each such puncture on your human surface, you acquire a global symmetry. So you had here a five-dimensional gauge symmetry, you have in the bulk of this five-dimensional theory and each boundary with such boundary conditions gives you a flavor symmetry which is equal to 5D gauge symmetry. This is very important. So punctures come with symmetry. If you don't have to put these types of boundary conditions, you can put Dirichlet boundary conditions, more sophisticated boundary conditions and then you can make smaller part of the gauge symmetry to be your flavor symmetry. So this is the maximal symmetry that you can put, that's why it is called the maximal boundary condition. But it is very important that you get this large gauge transformations as your flavor symmetry. And finally what we want to do is when we break these surfaces into pieces what we want to do is to glue them together geometrically. And what it corresponds in this picture, we just take two copies of this picture and we remove the boundary condition. We make the gauge fields dynamical again on this boundary. We just remove this freezing of the gauge fields at the boundary. So what this corresponds to is simply gauging this flavor symmetry, this GF that I call there. So that's how gluing geometrical gluing of surfaces becomes a gauging of some symmetry. And the symmetry we are gauging eventually we will be doing this procedure in four dimensions but the symmetry which corresponds to these punctures is the five dimensional gauge symmetry. Again it's a little bit confusing but needs to be understood. So you start from 5D gauge theory you freeze this symmetry at the boundaries you get some flavor symmetry and then if you want to get rid of the punctures you make this field dynamical again. So you make AMU, you remove Dirichlet for AMU and also you should remove Dirichlet for all, if you have other fields which have Dirichlet boundary conditions you need to remove Dirichlet boundary conditions for them, you need to introduce some dynamical fields which you have frozen at the boundaries and we will see it in the example of Easterine how it exactly happens. Questions about this? That's right, you just, yes you remove these boundary conditions you make this the things that you have frozen you now integrate all. And again since we have this full 6D in 6D story is complicated to understand because we don't know what the 6D theory is in terms of fields, it's some strongly coupled theory, but this is a theory of fields so when we will further reduce this to 4 dimensions this just will be basically a direct dimensional reduction some kind of a dimensional reduction to 4 dimensions and then again since you gauge this symmetry in 5D you need to gauge it in 4D this is kind of an understanding that you should have yes because we have here a gauge so the question is that that's always this undoing of Dirichlet boundary condition correspond to gauging some symmetries if you have frozen your boundary condition have frozen some gauge fields you need to make them dynamical again so you are gauging them if you know you have some theory without gauge fields and you put some Dirichlet boundary condition you don't need to gauge any symmetric so the question is we have two different theories here there is a 5D theory living here a 5D theory living here there are different 5D theories so what we are doing now when we are gauging these theories we are identifying the gauge field the gauge field at the boundary here the 5D gauge field at the boundary here and another 5D gauge field which lives here we identify them at the boundary this is the continuity you are talking about and we make it dynamical so we are done with these general statements so let me summarize what you need to understand in order to start really understanding compactifications so there are from this discussion there are very little things that you need to know there is there are not many things you need to know of course if you know more things you will be able to say more things in four dimensions but to derive some non-trivial results in four dimensions you basically need to answer several questions you need to take some six dimensional theory anything you want and you need to understand what is the symmetry and what are the anomalies we will soon discuss what anomalies are useful it will be very useful when you go to five dimensions this 6D theory in five dimensions is described by gauge theory so this you need to find so you need to find a 6D theory that upon compactification on a circle is described by fields in terms of Lagrangian by gauge theory with some gauge group G5D gauge and some metal okay and then in five dimensions the moment you understood what the 5D theory is you need to do two things first you need to study domain walls these duality domain walls and second thing you need to study is the boundary conditions so the claim is the moment again you found a 6D theory which has a 5D gauge theory description and you understand about this 5D description these two things you can just do a very simple thing you can take your 6D theory put it on a cylinder turn on some flux okay and then this flux in five dimensions will be engineered let me draw it here by some domain wall so the theory you will get in five dimensions a five dimension theory with a domain wall domain wall corresponds to the flux and then you just reduce it reduce to 4D that's the algorithm how you understand compactifications on a cylinder with flux as simple as that okay and now what we will do we will understand this program in a very particular example and see what are the consequences any questions? how much time do I have? oh so the question was to give more details about reducing this the 5D theory to 4D we are already in 5D so basically you should think of this this is the 5D direction this is the circle is the 6D direction so we are already in 5D so this is the picture we have the boundaries are four dimensional so we are compactifying this direction we are basically getting rid of this size okay and then we are only left with a four dimensional okay so in the five minutes that I have let me start give you basic facts about the Eastring theory and we will continue in one hour with more details so this was very very generic discussion so now I need to show you that it actually works in some examples so the example I have chosen because it's rather beautiful is this example of the Eastring so it's some six dimensional theory and again the only thing you need to know is that about this theory is that you understand we understand now all these pieces here and I will describe them to you now so first of all we know that this theory exists in six dimension and as already mentioned this theory has a very beautiful symmetry which is E8, a huge symmetry and it is E8 another thing you understand is that anomaly polynomial is known took a lot of years to derive this anomaly polynomial in rather sophisticated techniques and it was done very recently by Omori, Shimidzu Tachikawa and Kura so there was a lot of hard work which went into this computation but the bottom line is that we know not just what the symmetry is of this theory but also what are the different Hooft anomalies and we can package this information about Hooft anomalies in what is called anomaly polynomial so we are in six dimensions so anomaly polynomial is some eight form and you need to write the most general eight form that you can think of using all the characteristic classes that you have available for this particular theory and fix the coefficient so I will not try to the anomaly polynomial of this theory it will take the whole blackboard but just to give you a feel of it so there are some numbers that you need to compute and these guys computed them and then there are some characteristic classes that you can have so it looks something like that and plus many, many more terms so these are some characteristic classes that what you need to do to compute the anomalies is to fix these numbers so this task was done we will not discuss it we will just assume it is correct there is always a chance that somebody made a mistake but then you need to find the mistake so these are the two basic things that we know about 6D so this is the 6D part of the story and 5D part of the story is the thing I already told you is that if you take this particular theory this 6D theory and put it on a circle and turn on the polynomial which takes E8 to SO14 times U1 the theory you get in 5D is SU2 theory plus 8 hypers and all of what I said is rather complicated for example you see we have broken here E8 to SO14 times U1 what is the symmetry of the theory we get in 5D you have 8 hypers of SU2 so the symmetry that you get here is SO16 and there is another symmetry which comes from instanton so there is an SO16 times U1 symmetry that you see in 5D so the precise statement is that the SO16 that you see here is kind of accidental it's important that it has this SO14 times U1 group so the SO14 inside SO16 some SO14 inside SO16 is the SO14 that you got from U1 and this U1 which completes the SO14 to SO16 and one of these combinations of the two combinations is related to the KK symmetry to the Kalutza Klein symmetry that you have on a circle and another is this U1 that you want to find so this story this statement by itself is rather non-trivial but from here we can immediately answer one thing that we will have if we will start from 6-dimensional E-string theory and compactify it on a Riemann surface with punctures there are certain punctures which have SU2 flavor symmetry with proper boundary conditions there will be punctures that we will have SU2 flavor symmetry and starting from here we will continue