 sending six-dimensional compactifications of down to four-dimensional theories and some symmetry enhancements. So an important part of this program is to understand some building blocks. And we have obtained several building, couple of building blocks last time and rules how to glue them in field theory. So let me remind you that we are studying these compactifications of an instinct theory on a torus with flux. And we claimed that if we compactify on a cylinder with flux which preserves symmetry seven times u1, and these fluxes of the following form, one, two. Then the field theory one obtains in four dimensions is a very silly Wezzumino model of the following type. So it's just a theory of free fields with some super potential. And the symmetry that you see explicitly in this Lagrangian, in the Lagrangian which describes this quiver diagram is SU8 times u1, and the SU8 is some subgroup of u7. So this is any seven flux that we consider. And the fact that we don't see seven is because we have punctures. And the punctures break farther the symmetry from u7 down to SU8. Then we also mentioned towards the end of last talk that if you just take the same flux, same as seven flux, but flip all the signs, the theory should be basically the same. And what you need to do is just reverse all the charges. So in particular, we are reversing all the arrows in this quiver diagram. So again, as we mentioned last time, there is no physical difference between these theories if you consider them independently. But the difference will come when we will try to glue them together. When we will consider them as parts of the same theory, then these arrows will matter. And the way they matter is because we have slightly different gluing rules when we combine two theories of this type or say two theories of this type or such a theory and theory of this kind. So again, in order to define this factorization program, we don't just need to define what are the building blocks, but also how we glue them together. So geometrically, we can have two pieces corresponding to two different cylinders. Yes. What do you mean you flipped it? Yes. That's right. But again, the point will be how you glue. Do you glue with arrows pointing the same way or the opposite way? You are right that this picture is the same picture, but flipped. So again, these theories are physically equivalent. The only question is what you glue to what. And that's how we identify that the flux of these two theories is opposite or is the same. OK, so the gluing rules schematically, and we will see several examples today, are the following. So when we glue two cylinders, we gauge an SU2 symmetry corresponding to the punctures. So one of these SU2s. And we also add something. So in case when we glue two such theories together, in this case, two theories of the same flux, of the same sign of the flux, we need to add a certain field which we called phi, which is just a bifundamental of SU2 and SU8. And we turn on certain superpotential. And in case we glue such and such theory together, we don't introduce any field phi, but we turn on certain superpotential. And again, we will see many examples today. And finally, we make the use of these rules to derive theories corresponding to combatification on a torus with flux, which is just F, some number times this vector, which is an E7 flux. And the theories you obtain following these rules are just very simple quivers with this circular structure. So you have a lot of SU2s on a circle. You have an SU8 in the middle, flavor symmetry group. And then you have arrows like that. And the number of gauge groups is twice the value of flux. So twice the value of flux of E7 is the number of gauge groups. And what we claim is that if you compute the anomalies of this theory, then they exactly match what you would expect from six dimensions. And also, we saw that the structure of representation theory of the group E7 appears. In particular, all the protected operators of this theory form representations of E7. And we claim that somewhere on the conformal manifold of this theory, there might be a point, maybe a strong coupling, where the symmetry group enhances from SU8 times U1 to E7 times U1. OK, so what we will do today, we will just play very, very fun games. So the only thing we will need is just these building blocks, these LEGO pieces, and the gluing rules that we discussed. And we will construct many different theories corresponding to torus compactifications, which have various properties and various symmetries. And we will see why this picture is consistent, how it is related to dualities, and how it is related to symmetry enhancements. Yes? Oh, OK, so it's a very good question. So the question is, I just said that when we glue things together, then I decide what to put here is what, depending on what the flux of the theory is. That is true for these simple theories. In general, this is not how one should phrase things, because flux is something which corresponds to the torus, if you wish, sorry, to the surface itself. It's not something which doesn't correspond to the puncture. What's really the precise way to phrase it is that you need to understand what type of puncture you have. So I didn't want to go into it, because there are a lot of different subtleties in defining exactly what the puncture is. But since you asked, in this case, it's when you just complex conjugate, it's not just that you flip the value of the flux. You do something else. And for example, you can see that the punctures are not exactly the same of this tube. They are slightly different. For example, you see that there is an operator which is charged under the symmetry of this puncture, and it is in a fundamental representation of SUA. The operator which is charged under this puncture is in the anti-fundamental representation of SUA. So the punctures are not exactly the same. There are slight differences. So what you really need to do is understand what type of puncture you glue to what type of puncture. So there are many choices. You need to classify them. And in this case, it's actually not too complicated. And then the only thing you need to know, which puncture you glue to which puncture. The flux is not important. In this particular example, you can kind of trade defining what is the type of the puncture to saying which theory you glue to which theory. It's an excellent point. OK, so next thing I will do. So till now, we just combined theories of one type. We combine theories of this type to themselves to form theory which has an E7. But now let us discuss what happens if we combine some theories of this type and some theories of this type. So in particular, we will understand how the equation minus a half equals to 0 is consistent with what we just said. So the basic point is say we take many theories of this type. So we say we take, I don't know, F theories of this type. Then the flux which corresponds to this theory is F over 2. Then if we glue theory of this type to what we just did, then the flux which we should get is F over 2 minus a half. But we know what we can build the theory independently just by gluing F minus 1 objects of this type. We don't need to glue this object of this type. So what should happen is that one such triangle will annihilate such a triangle. So let us see how this happens. So this is a big picture. So I hope I will draw it nicely, which I doubt. So we have, say, such a theory. So this is a theory which we obtain by just combining several such pieces. And we can have many of such pieces. Then what I want to do, I want to take one such piece and glue it to this theory. So we've taken this tube theory. But now since it has the opposite flux, this arrow and this arrow are oppositely oriented. And then we have, here we have a flavor symmetry. All of these are SU2s. So this is a 2. And we have a 2 here. And then we have something like that. And then let us glue it to another block of theories, which, again, have the same sign of flux as this one. So say this has flux plus F prime. This has flux minus 1 half. And then we glue this thing to another piece which has flux F prime prime. But what is important for us, it is that it is a positive value of flux. So there is something like that, dot, dot, dot. So we glue two such things together. So this has flux minus 1 half. And this has flux plus F prime prime. So what are the rules of the game? How we glue these things together? So there are two things we need to do, as we mentioned. We need to gauge the SU2 symmetries. And in this case, since the lines are oppositely oriented, these are different types of punctures. Then we just turn on a superpotential, which is of the following type, as we discussed. Let's call it this F-right 1, this field F-right 1. And this FL2, this F-right 2, and this one FL3. So the superpotential, which we need to turn on in this case, is very simple. It's just phi R1 times phi L2 plus phi R2 phi L3. And then that's everything we do. These superpotentials identify the different SU8 symmetries of the different blocks. We gauge the symmetries, and we obtain some theory, which we'll draw in a moment. But what are these terms? Do I identify what, how do you call such terms? They have a simple name. These are exactly, these are mass terms. So when you add such terms, you basically give masses to these fields. You just give masses to these fields. So in the IR, what will happen is that we just remove all these arrows. So what happens is that in the IR, the theory we'll obtain is of the following type. So we have this thing, this thing, and then we trade this symmetry with gauge symmetries. We gauge these symmetries. And I missed one group. No, that's fine. But now what we can study is the dynamics of this theory, of this gauge theory. What are these gauge nodes? These gauge nodes are very simple. These are SU2 gauge theorists, SQCDs, with NF equals 2. We have four fundamental chiral fields, that these SU2 fields, and also these SU2 fields, four fundamental fields. Now the dynamics of this SU2, SQCD, with NF equals 2 was understood many, many years ago. This is a little bit funny theory. It has what is called quantum deformed modulized space, in particular, if you want to have a supersymmetric vacuum for this theory, you need to turn on a VEV, vacuum expectation value for some gauge invariant operator. And the superpotentials that we have in this theory, they basically force you to turn on a VEV. For example, if you first try to study this gauge group, then you turn on a VEV for a mesonic operator, which is this field times this field. This is some standard analysis of these types of QCD done by Seiberg many, many years ago. But if you turn on a VEV to this composite operator, which is built from gauge variant fields which are charged under this SU2, what happens is that this SU2 is higged. It disappears. You give a VEV, this SU2 is higged. And when you analyze the dynamics of this sector, what happens is actually that you can remove all of these with a single line. What you get in the IR is just a theory without these two gauge groups. This is standard dynamic. So SU2 is QCD with an f equals 2. But now this theory is exactly what we expect. You see, we had two triangles which basically annihilated each other. So we get exactly a theory which you just build from such pieces which corresponds to flux which is equal to sum of the fluxes minus 1 half. Some of the fluxes of the original pieces that we had minus 1 half. So this had to be true if our picture is consistent. If what we were saying is inconsistent, then gluing such a piece, building the same theory only from such a pieces or from such and such pieces would give a different answer. But we want a consistent story. And so it doesn't matter in which way we'll obtain a theory corresponding to the same surface with the same value of flux. And that is how it works here. So what guarantees that we have this consistent story is the dynamics, this simple dynamics of SU2 theory with two flavors. We did not put in this dynamics. But if we wouldn't have known about it, it should have been true. And in this particular case, this dynamics was known long before. But one can think of other cases in other compactifications where such properties of simple field theories appear which were not known prior to these types of analysis. So this is a very, very simple property which we can phrase as just half minus half equals 0. It is standard cyber dynamics. Questions. So another thing that we can do is now try to study theories which have different values of flux, not E7 type of flux, but flux which correspond to some other group. So we have many choices. And let me remind you what I already told you last time. So the flux for E7 is proportional to such vectors, vectors which have all entries which are equal to 1. Or by using the while group of E8, we can build other types of vectors where we have only two non-zero entries. And the flux is proportional to this. Then for SO 14, we have mentioned that we can do several things. For example, fluxes which have the form 4, 0, and all the rest zeros are all four 2s and four 0s. And for E6 times SU2, the fluxes in this basis have the form of 6 2s and 2 0s. So how would we build such fluxes using the building block that we have? And it's very simple. So let us start from the theory which corresponds to SO 14. Yes, to SO 14. And let us try to engineer flux of this form. And in particular, let us just divide everything by 2 and try to engineer something which has half of the flux which is written here. How would you do so with the building blocks that we have? And that's very simple. You just take one block with all halves. I hope I can count to 8. I cannot. And now we add to it a similar piece, but we flip signs of some entries. So we add to it something which has half in the first entries and has minus a half in the rest. So this is exactly a theory of the kind we have considered. And this looks something new. But actually, we can very quickly understand what this theory is. Remember that to understand the theory with all minuses, we flipped charges of all the fields that we had in this picture. So to understand this theory, we will just flip charges of part of the fields. So let me show how it goes. So this theory is that triangle. And let me write it in a slightly different way. How did I draw the arrows like that? Let me do it like this. So this theory I can write in the following manner. I didn't do anything. The picture there has an 8 and 2 SU2s. Here I just split the SU8, the 8 into 2 fourths. So you can think of that picture as just folding the upper side down. It's just another way to write the same quiver, just by splitting 8 to 4 and 4. Here I will need to do that, to split, to treat differently, two quartets of fields. And then what this guy will correspond to, let me write it in a very similar way. So let us think of this for four first entries here. So this will be exactly the same. So this will be piece like that. Down here, I want to flip the signs. I want to flip the charges. So I flip the directions of the arrows. So what we will get here is the following thing. The arrow here goes down. It will go up here. We will have a 4 and an arrow like that. And now we will want to combine these two theories according to our rules. So what we need to do is to gauge this SU2. So we gauge a diagonal combination of these two SU2s. And we turn on some superpotential. Up here, the arrows are oriented in the same way. So we need to add this field phi that I mentioned with some superpotential. So we need to add another field phi, which will be by fundamental of this SU4 and SU2. But here, the arrows are oppositely oriented. So we don't need to add any field, any extra field. What we need to do, just couple them together, like we did in that example. But again, since the superpotential we turn on is just this field, time this field, this will be a master. And what will happen is that these two fields will just must up and disappear in the IR. So the theory we get in the IR is very simple. It's given by the following quiver. So we have this gauge SU2, which we have in the middle. And then we have arrows going to SU4 flavor symmetry like that. But downstairs, the theory looks a little bit different because these two fields disappeared. So it looks like that. We have a four. And the arrows go like that. So what the claim is that this is a flux. This is an SO14 type of flux. And in SO14 language, the value of this flux is a half. This is let's call it FSO14. And the theory which corresponds to a compactification of tubes with that value of flux should be this theory. So now here you see there are what are the symmetry groups. So there is an SU4 symmetry and another SU4 symmetry. And if you analyze other symmetries that this quiver has, you find that there are two more U1s. And the way this goes is that the way that you see the SO14 representation is that when you, for example, will analyze the tori quivers that you will build from that. So this is a tube. You can now concatenate them together to form theories which correspond to tori. And then what you will see like in the previous case of E7 that all the protected states of this theory will form representations of SO14. And the embedding of the groups that you actually see in the quiver inside SO14 is as follows. So the 14 dimensional representation of SO14, which is the vector representation, is built from six one plus one six plus two singlets where these singlets are charged. They are singlets of the SU4 groups. So this is in the six dimensional representation of one SU4 and a singlet of the other SU4 and this is other way around. And then there are two singlets of SU4s but they are charged in some way under the ones. So that's how the group SO14 will appear. So everything you will see, every protected operator that you will see in appearing in this quiver will be either in the 14th representation, 14 dimensional representation or representation that you can get from it. Moreover, if you will compute conformal anomalies of the theories that you can get this way, the numbers which you will obtain are just A equals to twice square root times the value of flux and C is equal to five half times square root of F. So this theory, unlike the one here which was free, which was a conformal theory, that theory is not conformal and the R charges that you will obtain by performing A maximization that I mentioned last time will not be the free R charges. It will flow to some as CFD and once you analyze the same maximization you will get these anomalies. So again, you get a consistent picture. This is rather non-trivial that the anomalies which follow from this theory agree with a six dimensional picture. Questions? So it's rather miraculous that this group theory which follows from six dimensions, it's a very simple fact that you need, you can write this vector as a sum of such vectors. This very, very, very simple fact gives you produces, allows you to produce a theory which has very interesting symmetry enhancement from SU4 times SU4 and a bunch of U1s to SU14 times U1. Let me give yet another example. Keep this one. So we can, as we discussed, we can obtain a vector of fluxes which gives any seven symmetry in different ways. So this is one choice and this choice corresponds to the theory which we already built. But we can obtain a different vector of fluxes which looks like that and this vector of fluxes is related to this one by the while group of V8. So we can ask now, can we build something which has these fluxes? And if we can build something which has these fluxes it better be the same theory as the theory we obtained before. At least it should be an equivalent theory. So we play exactly the same game as we played there. Just we now will demand that two first entries in the vector will be the same and the rest will be zero. So we will have exactly the same equation with all the halves here but here we will have a half and a half and then six minus halves. So exactly the same game. So let me draw what you obtain from this equation. Let me write it down maybe. So a question which is one and one and then everything else is zeros equal to all halves plus two halves and then many minus halves, six minus halves. So again, exactly the same picture but now instead of splitting to four and four we need to split to two and six. So this is the picture we can draw. So we have a two and a row like that. Okay, this is again completely equivalent to this picture we just need to flip it down. It's just a different way of writing it. And then when we add the quiver which corresponds to this piece which is given by this drawing so it will be something which is oriented in the same way. Let me draw it like that. A two, two groups SU2 and then here it's oriented the opposite way. If we add such a piece exactly following the rules of the game that we have there the quiver we will obtain is of the following form. It will be SU2 and then we have an SU2 gauge group an SU2 flavor group. Then we have a two here a two here and something like that to have the arrows right it will have the following form. So it doesn't look the same as what we had before. So this theory should have the same we want to correspond it associated to the same type of flux to any seven type of flux as this theory. It's just different representation of the flux. So here it is represented by this factor. Here it is represented by this factor. They should be equivalent. Group theory tells us that they should be equivalent. They are just related by some wild transformation of E8. They look different but they should be the same. So why this type of theory is of the same type as this one? So if you look on the gauge theory which appears here on SU2 which appears here in the middle it's a very interesting theory. So what appears here is an SU2 gauge theory with an f equals three, okay? With three flavors, okay? And the dynamics of this theory is very simple. An f equals three means that we have six fundamental chiral fields. So I runs from one to six, okay? And again, more than 20 already, 25 years ago the dynamics of this theory was analyzed and it is claimed that in the IR this theory is effectively described by a bunch of free fields, okay? Which a map, which are basically the composite operators in this UV theory, okay? If you take QI, QJ, built from the composite operators, gauge invariant operators, then this corresponds, because you have an SU2 contraction you need to anti-symmetrize the indices. So what you get is 15 mesonic fields, okay? So the dynamics of this theory is very, very simple. And if you now put in this dynamics into this quiver what you will get is that in quiver notations these mesons have the following form. So you have mesons, the QIs that I wrote here are these QIs, these QIs and these QIs. So some of the mesons will look like that. It will be just aligned from here to here. Some of them will be aligned from here to here. Some of them will be aligned from here to here. And then there will be some singlets associated to the nodes. Let me not write them. So what you can do is trade all this piece, this gauge theory with some free fields, okay? And the moment you do this analysis carefully and you close the theory to the torus you obtain exactly the same time types of quivers you can build from that, okay? So this dynamics SU2 with an F equals three is responsible for this very simple fact that the vector 1.1 and a lot of zeros under while group of E8 is equivalent to this vector. So again, if we wouldn't have not known that this duality is correct we would have been discovering it. If our picture is consistent, this property should be true. Theories that you build from such blocks at least in the infrared should be equivalent to theories which you build from such blocks. So there has to be some dynamics which is responsible for this. What is the microscopics of these dynamics? How things actually happen? We don't know, we cannot derive from such picture. But again, this has to be true because of the consistency of the picture, okay? And finally, let me do another example. So what you can get from here and maybe I'm doing it a little bit intentionally it looks that what you obtain from this type of logic is something which is looks rather complicated, okay? You have big quivers, you have some funny properties of big quivers, why should we care, okay? So what we will do next, we will derive from these complicated quivers something very surprising about very, very simple theory. So a consistency of all these construction will lead us to say something about really one of the simplest theories that you can imagine. And to do so, we will first understand again following the same types of rules. What are the theories which correspond to this compactification? To compactification which should give us the symmetry which is E6 times SU2 times U1 that I never write. So again, it's exactly the same game that we played here. So now we will add two ones here, okay? And we will be interested in E6 times SU2 compactification. So what we need to do is add such a vector to a vector which has two more halves and then minus halves, okay? So the way we build such a quiver is as follows. So instead of four here, we will have a six and instead of four here, we will have a two. And the same way here. And what we will obtain is a quiver theory of this kind. It's very similar to the quiver that we had here. Instead, we switch between the six and the two, okay? And this gives us completely different theory because the number of flavors that the central, that this gauge node here feels is different than the number of flavors that the gauge theory here feels. Okay? And again, statement number one. If all of this is consistent, then the theories you build from such blocks, for example, when you form a torus, SU2 should have any six symmetry. And the anomalies should match what we predicted from six dimensions. And if you remember, the anomalies were predicted from six dimensions are of this type. A is equal twice square root of three times the value of flux. And C is five halves times square root of three times F. And again, I encourage you to do this anomaly computation and this is exactly the anomalies you will find, okay? Again, this theory is not conformal. It flows to some SCFT in the IR. And these are the conformal anomalies this theory will have, okay? And finally, one should understand whether one can see E6 representations for this simple model. And what happens is that E6 is built from SU6 times SU2. SU6 times SU2 is a subgroup of E6. And the way it goes is that, say 27 bar representation of E6, that composes as two six plus one 15 bar. Okay, again, this is an SU2. The first entry here is an SU2. The second is an SU6. Okay, this is how group theory tells us that representation should decompose. And if you analyze the supersymmetric spectrum of models you built from this, you all, what you will see is only representations of E6, simplest one of which is 27, which is the conjugate of this one, or 27 bar and the way you will see it that always whenever you see two six, it will be accompanied by one 15 with exactly same charges, same R charges, same charges under the other one global symmetries that this theory might have. However, so this seems exactly as complicated as other examples that we have discussed. But now we can do something really neat. We can try to build the simplest possible theory that one can build using this simple block. Okay, and one thing we can do, we can just do the following thing. This is a tube with some value of flux. So what we can do, we can glue this tube to itself. Okay, so what will be the theory which one obtains by gluing this tube to itself? The theory is of the following type. Okay, I just take this theory and identify the two flavor SU2s and gauge them. So what you will obtain is the quiver which has the following form. There is an SU2 symmetry here. Then there are two lines, another SU2. Then we have something like that and then we have an additional SU2, okay? So imagine it a little bit, just fold this picture onto itself and then identify these two SU2s and then do gluing with the rules that we have discussed and that is the quiver that you will get. It's a little bit simpler quiver. Now this procedure doesn't break the six symmetry. Actually one has to be very careful. It's not trivial since this tube has, in this case also the flux is half integer. So this is an illegal flux. When you have punctures, you can get away with these types of flux but when you have a closed Riemann surface, this flux turns out to be illegal and what happens is that some of the symmetry of this theory when you close it on itself is broken that what this fractional flux does. In particular the symmetry was E6 times SU2 times C1 so SU2 is broken but if you analyze the things carefully then E6 should be preserved and still the E6 should be built from this SU6 and this SU2, okay? So this gluing doesn't break the E6 property. So if we did things correctly, again if the picture is consistent, this more simpler theory has to have an E6 symmetry and again you can check it but now that we have a simple theory we can make it even simpler, okay? So if we think that the way the E6 appears is by taking representations of SU6 and combining them with representation of SU2, things which are not charged under SU6 and not under this SU2, we just can get rid of them. We just can do something to them and erase them from the theory. How do you erase fields from a theory? The same question I asked before. Sorry? This is more complicated way, a simpler way. Give mass. You just give mass. So what you can do, you have these bifundamental fields. You just can give a mass to them, okay? There is a cubic superpotential here so if you give mass to these fields you will generate some quartic superpotential and one needs to analyze the flow carefully and it's a little bit subtle but bottom line what happens if the moment you give mass to these bifundamental fields again, you just erase them and then you get a quiver of this type and again you should remember that there is a quartic superpotential which glues two copies of this field to two copies of this field such that you get a gauge invariant superpotential. So now it is an extremely simple quiver and you can actually make it even simpler, okay? It's the same type of property that we can use. If you look on this SU2, it has six chiral fields, okay? So this is an SU2 with an F equals to three. So what we can do, we can use the dynamics of SU2 with an F equals to three to trade this theory with gauge singlet fields with the mesons, okay? So let us call these fields qi and let us call these fields q tilde j. Then the superpotential, the quartic superpotential that we have is of the schematic form q tilde j, q tilde k, qi, qj, qk. This is the quartic superpotential that this theory has but after we are done with the dynamics of this SU2 with an F equals three, we can erase this part. This is just traded with these mesons and these mesons contribute to the superpotential. They couple to the theory with superpotential. So q tilde j, q tilde k is built from these fields. They are still there but the fields we had before are traded with this gauge singlet chiral fields. So there will be a cubic superpotential of this form and these types of superpotentials, gauge singlet fields, we are again denoting with an X. So what this X means, you just take the gauge invariance you can build from this q tilde. So this gives you 15 fields, sorry, 15 operators and you introduce 15 gauge singlet fields and you couple them through such a superpotential, okay? So now again, if our logic is correct, this extremely simple theory has to have an E6 symmetry, okay? But now it doesn't get simpler than this, okay? This is an extremely simple quiver. It's an SU2 with eight fundamental chirals, okay? So this is what is called SU2 with an F equals four but it's not just that. It's this thing with a superpotential with additional gauge singlet fields and the superpotential. This superpotential breaks the SU8 symmetry of this theory. If you look at it without a superpotential, there's two SU2 times SU6 and then there is some additional E1 floating around, okay? But this theory is very, very easy to analyze. Again, it doesn't get simpler than that. So one thing you can do, you can find the super conformal or symmetry assignments for this quiver and again, I encourage you to do it yourself. It's a very, very simple exercise to do. So for example, what you find is that these fields, the fields which are by fundamental of SU2 and SU6, have R charge, super conformal R charge, five ninths, okay, just the result of a computation. These fields have a super conformal R symmetry of one third and these fields have a super conformal R symmetry eight ninths. Yes, I think it's correct, okay? So now what you can ask yourself is what are the simplest gauge invariant operators you can have in this theory? So one thing that you notice immediately that is that since the super conformal R symmetry of this field is one third, if you build a gauge invariant operator out of it, it will have R symmetry which is two third, which means that the meson that you build from this field will be free, okay? So it will be not an interesting operator. But you can, in particular, it will be, it is a singlet, it's not just free, it is a singlet of this SU2 and this SU6 which we claim should enhance to E6 symmetry. But then you can look on other operators. For example, you can look on operator, let us call this field QA, this field you call QB, and this field, the Flipper field, the field, this gauge singlet fields, let us call them MA. And what you can do is look on these very simple operators of the following type. For example, QA times QB is a gauge singlet operator. QA is bifundamental of SU6 and SU2, QB is a bifundamental of this SU2 and this SU2. So this operator is in representation two six of the two groups. So it's a doublet of this SU2 and it's in six-dimensional representation of this SU6. And then we have this operator MA, which is gauge singlet, okay? It's not charged under this SU2 gauge symmetry, but if you analyze the charges, it has the same charges as this operator. And it is in representation one 15 or 15 bar. If you are more careful. And again, this very, very simple operator builds for you the simplest representation of SU6, which is 27 bar. Notice that this operator MA, which gives you this piece of the representation and QA, QB are completely different. They have completely different origin in the UV theory. This operator is a gauge singlet. This is a composite operator, okay? But the claim is that this theory in the IR flows to an SCFT where these two types of operators become exactly on the same footing. And they build for you representations of SU6, okay? So you can analyze the spectrum of this theory in full detail and find that everything you see, every supersymmetric operators you see are consistent with the enhancement of SU6. But actually this theory is even more neat than you could expect it to be. For example, what happens with this theory is that you can show by computing some supersymmetric partition functions that it flows to an SCFT, which doesn't have a conformal manifold, okay? Remember before that when we build theories which we claimed have a seven symmetry, we didn't see the seven in the Lagrangian, Lagrangian was conformal and we were saying the following words. We don't see the seven, but maybe there is a point on the conformal manifold where symmetry enhances. Here we don't have such election. If you analyze this theory again, it flows to a SCFT with no parameters. Just the conformal manifold is a single point. So if there is an enhancement, you should see it, okay? You don't see it in the Lagrangian, but it's fine because it's not a conformal Lagrangian. This Lagrangian flows to something. But if you consider certain supersymmetric partition functions, which tell you some very precise information about the fixed point, you should see whether the concert currents of E6 appear or not, okay? And there is one very useful partition function which is called the supersymmetric index that Leonardo has mentioned. I will not get into the details of it, but what you can extract from this index rather neatly. Again, as Leonardo was discussing, you cannot really see what operators your theory has. You can see some equivalence classes of operators. In particular, what you can deduce for N equal one theories is that this index contains exact information of what is the number of marginal operators your theory has minus the concert currents, okay? You just can look at this index and read off this information, okay? What is the number of marginal operators minus concert currents? You don't know from it what is the number of concert currents or what is the number of marginal operators, you only know the difference because of these recombination rules that Leonardo has discussed, okay? And if you compute the supersymmetric index of this theory, what you find is that this is equal to minus 78 and 78 is not a number. It's actually what you can compute from the index is the character of the representation of some operator. So what you, this 78, which appears in the index is something which you build from fugacities that you turn on for different fields that you have different symmetries that you have in your problem. 78 is exactly the dimension of a joint of a six. So what you can learn from this computation of the index that your symmetry is at least is six, okay? Actually to be more precise, you see minus 78 minus a one and this minus a one corresponds to concert current of an additional U1 that this theory has. So what you learn is that your symmetry is at least is six times U1. If it is a six times U1, there are no marginal operators. If it is bigger, then there might be marginal operators to cancel the extra concert currents but we don't have any evidence that the symmetry is bigger. So the minimal assumption is that this is the symmetry and there are no exactly marginal operators. So under very mild assumptions that there are no accidental symmetries for which we see no evidence. This is a proof that this very, very simple theory has any six symmetry, okay? So this is something you could discover 25 years ago. This is an extremely simple theory an extremely simple property but without all this song and dance it would have been hard to do and it was not done. So this is one utility of this approach. So it doesn't just give you known examples, known dualities but also can produce complicated properties of complicated quivers but also very interesting and surprising properties of this very, very simple theory. Now final thing I want to say is that it actually could have been discovered long ago and this type of property actually follows from just two things under the assumption that from the assumption that this theory has no conformal manifold and from cyber duality. That's the only thing you need to know, okay? And how cyber duality goes. So this theory say without these flip fields if you would just look at SU2 with an F equals four you can draw it in the following way. You have a four, an SU2 and a four. Now of course there is no difference between the different fundamental representation so we could write it as a two with an eight but we choose to split it and we choose to split it because cyber has taught us that this theory in the IR is equivalent to the following quiver, okay? A quiver which looks like that, okay? So we have the same matter content. SU2 with an F equals four is dual to SU2 with an F equals four. So it's a self-dual theory. It's self-dual, almost self-dual. To make it dual you need to add some gauge singlet fields which I draw like that, okay? So this is a known fact from cyber duality and now we can relate this to what happens here. We can write this quiver in the following way. We here don't have an SU4 and SU4. We have an SU6 and SU2. So what we can do, we can write exactly the quiver we wrote there in the following way. We split one of the SU4s into two and two and we add a four here. So this is just some splitting of this theory. We need now to add the bifundamental field. So the gauge singlet fields and the claim is that the gauge singlet fields have the following form. If you add them in this split version, so what you will get is that you flip, you add some gauge singlet fields for these bifundamentals and you add some gauge singlet bifundamental between SU2 and SU4. And for this to be precise, I need to add here another gauge singlet field which does nothing. It couples to the meson built from this simple field. It is not charged under a six, so it doesn't change the story. But now if you use the cyber duality rules, just use the cyber duality for this quiver with this additional gauge singlet fields, what you will obtain that the quiver you will get will be exactly the same quiver. The only thing which will change is that the gauge singlet bifundamental will connect not this SU2 and this SU4, but the other SU2 and SU4, okay? So remember that in this quiver, we have an SU4 and SU2, which actually combines to SU6 if you think of it carefully, and another SU2, so you have an SU6 times SU2 symmetry. After you do cyber duality, what happens? You switch the roles of these two SU2s. So it's like you break the SU6, you take out of it an SU2, and you put in another SU2. So what cyber duality does for you, it acts as an additional while transformation of SU6. So what, well, and this is an explanation of like an explanation from cyber duality why this theory has an SU6, okay? The fact that you get the same quiver up to an action of, up to some action which identifies in some non-trivial way the symmetries, by itself is not a guarantee that there is an enhanced symmetry. For example, we know that n equals four super Yang-Mills is of dual. If you do as duality, you get the same theory, but the duality doesn't just act on representations and so on. It also changes your location on the conformal manifold. The duality takes you from one value of a coupling to another value of a coupling. And Leonardo mentioned this property for the SO8 and equal to theories, and the duality group of that theory acts as a triality group of SO8, but it also acts on the couplings. So an additional piece to complete the proof from cyber duality here is the assumption that conformal manifold is just the point. So the only thing which happens is that you stay in that particular point, you have nowhere to go, and the only thing that the duality does, it reorganizes your representations. And for this thing to be consistent, the symmetry has to be any six. Okay, so this is my final example. So let me, how much time do I have? Okay, so let me stop here.