 Koliko cabo začaj, eleven muškival s presquisадцjefa in prepen Net маленьkih ved surname. Čest하고vu. Je pa vz IG o komparworkov 96. Mest, delivered, posledanost in mandקusov. teoriče, ki se potrebimo, da se prišli, da se prišli od 6 dimenzijanja. In ne znamo, da imamo nekaj referencijo na toga. Tuk je bilo izgleda na nekaj nekaj naprav, ki so vsega vsega kolaboratora v past 3-4 roj, v nekaj nekaj nekaj naprav So some subject which are covered in these papers, but the basics are detailed in the papers on the list. Ok, so let us start from some general considerations. So what do I mean by understanding dynamics of four-dimensional n equal one supersymmetric theories. Ok, so this is a huge subject, one can study many, many things. So in this first lecture, I will give you some pictures mainly, not many equations, but some pictures and outline of the program that we are pursuing and what type of questions we are trying to answer and what are the main approaches and what are the main challenges and then in the remaining lectures, the amount of technicalities will gradually increase. So this will be our plan. So today I start from general considerations. So again, this is a very big questions, four-dimensional n equal one, four-dimensional supersymmetric theories, occupied research of many, many people in the last 20 years. So what do I actually mean? What type of questions I want to answer? So let me start with very basic things. So, the way we think about quantum field theories usually is by starting from some theory, a conformal field theory in the UV. You start from some starting point. Typically this starting point historically is taken to be some free theory, which is given in terms of some weakly coupled Lagrangian. There is, you don't turn on any interactions in the beginning, you just have some collection of fields, and then you turn on interactions and some relevant deformations, and then you flow, you have an RG flow to a new fixed point, to a new CFT, which we'll call CFT in the IR. And this CFT in the IR can have a variety of behaviors. For example, it can be an interacting CFT, it can be a very non-trivial one, it can be a gap theory, there might be nothing there, you just might flow, start with some degrees of freedom, and end up in the IR with an empty theory. It can be a free theory, it can have free matter fields or free gauge fields, so there is some variety of behaviors that such a flow can lead to. And there are many interesting questions that you can ask about such energy flows. What we call a QFT is usually what happens in between these two CFTs. What the interesting physics which happens in between. And some types of the questions that you can ask are the following. So what, for example, can you learn about the IR fixed point, the IR conformal field theory, just looking at the UV starting point. In principle, the moment you define a UV starting point, you can compute anything you want. You can just solve the renormalization group equations in principle, and then reduce anything you want about the fixed point, but typically this procedure is very hard to do in practice. It's very hard to understand what type of theory you will get in the infrared. For example, one thing which will occupy main focus of these talks is what is the symmetry of the theory in the infrared. This seems like a very simple question. For example, you can start in the UV with a theory which has some symmetry, which we will call GUV. And by this symmetry, I don't mean the symmetry of the Frilla Grandian, but the symmetry of the theory once you already have chosen the vacuum and you have turned on the interaction, so that the symmetry in the beginning of the flow, if you wish. And then there is a question, what will be the symmetry in the infrared fixed point? What will be the G in the infrared? And typically the symmetry in the infrared, in many cases, it will be the same as the symmetry in the UV, but in some cases it can be bigger. Some degrees of freedom that were there in the UV that were transforming non-trivially under the symmetry in the UV might become massive, might decouple in the IR, and the symmetry might be bigger. So this will be one of the main questions we will ask. What is the symmetry in the IR and the symmetry if I tell you the symmetry in the UV or the theory in the UV? And there are many, many other questions. So let me list several questions. Some of them we will try to answer, and some of them we will just serve as guiding principles, as kind of questions which will guide our research, what we want to do. Question number one is a very general question. It's a very interesting question. And if I'm talking about questions, ask me questions. Feel a bit silly just talking. So question number one, which is a very tricky question, which we will not answer in these lectures, but we will, this is where we are heading to, is of the following nature. Is there some strongly coupled CFT? Somehow you managed to produce a strongly coupled CFT. Is there always a weakly coupled flow leading to it? So one thing that you can understand immediately from this picture, that although historically we are taught, that's how we are discussing quantum field theory in the classes, we are starting from free theory with Lagrangian, and then we are studying these types of theories, but the CFTs that we can obtain in the infrared can be strongly coupled, and once we understand that the strongly coupled CFTs exist in nature, well, you can ask yourself what happens if you start from this CFT and think of it as some new CFT in the UV, let's call it CFT in the UVB, and perform some relevant deformation and you flow to a new CFT in the IR, which again might have a variety of behaviors, it again might be interacting and so on. There is nothing holy about this free starting point. You can just start from the interacting CFTs, flow to some other CFTs, and then the natural question is, is any interacting CFT, any non-trivial CFT, accessible through such historically preferred starting point? Is any CFT, does any CFT has a weekly coupled theory or a free theory which flows to it after some deformation? So let me give you one example of such a theory which has such a flow, an example of a theory which we believe is interacting and it has a weekly coupled flow leading to it. Okay, so this is our first example, simple example. So let us take minimally supersymmetric theory in four dimensions, and again I will stress that all these talks will be about minimally supersymmetric theories in four dimensions. We will have higher dimensions, six-dimensional physics and five-dimensional physics entering into considerations, but the main interest of the talks will be n-equal one theories in four dimensions. So let us take maybe one of the simplest theories you can consider, and n-equal one SU2 gauge theory, SQCD, with five flavors. And to remind you what this theory is, this theory has an SU2 gauge group and the vector multiplet is an n-equal one vector multiplet, and then we have five fundamentals and five anti-fundamentals which we can denote by q and q tilde. So this transforms in the fundamental of SU2, this transforms in the anti-fundamental of SU2, but for SU2 these are the same representations, so there is no real difference, there will be difference for other groups. So one can start from such free theory and then turn on gauge interactions and flow to some theory in the IR and the claim is that this theory in the IR is some interacting TFT. Discussion of these types of theories was done 20 years ago. If one studies SUN gauge theories with NF flavors, so here we have a special case of n-equals two and NF equals five, then there is something called conformal window for some range of parameters n and NF. It is believed that the theories flow to an interacting CFT. If you are outside of this range, you might flow to a free theory. And the details of this conformal window will not be interesting to us, but this is an example of a flow where you start from a free theory and you flow to an interacting CFT. However, in the recent years we have learned that not all theories that you can consider are as simple as that. If we would not know of any theories, any strongly interacting theories which are not of this type, this question wouldn't have arisen. But there are examples of theories that one can consider for existence of which we can argue in various ways and for which we don't know of a weekly couple starting point in the UV to the flow to these theories. And many of these examples or the canonical such examples have a little bit more supersymmetry, n equal to supersymmetry. And the reason it is easier to argue for existence for such theories with more supersymmetry is that more supersymmetry is more constraining. Basically you can list all the possible Lagrangians with n equal to supersymmetry, with n equal to supersymmetry, in writing Lagrangians is the choice of the gauge group, choice of representation of the matter, but basically everything else is fixed so you can just list all the possible Lagrangians and study their properties. And then by more sophisticated techniques coming from different places for example string theory and so on, you can deduce existence of strongly coupled theories which should have n equal to supersymmetry, but they have weird properties. For example, some of these theories have symmetry groups which you cannot engineer using these very, very simple n equal to Lagrangians. For example, symmetry groups which are A6, A7, A8, these theories are usually called Minahan-Emešanski theories. There are theories which are called Argyros-Daglas theories. Argyros-Daglas theories for which some operators in these theories have some weird fractional charges which you cannot, just by looking at the Lagrangians that you can write, you cannot find the Lagrangian for which these charges appear. Some protected operators with weird charges and there are no Lagrangians that you can write which have these charges. So you start, if you want to discuss theories which have manifest n equal to supersymmetry, you insist on having all the symmetries manifest in the UV starting point, both supersymmetry and other symmetries you just cannot write any Lagrangian. So that's how we know that the question exists and moreover, they come from a very sophisticated type of flow. You mean, it's not, I would say, you need to do something fancy. You go on a Coulomb branch, some kind of a scaling limit and then you derive by consistency of the whole picture that there should be a CFT and it should have these types of properties. It's an example of the case where you don't start with a weekly couple theory and you flow. You tune parameters, you do something fancy. I agree with you that it is a little bit shaker in this type of examples. But again, it's not just that you start from free fields, you just turn on some interactions and you flow to these theories. And in recent years we have derived many, many more examples of such phenomenon and we will get to them maybe later in the lectures. So by now we know of existence of a large variety of theories with n equal to supersymmetry for which we don't know of an interesting, of a useful weekly couple starting point from which we can flow to those theories. Yes. You just can list all the possible agrangians with n equal to supersymmetry and ask what are the global symmetries of these theories and you just cannot find such cases. Okay. So this is one question we would want to address. Question number one. Question number two is of the following sort. Okay. So question number one is, is there a flow leading to a certain CFT? A question number two is of the sort how many different flows can lead to the same CFT in the IR. And again, we know of many examples and I will give you some where you have some CFT which is strongly coupled, see some CFT in the IR to which you can flow starting from very different agrangians. There is one CFT UV that you can choose. There is another CFT UV that you can choose. There are many different starting points from which you can start and flow to the same theory. Okay. This goes under the name of universality or IR duality. You start from theories which are different in the UV. They look differently but once you flow to the infrared you flow to exactly the same CFT. And one example that I will give you is again of this sort. So you can start from an SU2 theory with five flavors and flow to some CFT in the IR. This is our starting point in the UV and there exists another starting point which looks very, very different which is an SQCD with a different gauge group an SU3 gauge group again plus five flavors and which has some gauge singlet fields. Details are not important for now. Very different theory which nevertheless is argued to flow to exactly the same fixed point. This is an example of an IR duality which is called also cyber duality. Most of the cases may be of these types where discovered by Cyberg and his collaborators 20 years ago where you can relate SUN gauge theories with NF flavors to SUNF minus N gauge theories with NF flavors and some singlets. Okay. So there is another question that one can ask is what are the universality classes of flows? Starting from two different Lagrangians starting from two different theories in the UV is there a way to know that they should flow to the same fixed point? Here it's very hard to guess if I give you two very different looking starting points that they will flow to the same CFT in the infrared. Is there a way? Is there a mechanism? Is there an algorithm to tell us that two given theories will flow to the same fixed point? And in general there might be more than two theories flowing to the same fixed point. There might be some equivalence classes and of course you can have a very complicated network of flows when you start from several different theories you flow to another theory and then you start flowing from it to yet depending on which types of deformations you turn on your flow to a variety of theories and then some of these flows might be equivalent again so there is a very sophisticated picture of RG flows that we can have and there is a question of what is the organizing principle. So this is a question number two. The question number one we are not going to answer in the lectures but we will have some answers about question number two. At least some progress in answering this question. Now if you look at this particular example of duality it has also a very interesting property. As I already mentioned, this theory as you do with five flavors has five fields in the fundamental representations and five carol fields in anti-fundamental representation but fundamental and anti-fundamental are equivalent here so the symmetry that you have in the UV is actually SU10. So you can rotate all the ten carol fields that you have, five fundamentals and five anti-fundamentals into each other and the symmetry group that you have in the UV for this particular example is SU10. However for this theory since for SU3 there is a difference between fundamental and anti-fundamental representation the symmetry group is only SU5 times SU5 times U1. So the global symmetry with which you start in these dual theories is different. So the only chance for this duality to be true is that the symmetry that you obtain in the IR fixed point is consistent with both symmetries and it has to be at least SU10. And from the point of view of this description the symmetry should enhance in the infrared. You start from a smaller symmetry and the symmetry becomes bigger in the infrared. So this is an example of enhancement of symmetry. And there are many examples like that that are known so question number three that we will ask and again we will have some partial answers to given GUV what is GIR? How can we determine the infrared symmetry starting from the UV symmetry? Again I will not give you an answer of this type give me a group theory sorry a gauge theory with certain group gauge groups and certain matter content I will not be able to tell you what the symmetry in the infrared will be so build some kind of a framework where you can ask these types of questions. And finally let me mention two other questions which are a little bit more generic and again one can have them in mind while discussing these topics. So nothing is special about symmetry symmetry is something we like we understand well so we can ask questions about symmetry but we can ask more general questions which I already mentioned what about IR can we easily deduce from UV? Again in principle we can deduce anything the moment we have a Lagrangian but usually making this deduction is a hard problem so the question here is what easily can be done and the symmetries we will see in some cases can be deduced rather easily and yes, can you speak a little? Nothing breaks them all the interactions are consistent Sorry can you repeat the questions because most of the time we cannot hear So the question is why symmetries are not broken during the flow there is nothing breaking them there has to be something breaking them and nothing is breaking them What can happen is that suddenly nothing is charged under a symmetry so some symmetry acts trivially so some degrees of freedom that were rotated by symmetry they become massive and once you go below certain energy scale nothing is charged under that symmetry so you don't see anything non-trivially happening and again I'm not talking about yes, yes, yes, yes symmetry is symmetry I take the modern stance that there is no such a thing as gauge symmetry there is symmetry by symmetry I mean global symmetry all the time gauge symmetry is the way we describe the theory but global symmetry is some property of the spectrum of the operators Ok, so this is another question we can ask and finally a more general question which really is an important question and that we would like to answer and that question is can we classify all possible n equal 1 CFTs and say in four dimensions this is a big question so can we make a list can we build a store where on the shelves we have all the possible theories all the possible descriptions and say some experimentalist come to this store and ask for a theory which has a certain property and we give them a list of theories which can satisfy their needs is the store we have now starting from weekly couple theories some Lagrangian is it complete is it incomplete are there some beasts that some CFTs that can exist that cannot have such description if an experimentalist ask for a theory which has a certain symmetry or theories we will be able to give the experimentalist do these theories have manifest symmetries in the UV or we can give them theories which only have these symmetries emerging in the IR can we find such theories and so on so these are the big questions questions questions about questions so once we lay down the questions we can ask what are the approaches what is the approach we are going to undertake to answer these questions and there are two main approaches in the market what are the approaches so approach number one which we will not going to undertake is as follows we are talking about conformal field theories we are talking about flows between conformal field theories so conformal field theories are important in understanding QFTs they are the starting points so conformal field theories in addition to the global symmetries that we have discussed and supersymmetries they have also what is called conformal symmetry and this symmetry is very constraining and we can try to use this symmetry to fix everything almost everything as much as you can you can try to use symmetry and maybe some extra input like some minimal amount of input something about what are the types of operators like simplest operators that you have in this theory and try to fix everything about the CFT possible so this goes under the name bootstrap conformal bootstrap it's a very popular endeavor and I think you will hear more about it in Leonardo's talks later on so in this approach you kind of you center yourself on symmetry symmetry is important you take symmetry as your main guiding principle and you try to deduce from it everything you can but you don't care whether your theory has a weekly coupled Lagrangian or doesn't have a weekly coupled Lagrangian can you write it in terms of fields or you cannot write it in terms of fields this is an unimportant input in this approach all you care about is what are the symmetries of the problem so here you you make symmetries wholly and you don't care about Lagrangians this approach is not very useful for some of the questions we have asked for example we have asked questions about emerging symmetries if you start from one conformal field theory and you flow how the symmetry emerges to what type of symmetry you get in the infrared so we should not make too much use of the symmetry both because the global symmetry can emerge and because some flow is involved and conformal symmetry only is recovered at the end points so what we will do we will take another approach in a sense complementary approach and we will not take symmetry to be wholly so what we will do in some sense we will give up symmetry at least some parts of the symmetry we will not center our approach on deducing things by starting from symmetry but we will insist on having weekly coupled descriptions you will not see non lagrangian theories in this talk we will have always lagrangians and by studying these lagrangians and studying flows interrelating the lagrangians we will try to answer the questions that we have posed so these two approaches seem very different they are in a sense complementary so in the bootstrap approach what happens is that in this picture of the flow you focus on the starting and end points of the flow so this is approach number one you try to understand everything about the CFTs from the symmetries and what we will try to understand we will try to understand what are the flows interrelating different types of constructions so this is a complementary approach one caveat that I need to stress is that is of the following nature as I stressed already several times we will always have some supersymmetry and we will be in four dimensions the reason we will insist on supersymmetry is because supersymmetry gives us some technical tools to answer the questions we want to answer however once you understand that some symmetries can enhance during the flow also super you should be aware of the fact that supersymmetry can enhance during the flow and if you want to understand for example all the possible supersymmetric theories some of them might have a weekly couple description but this weekly couple description might be not supersymmetry supersymmetry will enhance so we will always assume supersymmetry we will make some progress using this assumption but you should remember this caveat we will always assume that there is a supersymmetric that there are supersymmetric flows leading to the theories we want to study and this is an assumption which just might be plainly wrong and before continuing to actually telling you what what will be the technicalities of this approach let me give you an interesting example of supersymmetry enhancement in four dimensions that maybe it's the first example in four dimensions that appear maybe I'm wrong maybe people who were around back then can correct me yes can you say again no did you ask about the bootstrap in bootstrap sorry Slavo if you can always repeat the question I'm not sure I understood the question but let me answer the question I understood so I cannot repeat it but let me know if I'm wrong so in the conformal bootstrap again as I have written here you always are stuck at the conformal point you don't have any flow yes that's an excellent point so it's not when you study a CFT you don't study the flow but you definitely can get an information where you can go from that point you can understand what are the relevant deformations what are the operators which you can add to the Lagrang to the theory to the action of the theory so that it will start the flow so you do so again these are not completely not overlapping approaches so there is nothing about the RG flow and in the approach we will undertake we will learn something about the fixed point does this answer the question yes that's a very good question so this is this this goes back to the question number one this is the question can you formulate such a condition for example that would be great so let me give you an example of supersymmetry enhancement that is extremely simple so we all know and love n equals 4 as super young mills in four dimensions it's a very useful theory in many many contexts and you might know that n equals 4 super young mills with SU2 gauge group maybe the simplest theory has a dual description this theory you can think of again as le algebra this is the same as an SO3 group n equals 4 super young mills has 3 adjoint fields adjoints for SO3 are the same as vectors so there is another description of this theory in terms of n n equal 1 gauge theory with gauge group which is SO4 plus 3 vectors plus some gauge singlets this duality was it's a special case of SOn cyber duality there is a duality between SOn and SOnf minus n plus 4 gauge theories that cyber has discussed and for a very particular case when the n is equal to 3 and you have 3 in the number of vectors 3 you have this duality so this is an example of enhancement of supersymmetry here you have a conformal field theory you have a conformal description of some fixed point and here you have an n equal 1 flow which flows to that theory supersymmetry here is only n equal 1 and it emerges it enhances to n equals 4 only in the infrared and we will have something more to say about this later on so, if we are giving up symmetry we need to have some other guiding principle which will let us make progress on answering these questions we cannot just give up symmetry and do something useful we need some other guiding principle so the guiding principle that we will have another guiding principle that we will have will be geometry and let me discuss now what I mean by geometry so what we will do we will discuss another way to engineer 4-dimensional safety so one way to engineer a 4-dimensional safety is to start from another 4-dimensional safety however, what one can do one can start from a CFT living in another number of dimensions so one can start from some theory for example living in 6-dimensions you can start from a CFT in 6-dimensions and then you can put this 6-dimensional CFT on some compact geometry and let's say this geometry is of scale L and then what will happen if you will consider this 6-dimensional theory on this compact manifold at energy scales which are much smaller than the energy scale given by the size of this compact geometry you will get an effective 4D theory you will have some 4-dimensional theory in the infrared sometimes this 4-dimensional theory will be an interacting CFT sometimes it will be free CFT the same words we said about 4-dimensional flows can be said also here here the relevant deformations where for example some relevant operators you can add to the Lagrangian you can give some vacuum expectation values to some operators here in this picture the relevant deformation is geometric you put your theory on some manifold which has some scale you break conformal symmetry by introducing this scale you flow to some effective theory which is effectively a 4-dimensional theory so this is just another way to engineer 4-dimensional theories and what we will be after descriptions of such flows starting from 4-dimensional Lagrangians let us assume that the theory we get by this geometric compactification is some interesting interacting CFT and what we will want to find is there a 4-dimensional Lagrangian which flows to the same CFT and in some cases we will find such Lagrangian and we will make a dictionary we will find a dictionary between this type of geometric compactifications in 4-dimensional Lagrangians and then the fact that the formations of the 6-dimensional theory you start with are labeled by some simple mathematics by simple geometry will give us some very simple tools to answer some of the questions we have posed no, we will not assume Lagrangian this is an excellent point the 6-dimensional theory so the question is is the higher-dimensional theory will start with a Lagrangian theory does it have a Lagrangian we will not assume that and this will not be true the 6-dimensional theories will be some strongly coupled theories they are very hard to study by themselves but the fact what we will study is not these theories by themselves but these theories compactified on some geometry and we will see that the fact that this geometry there fixes a lot of the properties of this CFT without knowing much details of this 6D this will be the magic of what we will do this starting point is complicated but it will allow us to find certain Lagrangians which will claim have certain properties and these properties will be labeled by geometric information and making use of this geometric information will be able, for example explain enhancements of symmetry of this or supersymmetry of this type more questions 6 dimensions will give us some rich starting point we will have a lot of there are several answers to that I will answer a little bit more in detail later on but one kind of a quick answer which doesn't go in any detail into the physics of this thing is that Riemann surfaces are rich enough objects, two-dimensional surfaces are reaching enough objects which will allow us enough different tools to answer the questions we are after. There are other questions for example there are other answers to your questions like we can at most go to 6 dimensions because we insist on supersymmetry and 6 dimensions is the highest number of dimensions where super conformal theories exist for example. But you can in principle ask this question starting in 5 dimensions so this is the general idea that what we will do and now let me give you a feeling why this idea is useful why thinking about things geometrically we can make progress and the idea is very very simple the idea is of the following sort so let us start from 6 dimensions with some theory T6D so we are in this situation we start from some T6D we put a theory on some kind of geometry and geometry again in our case it will be some two-dimensional surface some Riemann surface and we can turn on some flux flux will be some background configurations for vector fields the conserved currents of the global symmetry that the 6-dimensional theory can have so we make these choices we make the choice of the Riemann surface and we make the choice of fluxes and we end up in some 4-dimensional theory and what is the 4-dimensional theory we end with is determined by the choice of the 6-dimensional theory by the choice of geometry and by the choice of fluxes in particular even if you start with the same 6-dimensional theory you can end up in wide variety of 4-dimensional theory depending on what are the choices of geometry and fluxes you have chosen so there are many many many choices you can make and this leads to many many different 4-dimensional theories you can have and the magic which happens which allows us to make progress is that sometimes understanding what this 4-dimensional theory is this question factorizes and it factorizes in the following sense say you can think of this Riemann surface as being in some sense sum of two Riemann surfaces and the flux is sum of two fluxes and by sum here of geometry say literally mean for example sum Riemann surface for example a genus 2 Riemann surface this is your C but you can think of it as gluing together, a geometrical gluing together of two Riemann surfaces one is genus one with a hole, with a puncture and another of the same type and you just combine them together you geometrically glue them together so this will be C1 and this will be C2 flux which is supported on this genus 2 Riemann surface and we can separate, we can find some configurations where this flux is separated into f1 and f2 which are in turn localized on the two different pieces of this geometry literally f is equal f1 plus f2 and we will discuss this in more detail later on for a given Riemann surface of some genus and punctures the decomposition into further Riemann surfaces may not be unique yes, I will come to this in a moment I am saying just take any the composition you want pick the composition the compose the compose the flux in any way you want also this you can do in many ways we will discuss it in two minutes just do some kind of decomposition think of this Riemann surface and flux as just some of two components in some favorable situations which we will discuss in detail what you can do is you can first understand what happens in this flow when you study compactifications of the same 6D theory on this piece and on this piece we will bring them together already in four dimensions so first you understand what happens when you compactify this T6D on C1 with flux f1 you get some 4D theory then you get another 4D theory when you start from the same 6D theory on surface C2 and flux f2 these are different flows you start from 6D theory then you multiply on this surface with f1, on this surface with f2 you get some two different possibly four dimensional theories and then you combine them together in 4D and this combination of the theory concatenation of theories in four dimensions as we will see corresponds to gauging of global symmetries of some global symmetries so what will happen is that complicated questions let me phrase it in analysis and by analysis I mean as we discussed in the beginning understanding these RG flows understanding how for example anomalous dimensions behave how this flow behaves in solving the better function equations will translate into simple questions in algebra and geometry because of this type of relation this relation doesn't exist always not for any 6D SCFT and not for any choice of the data such factorization of the problem exists at least we don't have evidence that it always exists we have many examples that such thing happens and those examples we will discuss and using these examples we will be able to answer some of the questions for example one of the questions which will be easy to answer in such factorization will be enhancements of symmetry emergence of symmetry in four dimensions so how does it go so the symmetry let us assume that the theory in six dimensions has some symmetry which will cause g6D when we turn on fluxes this symmetry is broken explicitly fluxes again are just some explicit gauge fields we turn on in form under this global symmetry in six dimensions so we explicitly break some of the six dimensions symmetry when we compactify so the symmetry in four dimensions is determined by the symmetry in six dimensions modulo these fluxes so the commutant of the flux in the global symmetry will give us the global symmetry in four dimensions for example if you have simple group in su2 gauge group in six dimensions and you turn on the flux for the carton generator of this su2 say the flux will be equal to plus one then the symmetry in four dimensions will be broken from su2 down to u1 again we will discuss it in more detail later on so now what happens in this type of constructions is as following you can understand first compactification of this sort and say with flux which is equal to plus one and then the symmetry of the four-dimensional theory will be u1 you can understand another compactification of this sort where the flux is minus one the symmetry will be still equal to u1 in four dimensions but when you combine these theories together this combination should correspond to a compactification on a surface which flux of which is just the sum of these two fluxes which is zero and that means that the six-dimensional symmetry is not broken and that means that the symmetry that you should get in four dimensions is su2 but from this perspective this symmetry will be emergent this four-dimensional theory has a u1 symmetry this four-dimensional theory has a u1 symmetry you just take these two theories and you gauge some global symmetry that these two theories have and then by gauging the symmetry you flow to some CFT which is claimed to be the CFT obtained by performing this type of compactification and if this dictionary is correct the dictionary between this compactification data this six-dimensional theory and this four-dimensional theory is correct it has to be in the infrared of this four-dimensional Lagrangian will enhance so this will be the main idea of what we are going to do there will be a lot of technicalities involved but this will be the main idea questions, yes the topology changes so let me repeat your question and let me know if I repeated it correctly so you are saying we are taking two theories, two building blocks which have different topology and we are combining from them a theory of a third topology and then you are asking how this change of topology is affecting the theory that we will get it will affect the theory that we will get will be different like there will not what we will find is that there will be certain rules that we can write that if you have a theory corresponding to topology a theory corresponding to topology b then a theory which corresponds to topology a plus b and by that I literally mean just sum up the genera and sum up the number of holes and types of holes, holes can have different types just this very, very rudimentary information will determine what type of theory you get when the topology is a plus b more questions yes so you are asking if the enhancement is unique well you define so the question is if enhancement of symmetry is unique you start from two theories you combine them and you get a unique and the answer is yes because what you do you take two theories and they have very particular properties and then you do a unique operation you combine them by gauging some symmetry it's a physical process it goes to some CFT that CFT has a very particular symmetry so the symmetry in the infrared is unique and that unique symmetry we want to determine before starting how much time do I have three minutes three minutes before starting next time let me just make a conjecture which will guide again us through these considerations and we will conjecture the following thing that any four-dimensional supersymmetric theory can be obtained by these geometric flows so you start from any theory from some supersymmetric theory in six dimensions and then you flow down to and in principle this is a very special construction and it you can wonder why should it give all the possible four-dimensional theories this will be our working assumption this assumption might be wrong but we will assume that it is true and we will see that we will get a lot of different theories and a lot of different symmetry enhancements that are known in the past and some of them are new just by performing these geometrical considerations let me just give you the answer for this particular case so why this enhancement of symmetry happens in this n equals four-superion mills and here it happens exactly because of this typof logic so there are two ways to engineer these two theories the n equals four-superion mills with gauge group SU2 you start from some 6D theory which is called a 2,0 theory and a1 2,0 theory we will discuss it next time and you put it on a surface which is a sphere with three holes a pair of pans like that and then you glue together the two of the holes so the topology of the surface that you get is to be done with one puncture so up to the coupled free fields this theory is a 9 equals four SU2 as superion mills and in some notations that we will discuss this theory has a flux which is given by plus half we will discuss this next time but there is another way to think about this theory and another way to think about this theory is to start from a pair of pans which have a different flux same type of punctures but a flux which is minus a half and glue to it a tube which has flux plus one and by gluing as we will discuss we mean gauging some symmetry and in this case it's an SU2 symmetry so this theory which corresponds to compactification on a three puncture sphere will be just some combination of free chiral fields and we gauge one SU2 symmetry and the matter content will be just exactly as the matter content of n equals four superion mills module of free fields and if we do this type of construction it's just a different splitting of the same Riemann surface we will get two gaugings each one of them is SU2 so the gauge group will be SU2 squared SU2 squared is SO4 it will be exactly this type of Lagrangian and what more by understanding this type of consideration in this case you can generalize these types of duality so here there was a duality between n equals four superion mills with group which is SU2 to n equals one sqcd with group which is SO4 because of some group theory magic because SU2 is SO3 but the moment you understand this type of this type of relations you can generalize to other groups in particular this type of relation between this geometry and this geometry exists for any group and then you can devise a duality between n equals four superion mills with SU engage group to some other theory this theories will have n equal one supersymmetry only for SU2 for A when you start from 6d theory which is of this type A1 these building blocks are simple they are just free fields for other cases this will be some more complicated as CFT sorry I can argue for simply lace but I think it should be true for anything because it's just geometry you can make this breaking for anything also in 6d, yes that's right that's right, sorry, yes yes, I have nothing to say about not simply lace so this will be the power and these types of magic that's what we will try to understand in the remaining of the talks thank you