 OK. Can you hear me? Welcome. So for so, let me start saying that I've been a student at this school various times. So for me it's a great pleasure and great honor to be able now to contribute to this school. I will talk about some recent developments in our understanding of gauge theories with matter in three dimensions, two plus one dimensions, so two space in one time. And, well, of course, we live in three plus one dimensions, at least these are the dimensions that we have observed so far, but nevertheless it turns out, at least it seems, that gauge theories in three dimensions are relevant for our world. Just to give you an example, there are materials known as topological insulators, and if you look at the behavior at the surfaces of these materials, or more generally at interfaces between different topological insulators, there are some massless or gapless modes that propagate, and in certain cases these modes are described by three-dimensional gauge theories with matter. Another example is lowest land level at half filling fraction. This is a system that can be realized, for instance, if you take a thin film of material and you immerse it in a magnetic field, and once again defective physics can be described, it seems, by three-dimensional gauge theories. So, this fact for us will be just some motivations, so I will not talk about condensed matter physics, neither I will talk how these gauge theories can be obtained, nor what is the condensed matter phenomenology of these theories, our interest will be purely theoretical. So one of the main tools that I will try to develop in these lectures is the tool of infrared dualities, and this word is in the title. So, what do I mean by that? Well, I mean that we have two theories, theory B, it could be more than two, let's say there are two. These are two different theories with a different physics, different Lagrangian, however, if you look at the behavior of these theories at long distances, for instance, a correlation function at long distances, that means at low energies, so let me draw this arrow, by which I mean the RG flow, then they present the same physics. Yes, in fact, a special case of this could be that you have some theory, which is A, and you ask, what is the infrared physics, and maybe the infrared physics is weekly coupled, and then you write down some theory B, and then you say that this is the effective description of theory A, if this is weekly coupled, but more generally you are in this situation, in which maybe you don't have a weekly couple description, and yes, this precisely expresses the fact that they are in the same universality class. Now, of course, this infrared duality is particularly useful, or nice, is this infrared physics, is some non-trivial CFT, because then this equality of these two theories, which is not really an equality, because in the UV these theories are different, and then becomes, we can go lower energies, and this equality becomes more and more accurate. Now, we will not be able to prove this duality, so we will conjecture this duality, and we will do some number of checks, you will tell me probably if you want, one will have to run lattice Monte Carlo simulation to check these dualities, but what we will see is that if these dualities are correct, then there is a number of very interesting physical implications that one can derive from them. Yeah, so the statement of duality is that I have two theories, which are really different in the UV, this is what I mean by infrared duality, it's not an exact duality, so if you compute correlation functions or scattering processes in the UV, they are different, however, if you look at long distances, or low energies, low momenta, then they are approximate, asymptotically the same, so in particular if you go in the CFT limit, correlation function of very long distances, those are the same. We will not have an exact dictionary, because that is very hard to find, but that's the idea, that there should be a dictionary between operators here and operator here, such that if you compute correlation functions, you compare them using the dictionary, then they have the same behavior. And yeah, so the dictionary is part of the duality, and in general it's hard to find a complete dictionary, in our example we will be able to find a dictionary between some simple operators, if you want the lowest dimension operator with certain charges, but for instance the operators do not have specific quantum numbers, then it's really hard. But there is a lot that one can do. Now the dualities that we will be talking about, in fact are similar to supersymmetric dualities, and some of you might be familiar with supersymmetric dualities. We know a lot of supersymmetric dualities in three dimensions, but also in four and two dimensions, in fact they make a very big story, very big and intricate story, and in fact it turns out that the huge amount of knowledge that we have developed over the last many decades about supersymmetric dualities will be a very, has been a very important guiding principle to formulate these dualities. Now in the case of supersymmetric theories and supersymmetric dualities, we can run extensive and very convincing checks of these dualities. In fact we are convinced that those dualities are correct. Essentially because of supersymmetry, there are a lot of quantities that we can compute exactly and not perturbed, but they will be a strong coupling, for instance with localization techniques, so with some other methods. And every time that we compute these quantities on the two side, we find a perfect match. Of course this quantity should be sensitive only to the infrared physics because, as we say, in general the theories are different in the UV, but of course this gives us a great confidence in these dualities. Now here in this context we will not have supersymmetry, so this will be a non supersymmetric context. We don't have these very powerful tools at our disposal, so it will be much more difficult to run powerful checks, but nevertheless there is a sector of operators or quantities, I should say, in these theories that are protected in the sense that can be computed exactly even at strong coupling. And these protected sectors or quantities have to do with global symmetries and anomalies, so we will touch on that. OK, so after these introductions to the plan of the lectures is the following. So I will start discussing some aspects of Saint Simon's theory. And here we will see the first example of duality, which is a level rank duality. In fact this duality will turn out to be not an infrared duality, but an exact duality. Then we will discuss some dualities in simple abelian theories with matter, and in particular I would like to review a particle vortex duality in some generalizations. Then we will go to more complicated Saint Simon's matter dualities. We will say something about anomalies, more specifically tough anomalies, and then depending on how much time we have we will discuss various physical consequences, implications of the dualities. Any questions so far? So I will start discussing some aspects of Saint Simon's theory. And now this first part will be a bit technical, but we will develop some tools and some concepts that will be useful in the following explorations. So I hope I will not lose your attention, but in case in the following parts it will be more and more physics, if you want. Moreover I will just cover some very specific aspects of this theory. This is extremely rich, extremely interesting theory. We could spend months talking about this theory, so I will just highlight some aspects. OK, so we want to discuss gauge theories in three dimensions and there are two kinetic terms that we can consider. One is the Young Mill's kinetic term. So let me use the language of forms to write this kinetic term. So here f is my field strength, which is written in terms of the connection. In my convention the covariant derivative will be the standard derivative as commutator with the connection, or anti-commutator when it acts on forms. This is the odd dual, constructed with the metric. And this is the gauge coupling that in three dimensions has dimension of mass. And there is another term, which is the Saint Simon's term. Here there is a trace, generally it's a non-Abelian group. OK, so the first question that I would like to ask is what are the degrees of freedom in this theory? And we can try to answer these questions semi-classically. So we compute the equations of motion. So if you compute the equations of motion, you'll find something which is first order in the field strength. It is Maxwell equation in this theory. Then we can take this star, we apply this star to this equation. We use the equation again because then we get this star here. We use the equation, we also use the Bianchi identity. And we can reduce this to the following equation. So this is the Laplacian. The Laplacian of F. OK, so what are the... So we're interested in the semi-classical degrees of freedom. So we solve the linearized version of this equation and we look what are the solutions. So if you write the solutions in terms of the connection, we find two types of the solutions. One is waves. And the mass of these waves is g square k over 2 pi. In particular, even though here we have gauge theory, we have not broken gauge invariance, but this dynamical degrees of freedom are massive. In fact, this theory is sometimes called topologically massive young meals. But this is not the only solution. There are also zero modes. If you have the flat connections, so for solution for which F is the field strength is zero, well, since F is zero, these solutions have zero energy. And so we have these two types of solutions. Now, if you are interested in the pure Chen-Simon's theory, we want to get rid of this term, so we send g to infinity. And what happens in this limit is that this degrees of freedom becomes infinitely massive. We can forget about them. So the dynamical degrees of freedom disappear. We are like with these modes, but in fact these are not local dynamical degrees of freedom. If you wish, because they have zero energy, and so we cannot kick them and give them some momentum and obtain something which is some local dynamical degrees of freedom. We only have these modes, which have zero energies, and so in particular the theory that we remain with is gat and is topological. So, yes, sorry, so now I will go there and I'll start big again. Thanks. Thanks for reminding me. So what do we mean that the theory is topological? H is the emiltonian, the energy. So these are zero energy modes. So what do we mean that the theory is topological? Well, what we can interpret, we can say this in various ways, so one way to see that, to say what does it mean, is that the theory does not depend on the metric. And this is clear classically from this action. This action classically does not depend on the metric. This piece did depend on the metric because in order to construct the object dual we need the metric, but not in here. We only use these forms. So the classical theory does not depend on the metric and in fact also the quantum theory does not, and we will say a little bit more in a minute. The theory is gapped because the dynamical degrees of freedom are infinitely massive, but still it is not trivial. In fact, it is far from trivial. It is not a trivial gapped vacuum. And the reason is that even though we only have these states with zero energy, we can have some degeneracy. We can have some number of these states. And in particular, the number of states that we have depends on the topology of space. So if we take this theory and we put it on a sphere, then there will be a single vacuum, a single gapped vacuum. This, if you want, is trivial. But if we take a more general topology for space like a torus or some remote surface, then this illbe space will have some degeneracy that will be multiple states, and so the theory starts becoming more interesting and more complicated and far from trivial. Okay, so what are the observables in this theory? Well, there are no local observables because we can go back to the equation of motion and if there is no, this Young-Mills term, f equal to zero. So the field strength is zero. We cannot construct local gauge invariants. So there are no local observables. But there are some non-local observables, which are very interesting. These are the Wilson line operators. So these are operators that are constructed along a line. So we need to pick some line, some close contour gamma. Then we construct these operators in the following way while we can integrate the connection. This is what we have. So we can integrate along gamma. This, more precisely, is patored exponential. It's such a way to make it gauge invariant. And in order to have a gauge invariant, here we have to take a trace. And when we take the trace, we can choose a representation of the gauge group. And so this operator here, in fact, is a function of the contour gamma and is a function of this representation. In fact, we can think of these operators as the word lines of some very massive particle that was created by some field at very high energy. And some theory that maybe flows to this topological theory. And this is clear from here, because this is the word line action for a theory that is coupled to A. Sorry, for a particle that is coupled to A, but this particle is very massive. So these are the observables in the theory. And so what we can do is to compute correlation functions of these observables. And in particular, we mainly talk, but not only about the Euclidean formulation. So in the Euclidean formulation of the theory, we go on a manifold, which is Euclidean, some three manifold. Then we can take some close three manifold. And we can draw these lines, or maybe some linked lines. And then we can ask what is the expectation value, what is the partition function with this insertion, what are the expectation values. Is some representation of the group. Because A, I mean, we can contract A with any generators of the group in any representation. And then we take the trace in the given representation and we get a different answer. In fact, in principle, a different operator. So these are the objects that we can compute in this theory. So let me make a couple of remarks. So the first thing is that we said, well, the theory is topological because the tensimons functional is topological, does not depend on the metric, just you write it without the metric. But in fact, this is a gauge theory, so we should do gauge fixing to quantify the theory and the gauge fixing action is not. So we might be worried that, okay, maybe classically the theory looks topological, but then we have to regularize it and do the gauge fixing. So maybe we lose the fact that the quantum theory is topological. But in fact, it turns out that with addition of suitable counter term, local counter term, and this is what we always do in quantum field theory, we can keep the regularized theory topological and so the quantum theory is topological. Up to a small remnant of this fact and this remnant is called the framing anomaly. And what this framing anomaly is, is the fact that we need to specify some extra data if we want to associate a well-defined number to the partition function in this theory. We'll not bother exploring what this quantity is because we will not need it, but we'll have to choose some framing and for a given choice of this extra data, we'll be able to associate a well-defined number to the partition function. If we change this data, we will get a different number, but the nice thing is that if we know how we change this data, we know precisely how the partition function changes. So this is almost as good as saying that we didn't need any other data. And in particular, the only change in this partition function is a phase. We get a phase that depends on how many units we change framing times some number. And this number C is called, in this context, the framing anomaly. Now, there is another type of framing anomaly which is related to this one. And this other type is related to the lines that we inserted. And this other anomaly can be understood in the following way. So these lines are defined as some exponential of these operators which are the connection. Imagine we want to compute a correlation function of these lines. These lines, we can imagine that we expand these exponentials out. And so we get a summation from the Taylor expansion where we have products of these integrals. And we have to integrate these operators A along the contour, but then we have multiple. When we expand, we get things like this, the products. And when we do these integrals, some point, two operators collide at the same point, and this gives us divergences in general that we have to regulate and cure. And so here we need some extra data and this extra data is the framing of these lines. Now, what this framing at the end of the day is is the following. So you have your line, and what you have to do is to attach a small ribbon to these lines. So imagine that there is a small ribbon that you have to attach it and only when you do, you attach this small ribbon, then you can get a well-defined number after the correlation functions. Now the reason why is precisely this ribbon is precisely what I described here. So the problem was that when you do the integral, at some point there are two points that collide and so what you do, one contour, you leave it along this line, the other one you move it a little bit, off in this direction, but you have to tell me in which direction you move it. So that is the reason. The point is that there are different choices for this ribbon because I could have chosen something like the following, which the ribbon goes around, and if you compute with these two different regularization of these two different framings, the difference in your correlation functions is, again, a phase. Let me write it here. So if we compute the correlation function of these operators with this prescription and we compare it with this other prescription, the difference is a phase. Again, there is an integer number that tells us how many twists we made in this ribbon, and then they have h, and this h is called the spin. This is the spin that we associate to this line. Of course, from this definition, you see that this spin is only defined in mode one. So to each line we can associate this spin. In fact, we can understand physically why there is this spin. If we think that these lines really are some very massive particle, if this very massive particle has an integer spin, then the correlation functions are single-valued. But for instance, suppose that particle was a fermion, then if you rotate by 2 pi around the particle, you get a minus, and then, in fact, if the spin is 1 half, you precisely get a minus. More generally, in three-dimension, you can have fractional spins, and so in general you will get some fraction here. But in the low-energy topological theory, you are only sensitive to the spin mode one. OK? OK, any question? Yes, so what I have in mind is that, OK, I'm computing some generic correlation of these lines. For instance, I could have a line that goes like this, another one that goes like this, maybe there is some knot, something like this. For each of these lines, if I want to get a well-defined number, I should attach this small ribbon around each of the lines. OK, it's a small ribbon which is on the side and goes all around. After you do that, you get a well-defined number. But if you change and you say, OK, I choose some ribbon, my friend chooses another one, which has a small twist, then my friend will get the answer, which is almost the same, but it differs by a phase. OK, and if I look at how the answer changes when I do this twist in the ribbon, I can extract the spin of this line. Well, it's an anomaly in the sense that, classically, you didn't need this information, but when you go in the quantum theory, you need some extra data, if you wish. In the same way as the framing anomaly, you needed some extra data that was not obvious from the beginning. In these senses, anomaly. You can probably express it in some other way. Well, what it is, is that you have to give me... So, there is a line, there is a normal bundle, which is a plane, if you want, but you just look close to the line. You don't need to think about the old manifold, and you have to give me some non-venison section of this normal bundle. So, this will be the mathematical way to think about it. Physically, it's just a very small ribbon that only has to be in a neighborhood. I mean, the theory is topological anyway, so it doesn't matter if the... No, no, for each line, for each line, maybe you have just one. In general, there are no intersections, first of all. OK, this line do not intersect in this picture so far. And for each line, you have to attach this ribbon that goes all around. I'm just saying, if you don't do that, you cannot associate a well-defined number to the correlation function. You need to do that, then you get a number, because you do this regularization, in which you do this point-splitting regularization, essentially. And if you make a different choice, you get this face. Don't look convinced. Yeah, so I told you that normally it appears because when you compute these correlation functions, you expand out this, and you integrate along a contour. So you have this contour, and when you expand out, you have products of this... You use Taylor formula, right? This explanation is one plus the integral, plus integral, integral, and so on. And when you do have two of them, each of them is integrated along the path, at some point one is x, the other is y, and you integrate, at some point x and y collide, then you get divergences. You have two operators at the same point. I mean, this is an integral formulation, you have operators that collide in point, and you get these divergences. This is a path integral point of view. OK, I didn't want to spend one hour on this, but... Ha-ha-ha! So the origin of the framing anomaly, just for the three manifold. So the origin is that... As we said, so the classical action is... Some action is independent of the metric, but the gauge fixing action is not. Then you need to add some counter terms to make the theory topological, but these counter terms depend on some choice, which is a trivialization of the tangent bundle. But we will not need this, so if you want, we can talk more in the discussion session. OK. Also because we will not need this framing anomaly, we will need this pin. OK, so let me say a few things about the transhumance functional, just as a classical functional. So we said, OK, the transhumance functional, let me write it once again. So this is a function of your connection on some three manifold. Let me write it in this way. I don't write wedge. I don't understand it there. Fine. So first of all, this functional is written in terms of the connection. So if we want that this makes sense, the connection should be well defined on our three manifold. Now, we have a gauge theory. So this gauge theory has some gauge bundle, some principal bundle for gauge group G. So E is some principal G bundle. Now if this G bundle is trivial or parallelizable, then you can find some connection, which is just a globally defined one form on your three manifold, and we can use this formula. If the G bundle, the gauge bundle is not trivial, in general, we can't. So we'll come back to this in a moment. So for now, let's assume that we are looking at some gauge bundle, which is trivial. So is this functional really gauge invariant? Well, we can try to do some gauge transformation. So we pick some function omega on the three manifold. Is our gauge parameter for gauge transformations. We transform A, and we look just to the computation, we see how does this functional change. And in fact, it does change. So if we compare the function on A omega with the functional on A, it is not zero, the difference, but is equal to, and I invite you to do this computation, is equal up to some factors, 2 pi k, to an integer, the functional of the gauge transformation. And what this is, is the winding number of this map omega. So this was a map from m3 to g, and there is some winding number. Here if you assume that this g is simple, otherwise there would be multiple possible winding numbers. But there is this winding number that counts how many times this map rops inside g, and it's not zero, but it is an integer. And so, well, this functional is not really gauge invariant. We do, I mean, if we do an infinitesimal gauge transformation, this winding number is zero, and then this is zero, but otherwise it's not zero. However, so it looks like we are in trouble, however, when we define the quantum theory, so the pat integral, what we have is the exponential of the action. And so what we really need to be invariant is not really the action, but is the exponential of the action, OK? If this is well defined, this is good enough for the quantum theory. And so you see that if k is an integer, then this object only jumps by multiples of two pi, and then this exponential is well defined, because even though the function shifts, the exponential does not. And so we find a quantization condition, so k should be an integer, OK? If k is not an integer, this function is just not defined, even if you want in perturbation theory, in the sense that here we are assuming that the gauge bundle is trivial, the problem is not well defined, OK? What if the gauge bundle is not trivial? In this case, we cannot really find a globally defined one form, which is this connection, so this function does not make sense. And we need some other definition. And the other definition that will be important for us in the following lectures is the following. So we start with our three manifold with some gauge bundle e. And then we extend this manifold in the bulk. What does it mean? That we find some other manifold, some other four manifold, y4, with the property that the boundary of y4 is m3. And then we extend the bundle that we have on m3, we extend it on all y4. And so we have this system, what is this geometry, and we define, which in Simon's functional, as k over 4 pi, in the integral now on this y4 of trace f with f. Now, well, this definition is clearly gauging variant, because f is gauging variant. Moreover, if we are in the previous case, it reduces to the previous case, because if this bundle was trivial, and then when we do the extension also this bundle is trivial, so we can find a globally defined connection, and then you can easily check that this is a total derivative, so you use torque theory, and in fact it's the total derivative of the transformant term. If you want the transformant term is defined in such a way that its differential is given by this object here. So it reduces the definition, but it is more general, because it's varied when the bundle is non-trivial. Well, I mean, so the question was why transformant theory exists in odd dimensions. So if you define transformant theories like something which is written in terms of a, but not in terms of just f, then yes, if you want, these are the descendants, Vess and Zumino, of powers of this f. Now, you might give the name transformant to something even dimension. There are these things, which are related to discrete groups as opposed to continuous groups, so we will not enter into that. Okay, so we have this definition. However, it looks like this definition depends on some choices that we made, because we have to choose this manifold and to choose the extension of the bundle. So we might worry that it's not really a good definition, it depends on extra data. And in order to understand this question we use a standard trick, which is the following. So suppose that we made some choice and suppose that now we made a different choice. Now, this is some Y4 with some E tilde. We made some different choice. Okay, the bundle is the same, but now here we choose some Y4 prime with some E tilde prime, and we get the same. So we want to know what is the difference when we evaluate this function on these two configurations. So what we can do is that we can take this geometry, we do some party or orientation reversal, and then we can glue it here, because they have the same boundary. So we can say that the result is the integral over Y4 plus Y4 prime bar, this is orientation reversal, because the boundary is the same. And now this object here is for pi, the integral on this manifold, which now is a closed manifold, and this object is not zero once again. However, we recognize that this is the instanton number, or if you want, it's the second-gen class or the first-pontriagin class, depending whether the bundle is complex or real. And so in particular this is quantized, it's not the generic real number. This is a multiple of Z, sorry, a multiple of two pi, and so we are in the same situation as before. This is really not well-defined, and when we take the exponential, it is well-defined, and provided once again that this K is integer, this is quantized. No, no, no, no, it's FJGF, so this is, if you want the theta angle in four dimensions. If I write it in components, this would be F mu nu, F rho sigma, epsilon mu nu rho sigma. No, it's not the young Milster, this is topological. In fact, this is an integer. So questions on this? So here, okay, there was a small assumption that we made, that in fact given a manifold and a bundle we can extend the manifold and the bundle. This assumption is not correct in general. There are cases in which we can always extend the manifold, but there are cases in which we cannot extend the bundle. But there is a way to cure this problem, this has been done by a diagram in Witten. The answer is very technical, so I will not go into that, but for us it will be enough to assume that this is possible to do and when it is not, there is a way to cure it. So now we have these theories, we understand how the computation of the loops is hard because this is not a trivial theory and in fact I will describe a piece of the story, a very small piece of the story now, but not the old story. But the important thing is that since this is, if you want, this is a topological theory it can be solved. In this theory we can compute everything we want. But I will not describe, as I said, we can spend months in starting this theory. But I will describe, I will try to describe a small piece of the story. In particular, there is an obstruction, I mean it's a topological condition, I really don't want to spend 10 minutes in this. There is some topological obstruction that you can get using group-comology that tells you when it is not possible to find a man for the extents, the bundle and when it's not possible, there is a way to cure it. It's very technical and for us it will not be illuminating although it leads to very interesting physics when one has discrete groups but we will not go there. Other questions? So somebody asked about how do we compute correlators and I would like to say, before going there we should ask what are the observables in the quantum theory because we only said what the observables are in the classical theory. The first question that I would like to ask is what is the spectrum lines particularly in equivalent lines because the construction that we had before was classical. It's not obvious that different lines have different definition in fact turns out to be the same operator. So we like to understand what are the inequivalent lines and their spin. And the reason why I would like to look at this question first of all because this is the first question before going to the correlation function we want to understand what are the objects that we want to compute correlation functions of but if you want from the physical point of view as you said these lines in fact they are representable line of very massive particles and these spins are the spins of the fractional part of the spins of these particles and so in fact the condensed matter literature these are the anions so we ask in this theory what is the spectrum of the anions so this is a very physical question to ask. Although I will not go to the derivation of this statement but a very convenient and efficient way to make computations these sort of computations is to use a correspondence for us will be is some computational tool a correspondence between 3D and Simon's theory and two dimension like Karel algebras and this was discovered by Whitten and then More and Cyberg although maybe clues of this were before so what is so we talk about in Simon's theory what is this Karel algebra well you should be familiar with this if you study two dimensional conformal field theory because essentially in 2D CFT we have some spectrum of operators we have some fields and these fields in general we have some that depends on z and z bar and they have two conformal dimensions in h bar and now there is a subsektor of these fields in which one and two sides let me say on these sides the dimension is zero let me say that these fields are holomorphic so this Karel algebra is the sector of holomorphic fields we can think of it in terms of holomorphic fields in some CFT and so this Karel algebra will be some set of fields they are holomorphic and they have some dimensions in particular among these fields we always have the identity we always have the stress tensor but then we can have more and what will be important for us these fields are mutually local so what does it mean so these fields since we are in a CFT so there are OPE and of course this OPE is holomorphic already so this OPE is in Thomas lectures but we probably saw it before we have some holomorphic OPE and we want these OPEs to be single valued we will define functions of these complex variables so in particular all these pins should be integer in this way the correlation of the OPEs we will define the single valued and the theory can be modular invariant so in particular for us what this means is that all these pins are integer now once you have a Karel algebra this Karel algebra here there are irreducible representations let me call them h lambda and essentially these irreducible representations you take some primary of this Karel algebra and then you construct all the descendants this is the infinite dimensional representation and in particular we will have some dimension so h lambda will be the dimension of the primary in this representation ok, so let me make this a little bit concrete so first of all where does this correspondence come from and I will give you an example so one nice way to derive this correspondence and I will just give you the physical picture is the following so you take a very special manifold which is a disk times time ok, so this is a manifold with boundary and then you quantize in Simon's theory on this manifold for instance you do canonical quantization we will find some Hilbert space and this Hilbert space turns out to be precisely the 2D Karel algebra you have to use operator state correspondence so you can map states to operators in two dimensions but this Karel algebra arises in this very simple and physical way in fact we can do something else we can also insert a line here h lambda and repeat this quantization and when we do that so if we insert the w lambda and again we compute the Hilbert space now what we get is in fact one of these representations dimension of the primary of this representation mod 1 is precisely equal to the spin of this line so I will not go through this computation it is rather long and technical and difficult but I just want to state this correspondence that as I said it becomes at the end of the day a computational tool so so from the so we associate this tradition Simon's theory to a given Karel algebra and moreover we associate lines so a line w lambda which with spin h to some irreducible representation with dimension so I know that this is fast but the point that I want to make is that now that we know about this correspondence what we can do we can take our favorite textbook on CFTs or Karel Algebras we can take if you want the yellow book of Di Francesco and then we can find what we want about the Chen-Simon's theory particularly we can find what is the spectrum of the lines, the spin and of course there is more in Chen-Simon's theory there is fusion, braiding and so on but everything is contained in this Karel Algebras so we can read off everything from there so let me give you one example so suppose that we take and this is one of the simplest examples so we want to study at some level K where G is a simple connected and simply connected SUN spin n your spin n and so on you know these groups so what is the corresponding Karel Algebra the corresponding Karel Algebra is just the enveloping algebra the smooth current algebra or if you want this is also known as the affine the algebra at level K so this is something that we know very well and so as I said we can open the yellow book of Di Francesco and we can read off what are sorry compact and we can read off what are the lines and the spin in this theory in particular how what are the representations in this theory and the representation of this Karel Algebra or affine the algebra we want to classify the representations so we start with the dinking diagram the extended dinking diagram in this case so let me do for instance SUN we have to extend the dinking diagram this will be the dinking diagram of SUN we add one more node let me say N plus one so if the rank is N and then if you want to classify the representation we have to provide non-negative integers these are the dinking labels so we have to choose some lambda 0 lambda 1 and so on up to lambda N there should be no negative because it's only the dominant weight that give your representation however in this extended lambda 0 is fixed in terms of the other ones lambda 0 is K minus minus lambda this product between the weight that we are using the highest weight of the representation and is the highest root but this should be non-negative and so we get a constraint that this lambda theta should be in so in particular these are called integrable representations the integrable representation in this SUN plus one K-Chang-Simon's theory that means the lines here corresponds to representations of this group such that they satisfy this bound and in particular they are finite there is a finite number of them so even though classically we thought that there was an infinite number of operators we didn't have these representations but in the quantum theory it's just a finite number ok ok, I should probably stop here as it means that it's a transformational theory classic that does not depend on the metric it's completely invariant under different morphisms even though it's not a gravitational theory in the sense that you integrate where the metric it does not depend on it however, so classically this is so but when you do the quantum theory you have the standard thing that you find with anomaly that you need to regularize that you can break this symmetry because gauge fixing requires a metric and you almost recover it with the suitable choice of local converters but not completely because the final answer that you get still is not for the most part does not depend on the metric but there is this dependence on the framing that I didn't say what it was because it's not particularly relevant for us this choice of the trivialization of the tangent bundle but the answer depends on this extra data so this is the symmetry that is broken so we can start I think we have the lecturers of the day Francesco Tom and Guillermo is there and Marina so you can start asking questions just can you go through the last step of this song