 Today I'll continue my discussion of particle vortex duality and then we'll venture into more modern territories, where I'll teach you a little bit Schoen-Simon theory and we'll discuss how particle vortex duality is modified for Schoen-Simon theories. Since we have two more minutes and I see people are flowing in, maybe I will start only, because there's a constant flow, so I'll just, we have two more minutes anyway. We can start with some questions, if there are any questions about the previous, I won't start before 9th, yes, yes. The question is if we are really 100% sure that this is a conformal filter, meaning that this transition is second order, because the question is if the lattice can distinguish between very weakly first order transitions and second order transitions. So the answer is that in general a very, very weakly first order transition can look arbitrarily, much like arbitrarily close to a second order transition, and therefore you can never be sure and there are like well-known examples from the, I guess 80s or 90s, lattice models that have such a long correlation lens that you might think it's second order, but it's in fact first order. And sometimes this correlation lens can be like over thousands of lattice sites. So the answer is that we are not, I wouldn't say that we are 100% sure, because the lattices are always finite, but in this case there would be, I guess, little doubt. There isn't being the following. There isn't being that there are also ways to establish the existence of this conformal filter using the epsilon expansion or using the bootstrap, and you get some numbers for the critical exponents. And if you look at the lattice, then it gives the same numbers, and this precision now is over a few digits, so it seems to be in the same universality class that can be seen in the bootstrap and that can be seen in the epsilon expansion. So for this reason you are quite sure, yeah. In other cases for very weak first order transitions, like the lattice may give you some numbers, but these numbers keep changing as you increase the lattice, and also you don't find this kind of universality classes from the bootstrap, so. Any other questions? I'll try to adjust it somehow. Yeah. Any other questions before I begin? So should I say everything again? Any other questions before I begin? Yes. Okay, so the question is about treating lambda as a perturbative parameter. So here I want to just make some general remark, since I have, like, anyway, I started with some, there's one general remark, and then I'll get back to your question. Suppose that, you know, when you study quantum filter, you often study this model. This is like the canonical example in pesky and shredder for some model which you can treat with perturbation theory. Now in four dimensions, this makes sense because we know sort of retrospectively that lambda becomes a, sorry, that lambda becomes weaker and weaker as we go to longer and longer distances because there is a beta function for lambda and lambda equals zero is an infrared, lambda equals zero is a good attractive fixed point in the infrared. So in four dimensions, when you do perturbation theory for this model, your predictions at long distances become more and more precise, excessively precise because of this fact. But in three dimensions, it's not going to work. Suppose you try to do perturbation theory for this model in three dimensions. So we can understand what's going to happen from dimensional analysis. That should explain why you cannot analytically understand the region around this model, at least with existing techniques. So the dimension of lambda, let's start from the dimension of the field. In three dimensions, the dimension of phi is just a half because then you get two derivatives and two halves, so you get three. And therefore, the dimension of lambda is one. Now suppose you computed the correlation function, you try to compute the correlation function phi of x, sorry, phi of x phi star of y. Okay, that's a correlation function that's supposed to be non-vanishing because it's like phi and phi star. And you can try to compute it using perturbation theory, like in pesky and Schroder. So you would have a propagator that would be your first approximation. So how does the propagator look like in three dimensions? Since phi has dimension a half, the propagator has dimension one. And so the propagator, this is the propagator. That's gonna look like one over the distance to absolute value, in absolute value, okay? So that's how the propagator is gonna look like just from dimensional analysis. And then the next order correction would be maybe some kind of loop like this. That's gonna be the next order correction, but you can also estimate how it's gonna look like from just dimensional analysis. And like in four dimensions where there are logarithms, here everything is fixed by dimensional analysis. So this is gonna look like lambda squared over now we need some power of x and y. So first of all, the dimensions of the fields. So the overall dimension still has to be exactly one. But now lambda has dimension one. So we need several more powers of x. So one way to write the answer would be to put x minus y in the denominator. But then x minus y, a school word in the numerator, okay? So that's gonna be just from dimensional analysis, that's how it's gonna look like. And it will continue in this fashion. And what you see is that every diagram is more important at long distances than the previous diagram. So as you take the distance to infinity, which is the long distance limit, or this is the limit in which we can understand the vacuum. Or the conformal field theory or the second order, first order transition question. So these questions are about long distances. So as we take the long distance limit, every term is more important than the previous one, and therefore perturbation theory is entirely useless. Unlike in four dimensions, where using the beta function techniques, you can resum the logarithms and show that perturbation theory converges at long distances, because it's an infrared free fixed point. So the reason that it works in four dimension is really detailed. It's because of some sign in some beta function where you resum logarithms. But in three dimensions, there's no way, it just doesn't work. So clearly, normal perturbation theory is completely useless here. And that's why we cannot understand, we have to resort to the lattice to tell us whether it's a second order or first order transition. And we need to use other techniques to understand the critical exponents of this O2 model, the scaling exponents. We cannot do it analytically. Any other questions? Yeah, okay, I'll say again, okay, so this has nothing to do with that. What this is saying is that perturbation theory is useless. It doesn't say anything about whether lambda is relevant or irrelevant near the fixed point. In fact, it says that you cannot learn anything about the fixed point from ordinary perturbation theory. Before I answer your question, just one more remark. Here I did not have a mass, but suppose I did have a mass. I told you something I told you before, in the previous session, was that if you are far away from this fixed point, then you can do perturbation theory. And that's how we were able to determine the asymptotic phases, even without the lattice. And the reason that that works is that if you have a mass, then these propagators do not look like that. Instead of some power loss here, you would get exponentially decaying functions. And the exponential, so instead of x minus y squared, you might get something like e to the minus mx minus y. And if you replace this by an exponentially decaying function, at long distances, you're good. Because then the long distance limit exists and you can take it and you can do everything in perturbation theory. Maybe you should think about it as a little homework exercise. It's just something to think about. Of why in the presence of a mass, this issue doesn't plague the problem. Large masses, you can really make progress. And it's essentially, for the technical reason, it's dead. And the conceptual reason is that the model becomes the model trivializes at energies much above the scale set by lambda. Now to your question of why we have only one axis here. In general, in quantum field theory, we have many coupling constants, say g1 and g2. Right, and there is some kind of our g flow and there may be fixed points. So there may be some fixed point. And there should be a fixed point. Now if the lines all go towards the fixed point, we call both g1 and g2 relevant parameters. But it can also be that the fixed point would look like, in some direction, the lines would go like this, but in some directions they would go out. So there might be a flow like that, okay? So I'll just draw it carefully, something like that. This is another option for how the lines may look like. So one direction will be attractive, namely this one. But one direction will be repulsive, namely this one. So the repulsive direction is called an irrelevant direction. And whether or not the direction is repulsive or attractive depends on the eigenvalues of the scaling operator of the conformal field theory. So this is the dilatation operator. So to understand if some operator is relevant or not, you just have to compare its dimension with three. If it's smaller than three, it's relevant. If it's bigger than three, it's irrelevant. So this is relevant and this is irrelevant. Okay, so if some direction is irrelevant, it means that it's a repulsive direction. Sorry, that it's an attractive direction. So I may have said something wrong. Attractive directions are irrelevant and repulsive directions are relevant. Write it down for the record. I may have misspoke. Attractive means irrelevant and repulsive relevant. So when we draw the face, so when we discuss the number of relevant perturbations of some conformal field theory, we only count those which are repulsive because they may take you somewhere else to a new phase. And indeed, when you add this mass forward, which is irrelevant perturbation, they take you to new phases, like a compact scalar, a goldstone boson, or a trivially gap phase. But in this O2 model, when people computed the scaling dimensions, let's say on the lattice or on the bootstrap, they found that the coefficient of pi to the four is an attractive perturbation. So if you change lambda, you're not gonna get new phases because it just decays to the same. You can change lambda, but then the RG lines will take you back. You won't get a new phase because the lines converge towards that phase, okay? It's just a numerical fact that the O2 model has just one relevant perturbation which preserves all the symmetries. Any other questions? That will be the case in fact for all the models I'll study today. In all the models that we'll study today and tomorrow, it's believed that pi to the four is irrelevant. Yeah, any other? Yes. Well, they have some thickness which some characteristic thickness, but if you go to very long distances, then they look like they're one dimension one objects. Is that the question? Okay, it's a long distance approximation that they look co-dimension one. That's right. So yeah, if you wanna be very gross, you could have added here also pi to the six. Okay, now let me just say a few words about it, but then it won't be for the models that we discussed today and tomorrow. It is indeed true that if this was relevant, so the number of, okay, let me just say something general and then I'll get back to it again. So when you try to look for a fixed point, the number of couplings that you have to tune is the number of relevant perturbations. So if you're an experimentalist, the number of knobs that you have to tune is the number of relevant perturbations. The number of knobs is the number of relevant perturbations. And that's again because you don't need to tune couplings which are attractive because they anyway go to where they need to be. But the ones that take you away from the fixed points, namely the repulsive one, the relevant perturbations, they need to be tuned. Otherwise you won't end up at the fixed point. In this example, for instance, if you don't tune your location on the horizontal axis, sorry, on the vertical axis, you might end up elsewhere because the lines go outside of that point. So the reason that I'm not writing lambda prime five to the six is because both this and this are irrelevant in this particular model and also in all the models that we'll study in this course. And so we don't need to write. We can also write five to the eight in principle dead male and so on and so forth. But in this model, there's just one symmetry preserving relevant perturbation. So that's why I'm not writing it. However, there are other fixed points, perhaps where this coupling becomes relevant and you might want to study them. But for today, it won't be an issue. It's just a numerical fact that there's one symmetry preserving relevant perturbation. And that's why I truncated the Lagrangian to five to the four and that's it. Any other questions? Okay, so let's get back to speed. Yes? That's right. It's relevant at which fixed point? It's relevant in the fixed point around of the free scalar field theory. Another interesting fixed point in this, an interesting fixed point is just the free field theory where you don't have any masses or lambdas. This is a good conformal field theory, just one free complex scalar. That fixed point has two relevant perturbations which are symmetry preserving, which are five squared and five to the four. That's why in order to get to the free field theory, we need to tune how many knobs, two. We need to tune both that and that. But there is a better fixed point in the space of fixed points of a complex scalar, which is the O2 fixed point. In that fixed point, there is only one relevant perturbation, so we need to make less tuning. So if you're an experimentalist and you have a system with O2 symmetry, you're much more likely to end up here than in the free field theory model because the free field theory model is more fine tuned. It has two fine tunings in the language of particle physics phenomenology. We need to make one fine tuning to get to O2. Indeed, we just need to change the mass. So if we fine tune the mass, we landed the fixed point. But to end up at the free field theory point, we have to make two fine tunings. So if you wanted to make this a little bit more manifest, you would add another, I don't wanna destroy this drawing, you could add another axis, let's say lambda, and repeat the phase diagram. So you would say, oh, okay, here it's trivially gapped. You know, in this whole region, it's trivially gapped. And, well, lambda should be positive. I don't wanna discuss negative lambda because that will be a sick model. So here it's trivially gapped and here maybe you have number goldstone bosons. But here there will be an O2 model, here there will be an O2 model and here there will be an O2 model. That's why I'm saying I'm not including lambda because it's gonna be O2. The transition between this side and this side, if you scan horizontally, it's always gonna be the O2 model because lambda is attractive. But what you said means that if you go, sorry, if you go, so there will be O2 everywhere here, but if you go to this special point and it's only one point, it's co-dimension two rather than co-dimension one. So here there is O2 everywhere except for this one special point where you get free field. That's why I'm not drawing lambda. I'm sitting at some generic lambda which is non-zero and then I hit an O2 fixed point when I traverse this, the phase diagram horizontally. Except if I was here, I would go through free field theory but that would be a very special case, okay? So yeah, that's the technical meaning of irrelevant. That basically there is another axis but nothing happens on that axis except for maybe one point. Any other questions? So we get back to this model. There are these two phases, the symmetry broken phase where the Goldstone boson is gonna be denoted by theta and the O2 fixed point and then there's a trivially gap phase. I just wanna make a few remarks about it and then move on to gauge theory to the duality. So we have an operator in this model which is phi. Here I put a little head over phi to indicate that now it's an operator that acts on the Hilbert space. So in the broken phase, in the number Goldstone phase, if phi acts on the vacuum, we get a Goldstone boson, namely the theta particle. By contrast, in the unbroken phase, if you act with phi on the vacuum, you get a massive particle which is stable because this massive particle is protected by the O2 symmetry. It's just an ordinary representation of the Lorenz group which would look like a one particle state in the theory in the O2 representation. Okay, so you just bury that in mind. If you read the literature on superfluids, which involved this model, you'll also see another very important character in condensed matter physics. It's an object that's not yet on the blackboard, but it will be very important soon when we discuss gauge theory. So there's another character in this movie about the O2 model, which is the vortex. And I wanna talk to you a little bit about this vortex. That's something that condensed matter people call, that's what they call the vortex in the superfluid. And there's a huge amount of literature about it, but it's an extremely subtle object and it's not very often discussed in the literature. So I want you to pay attention because the situation with this vortex is very subtle. So what is this vortex? So first of all, the vortex is never discussed in the unbroken phase. The vortex is an excitation that people discuss in the broken phase. Okay, so it's supposed to exist here in that broken phase, it will be quite murky to try to define it. So what you do is the following. You pick a point, let's say X zero, and you try to construct a field configuration, which minimizes the energy, such that around X zero, the theta field, which is now a periodic field, which is a periodic field because it started it's life as the phase of five. Under goes the monogamy. So theta under goes the monogamy around this point. So theta goes to theta plus two pi when we go around X zero. So this continue because there is such a monogamy, it goes on forever. Any distance from the point X naught, there will be some vorticity. That's why it's called the vortex. It leads to some vorticity in the flow of five. Now here I'm describing it with theta, but I don't have to. I can describe it with five in the broken phase. Five is just some, five is given by some real part times e to the i theta. But in the broken phase, the mass squared is negative and the modulus of five is more or less stable. The modulus of five stabilize that v squared where v is some parameter related to this f pi squared here, which is the coefficient of the action. So far away from the vortex, this is approximately, this is far away from the vortex. This is gonna be approximately f pi times e to the i theta because the modulus of this field is stabilized to be more or less at the vacuum and theta undergoes this monogamy. So you can really think about this configuration as the phase of the complex field that we started from undergoing a monogamy. Now, if you look near the core, near the core, this description is not very good because near the core, it makes no sense. It's a singular configuration. You cannot have this thing winding near the core. So what happens near the core is that five just goes to zero so that the configuration is smooth. Near the core, the whole field goes to just zero because that's the only smooth way to cap it off this solution. Or more pictorially, if you imagine this, the potential in the broken phase is like a Mexican potential because the mass core is negative. And so what happens around the vortex is that you go around the Mexican potential but when you get closer and closer to the core, you climb up this hill until you go to zero. That's zero. Okay, so you have something that's just swirling around but when you get very close to the core, it goes up to the top of the hill. But away from the core, it's just swinging in the bottom of the potential because that's what it wants to do. So if you look at papers on superfluids and on this model, you'll see that these things appear in the lab. They see vortices. Nobody in his right mind disputes the fact that they exist. They can be seen experimentally, this vortices. But there is a subtlety about this vortices which is that their energy is infinite. So let me try to explain why. Let's try to estimate the total energy of such a vortex. Let's assume that you have one isolated vortex, no anti-vortices and space is infinite. So the energy is given by an integral over x. Now it's a two-dimensional integral since we're integrating just over space. And I'm just gonna try to estimate the contribution far away from the vortex. Just to see what is the contribution from far away from the vortex where the particle where this phase just goes around the circle. So the contribution is just gonna be d2x and then we have the gradient of the field theta squared. So we don't need absolute value, just that, okay? That's the kinetic energy stored in the vortex. And this is d2x. Now this only has dependence on the angular variable. So we have one over x squared and then we have a derivative of theta with respect to the angle. But we can just assume that the derivative of theta with respect to the angle is one. Since as we go around the angle, the polar angle, this is the polar angle. As we go around the polar angle, theta goes to theta plus two pi, so this is just one. So therefore we get d2x over x squared and this is the logarithmically divergent in the infrared. I'm not talking about the UV divergence. There is no UV divergence because near the core, the thing is perfectly regular and finite. This is an infrared divergence. So this is like the logarithm of the size of space. So L is like the size of space. So it's gonna be the logarithm of the size of space or L is the size of the box. So what does it mean? What this story means is that there is no way to create a single vortex. It just it costs infinite energy and it's an infrared divergent object. Now in condensed matter systems, it's often in a box. So it's a finite large number, but it's a logarithmically large number and it still can be created with some finite investment of energy. Second of all, sometimes you see vortices and anti-vortices. When you have a total number of vortices and anti-vortices that combines to zero, you can have them separated. So you can have a configuration where there's a vortex here and an anti-vortex here with opposite swirling and then they glue to each other in some way, okay? They glue in some way. And that's finite. If you compute the total energy here, it would be finite. And we'll do this computation now. We can try to understand what happens to two vortices, vortex and anti-vortex when they are together, okay? So when these things appear in the lab, it's always in the context of finite space or in the context of additional vortices that are hiding somewhere. But in quantum filtering, strictly speaking, there's no such thing as a vortex particle. It doesn't exist because it has infinite energy. In the thermodynamic limit, there is no such thing. I'll just summarize it. There is no such thing as a vortex particle. It's an approximate concept that applies in a box or when there are other particles. But let's do a small exercise. This is to motivate what I'm gonna do next. Let's try to compute the energy in the system of one vortex and one anti-vortex that is very far away. Now that's gonna be finite. So let's have a vortex here. So that's gonna be clockwise. And then let's have a vortex here and that's gonna be anti-clockwise. And then the fields are supposed to reconnect in some way. Now the total vorticity vanishes and the field is gonna be regular if you are very far from the two vortices. There is no vorticity very far from the two vortices. And let's assume that the distance between them is L. So now L is not gonna be the size of the box, but it's gonna be the distance between the vortex and anti-vortex. So by the same computation as we did before, it's clear that the energy in this configuration goes like a logarithm of L. Where L is now the separation between two entities, more or less well-defined entities with a center. Which force does this derive from? If you wanted to understand the force between the vortices, which force gives rise to a logarithmic energy? What is the force that leads to energy that goes like log L? One over L, right. The force that gives rise to such an energy is one over L. And now here, this should ring a bell. Do you know of another system in two plus one dimensions which has forces that go like one over L? Does anybody, right, somebody said Coulomb, correct. So that's the starting point for the duality. That there are these objects which are called vortices, but they're not really well-defined because they have infinite energy. Yet they kind of exist and they can be seen in the laboratory and they have a Coulomb force, okay. So in fact, if you have charges, we'll discuss now charges. We'll switch to gauge theory. And the same comments that I made for vortices, they apply it for charges. Only when we have charges we say that we are not allowed to observe charges because of gauge invariance or whatever. But here it's very specific. There is no gauge symmetry. And the gauge symmetry almost like appeared out of the blue. So the idea here is that since the force goes like one over L and it reminds us of the Coulomb force, we can say that somehow the gauge symmetry appeared out of the blue. The idea is to try to interpret these vortices in a completely dual language with different fields as the charges, as ordinary charges under in electrodynamics in two plus one dimensions. And in fact, the idea would be to not just make it work in the broken phase, but to make it work everywhere. So that's what we're gonna do next today. Okay. Are there any questions before I switch to gauge theory? Yes. I just couldn't hear anything. What you're asking about, okay. In the original model, there is O2 symmetry. In the broken phase, the SO2 part of the O2 is completely broken. So as we said, O2 is broken to Z2. This symmetry is a, this is spontaneous breaking. This is not explicitly breaking. This is spontaneous breaking of O2 to Z2, or SO2 to nothing. This unbroken Z2 acts on theta by conjugating theta to minus theta. So theta to minus theta is the unbroken Z2. Any other questions? Yeah. F pi squared is just a dimension full coefficient. So that this action makes sense with a dimensionless two pi periodic scalar field. So F pi squared has dimension one. In condensed matter literature, it's called the stiffness constant of the superfluid. In particle physics, it's called the decay constant of the pion. That's why I call it F pi squared. It reminds us of the pion, of the decay constant of the pion. Any other questions? Yes. What do you think? It's a great exercise. No, take the analogy with electrodynamics seriously. Attractive, right? So two vortices are repellent, two vortex and anti-vortex, they attract. Well, it's obvious that they attract because it costs a lot of energy for them to be separated. But if you brought them together, you could just annihilate them because it's a vortex and anti-vortex. If it was vortex plus vortex, it wouldn't make sense to bring them together because you just get a double vortex. But for these guys, they obviously want to come close and to vanish into thin air. So it's an attractive force. Any other questions? It's very easy to show it by just solving the classical equations. It's a very nice classical PD exercise to try to construct approximately dissolution and check the energy. Okay. So now we switch to gauge theory. This analogy is very tempting. And so people have started studying gauge theories in relation to this model. So gauge theory in two plus one dimensions is quite different from gauge theory in three plus one dimensions. So I'll start by just doing free gauge theory, free U1. So that's free U1. Let's start by doing free U1 gauge theory. Now there are no dynamical charges. We may have some probe charges, but no dynamical charges. Let me just make sure that my conventions would be compatible with what I need. So free gauge theory in two plus one dimensions has the following Lagrangian. And AMU, the U1 gauge field. It's the same Lagrangian as in three plus one dimensions, but its physical consequences are quite different. So here there are some, there are many things to say about it. Well, let's start from the behavior of probe charges. And then we'll discuss the content of the quantum theory. And then we'll discuss the monofil operators. And then we'll add matter fields and exhibited duality. So let's start from probe charges. So the charges that we're going to discuss now are non dynamical, meaning there are classical charges that we can put in and we dictate how they move. They don't fluctuate. So there are a couple to the gauge theory as always through some matter current. But this matter current is classical. That's what it means for the charges to be probe charges that this object is entirely classical. It's a classical function of space time. It's not fluctuating and it's not subject to pass integration. Okay, so we can put some positive charge, let's say plus E and we can put some negative charge minus E and of course since the Maxwell equations, I mean the Euler Lagrange equations here are the same as the Maxwell equations. The Maxwell equations or even some static version they're off would lead to a force that goes like minus one over L so that you want to decrease L. And if the charges are alike then the force would be with a plus sign. So plus minus according to whether it's plus minus, it's plus plus, so if it's plus plus we get this and if it's plus minus we get this. And if it's minus minus of course we also get a plus. So this is the force and therefore the energy is logarithmic exactly as before. But this has a dramatic consequence. This is not, this makes these models in two plus one dimensions entirely physically different from three plus one dimensions because there is confinement already classically. So there is already classical confinement in two plus one dimensions. Don't even need to do quantum stuff to see confinement. There is already confinement classically. Unlike in three plus one dimensions it's impossible to create an isolated charge in two plus one dimensions even classically because of the exact same logarithmic divergence that we started here. Remember that the energy that is stored in the electromagnetic field, the energy that is stored in the electromagnetic field is this squared x times the electric field squared. So if you have one isolated charge this is gonna be this squared x one over x squared. And this is logarithmically divergent in the infrared of course. It's also logarithmically divergent in the ultraviolet but we don't care about ultraviolet. The ultraviolet is just like the core of this charge. And at the core who knows what it is, right? At the core there could be something complicated and maybe the divergence is resolved. But infrared divergences are incurable. They're much worse than UV divergences because infrared divergences are in a region where you are supposed to control the physics. And so if they exist they're incurable. Well UV divergences are always curable because they pertain to some details that we haven't specified. Well infrared divergences are robust. And so there is classical confinement. I'll write it again. And therefore the only observable objects have net zero charge. And that's already something that's in good analogy with the story about vortices and anti-vortices. That's comment number one about electrodynamics in two plus one dimensions. Notice that in three plus one dimensions which is the universe, our universe, we put D3X and therefore this is perfectly convergent in the, we put D3X but then this becomes one over X to the four and it's perfectly convergent in the infrared. However the divergence in the ultraviolet. That's what happens in our universe, right? In our universe if you do the same computation you get one over X to the four. And this is convergent in the infrared but divergent in the ultraviolet. But as I told you, we don't worry about ultraviolet. This diverges in ultraviolet is just the self energy of the electron that's regulated by quantum fluctuations. Who cares? There is some Compton wavelengths that regularizes it. So two plus one is really very different conceptually. It's not just that the Lagrangian is the same but the physics is totally different. Okay, that's comment number one. Comment number two about two plus one dimensions is the number of degrees of freedom. Now anybody remembers how many degrees of freedom does the photon have? In three plus one, how many in three plus one? How many degrees of freedom in three plus one? Two, what about two plus one? How many degrees of freedom is the photon supposed to have in two plus one dimensions? One, right. So we have one massless degree of freedom, right? That's what this model describes. One massless degree of freedom. Now because it's just one massless degree of freedom we know another model that has one massless degree of freedom. We just wrote this model down. There is another model that has one massless degree of freedom which is just this. Free scalar field, theta. Now it was compact, I'll get to this in a second. The compactness, just forget about it for a sec. I'll get to the compactness in a sec. But just as far as counting degrees of freedom goes and the dispersion relation, the photon in three plus one dimensions has electro and magnetic fields that are fluctuating together. But in two plus one dimensions there is only one depolarization. And so it seems very similar to just a free relativistic scalar field. And in fact it's exactly the same. That's another piece of this duality story that just one massless photon without any propagating charges is exactly the same as the free scalar field. So let me just write it down carefully and explain why. So I'll show you the derivation. Now I'll show you the derivation of this fact. Let me just write down the claim and then we'll do the derivation. Claim, the massless photon, that's also very different from three plus one. The massless photon in two plus one dimensions is the same as the free real scalar field. There is this issue of compactness which I skimmed over. I'll get to the compactness soon. But the field content is just a free real scalar field. So how do we prove this? So let me just sketch the proof. So we define as usual Fp nu, Pd mu, a nu, anti-symmetrized, sorry. The pass integral that we started from the gauge theory pass integral was a pass integral over all the possible gauge field configurations, module gauge transformations of the exponential of Fp nu squared with some one over two g squared. I'm not gonna do this derivation with all the factors because they're not crucial, it's just, it's a conceptual little idea. So this is the pass integral of gauge theory and we can compute many things here. And this pass integral is kind of tricky because we're integrating over the gauge field but what appears in the exponent is the field trends. If we could replace the pass integral over the gauge fields by the pass integral over F, then this would be a Gaussian integral and it would be boring, right? So the idea is to try to make it look like a Gaussian integral. Why aren't we allowed to do it in the first place? We're not allowed to do it in the first place because F mu nu is the constrained object. We cannot say that this is the same as the pass integral over all the possible Fs because F is constrained. What is the constraint that F obeys? F obeys a constraint which is the Bianca identity. So if we take rho, mu nu rho, this vanishes. And this is exactly the reflection of the fact that F is a derivative of something. That's why we need to integrate over that something rather than over F directly. So it obeys this Bianca identity. So the idea is to try to enforce the Bianca identity by a Lagrange multiplier. And then maybe we can do a pass integral over F. So the idea is to produce another field B and another and integrate over F mu, okay? So we can do an integral over F mu but we need another Lagrange multiplier field B. So now the exponent looks like F mu squared as before with one over two G squared and an integral. But now there is some B which couples to F mu nu like this with epsilon mu nu rho. Or I could even replace this B by a derivative of a scalar. So let B be a scalar and I'll put the derivative here. That will be even better. So if you move the derivative to act on F, so now F is un-constrained because we're pass integrating over F. If you move the derivative to act on F and you integrate over the Lagrange multiplier you get a delta function. And the delta function would be that of the Bianca identity and you'd get a pass integral over all the possible field configurations that obey the Bianca identity. And that's just gauge theory. That's a rephrasing of what is gauge theory. So that's a useful trick. When people do T duality in sigma models, mirror symmetry and many other little such things they rely on such a stupid trick on trying to replace the Bianca identity with a Lagrange multiplier. So one side is clear. If we first pass integrate over B we get back what we wanted. But now we reverse the order of the derivation and we pass integrate over F and then we get some theory for B. So now we do the pass integral over F first and we get some theory for B. But now the pass integral over F is entirely trivial because F appears quadratically. In fact, there aren't any derivatives for F. There isn't even propagation for F because F now appears like an auxiliary field, so to speak. So we just complete it to a square integrate and we get E to the d mu B squared. With some coefficient here that would be like G squared. So that's a formal proof that they, yeah, there is a question about compactness. I'll get to it soon. I'll get to the question of compactness soon. Just give me a sec. But if you just ignore the global issues about this compactness, this is a formal derivation showing that the local properties of a gauge field in two plus one dimensions, U one gauge field are the same as that of a scalar field. This derivation does not work in three plus one dimensions because you cannot use this term. So in three plus one dimensions, it's not true. If you try to repeat this thing in three plus one dimensions, you'll get the statement of duality. Self electromagnetic duality that Maxwell found. The electric field and the magnetic field look the same. But in two plus one dimensions, this is what you get. Now there is a question of duality. So of the compactness of the scalar, which is very important. Remember that here the scalar was compact, so it better be that this scalar B that we use there is also compact. So let me show where the compactness comes from. When we solve the equations for the field F, if you just complete it to a square and you solve for the field F, what you get is the following equation. F mu nu, one over two pi. The epsilon mu nu rho d rho B. That's the fundamental identity here. Sometimes this is called Poincare duality. Sometimes this, yeah. So this is sometimes called Poincare duality. So that's the dictionary between the scalar field B and the gauge field F. That's how the duality works. So this equality is not really equality, it's more like duality. An important fact is that this gauge theory is a U1 gauge theory. So the fluxes are quantized. In U1 gauge theory, the flux through a two surface must be quantized in integer units. So the left-hand side is quantized and therefore the right-hand side needs to be quantized. And the compactness comes from this quantization condition. So for U1, so if you are doing a U1 gauge theory, theta needs to be identified with theta plus two pi. This massless real scalar field. Some people like to do what they call R gauge theory, which is a non-compact gauge theory. In non-compact gauge theory, then this is not true. So this really depends on the global properties of the gauge theory. And the way to understand the periodicity is by looking at this equation. By requiring that the two sides are properly quantized, one integrated over two manifolds. So the compactness is a little bit more mysterious, but it comes from the quantization of the flux. Okay. Okay, now we're discussing the U1 gauge theory in two plus one dimensions. We just argued that it's the same as a free scalar field. So we have this equation, that one over two pi F minu is epsilon minu rho d rho theta. And this is our, but now there is something really interesting, that one should understand. Remember that in the language of the massless scalar field, we just have a compact scalar that sits on a circle. And it's claimed to be the same as a free gauge field. But this appeared in the Goldstone phase of the original model. So this corresponds to symmetry breaking. When you see a compact scalar, you think about it as the arising due to symmetry breaking. And the question is, which symmetry are we talking about? Because in the gauge theory language, which is the symmetry that's supposed to be spontaneously broken? We're basically claiming that the free gauge field in two plus one dimensions is the same as the Nambu Goldstone boson. But Nambu Goldstone boson for what? Does anybody see a U1 symmetry here? It's not obvious, right? So the idea that the gauge field in two plus one is the same as the free, as the Nambu Goldstone boson for some U1, for U1 global symmetry. But which U1? There is a U1 gauge symmetry, but that has nothing to do with the U1 that we're interested in. We're interested in a global U1. So indeed, it turns out that there is a U1 symmetry. No, no, it's an ordinary U1 symmetry. So there is a U1 symmetry in free electromagnetism in two plus one dimensions. And the conserved current is this, mu, nu, rho. Consider this expression, or F mu nu is made out of the gauge field. Oh, this should have been A. I made a mess. Consider this gauge field, A mu, of which the field transfer, of which the field transfer is made. So F mu nu is just the mu and nu anti-symmetrized. So this looks like a nice conserved current. Indeed, it's conserved by virtue of the Bianchi identity. Right, that's nothing but the Bianchi identity. So that's another big difference from three plus one dimensions that just the Bianchi identity leads to an ordinary U1 symmetry. And the nice thing is that this symmetry is broken in the vacuum. It turns out that this mysterious symmetry of the free gauge field is broken in the vacuum. And that's made manifest by this dictionary because the model is equivalent to a Nambo-Golston model. So this symmetry is called in the literature magnetic U1 symmetry. That's how people call it in the literature. It's called the magnetic U1 symmetry. That's an important piece of terminology. But in two plus one dimensions, every U1 gauge field comes equipped with a magnetic U1 symmetry that has to do with the Bianchi identity. And furthermore, it's spontaneously broken in the Coulomb phase. That's another phrase that you have to put in your brain. The magnetic U1 symmetry is spontaneously broken in the Coulomb phase. The Coulomb phase is the phase where the gauge field is propagating. Okay, good. And now the last thing that I wanna discuss is the question of the order parameter for this breaking. Whenever we have a broken symmetry, we need to identify an order parameter. So that's the last thing I wanna say about free gauge theory. The order parameter for the symmetry breaking. If you have a compact scalar field, like theta, it's very easy to write an object that transforms linearly under the spontaneously broken symmetry and that has an expectation value in the vacuum. So in the language of theta, it's really easy to write the answer. It's e to the i theta. e to the i theta is a nice local operator that transforms linearly under shifts of theta by a constant. It transforms by phase. And this is the order parameter. But it would be really nice to be able to write this order parameter using the original gauge field, a. So the dictionary between the two models does not allow us to do that yet because we only have a dictionary that tells us how to relate derivatives of theta to the field strength. But how do we relate the exponent of theta to something? And the answer is that it cannot do it really easily. There is no simple expression that allows you to write it down. Instead, there is an abstract object that's introduced, which is called the monopole operator, m. And this is what maps to e to the i theta. So U1 gauge theory has local operators which are called monopole operators. They are charged under the U1 magnetic symmetry. And they are the order parameters for spontaneous symmetry breaking. So you see that the phenomenology of two plus one dimensional gauge theory is very different from three plus one. There are all these bizarre objects that appear. And now I just wanted to tell you how this monopole operator is defined. So the hint, so the way this monopole operator is defined, let's say we have a monopole operator as a point x, x zero. So we take a small sphere around this point. We pull it out of space. This is how the construction of the monopole operator works. And there is no way to construct it using the elementary gauge field A. You cannot write it explicitly. You have to define it through these boundary conditions in space. So we take a small, small sphere, we take it out of space, and we impose that the integral of DA over this small sphere is one, is two pi. That's the minimal allowed flux. So you take a tiny, tiny sphere and you impose that there is like flux through that sphere. And that defines a local operator that sits at the center of that little sphere, x naught. So that's how this monopole operator is defined. Now, it's easy to understand why it works. Because the whole point of this monopole operator is that it's charged under U1 magnetic. Charged under the magnetic U1 symmetry. Sometimes it's called U1M for magnetic. And indeed, the magnetic symmetry, as you remember, was due to the Bianca identity. Now, in the presence of a monopole operator, there is a delta function. Because the Bianca identity is violated by magnetic charges. Remember that in Maxwell's original equations, this was vanishing. But if you had magnetic monopoles, then there will be some delta function here. And this is just from the Bianca identity. So similarly, when there is a magnetic monopole or a monopole operator, this has a delta function on the right hand side. And that just means that the monopole operator carries U1 magnetic charge. Or carries charge under the U1 magnetic symmetry. So our dictionary now is complete and we can go to a break. I'll just write the dictionary, summarizing it, and then maybe questions and we'll go to a break. And the next, after Kiriakos' talk, we'll then do the duality. So the dictionary. That's the dictionary. The Lagrangian of T mu, A nu squared, anti-symmetrized, one over two G squared, is quantum mechanically the same as the Lagrangian for a compact scalar field. So this is one piece of the story. So a photon is the same as a compact scalar field. That's entry number one. Entry number two in the dictionary is that the field strands, namely the electric and magnetic fields, they map to derivatives of theta. So they map to a epsilon mu nu rho, d rho theta. The U1 magnetic symmetry maps to the spontaneously broken U1 on the scalar side. And finally, the most mysterious part of the dictionary is that monopole operators, which create magnetic flux at some given point in space, they map to e to the i theta. And the last thing I wanted to say about these monopole operators is that therefore they condense in the vacuum. So in the Coulomb vacuum, the monopole operator expectation values are in on zero. So there are the order parameters for the spontaneous breaking of the magnetic symmetry that's generated by the Bianca identity. So this is the dictionary between free gauge theory and the free compact scalar. Just a second. So this leads to some good understanding of the broken phase of the O2 model. We see that the broken phase of the O2 model, namely the Goldstone phase, can be rewritten using a gauge theory because these are the ingredients. We had one more entry in the dictionary, which is actually nice. Let me just write the last one, I forgot about it. So a vortex for theta, what does it map to? Somebody. What does the vortex map to on the gauge theory side? Electric charge. So this gives us some really good understanding of the broken phase of the O2 model. We see that many ingredients that we encountered before have a gauge theory counterpart. Completely different language, but it's the same. Now the challenge is to try to patch up the whole phase diagram, not just the spontaneously broken part, and that's what we'll do next. Okay, so there was one question, yeah. Which zero mode do you mean? Which are the vergences? Like what the verges, can you say? It doesn't diverge, why? The conjugate of theta is d theta, I guess. So the two point function of d theta is perfectly finite. No, no, the two point function of d theta according to this dictionary is the same as the two point function of f mu and it's perfectly finite. At separated points, maybe you talk about, maybe you have in mind mental space. In position space, it has a perfectly finite two point function for any x and y. No, but it's perfectly okay. If you take x and y to infinity, no problem. It's perfectly convergent. For a compact scalar, it has absolutely no, there are a little bit of difficulties with the compact scalar. If you do more exotic computations like entanglement entropy, there is a well-known difficulty with this model when you try to compute the entanglement entropy, then indeed you encounter some sort of divergence. But for normal correlation functions involving monopole operators, the field trends, anything of that sort, it's perfectly okay. For the entanglement entropy, we can have a separate discussion maybe in the evening of how to define entanglement entropy or the three-sphere partition function of that model. That's a very subtle issue, but for all the local questions, it's perfectly finite. Maybe we can postpone your question to the discussion this week. Yeah, let's do it in the discussion segment. Any other, yes. Well, in four dimensions, if you try to repeat the same exercise with the Lagrange multiplier that enforces the Bianchi identity, it also leads to something very interesting. It leads to S duality. In two dimensions, it leads to T duality. In one plus one, it leads to two duality. In two plus one, it leads to this duality. And in three plus one, it leads to S duality. In five dimensions, probably it's useless, but I'm not sure. Okay, yeah. I think it's even more useless. I think there is a discussion session. We can postpone all questions to the discussion. So we'll stop here and let's start the speaker.