 Where are you sitting? I can't find the place. Where are you sitting? I'm sitting. I always prefer a place where I can write. So I'm sitting in the front. OK, let's get started. I think we have kind of a packed program. So the next lecture is by Zohar Komargotsky. He will tell us about quantum fields in three and four dimensions. It's OK. It's working, right? Is it working? OK, thanks for coming to my lectures. My lectures will be about 3 plus 1 and 2 plus 1 dimensional dynamics in quantum field theory. I'll try to cover several topics, but not exhaustively, more rather true examples, because the topics are too extensive to cover exhaustively from first principles. So rather, I'll give you a flavor of some new ideas and new tools through examples rather than through an exhaustive coverage of these concepts. Now, of course, you're more than free. And you're encouraged to interject and ask questions whenever you have any questions about anything that I'm going to discuss. And furthermore, if there are any other comments between the talks, after the talks, whenever you see me, also please feel free. Also, there are many experts in the audience, so if you stumble me with some question that I don't know, I'm sure somebody will be able to help. OK, so let's start. I wanted to start with some general motivations of various ideas that you might not be familiar with if you haven't studied this topic before. And then we'll start covering some more concrete material. So the motivations are quantum phase. I'll discuss a little bit about quantum phase transitions. I'll discuss some conceptual ideas about quantum phase transitions that are interesting. Then I'll discuss a little bit the connection to young military. I'll discuss a little bit finite temperature, finite temperature aspects of young military. I'm not going to discuss any of these topics exhaustively. Just a bunch of motivations for physical problems or these kind of topics that I'm going to cover are useful. So let's start with the general concept of phase transitions, or quantum phase transitions, or phase transitions more generally. So in these lectures, I'll be using some terminology that I want to now define. So this is going to be a little bit of terminology that will be useful. So suppose you have a quantum field theory. In any number of dimensions, say d plus 1. And suppose you are interested. So some people are interested in the massive excitations. People who look for new resonances at the LHC obviously are more interested in the massive excitations, massive particles of quantum field theory. But a simpler problem that we might have a better chance of solving is to understand the vacua, just the deep, deep long distance limit of quantum field theory. So we're going to discuss the long distance limit, or the low energy limit. Quantum field theory could have many types of vacua, many types of long distance limits. And there is some terminology that we need to all agree on when we discuss these topics. So the question is, which kind of long distance limits could there exist in principle? So maybe just listing all the possibilities is kind of too much. So I'll list the main options. And these main options have some terminology associated to them. So the first option that you are all familiar with is that the vacuum is gapped and trivial. So if you have some vacuum in QFT, everything is massive. And there are no interesting observables that one can measure at long distances. Then the vacuum is said to be trivial and gapped. Gapped means that there are no particles and trivial. This word trivial would become apparent in a second. The next option that you are probably very familiar with is when the vacuum is gapless. When we say that it's gapless, there are actually two distinct options that appear very commonly in applications. The first option that appears in applications is when there are Nambu Goldstone particles. Nambu Goldstone particles appear when there is a spontaneous symmetry breaking of a continuous symmetry. So here this is associated to spontaneous symmetry breaking of a continuous symmetry. That's one instance when we are, I mean, this is what happens in QCD when the math list, what's the so-called Keier limit. The other situation when there are gapless modes is when we have a conformal field theory. Non-trivial conformal field theory. It could be trivial, but it could also be non-trivial. Conformal field theory. Now conformal field theories don't appear if you have a Hamiltonian with many parameters. You generally don't expect that there would be a conformal field theory for every value of these parameters. It might appear in some co-dimension space of the parameter space. And these points where it appears are usually called second-order phase transitions. So this is usually called the second-order transition because it appears in some subspace of the phase diagram and it's a rather smooth transition. Unlike the one that I'll mention soon. But there is yet another option that will actually play an important role in these lectures, which is that it's a topological field theory. So it's gapped, but non-trivial. Non-trivial. So this is the option that I wanted just to spend a minute on because it might be the less familiar one. It's not covered in at least the standard books that are used in graduate school in higher-G physics. So this should be contrasted with the first option, which is trivial and gapped. So by non-trivial and gapped, what we mean is that the Hamiltonian, in fact, doesn't have any, there are no propagating degrees of freedom which are at zero mass. So there are no masses propagating degrees of freedom. That's what the word gap stands for. But there's still something that one can measure at long distances. Even though there are no light particles, there is still something to be measured. So the idea of non-trivial is that we can measure something even though there are no light degrees of freedom, even though there are no massless degrees of freedom. And we can measure this something without investing any energy. Of course, if we pump up the energy, we can measure whatever we want, but the idea is that we can measure something without pumping any energy. So we can measure something without pumping any significant amount of energy. So how can it be that there is something to measure if there is no, how can it be that there is something to measure without even though there are no massless particles? The idea is that there could be non-trivial statistics, so to speak, in the infrared. So there could be some excitations that have non-trivial aaron of bone phases or braiding. Even though there are no massless particles, there could still be aaron of bone phases that we can measure. So what do we measure? We measure aaron of bone phases or generalizations of aaron. Of course, there are non-trivial generalizations. So by aaron of bone phases, I mean in some generalized sense. Aaron of bone phases. The simplest known example of non-trivial aaron of bone phases that we can measure is, of course, from two plus one dimensions. But there are also three plus one dimensional examples. So the simplest example is in two plus one dimensions. The theory could be completely gapped, but there could be anions. So there could be particles, heavy particles that's called in one and two. So these particles would be heavy and unrealistic and you can just move them very, very slowly. Now in a trivial gapped vacuum, you take two heavy particles, you move them slowly, you don't see anything. But if there are any ions, by moving one particle across another, you can get a non-trivial phase, which is, let's say, delta one, two. So the wave function could be multiplied by non-trivial phase, in the simplest cases. And that phase is measurable, even when the process is very adiabatic and you don't invest any energy. So these are measurable observables. So this is a measurable observable that does not require pumping any energy. So this is our measurable observable. I'll review the concept of anions, but of course, anions in this form are special to two plus one dimensions, because if we take space to be R2, that's where we can braid particles across each other. In three plus one dimensions, we cannot just braid simply ordinary particles across each other, because you can always contract the loop by lifting the particle off the board. But in two plus one dimensions, this can be realized. So these are three distinct vacuums that you could encounter in continuum quantum filter to infinite volume. And on top of all that, there is also the concept of super selection sectors. So on top of that, we have the concept of super selection sectors, which I want to explain now. But before I go to super selection sectors, are there any questions about this terminology? Any complaints or? Okay, so the concept of super selection sectors is that if you look at the vacuum of quantum filter, there could be several vacuums that are exactly the generate, let's say. This could be some energy or potential, and this could be some field, some field space. There could be several vacuums that are exactly the generate. This can happen in quantum filter due to the breakdown of a discrete symmetry, for instance. So this could happen, this could be enforced, such as the generacy could be enforced. Such as the generacy could be enforced by many different ideas, such as a discrete symmetry, which is spontaneously broken. And I continue symmetry, which was discussed here. So this could be due to discrete symmetry, supersymmetry, like even in boiling water, any first-order phase transition. Any first-order phase transition, by definition, leads to a degenerate set of vacuums because what is a first-order phase transition? A first-order phase transition is such a picture, right? Two vacuums play this game. So at some point they have to become degenerate. And so such degeneracies can just appear accidentally in parameter space, in some co-dimension of parameter space where two vacuums become degenerate. So when we have this situation, that there are several vacuums, several super-selection sectors, more precisely, each one of those has its own properties. They don't talk to each other in quantum filter because what we do in quantum filter in infinite volume is we fix the boundary conditions so that we live in one of those vacuums. One has to do a little bit more work to try to get this vacuum to communicate in continuous quantum filter. Remember, in quantum mechanics, the wave function spreads. Let me just write it down. In quantum mechanics, the wave function spreads because in quantum mechanics there is no space. It's in zero plus one dimensions. But as soon as you have any space dimensions, you have to specify the boundary conditions for these fields at infinity. And once you have specified those boundary conditions, these don't talk to each other. So that's why they're called super-selection sectors. One has to do a little bit more work to get them to communicate. I'll give you one example soon. But so each one of those could have its own properties. So you could encounter situations where this is trivially gapped, this is non-trivially gapped, and this has a CFT. Such things exist. So this classification pertains to each super-selection sector on its own. And in fact, this could have goldstone bosons. So there could be like a little Mexican head here. So there could be a goldstone boson here. There could be a CFT here. There could be a topological filter here. This, I haven't said that piece of terminology yet, but when such observables exist, they are described by topological filters. The whole subject of topological filters, it's a claim to fame, is that it allows you to classify all the possible phases that can be measured long distances in gapped Vakwa. So the subject of topological filter is really about the classifications, the classification of all such possible a long distance phenomena, like a R-on of bomb type phenomena, and generalizations they're off. Okay, so when we discuss quantum filters, we try to first understand how many Vakwa there are. That's the first property of the vacuum, the first property of the long distance theory, how many super-selection sectors are there. That's question number one. Second question is, what are the properties of each of the super-selection sectors? Is it massless? Is it massive? Is there a topological filter? Isn't there a topological filter? And the last thing, okay, so this is what we do when we try to discuss the long distance limit of quantum filter. Then there's one more thing that you can do, which is a little bit more sophisticated, which I also want to explain generally. You can try to get the super-selection sectors to talk to each other, okay? Instead of just discussing each super-selection sector on its own, you can try to get them to talk to each other. So this is very intuitive to understand in the language of first-order phase transitions. In the first-order phase transitions, like when you have the liquid phase of water and the vapor phase, there is a layer that separates them. So this is like an equilibrium situation where there is a layer between the two phases. So this layer is like, this could be water, liquid water, could be liquid water, and this could be vapor, but you can construct a layer that separates them. So we can do the same in any quantum filter, which has several super-selection sectors. We can construct layers that separate these different super-selection sectors, and these layers could have interesting physics on their own. And this is also part of the long-distance theory that is interesting to study. So the layers in the technical, well, the technical term for these layers are called domain walls, and they exist whenever there are distinct super-selection sectors. So they exist, they always, they exist whenever there are, whenever there is more than one, more than one sector, super-selection sector. So whenever there is more than one, you can always construct these layers. And the way you do it is let's say, let's call this vacuum number one, and let's call this vacuum number two. So what you do, so what you do is that you, you just take this space, let's say that this is space, and you just decide that in one direction of space, you have vacuum number one, and in another direction of space, you have vacuum number two. If you're a lattice person that's very easy to arrange, you have some fields, and there are several super-selection sectors for these fields, and you just impose the boundary conditions that when you go to infinity in one direction, you have vacuum number one, and when you go to infinity in the other direction, you have vacuum number two. So now the whole system has to find an equilibrium, minimum energy state. But now it cannot be in any of the vacuum one or two because the boundary conditions on the two sides are incompatible. So what the theory does is to build a layer exactly like in the water vapor transition. There is a layer, and this layer is called the domain wall, and this domain wall lives in one dimension less naturally. So this lives in a deep minus one plus one dimensions, and there could be interesting physics on the domain wall. In some situations, there are interesting excitations that are trapped to the layer that cannot extend to the bulk. So sometimes there are trapped excitations on the layer. Excitations on the layer. And when there are such trapped, interesting excitations on the layer, we can try to study their physics, and then we're back to this terminology. We can try to understand if these excitations on the layer are trivially in-gap, non-trivially in-gap, and so on and so forth, so we can play this game. Also for the physics on these layers, okay? So these are the objects that will appear in many different examples and in many different guises. Now what I want to show you is one example where these things come to life, which is just ordinary young mill theory. So I'll give you some facts about ordinary young mill theory, which employs some of these ideas. Are there any questions about the terminology? Or anything else? Okay. Say again. So how do you construct these domain rules? As a lattice person, you just impose the boundary conditions at infinity. So what lattice people do when they try to understand the properties of some vacuum, they just impose that at infinity, the field goes to some values, which they find before to be the vacuum. And then the system does whatever it does. Here they don't impose anything in the bulk. They just say that in one direction in space, phi has to go to the values of the fields at vacuum one and in the other direction to the values of the fields in vacuum two. And then the system dynamically finds its best configuration. And this best configuration tends to have a layer which is pretty well localized in this space, in this direction. Because I mean, if the layer was too big, you would have to pay a lot of energy because for a huge amount of space, you would be outside of the vacuum, right? The system really wants to be as much as possible in one of the vacuum one and two. Because those are the lowest energy states. So if this layer were the transition between vacuum one and vacuum two happens was too big, it would not be very energetically efficient. So the system tries to stay as much as possible in each vacuum and then make a rapid transition. And this will have some typical tension. It will cost some energy to make this transition, but it will be pretty well localized in general. You must pick up either becoming deaf or it's just that you don't. How does the system decide where exactly this domain wall is created? It's a fantastic question. So I didn't want to say it right away because it's a, it's a fantastic question. So the question is, how does the system decide where to put this layer? So how does the, how does water decide where to put this layer? What do you think? A lot of the system decided where to put this layer. So you see the full Hamiltonian or the laws of physics in the system they have invariance under translational symmetry. Now if you are in each, if you are in some vacuum then you can construct configurations which are perfectly invariant under translational symmetry because you can be in the vacuum everywhere. So it's a perfectly homogeneous state. But these boundary conditions are such that you cannot really be in a translational invariant state because you must make a rapid transition between vacuum one and vacuum two. So these layers break spontaneously translational symmetry. So such layers always break spontaneously a translational symmetry. Okay, that's fact number one. That means that there's always some physics on the layer that is non-trivial. Dynamical Goldstone bosons on the layer due to the spontaneous breaking of translational symmetry. So the layer doesn't have to be straight. Let me just draw a projection of it because it's harder to draw it with dimensional plane. So the layer doesn't have to be strictly speaking straight. It could be a little bit like that and it would cost a tiny bit of energy like for any Goldstone bosons. You can excite them but it costs a little bit of energy. Okay, so the layer always has some massless degrees of freedom that allow it to fluctuate. Condensed matter people call these degrees of freedom phonons. But here these phonons are somewhat special because they come from the spontaneous breaking of translational symmetry in a system that started its life as a relativistically invariant system. So there are also some constraints on the Lagrangian for these phonons from boost. So here is your first exercise. It's actually a somewhat hard exercise but it's very educational. So the number of phonons is of course the number of transverse directions. So let i equals, let's say, in this situation that I'm discussing layers, there's always just one transverse direction. So let x be the transverse direction, be the field on the layer which parameterizes the excitations of the layer in transverse directions, which is the phonon. So this is the field which is the phonon. So we have to write some Lagrangian for this field. So we have to write d dx. Now this is d dx, not dd plus one x because the layer lives in d minus one plus one directions. So we have to write some Lagrangian for this field which is a function of this liter coordinates x. So let me just put here some symbols so that it doesn't get confused. So we have to find the Lagrangian for this field that would correctly capture the spontaneous breaking of translational symmetry and correctly reproduce the boosting variance of the system. So it turns out, so this Lagrangian is of course in general not something we can compute. And there are complicated corrections which depend on derivatives of x. But there is a theorem that you can prove, this is your exercise. So you can prove if you only assume that the Lagrangian is a function of derivatives of x, let's assume that the Lagrangian is only a function of derivatives of x. This is a natural assumption because translating x by constant shouldn't affect anything. This is the statement that you made in the beginning that the layer doesn't know where to be. So translating x by an overall constant shouldn't be a big deal. So the Lagrangian is expected to be only a function of derivatives. So your exercise is to prove that the only possible Lagrangian is the square root is the determinant of the unit matrix in d times d directions plus d i x d j x, where you take the determinant over the indices i and j. Now have you seen this Lagrangian before? Has anybody seen this Lagrangian before? Nambugoto. It's not born infill, it's just Nambugoto. This is called the Nambugoto Lagrangian. So the Lagrangians of d-branes in string theory always look like that, but the content of these Lagrangians is nothing more than Lorentz invariance. That's a point that's not stated in many string theory books or, but it just follows from Lorentz invariance. That this is the only possible form and you can prove it yourself. It's a hard but not undoable exercise. Now this is not the full Lagrangian as I told you. There are some corrections that could depend on second derivatives of x which we don't know, but this part is universal and the coefficient here is the tension. That's the tension of the layer. Does anybody know the tension of the layer in water vapor? Does anybody remember? I think it's like 500 calories per gram. So it's an object with some natural units that measures how much energy you gotta pay to create this layer, okay? Fine, so that was, now there was another question. Yeah, I mentioned here briefly that supersymmetry leads to super selection sectors. So the question is why did I mention supersymmetry here? It's a fact of life that in many supersymmetric models there are many that generate VACWA. The fundamental reason being that without supersymmetry the energy of the vacuum is not a very well defined concept. You could shift it by an overall constant and in general VACWA could move like that. But in supersymmetry there is like a special value of the energy which is zero in some convention, some natural convention. And so VACWA cannot play this game if they start at zero. They can only move horizontally and so it's common that there are degeneracies because they cannot move like that if they already touch zero. Any last questions? Okay, now I just wanna give you an example for how natural, how do you get naturally from the point of view of particle physics, a eniance or topological field theories. Now in condensed matter theory, eniance and topological phases and two plus one dimensional quantum filters they arise naturally in materials like quantum hole droplets and others. But in high energy physics it's a little bit, in high energy physics there are other constructions that also lead naturally to eniance and I just want to mention quickly one of those. It would not be pedagogical, it's just an example that you would bear something in mind maybe in the future you could learn about it if you wanted. And then I'll get to some more concrete discussions. So I want now what I'm gonna do now is I'll give you an example from high energy physics. So young mill theory in three plus one dimensions where these concepts come into play. Where these concepts are important, okay? Where these concepts are important and they appear non-trivial. Okay, so this example I'm saying again I'll just give you facts about this example and then you can read about it. But it's just like so that you have some idea of where some of these models could come from, the models that we'll discuss later. Okay, so let's consider SUN young mill theory. No supersymmetry and no matter fields. Just pure young mill theory. This is just SUN gauge theory. So question number one. What do we know about the vacuum of this theory? Does anybody know? What do people expect about the ground state in this terminology that we already explained? What do people expect generally from the vacuum of this theory? Gap, but we had two options for gapped. Trivially gapped or non-trivially gapped? Non-trivially gapped. Okay, why? So it's supposed to be trivially gapped. There is a clay price for that. Yeah, that's the million dollars price by the Clay Foundation. This vacuum is supposed to be unique and trivially gapped. Now, the gentleman here mentioned instant on VACWA. Well, I didn't put a theta term yet. This is a theta equals zero. I'll put a theta term in a second. The instant on VACWA that you mentioned, what they do is that they introduce some additional kind of metastable states, but these are not genuine super selection sectors because they're not degenerate. Okay, so these are not VACWA. If you wait for long enough time, this will decay. So these do not count when we do the strict infrared long distance analysis. Because if you try to set up the boundary conditions to be in this VACWA, there will be huge bubbles of the correct VACWA that pop out and they will eventually wipe out the whole thing. So this is what's called instant on VACWA, but they don't affect the long distance physics and the vacuum is trivially gapped. There is a unique trivially gapped vacuum. That's what everybody expects. Now, something a little bit more interesting happens when we add a theta term. There is the strong CP problem which is associated to the theta term and the theta term also leads to non-trivial, of let me just put an epsilon tensor here. Okay, so this was the picture theta equals zero. Theta equals pi, something a little bit more dramatic happened. There is a special value of theta which is pi or something a little bit more dramatic happens. As I said, I won't try to explain why. I'm just telling you a fact that could motivate you to study and so at theta equals zero, we said it's trivially gapped. At theta equals pi, it turns out that what happens is that there are two VACWA, each of which is trivially gapped. At theta equals pi, we have two VACWA which are exactly degenerate and they're both trivially gapped. So this is what happens if you take Yang-Mills theory theta equals pi. You have two VACWA which are each of which is trivial. There are not apological filters. There are not apological filters in them, but there are two. In fact, this is related to the instant on VACWA that you mentioned. What happens intuitively is that when you change theta, these things move a little bit and it's exactly theta equals pi. This and that guy become degenerate. Okay, so theta equals pi is like a first-order phase transition where these two VACWA change. So you could think about it as a first-order phase transition. So this is like a first-order transition. Okay, so this is the fact about Yang-Mills theory. Now you can ask what happens when you construct a layer, right? This is where it becomes really interesting. Let's call this VACWA one and two. Since they're both exactly degenerate and trivially gapped, we can construct a layer. This is something that people could do on the lattice in the future and they could test what I'm telling you about it. So we can construct such a layer and we've already learned that this layer has some phonons so it allows the layer to fluctuate in the orthogonal directions. That's boring. That's just from translational symmetry breaking. It's a universal property of any layer. Question is if this layer has something more than just, something more than just this phonon. And so let me tell you some things about it that this is like an invitation to study two-plus one-dimensional dynamics. So the physics of this layer, the physics of this domain wall, it has two components. It has two components. The first component is what we just discussed. This is the phonon, which is described by the Nambu-Goto action. The second component is much more surprising and it cannot be understood from symmetry breaking alone. It's more exotic. It turns out that there is a topological filter and this is much more mysterious in two-plus one-dimensions because the layer is two-plus one-dimensional. Two-plus one-dimensions. So now I'll tell you what this topological filter physically means and then we'll start discussing something more concrete. So what happens on this layer is that there are various objects that behave like enions. So if you take them across each other, you get some funny phases, a run-of-bomb-like phases and you could ask, what are these enions? In terms of the underlying degrees of freedom which are the SUN gauge fields and maybe some heavy quarks that are probe quarks. It turns out that these enions started their lives as quarks. So the quarks, when you take a quark in three-plus one-dimensions, it's confined so you can never observe a quark in isolation. So quarks in three-plus one are confined but when you bring them close to the wall, they become enions. So even though quarks started their life as objects with a spin which is integer or half integer, depending if your quarks are scalars or fermions like in the universe, when they bring them close to the wall, they acquire fractional spin and they behave like enions and they de-confine so you can observe them in isolation. So they have fractional spin and they de-confine. So you can observe them in isolation. So this is one construction within just pure young milister that naturally leads to topological order, topological filters on the wall and to very interesting physics associated to the statistics of quarks. And some interesting phenomena in young milisterry. So my goal would therefore be to study more systematically two-plus one-dimensional systems like the one that appears here on the wall which involve initially just ordinary phase transitions but then also enions and such things. And many of the systems that we'll discuss can be constructed starting from QCD like yours and starting the physics on some layers or domain walls. Correct, these are probe quarks. So if you want, I can add heavy quarks. If you want this to be, if you want what I'm saying to be, sorry, you asked the question? Yeah, if you want what I'm saying to be perfectly concrete, just add heavy quarks here. Okay, heavy dynamical quarks. So they don't fluctuate, but you can excite them and then check their statistics. So if you like a top quark, so if you have esoteric gauge theory with a heavy top quark, you cannot separate the top quark. As you know, if you try to separate the top quark, there is always like a buildup of pions around it and there's like, it fractionalizes or hydronizes into mesons and various heavy states. So this would be like that in the bulk. But what I'm saying is that if you bring it close to the wall, then it acquires fractional spin. And the fractional spin can be even computed. It turns out to be one over two end. So it acquires fractional spin and you can then measure it in isolation and you can take a top quark around another top quark and just measure this one over six the fractional spin if n is three. So this is a fun fact about Yang-Mills theory, which is supposed to be just a motivator to study a little bit more systematically two plus one dimensional systems. So now I'm gonna jump in if there are no more questions and I'll start with the simplest possible two plus one dimensional dynamics, which is the O2 model. Yes, could you just pick up? Okay, so the gentleman is asking a very nice question of why are the quarks de-confined? So in the eighties and the seventies people came up with phenomenological models for confinement, people wanted to understand why quarks are confined in space, in three plus one dimensional space. And what people said was that there's something that's called monopole condensation. It's like a mythical object that you cannot prove exists, but that's what people say causes quark confinement in three plus one dimensions. However, if you ask them about this situation in the seventies or eighties, what they would have told you is that here in this vacuum, the monopole condenses and that's why quarks are confined, but in this vacuum, the dion condenses. That's a phenomenological picture that they would have told you in the eighties. And so here quarks are confined and here quarks are confined. But then here they would be conflicted because you cannot condense both, they are mutually non-local. So what they would have told you probably, I'm just putting words in their mouth. What we are saying now about this is that since here the dion condenses is here the monopole condenses, it's here neither of them can condense and therefore there must be de-confinement, okay? Because you cannot condense them both simultaneously, they are mutually non-local, so instead none of them condenses and so there is de-confinement. It's like you're forcing the monopole condensation to disappear on the layer and therefore the quarks get liberated, okay? Yes, so the question is if there is a domain, well if there are, I think, well I might be over, I don't, whenever you have two vacua which are exactly the generate, there is always a way to make them communicate. Now it could be that the barrier between them is infinite and then you'll get a slightly singular object. But if the energy barrier between these two vacuos is finite, then there must be a way to make them communicate through this layer. I mean, I think about it on the lattice that you can just force the lattice to be in one vacuum on one side, at infinity and in another vacuum on the other side. So what is the lattice gonna do? It's not gonna tell you I don't wanna compute. It's gonna compute and find the lowest energy state which must look like that. This word in the past people thought about them as sort of what causes confinement. Monopole condensation or diamond condensation is what's supposed to trigger confinement. It's a qualitative phenomenological picture. There is no order parameter here. It's not a rigorous order parameter for anything. Any other questions? Okay, so now I wanna jump in. My plan is to first cover the O2 model which is also called the model of super fluidity or the XY model in two plus one. Then I will cover particle vortex duality and then we will add topological filters. I will cover a little bit of topological filter, just the basics. And then we will discuss generalizations of particle vortex duality, generalizations of particle vortex duality with topological filters, with enions, okay? So we'll discuss generalizations of particle vortex duality where neither the particle nor the vortex are actually particles, they're enions. So it shouldn't be called particle vortex duality, it's like enion vortex enion duality. So with enions. And eventually I'll make contact between that and this. So I'll give you a more concrete picture for the dynamics of the wall of the SC2 gauge theory with one quirk using this story. So we'll make contact between these two pictures. And if I have time, I will also discuss non-abelian enions and the dynamics of three two plus one dimensional QCD. Non-abelian enions and dualities. So let's start. I'll just do a little bit of the beginning, give you a little bit of homework, exercise send and we'll continue on Sunday, oh, sorry, on Monday. Okay, so I'm gonna, for the duration of these lectures until the very end, when I'll discuss maybe this and the connection to that, I'll be in two plus one dimensions. Okay, so it's gonna be in two plus one dimensions for the rest of these lectures. There isn't that we focus on two plus one dimensions because the interesting stuff has to do with this enions and enions exist only in two plus one dimensions. There is also topological, there are also topological filters in three plus one dimensions, but they're much harder to come by. They're not completely well understood yet and so on. So the simplest known examples are in two plus one dimensions at the moment. We're now starting from the O2 model, which in the literature is also called the XY model. I'll tell you in a second why. So the Lagrangian, it's the simplest possible Lagrangian you could write for a single complex scalar field. So phi is a complex scalar field. So let's say, let's write, well, let me first tell you why it's called the XY model. It's because of the deep fact that if you have a complex scalar field, you can decompose it like this as a sum of two real fields, which are called X and Y. So that's why people call it the XY model. Anyway, the symmetry of this model, does maybe anyone want to contribute? What is the symmetry of this model? Well, yeah, it's U1, SO2, but the reason it's called O2 is because it's the symmetry of the model. So the symmetry is O2, SO2 is obvious and the O just takes phi to phi star. Okay, it's generated by rotations of phi and by complex conjugation of phi, which together form the group O2. Now, it's interesting to study the phases of this model. Okay, this model is super important because it describes super fluidity in condensed matter and in behind-energy physics, it appears in so many applications, it's hard to count. This model is a very important universality class. Now I'm gonna draw the phase diagram. I'm always gonna do it like this kind of fashion. This is gonna be M squared. And M squared can be huge and positive or huge and negative. So of course, this model has a parameter, lambda, which has dimensions of mass. This sets essentially the scale, the energy scale, where the model is strongly coupled, okay? So this is the strong coupling scale because if you're at energies below lambda, these interactions are very strong by the Wilsonian general paradigm below. I'll just write it down. So lambda is the scale of strong interactions, scale of interactions. If lambda was zero, this model would have been free and we could have solved it at all energy scales easily. It would be a free field theory. But since lambda is present and this is a quartic non-linear vertex, the model develops strong interactions at a scale lambda. So if the energy is much, much smaller than lambda, we're in trouble. We don't know what to do. But the energy is much above lambda, we're in good shape. This is easy and this is hard, okay? So the trick to understand these models is to think about this M squared. Because if M squared is huge in units of lambda, we don't need to study the model below the scale lambda because already at M squared, we can integrate out phi and forget about it. So if M squared is much bigger in absolute value, than lambda, we can understand the model because the dynamics of the model happens at very high energies much before the model becomes a strongly interacting. So when we draw these phase diagrams, we're always gonna try to study the weak coupling limits of these phase diagrams and then try to patch it up. So it's like a patchwork. You have to understand the easy limits and then try to understand what happens in the hard limits when the mass is very small. So of course here, there are dragons. We don't know. This is when the mass is small. We cannot compute anything and it's up to conjectures or up to the lattice or up to maybe future developments to decide, but we can say what happens outside of this region which is tough. So what happens when the mass is huge and positive? Mass squared is huge and positive. Well, when it's huge and positive, the potential has one minimum. I should have said this is very crucial. Otherwise the model is kind of sick. So when M squared is huge and positive, the potential has one minimum. Everything is trivially gapped, obviously. There is nothing. And so there is a trivially gapped vacuum here. One trivially gapped vacuum. But when M squared is huge and negative, when M squared is huge and negative, there is a more interesting phase because the potential now has a bunch of minima which are exactly degenerate. I'll add this Mexican head. So in this limit, the O2 symmetry is spontaneously broken. Here there is spontaneous symmetry breaking. But the O2 symmetry is broken to Z2. So only this complex conjugation symmetry remains, but the SO2 part is gone. And therefore we have a number of goldstone boson. And so this is the picture. We have this Mexican head kind of picture. And we can write the effective Lagrangian in this phase easily. The effective Lagrangian is just that of a single Nambu goldstone boson. So the Lagrangian is just D, let's call the angle theta. D theta squared, up to some coefficient. Okay, where theta is the Nambu goldstone boson. It's a compact phase. Compact target space, which is isomorphic to S1. Now this question of why there is Z2, it's a homework exercise. I want you to convince yourself that O2 is not broken completely. Rather, it's broken to Z2. So this is homework exercise number two. Why is O2 broken to Z2? Why is there Z2 symmetry that's unbroken? If anybody wants to say the answer now, it's also acceptable. Okay, so even if you don't know how to compute, I have one hour, exactly right. So if you don't know how to compute in this region with the question marks, which we don't know, you cannot say much, but you can say something very important, which is that clearly there must be a phase transition. Because here you have a Nambu goldstone boson. This vacuum is gapless, but this vacuum is gap. So there must be something here. If somebody told you that there is nothing here, it would be a contradiction. Because there must be some point where the two descriptions are in conflict. So there must be either a first-order transition or a second-order transition. So there must be something. So there must be something in the region with the question marks. There must be some sort of phase transition. You can establish it just from the semi-classical analysis. And here I'm gonna tell you the answer. That's what we believe from the lattice. We cannot prove it, but we are, the lattice has assertively established that this is a second-order transition. Namely, it's a conformal filtering. And this conformal filter is called the O2 Wilson Fisher conformal filter. It's a conformal filter about which we know quite a bit from the lattice and from the bootstrap. So the phase diagram can be summarized as follows. There's one special point where there is a conformal filter in two plus one dimensions. It's a non-trivial conformal filter, by the way. So it's a non-trivial conformal filter, meaning it has non-trivial scaling exponents. And here there is a Nambu Goldstone phase, which we'll study now in detail. And here there is a trivially-gapped phase. And this is M squared. So that's what we believe happens in the O2 model as a function of M squared. Now, notice that I did not vary lambda. I only studied as a function of M squared, but I did not study what happens as a function of lambda. And the reason being that lambda is irrelevant. Lambda is something that just flows and it's fixed at the conformal filter and it's not a relevant perturbation. If you study the spectrum of this conformal filter, it has one relevant perturbation, which is the mass squared, but lambda is not a relevant perturbation. So that's why we don't vary lambda when we draw this phase diagram. It's irrelevant in the technical sense. It's just an irrelevant operator at long distances. You could ask what happens to the vacuum if I change lambda, the answer is nothing. That's what Wilson taught us, right? Irrelevant operators don't change the answer for the long distance physics. So that's why I don't vary lambda because from this whole story we learned that lambda is irrelevant. But M squared is relevant and indeed changing M squared changes the vacuum dramatically. You could say that M squared is irrelevant here and you would be right. If you change M squared from this to that, it doesn't matter. The long distance physics remains trivial. But M squared is very relevant around this transition because it makes the long distance physics change discontinuously. Okay, I'll just say what I'm gonna do tomorrow, one minute. So tomorrow we'll study some interesting properties of this phase. We'll study something, we'll study vortices a little bit in this phase. And then we'll start discussing duality. So tomorrow we'll do a study of this phase and duality. Vortices? Okay, let's thank Suar.