 Okay, so let's begin with the third lecture of Ashwin. I also should make an announcement that as you have probably noticed, there is no Wi-Fi today because of maintenance. Okay. Okay, so finally we are ready to leave one dimension. So we'll be talking about dualities and two plus one dimensions today, but we'll be really using a lot of the lessons that we learned in one dimension to kind of organize the discussion. And in particular, what I wanted to do today is to discuss one of the earliest dualities that at least in the condensed matter literature that was known in two dimensions, the duality between bosons in two dimensions and two spatial dimensions and water states. And it turns out this is extremely useful. And what I'd like to do is I'd like to describe this duality, but I want to only tell you things that actually have some kind of a conceptual application. So I want to emphasize a couple of applications over here. So this duality allows you to understand phase transitions and even some phases, a simple way to understand certain kinds of phases. So it's played a very important role in understanding certain kinds of phase transitions. I'll describe them. Those phase transitions were not accessible using more conventional techniques like the epsilon expansion, but because of this duality, we understand them now. And it also gives you a very powerful tool to look at new kinds of phases. This has really been, I think half my career, I've just worked with this duality and it really opens up a completely new world, a different way of looking at things that are extremely hard to look at if you insist on working with the original boson degrees of freedom. So just to remind you in one dimension what happened and how that's going to translate to the physics in this two-dimensional context. So in one dimension, what we did was we looked at a model with Z2 symmetry and we developed this duality between two bosonic descriptions. On the one side, you had the spin flips or the charges and on the other side, you had the domain walls. Those are the two degrees of freedom that we exchanged. Now, if you want to go up to two plus one dimensions, if you want to really pursue an analogous line of reasoning, you need to change your symmetry. We said that with just Z2 symmetry, the domain walls are no longer particles. You could consider a Z2 symmetric model in two dimensions, demand a dual description. You just wouldn't get a simple theory like the one we got over here. That's an interesting question for another day. But to really keep this parallel, it makes sense to change what the global symmetry is. So again, we're going to look for a duality between bosons and this will be a duality between charges and vortices. So vortices are going to be point particles in two plus one dimensions. We can develop a theory either in terms of charges or in terms of vortices. So one important distinction, if you think about these systems, they're both particles in one dimension. So the charges have short range interactions between them and the domain walls also only interact at short distances. So if you recall why we got this perfect duality between them, part of it had to do with the model that we looked at. We looked at a very special model, the Ising model with just nearest neighbor couplings. You could look at a more general model which just has the symmetry. You can still do this duality and you can describe the system in either of these two variables. They won't look exactly the same, but morally speaking, they're both described by very similar theories. Essentially in both cases, there are some real scalar field that have some short range interactions. So that's where really the self duality came out. Now it'll turn out that charges and vortices are both particles in two dimensions. You might think that maybe we'll also get a self duality over here. Turns out that's not quite true. So when charges interact with just, when particles, I have to be a little bit careful about what I mean by charge. So I mean U1 global charge, not electromagnetic charge. So these are particles that carry the U1 global charge. They have short range interactions, but it'll turn out that vortices, they interact more like electric charges in two dimensions. They have long range interactions. We'll actually calculate the interaction potential between a pair of vortices and the interactions are long-range. Can they will be mediated by a gauge field? So there won't be a perfect, there won't be a self duality over here. These will actually be distinct theories. If they look a little similar, but they're not exactly the same. And actually that's a lot more useful than having self duality in many ways. Because knowledge of this theory will give you some information about this one and vice versa. Okay, and that's where really our first application is going to come about. We're going to see that the transition where you condense vortices, that transition is very difficult to describe using conventional perturbative techniques. Okay, but by this duality, you have a prediction for what that transition should be. It's something that's related to just the Wilson Fisher fixed point and a lot of numerical studies that actually show that. Okay, so we'll get to that towards the end. Okay, the other thing that we'd like to do is we said that in one plus one dimension, so to continue writing this table. Okay, so we saw that there was a description of free fermions. Okay, so over here we had this my run of fermion and this eta was a combination of the charge and the defect. Okay, and you can demand to know what happens in two plus one dimension. Is there a way to describe a charge fermion in a similar way in two plus one dimensions, but based on the variables in this theory. Okay, and we'll see that there's a very analogous kind of situation over here. We can write down this free fermion in terms of some fields, which are essentially the bound states of the U1 charge and the vortices. Okay, so it's again a very analogous construction and we'll see the way in which we can implement this binding is using a Churn-Sieman's term. And the third thing that I'd like to do, that'll have to be the last thing I do in the next lecture today, is to actually come up with a something that's unique to this two plus one dimensional example, which is a fermion fermion duality. Okay, and again this will have an application which I'm hoping to cover. It actually has made some very nice predictions about the physics of the Halffield-Landau level. Okay, this is a very central problem in the Quantum Hall physics. And it's been worked on for decades, but after we understood this duality, you could make some new predictions about this Halffield-Landau level that were not obvious before. And at least there's numerical support that indeed this is on the right track. Okay, so again I'm hoping to get to that at the end of the next lecture today. But that's the plan. Okay, so before we start, there were a couple of questions before at the end of the last class. One question was, what is the relation between the kind of models we're looking at in this series of lectures and finite temperature statistical mechanics models? And the simple answer is that we have been looking at zero temperature quantum models. We're looking at zero temperature quantumizing model in one plus one D. And there's a very precise mapping at least for the universal quantities, the long distance properties, at finite temperature, but now in two dimensions. And this is stat mech, there's no time. Okay, so roughly speaking, the coupling constant in this theory, G, gets mapped to the temperature on this right hand side, a statistical physics model stat mech ising model. Okay, so this was the one that Onsaga solved exactly. We also got an exact solution over here. The two are related, essentially the time direction over here gets mapped to the extra space dimension over here. Okay, so the dimensions X and T when you do Euclidean time. Okay, so this is probably obvious to many of you, but I was asked this question more than once. So let me just make that precise. For every statistical mechanics model, there's a quantum and lock. Okay, in one lower spatial dimension, which is sitting at zero temperature if you're in the thermodynamic limit over here. It turns out the interesting thing is the converse is not true. There are quantum models that have no statistical mechanics and lock. Okay, you can go to one higher dimension, but in some cases, the weights don't look like positive numbers. So this mapping is always goes from a stat mech model to a quantum mechanics model. And similarly, I'll be talking now about the zero temperature physics of a two plus one D U one model. Sometimes I'll call it the XY model because you can think of this as a magnet, which has spins that live in the XY plane. And again, there's a relation to the three dimensional statistical mechanics of an XY and this is at finite temperature. So there are connections to statistical physics, but I'll somehow prefer talking about zero temperature quantum, just a matter of taste. I like to do Hamiltonians. If you do Hamiltonians, you kind of buy yourself a dimension. Okay, it's a little bit easier to write things down in one lower dimension and then use the Hamiltonian formulation. Okay, but that's just a question of taste and you can, when you write Lagrangians for these problems, they look exactly like the stat mech Lagrangians. They're very different systems. Yes, that's right. That's right, yeah. And also once you start talking about, for example, this fermion, fermion duality, there are really no stat mech and logs of that once you have fermions, where on both sides you have fermions. The physical degrees of freedom are fermions. So then, yeah, that's another thing to be kept in mind. Yeah, so for example, yeah. Okay, so the question is, are there examples where you can write down a zero temperature quantum model, but there is no analogous statistical mechanics model in one higher dimension. Okay, so one example is the spin a half, spin one half Heisenberg model in one dimension. Okay, so the spin a half is on every side and we call it the chain because that's one D. Okay, so you can write down a field theory for this model in terms of a non-linear sigma model. So there's some vector m. So there's a Lagrangian, which is looks like dm squared. That and itself can very easily be mapped to the side, but it turns out it also has a topological term. There's a term that looks like it has various, actually I think I just need the topological term over here. Some theta sitting over here. Okay, and it turns out that this term, it's like a various phase term. It has a linear derivative in time and there's an integral over the time. Okay, and there's a factor of i. It gives you different phase factors for different topological configurations of your vector field. Okay, so now let's try to continue this to Euclidean time. Okay, so t becomes itow, t becomes itow, but the i's cancel out. So even in Euclidean time, you have a weight for the different configurations that's complex, okay. In fact, this is just, ultimately it turns out we have a plus or minus one, but the Boltzmann weights, the thing that you would like to identify with Boltzmann weights are no longer positive. Okay, so this, you can call it a Statmeck model, but because the weights are non-positive, it doesn't obey the usual axioms of statistical mechanics. Okay, so this, at least in this basis, it does not have a statistical mechanics kind of extension. And those are some of the most interesting problems actually where you cannot go into a Statmeck model in one higher dimension. Okay, so if there are no more questions, let me proceed with talking about the U1 model and implementing this duality. Yes, so we have some, a model with, let me call psi being the operator that could be creating bosons. There's some U1 charts that's conserved. You could think of this as just boson number. And under this U1, suppose you have some symmetry operations parametrized by, it's a continuous symmetry over here. You rotate this by a phase. Okay, so these are U1 charged. And what we'd like to do, just like with the Ising model, we can write down a phase diagram for appropriate microscopic model. So I won't actually bore you with the details. I'll put it up in the notes. But imagine that you have some bosonic field which has got a kinetic energy. There's some, okay, so in this Lagrangian size, a complex field. And we can obviously see that there could be two phases. So I draw the phase diagram. It's a function of r, depending on the sign of r. If r is positive, as usual, I have this field to be gapped. So this is a gapped phase. If you like, it's like the vacuum. There are no bosons in the ground state. But then I take r negative. I end up condensing these bosons and giving the Bose field an expectation value. Okay, so very much like our Ising model, there's a symmetry broken phase, and there's a gap phase. Okay, the distinction over here is because this is a continuous symmetry. In the Bose condensed phase, there are actually gapless excitations. Okay, so in the condensed phase, the Goldstone modes, and this simply comes out because you have this large number of different equivalent vacua. So you have psi being condensed, but this could be any amplitude times a phase. Okay, and as long as it locally has the right amplitude, this is a ground state, but different parts of your sample could have different ground states. So as a function of position, you could have phi and as a function of time as well. This phase could vary. So you can write on an effective theory just for the phase. You've taken this into this expression. It'll have the minimum potential energy. It's just the phase that's varying, but this term will pick up some additional action. Okay, so there'll be some additional effect. There's an effective Lagrangian describing these fluctuations of phase. You take the derivative on this field psi. It doesn't act on the amplitude that's constant just on the phase and you get basically d phi squared times some overall coefficient. Okay, usually there's a factor of two that comes in. Maybe I should put a factor of two here. And this rho s is just modulus of psi not squared. Okay, that's the strength of this condensate. And of course, if you look at the equations of motion over here, you'll have a gapless mode. It's energy will be proportional to the momentum. It's just this relativistic kind of mode. So that's the ordered phase over here as a Goldstone mode. Okay, so that's one difference from the ising model case. Okay, so here the second distinction is in terms of the topological defects. So the one was the Goldstone modes. That's one property of the condensed phase. The second is the topological defects. Okay, so again, here you're looking at low energy configurations. So you take a loop in parameter space. This is a function you have. If you can parameterize ground states by just the phase. So we're gonna demand that you have exactly the right amplitude and the phase can wind around. And that's a good way of getting a low energy configuration because the only contribution to the energy is coming from the variation in space of this configuration. Okay, so if I look at it, just take a snapshot. In any little region, it just looks like the ground state. One of this whole family of ground states. So there's no energy cost coming from this term. All of it is gonna come from the potential term. Okay, so locally it looks like a ground state. And the question is, can you put down this collection of ground states in such a way that it's hard to unwind them? Okay, that they're really topologically stable. So we want to look at how this phase can wind. And the important thing is that the phase itself is not physical. Okay, so the phase and the phase plus any multiple of two pi is exactly identical. Okay, there's a redundancy in terms of what phase you assign. The only thing that is physical is exponential of the phase. Okay, that's really the physical quantity. You can think of this very pictorially. If you have a magnet, if you're really just describing the state of spins in a magnet, we have them living on a circle. And we can specify the angle. There's some ambiguity in specifying this angle, whether you can add a two pi to it. But the exponential of the phase is simply the location on the circle of the spin. Okay, that's completely physical. This is really the physical quantity. Okay, but it pays to actually talk in terms of this unphysical quantity as long as you remember that it doesn't have to be single valued, for example. Okay, there are two completely equivalent configurations, phi and phi plus two pi, for example, that describe exactly the same physical situation. Okay, so for example, when you go around this loop, you know, if this was a physical quantity, you'd have to come back to yourself at the end of this procedure. But now because this is not a physical quantity, you could accumulate some phase and you could come back, it's physically the same, but you can be off, but you can only be off by this amount. Okay, and if you're off by multiples of two pi, you're actually back to the same configuration. Okay, so in other words, if I were to take the integral, the line integral of gradient of phi, if this was a normal function, the line integral around the closed loop should be zero for any physical quantity for a function that's single valued. Okay, but phi is not such a function. You're forgiven if you come back by multiples of two pi. Okay, and these correspond to topological defects. They're called topological defects because you can sense them very far away from the origin. Okay, so you have a defect somewhere here. You can go very far away and just measure what the phase is doing and you can figure out that inside somewhere there's got to be a topological defect. You just do this integral by measuring the phi on going around it. And these are of course the vortices. Okay, so you have vortices that they're classified by an integer. Okay, you can think of this as the charge of the vortex. Okay, so a charge one and a charge minus one vortex, you put them together, they can annihilate and they can give you the smooth configuration. So there's a notion of vortices and anti-vortices and you can add vorticity and get higher vorticity and so on. Okay, so let's calculate the energy of one of these vortices and a couple of interesting important points that come out. Yeah, so to sort of set up this calculation, I want to think about my system as living on a disk with a certain radius, let me call it big L. Okay, so this disk is supposed to model the separation between a vortex, anti-vortex pair. Okay, so let's say I have a vortex at the center of this disk. I have an anti-vortex somewhere here. Yeah, and if I want to look at the interaction between this pair of objects, what I can do is I can calculate the total energy of the vortex, add it to the anti-vortex. That's a rough estimate of the total energy of this configuration. It'll tell you how much energy you need to put in to drag a pair of this vortex, anti-vortex, apart. Okay, distance L, two L apart. Yeah, so for that we just use this energy over here to calculate the energy of this vortex. The energy in those variables is gradient of phi squared. So in the radial direction, let's say there's not much variation, all of the variation is going to be around this, on going around. So the vortex configuration I want to think about is phi is essentially the angle theta that you subtend as you go around. Okay, that's a unit vortex. So when I take the gradient, I have d by dr and I have one over r d by d theta. It's only the second term, it's going to give me any energetics. So the energy density, so we just take the gradient of the phase where it's winding by the angle, that's the energy density, and then I've got to integrate it over this disk to get the entire energy. Okay, so let me not introduce another. You want to do this to get the energy of a vortex. So you can do the radial integral, that's easy, sorry, the angular integral. That gives you two pi r. And at this point you have to do the radial integral. And you see that in order to define this energy, not only do you need an upper cutoff over here, the system size L, you also need a lower cutoff. And essentially that means you need a certain region, a little region around the origin where the amplitude itself will go to zero. Okay, so we're saying that this phase winds around as theta. So your wave function looks, your amplitude looks like e to the i theta. And of course the theta is winding around and as you get to the origin, this becomes ill-defined. You cannot have this winding around just at the origin, just at that one point. The angle is ill-defined. So the only way this can make sense of is if this amplitude psi itself goes to zero. Okay, so we want that near the origin, sine out itself goes to zero for some radius that's within a core radius, RC. You need that kind of regularization for these objects to exist. Okay, so there's gonna be a lower limit RC. In our lattice models, there's no problem because this RC is just gonna be the spacing between the lattice sides. Okay, so if you have a lattice model and you have, and you define a complex scalar field, there are always gonna be water scenes. Okay, so the subtlety is when you define a continuum model, you have to be a little careful too so that you allow for the possibility of water scenes. It's something that you explicitly need to ensure. There are actually two kinds of models you can define. They look superficially the same. You could write on a model that's just this, gradient of phase squared, and you may be committing to a model that has no water scenes. If you do not allow for events where the amplitude drops to zero at some points, it's a very small energy cost you pay over there to drop the amplitude to zero, but then you can make these kind of vortex defects. Okay, so it's important to keep in mind that in continuum theories, you have to define it appropriately to get these topological defects. And you can miss them if you have the wrong kind of regularization. That's right. Yeah, you pay that energy. So the total energy of the vortex is gonna be the energy that I calculate now plus some constant amount that's related to that core energy. And you'll see that this one will dominate at least for dilute water scenes. Okay, so let's do this integral. So it's just one over R. So this is just pi rho S log of L divided by R C. That's the energy of this vortex of within a disk of radius L. But I want to think of this as a proxy for the interaction energy between a pair of water scenes. Essentially, it tells me that the potential between a pair of water scenes is going as log of the separation. Okay, so I can think of this as the distance to the anti-vortex where the field will become uniform. It scales as log of L. So it's essentially the potential between them is scaling as the log of the separation. Okay, so that's exactly the electrostatics of charges that are confined to two dimensions. Okay, so if you have electric charges where the field lines do not leave the plane, then the potential between a pair of charge and opposite charge will simply be the log of the separation. Okay, so this is 2D electrostatics. Okay, so this is to motivate for you. You know, we're not gonna derive the theory for a sign. You can do that, and I have some notes on that, but I won't have time to do that. I want to motivate the form of this dual theory. So the water scenes are also point particles, so they'll also be described in terms of some complex field sign. Now, but I want to take into account the fact that they behave as though they are charges in two dimensions. Okay, so unlike this field over here, this field simply has short range interactions. The interactions between the particles are summarized by the side of the fourth term. Essentially, particles interact only when they sit on top of each other. Okay, but now we want to modify this, the dual theory in terms of a vortex field, which will behave like charges in two dimensions. And the simplest way we can do that, that's consistent with like Lorentz invariance, is to just add a gauge potential and give that gauge potential a Maxwell action. Okay, so that's the first, you know, kind of motivation for writing down a dual theory that involves gauge fields. Yeah, let me give you another motivation, which is, so we have these objects, these water scenes and anti-water scenes that whose number is conserved. Okay, there's a topological conservation number. There's a topological quantum number. The vortices, if the vorticity is a topological quantum number, it's conserved. We'd like to have a model that captures that. Okay, so it's very much like electric charges. If you think about, you know, electrostatics, for example, you have this kind of defect in the electric field. Okay, if you have a charge, that's a source of electric field. You can just think of it as a kink, a point from where all the electric charges emanate. We call such things electric charges. Okay, so you can either just look at topological defects of the electric field and call that object an electric charge. In which case you're, and you assign a Gauss law to keep track of these electric charges. And of course we know that, you know, this is like a topological quantum number. There's no global symmetry associated with these charges. Charges are conserved not because of global symmetry, but really because they are topological defects in the electric field. So they have points where everything comes out. You can't get rid of that unless you get a point where everything comes out. And then you can make them smoothing them out. So this is a good model of keeping track of some other topological quantum number, like vortices. Okay, so what we'll do is we'll introduce a electric field, let's call it little e. Okay, and we'd like to write down a field so that its divergence is the density of vortices. Okay, so that's another reason why you might imagine that some gauge fields will come into play when you try to keep track of this topological quantum number. Okay, so in fact we can solve for what this electric field is in terms of things that we already know. Okay, so we said that there's a phase field phi. Can the integral of the phase field, so this is the density of charges of vortices that's inside the area that's swept out by this cup. So I do the integral over here and this is the area inside. Okay, so I can convert this by Stokes theorem into a, okay, sorry, so this was two pi times the, okay, so let me use one over two pi times curl of grad phi. Ds is, okay, so in other words, the candidate for my electric field is the following. I can take my, okay, so we want that curl of grad phi over two pi is divergence of the electric field. Okay, so in order to make this equality, okay, so of course usually the curl of a gradient is just zero, but in this case it's non-zero because these are, this is not a single valued function and you can get this non-zero curve. Okay, so if you solve this, you find that the electric field, the ith component is the anti-symmetric tensor. So epsilon one two is one, epsilon two one is minus one, all the other is zero, times the derivative acting on phi, okay, divided by two pi. Okay, so in fact you can write down the field strength, I guess, more generally the relativistic generalization of this will be, let me have it up there, yeah, sorry. So this is always, there's like a Z hat. So I'm using 3D notation for the curl, but then you take the dot product with Z hat. Okay, so you can see for example this term, this kind of term that gives you gradient of phi squared kind of energetics is going to simply be a Maxwell term. Okay, just the square of this f squared. Okay, so with all this motivation, let me just try to write down what we think the dual theory would look like. So unlike the Ising case, we didn't have that much microscopic direction over here, but I feel we have some, we have developed some intuition. So let's try it on the vortex Lagrangian. Okay, so there'll be a term that involves the psi, let's call it vortex, that's going to be our vortex field. And another term which is the Maxwell. Okay, so the scheme will be that let's guess, let's guess the answer, and then we're going to check it. Okay, we're going to check if it reproduces this phase diagram. And if it does, then it gives us some belief, then we'll do another check. We'll check if it gives the right responses, and then we'll be sort of, you know, just satisfied and then start looking at applications. So this will be, again, a five to the fourth kind of theory. So don't take these coefficients very seriously. Maybe I should be putting in some coefficients in front of these terms. So there is a gradient term, but now it comes as a covariant derivative because I have them. These vortices are not local objects, if you like. They actually, you create a phase field everywhere around, and that's, you know, these non-local objects could appear coupled to a gauge field. And we have argued why these vortices are like charges. Therefore, they should carry gauge charge. And then there's the usual, let's call it R prime, so I have vortex and some lambda prime side of the fourth, and then I have, this is my vortex, and then there's a Maxwell term, which is essentially this elastic energy of the rotations of this phase. And let me call that one over two kappa. I can just use what I had before. I know it's gonna be rho s divided by two. Mu nu, nu where in terms of the vector potentials, just to be completely clear, f mu nu is delta mu a nu minus delta nu a mu. Okay, so this theory we're claiming is dual to this one. Can, yeah. Okay, can you count the degrees of freedom? How did you do that? Yeah, but I guess maybe the point is that a gauge field, you shouldn't count, you know, when you have a gauge theory, it's a question of whether you should count that complex field as really a complex field, because it's coupled to the gauge field. For example, you cannot directly measure psi itself. Psi v cannot be measured, it's not gauge invariant. Yeah, so you'll see that actually this is, you know, so we know that in this theory, there are actually vortices, and the vortex degrees of freedom are represented by this. So, you know, in some sense, we are just shifting the focus to something else. And it should, you know, if you have more degrees of freedom, you'll see it as some extra mode or something like that. You know, unless it's always gapped, in which case it's probably okay. Okay, so anyway, let's check if the phase diagram works out. And it turns out it will, of course, but in a very strange way. The way in which it works out is extremely strange. And it tells you a lot about, you know, some very basic things like photons and two-dimensions and goldstone modes. Okay, it turns out they're equivalent. Okay, so let's do the easy one. Okay, and it turns out the easy one is actually this condensed phase. Okay, so the condensed phase over here was to take R negative and condense this boson field. Okay, so over here, any suggestion for how we can get the same phase? Okay, so my first instinct would be to take R prime negative as well, but that's wrong. Okay, so you might think that if you condense that, you should condense this as well, but in fact, there's sort of an opposite relation between them. If the boson is condensed, the vortices are not. And if the bosons are gapped, the vortices are condensed. Okay, so R negative over here, the condensed phase of the bosons, I want R prime positive, okay, and have these vortices to be gapped. Okay, so the vortex field, the Psi V is gonna be gapped, has no expectation value. So I can essentially ignore that part of the theory. Okay, these particles are up at very high energies. They may carry gate charge, but they're up at high energies. And if I look at the low energy dynamics, the only thing that's left are the photons, okay, are the gauge fields. Okay, so this line of reasoning, so the original condensed phase, okay, which corresponded to R less than zero in the original theory, which had goldstone modes. Okay, over here, we'll correspond to R prime greater than zero, vortices gapped. Okay, but at low energies, you have these photons that are free to propagate because there's no gate charged matter hanging around. They're all up at high energies. So the photon will propagate, this is a Coulomb phase. Okay, so for this to match, you have to identify the photon over here with the goldstone mode. Okay, that's the only way this is gonna work. So the goldstone mode is the phi, and the photon is just the Maxwell action. Yeah, that's right. And that makes sense because they are, from this reasoning, we already had this connection. That's right, yeah, yeah. So if you didn't know that part of the discussion, you could still come up with these two effective theories and ask how they correspond. Okay, so it turns out a goldstone mode in two plus one dimensions can be represented as a photon and vice versa. So it may seem okay, they're both gapless, what's the big deal, right? But it turns out you cannot do this in three dimensions. So this is, it's a very special feature of two dimensions. If photons are not interchangeable with goldstone modes. If photons have two polarizations, for example, in three dimensions, you can't do this kind of duality in three dimensions. It's very special to two dimensions. Three plus one, one plus three. Okay, so that's one phase that we got. That's good for our theory. What about the other phase, which is the gap phase and the original construction. So this is gap bosons. How do I describe it in terms of the vortices? And of course, the only other option I have is to take this r prime negative, condense the vortices. You know, that sounds like trouble. We want to end up getting a gap phase, but I'm condensing these vortices. That sounds like that's gonna give me a goldstone mode. And I also have the photon. Okay, so it looks like I'm gonna get two gapless degrees of freedom, rather than not. Okay, but all of you I'm sure are familiar with this problem. This is the problem of the Higgs. This is the Higgs problem. And of course, we know that it's solved. The gapless photon eats up this goldstone mode, and you get a gap to all the excitations. Okay, so by the Higgs mechanism, this is also gapped. Okay, so this is the Higgs phase. Okay, so there's no low energy excitation, no gapless excitation. Okay, by the Higgs mechanism. Okay, so the photon gets gapped. There is no goldstone mode, because you never really broke a symmetry. There's no global symmetry of the Psi V. They're actually a gauge. It's part of this gauge symmetry. So you don't expect a goldstone mode. That's one way of thinking about it. And the photon is gapped because you have this condensate of these Psi Vs. Okay, it's trying to propagate through this high density of gauge charge. And of course, it requires a mass. Okay, so there's another nice thing that you can do is, you can sit on this gapped phase. Am I out of time or something? You can sit on this gapped phase and you can ask, what does the boson look like if I'm using this to describe the phase? Okay, so we're talking about a phase where the bosons are gapped. I found a different equivalent description of that phase in terms of essentially what is a superconductor. I have a gauge charged object that condensed. It's coupled to an electromagnetic field. Okay, so what does the boson look like in this language? Okay, so let's say I'm in the phase where I give an expectation value to Psi V. Okay, these are within quotes because this is not really a gauge invariant object to begin with, but we kind of know what we mean, right? This is like the Higgs phase. You condense this object and let's say you're on that phase you want to understand what the bosons are. Does someone know what the bosons look like in this language? So I have a condensate of these vortices. Okay, so I have Psi V, which is condensed with the usual caveats. And you can imagine making a vortex in this. So this is also a complex field. And you can try to make a vortex in this. So there's a vortex or the vortex condensate. So let's imagine the Psi V winds around, makes a vortex. Okay, now of course, this carries gate charge. So if I write this down as Psi naught V, V to the I phi V, some phase of this vortex condensate, if I write on an effective action for the phase, you'll get gradient phi V minus the vector potential that is coupled to. So different from the previous case, there's now a vector potential over here. And you'd like to minimize this energy. Okay, so previously we said that when you're trying to minimize the energy of a vortex, you have to integrate this gradient phi squared. But now you can cancel it off against a gauge potential. Okay, so you can actually make this vortex into a finite energy object by supplying the right amount of gauge flux. Okay, so if you have at large distances, you have the grid grad phi V minus A is just zero. If this thing itself goes to zero at large distances, this integral is just zero. Then this object becomes something which has finite energy. Cannot an energy that grows logarithmically. So for this you need that the integral of A itself is some two pi times some integer. Okay, so you can make this vortex, you can make it finite energy by supplying some gauge flux. Okay, and this is quantized to integers. Okay, so the excitations of this fluid, the vortex excitations, they are also quantized, but they only carry finite energy. Okay, so that sounds a lot like the boson excitations. Okay, so the boson excitations come quantized, you can have one particle or two particles and so on, and they only have finite energy interactions between them. Okay, so this is very likely just the bosons of the original model. Okay, so there's a completely dual description. If you want to, someone gave me this theory, I can say, oh look, it has certain vortex excitations. I can try to build up a theory in terms of the vortex excitations of psi v, but I find those excitations are only short range interactions, so I could go back and write down this theory for that. Okay, I would write down a theory again in terms of particles described by some complex field, but no longer range interactions, I'd only have this contact interaction. Okay, so it's kind of self consistent, you can go either way just by making these general arguments. Okay, so it's sometimes kind of convenient to keep track of the boson density of, keep track of this global U1 charge by introducing an external electromagnetic field. Okay, just like a source term that keeps track of it. So just to keep track of the global U1, let me just add grad minus I big A. Okay, so A is just some external field which couples to the global U1. And the question is, how does that appear over here? Okay, so now there should be an extra term that keeps track of this external vector potential. And it's easy to guess what that is. We just said that the flux of little a, the integral of little a on a line integral, which is actually the surface integral of curl of little a is the boson density. Okay, so this additional term, let me just write it down, is just the flux of that. So it's delta nu a lambda divided by two pi times epsilon. Okay, that's the way in which it will couple. Every time I have a flux of two pi over here, it's like a unit boson charge which couples to this external electromagnetic field. Okay, so this is sometimes called a BF term. Not very sure why, but this is an A and D, A little F term. And I think most of what I've done for the last few lectures, I've dropped factors of two very liberally. This is one place where you cannot drop your factors of two. You've got to get your factors of two right for these kind of terms. Okay, so now with this, we can actually go a little bit further. So how much time do I have? Okay, not too much time. So let me just tell you about the applications. Okay, so one of the applications that I said was in terms of the phase transition. So we identified the two phases and now we can look at the critical point and we obtain a duality for the critical point. Okay, so the duality for the critical point is we're going to set R equal to zero over here. Just to raise this term, we have tuned to the critical point. That's one theory. This is just a Wilson Fisher critical point. We call it Wilson Fisher. Okay, and the dual critical theory is, again, tuning to the point where these vortices are just massless. You know, colloquially we'll do that by setting this R to zero. And this is a different critical point. Okay, it's called, it's a superconductor called 3D superconductor critical point. Okay, and if you believe the rest of the story, you might also guess that the critical theories of these two are the same. Okay, just two equivalent ways. You can either take a complex scalar coupled to a gauge field and make that gapless just at the point at which it will undergo the Higgs transition. Or you can think of a scalar field that does not couple to any long range forces, no dynamical gauge fields, also undergoing a condensation transition. And the condensing is happening in opposite directions. This condenses from the right to the left. The other one condenses from the left to the right. Okay, it goes from gap to condensed in opposite directions. Okay, so it turns out that people had studied this theory before and they had tried to do an epsilon expansion. Okay, so there were parallel efforts in both condensed matter and field theory literature. And what they found is that if you are near four dimensions, there is no fixed point. Okay, so this is called Alperin Maw. But this guy, this is 3D superconductor. Okay, it's actually the, let me just call it the superconductor. If you're close to four dimensional statistical mechanics models or one plus three dimensions. So if you're either at four dimensions or very slightly away, it looks like a first order transition. Okay, and you can make it second order by having multiple flavors of this Vortex field. So you have n Vortex greater than or equal to 365. Only then do you find a continuous transition. Okay, so basically the epsilon expansion is really kind of screaming at you that this is not gonna give you a fixed point. Okay, but of course three dimensions is a long way from four dimensions. If you drop down in dimension, we now believe that there should be a continuous transition, which is completely opaque in this way of writing the problem, but which is captured by this dual theory. Okay, so people have done numerical simulations of this model. Just take this model on a lattice, simulated numerically. And what they find is that it does indeed seem to go to a fixed point where the exponents are exactly the same as the three-dimensional X, Y model, okay, which is the same as this there. Okay, so there's still more work to be done in numerics. What you actually want to do is you want to go to the limit of very small charge. So that's really like very large row S. And see that along that entire line of parameters, you flow from the lattice scale to that fixed point. Okay, it's not quite been done, but there are many models where you, you know, many different studies. You take this kind of model on the lattice with some finite coupling strength and it seems to be given by this kind of criticality. Okay, this is a very important question because we have superconductors in three dimensions. Okay, so you think about it as a classical statinic problem. If I give you a superconductor in three dimensions, I heat it up, you lose superconductivity. That physics is described by this model with the fluctuating gauge field. And you'd like to know what is the universality class of that transition. And, you know, it appears that again, this is really the right theory. There's no earthly reason why you would think about this. You know, if you were just talking about charge particles that have Coulomb interactions, but this is really the theory of the Vortex lines, which actually have short range interactions. Okay, that's what they describe by this thing. Okay, so that's one application. So how much time do I have? I'm out out of time. Okay, so just a word about the other one, which is in the notes. So here what we did was we condensed the Vortices. You can ask what happens if I condensed not one, but a pair of Vortices. Okay, so that's like condensing. It's like the Higgs mechanism, but not in the fundamental representation of this gauge field, of this gauge group. It's actually some charge two object that condenses. Okay, so that's very easy to describe in terms of these Vortices. It turns out in terms of bosons that gives you a very exotic phase. Okay, so if you condense pairs of Vortices, it turns out that if you go through the same analysis, the defect that you can make in that condensate actually has half the Vorticity that it previously had. Okay, so you run through this logic. It tells you that the excitations in the Psi V squared condensed phase actually carry half charge of the global U1. Okay, so it's like if you start with a model of bosons, you get excitations that are like half of the boson. And so that's an exotic phase that only occurs with very strong interactions between the bosons. It's usually hard to describe in terms of the original bosons. And how do you fractionalize one boson? It's really a many body effect, but it's sort of trivial to describe in terms of these dual variables. Okay, so things that are hard in this language are sort of trivial in this other language. Okay, many things. And that's why it's particularly kind of interesting. Yeah, and another open question. We just talked about some, this distinction between zero temperature and finite temperature in different dimensions. It turns out that these two phases, this is like a insulator phase for the bosons. The bosons just do not conduct. This is like a superfluid. And in between, you get a metal. Okay, so there's a metal right in between which has a finite conductivity. And there's a lot of interest in trying to calculate this conductivity. It's very hard to do theoretically because you have to use real time other than this imaginary time formulation. Okay, so there's a lot of interest in doing that and there's some hope that using this dual description you get some constraints on it. Okay, so there's a similar question for this dual theory. It gives you a different answer, but that's related to the resistivity of your problem. Okay, so anyway, the main message is that once you have this new set of variables, it's like you can explore a lot of new physics that's very hard to do in the original variables. Maybe I'll stop there.