 I'm flying an easy jet, but let's not finish it. Okay, please go on. Okay. I would like to continue today the discussion of the response of Sharon Simons' quantum field theory to classical sources, the derivation of anionic statistics, and then give you one modern generalization of a particle vortex duality in the presence of anionic vortices. That's my plan for today. For this course, I've planned, I mean, much more. I wanted to get to non-abelian gauge theories and the connection to QCD and domain walls in QCD. But anyway, there were so many good questions and intermediate discussions that I don't mind finishing just 50% of what I planned. So the action, I'll just write the action and then ask if there are any questions about yesterday. So J mu is a classical field. F mu nu is a fluctuating field. And the action from which this is derived, this is called the Sharon Simons action. It's given by mu nu rho epsilon mu d nu A rho plus A J. J is classical. So we're going to analyze it a little bit today. It's a very fun system to study. It behaves in a very different fashion from electrodynamics. So now, are there any questions about yesterday that were not yet discussed? So let's just understand the response of the Sharon Simons gauge field to electric charges. Let's say static electric charges. Yeah, yeah, there's some minus sign. I'm being a little bit schematic here because it doesn't really... It's not going to really make a difference. So I was a little bit schematic. Thanks. Now, notice that the Bianchi identity, which is a necessary property of the field strands, is implemented here by the conservation equation for the classical source. So even though J mu is a classical source, it must obey the continuity equation like any other classical source in electromagnetism. It must obey the continuity equation. Otherwise, the system is inconsistent. Okay, so if you write these equations in components... So, I mean, if you just write that mu equals to zero and mu not equal to zero components of this relativistic equation of motion, what you find is that the magnetic field is given in terms of K over 2 pi B. This is given by the charge density. So that's a very weird response theory. You put electric charges, and instead of them generating electric fields, they lead to a magnetic field which is moreover linearly proportional to the charge density. So it's completely localized on where there are sources. Where there are no sources, there are no magnetic fields. And the response for the electric field is likewise weird. It's going to be K over 2 pi epsilon ij eij equals ji. Equals ji. So let me just talk a little bit about it because these equations are very important and they appear in some beautiful applications in physics. So first of all, as you remember, there were also many discussions about it in my lecture. People asked about this additional term that I dropped. You remember that described the propagation which had two derivatives. So this term that I dropped had waves and actual propagation of electromagnetic fields. But now there is no more propagation. You see that the magnetic field and the electric field are entirely fixed once you fix the classical sources. So there are no waves. And so for instance, one interesting configuration to consider is just time independent distribution of electric charges of a unit charge. So I'm not going to write the factor e here. It's just going to be a bunch of delta functions. And it's going to be time independent. So we're allowed to just put charges wherever we want. We stick them and let's say that all of the charges are positive. So the pictures that we have are two-dimensional space and we just put charges. So what this equation implies, given that there is no current flowing, these charges are fixed so there is no current, it just implies that there is a magnetic field pointing through each of these particles. So these electric charges became solenoids. So what the Trem Simon's term does is to address each electric excitation with the magnetic field. It's a very weird phenomenon, but that's what the Trem Simon's term is about. So electric charges get addressed with this magnetic field and this would be the source of the different physics that Wilson described. Now, I want to say some things about the term that would drop. The term that would drop was like an ordinary Maxwell-type term. And so if I didn't drop it, of course, an electric charge would create an electric field because if k was zero, for instance, then this term that I dropped would be the most significant one and that would lead to an ordinary electric field emanating from a charged particle. But when we had the Trem Simon's term, as I gave you an exercise yesterday, when we had the Trem Simon's term, the electric field gets a mass. So there are no more mass-less propagating degrees of freedom. And in fact, if you were to include this term, you can do it as a homework exercise. You'll find electric fields which quickly decay. Exponentially decay. So the picture is that it's only true at long distances that what you see is this magnetic solenoid. At joy distances, of course, you would see the usual response theory to electric charges. You would see some electric fields. But they propagate only over exponentially small distances given by the scale of the mass of the massive spin-1 excitation that some of you worked out. Some people told me that they worked out the propagator correctly and it sounds good. Okay. Now you can furthermore write down the formula for the electromagnetic field A. By the way, I've used the lower case notations for the dynamical gauge fields. So let me stick to that convention. I want to use lower case notations. So you can also write the formula for the vector potential A mu. That's an easy little exercise. I'll just write the answer and your job is to verify that it's correct. So this is going to be a sum. Let's write it down in the gauge where A0 is 0. So this is going to be a sum over epsilon ij xj minus x0j squared. Okay. So this is the vector potential. And it's very illuminating to try to prove that this vector potential is essentially a derivative of something and you have to find what off. And so this means that this is almost a pure gauge, but it's not quite a pure gauge. So what you did, you remember from your graduate courses that if you have a magnetic solenoid, the vector potential outside of the solenoid is pure gauge, but it's not quite pure gauge. It's almost pure gauge. It's a singular pure gauge. You cannot get rid of the vector potential because there is some interesting autonomy around it. So this is almost a pure gauge, but it's not. You can almost write it as a derivative of something. You can try and see what white fails. Yes? If you try to remove this gauge field by a certain gauge transformation, then this gauge transformation would not be single-valued, precisely, very good. So another way to say it is that there are, in fact, holonomies. So an interesting problem is to try to compute the holonomies around these charges. The holonomies are given by exponentials of i integrals of a over some closed curves. This is the fundamental holonomy, but we can also define a more general holonomy where you put some integer n. Okay? Yes. There's no i here, of course. I made a small mess. So x0, this one. Yeah, this is extremely... This is the J's component. It's sum over the particles. So that's called this index p. So there is another index p here. And this is the sum over the particles. Thank you, yeah. This was completely butchered. This equation was completely butchered, unfortunately. Okay. Very good. So now I wanted to just mention the holonomies, and then this is the gateway to understanding the fractional statistics. So the holonomies are very easy to compute. You have a gauge field configuration, which we already spelled out after Atish has corrected the formula. So we have some gauge field configuration here, and we can take any closed curve, okay, and compute the holonomy. Now, because this system is a billion, the holonomy over every closed curve can be written as a simple sum over the holonomies of these particles that are encompassed by the curve. That's generally not true. But in this case, everything is a billion and everything is easy. So you can just write it as a sum or as a product of single particle holonomies. So the answer that we find for, the answer that we find for W1 is going to be proportional to 1 over k. So let me just write it with the correct coefficients. We get e to the 2 pi i over k. So that's the crucial thing, that there is a non-trivial holonomy around each fundamental charge. Now, of course, you could have chosen your charge distribution such that these particles will have charges q. You could have put here a qi if you wanted. And then the holonomy would pick up a phase q here. So that's obvious because everything is linear. Okay, so now let's get to the important thing, which is the statistics. And this is the key to the statistics. So let's draw this picture again where we have these charges, which have a little magnetic field. I'm drawing this arrow, even though the magnetic field is a scalar field in two plus one dimensions. The magnetic field has no directions to go, but I'm still drawing an arrow. So now the quick question is, what happens to the wave function? Classically, of course, this is all there is. There's some vector potential, there are some holonomies, and that's it. But quantum mechanically, it's interesting to ask what happens when we take one of those heavy probe particles around another one. There's a certain wave function, and we can ask what happens to the wave function. So here one has to appeal to the Ahronov bone effect. Classically, this question makes no, I mean, there's no interesting content in this question. Classically, when you drop particles around each other, nothing happens, right? Because classically, this particle couldn't care less that there is a magnetic solenoid inside. But as you remember from quantum mechanics, when there is a little magnetic solenoid, then you pick up a phase. So Ahronov and Bohm told us the answer. Ahronov and Bohm found that if you take a particle of charge one and you drag it around some object, the phase that you pick is W1. So in general, dragging a particle of charge n anything, around anything gives Ahronov-Bohm phase, which is given by Wn. So to understand the phase, we have to compute first the autonomy around that particle. And that depends on the charge of this particle and the charge of that particle, which I have so far omitted. But let's call the charge of this particle n, and this is n prime. So therefore, the phase, the Ahronov-Bohm phase, will be, let's call it delta nn prime. This is going to be the Ahronov-Bohm phase when you drag one particle around another particle of charges nn prime. So first we need Wn. Wn is obtained by taking this formula and raising it to the n's power. But then we have to remember that now we're computing it around the charge n prime. So we have to also raise it to the power n prime. So we just get the previous phase multiplied by nn prime over k. And this is the important formula which we're going now to utilize. Okay? So let me just explain what this formula means. So this is called braiding phase. If you read the literature on topological filters, this is called the braiding matrix or the braiding phase. This is some piece of the data that specifies topological filters, these braiding phases. Abelian-Chern-Simons theory at level K is saying this is the answer. But this has many interesting consequences. This is just the phase that the wave function picks. But there is now some consequences for statistics. You remember that if you have two fermions and you exchange their roles, you get a minus sign. If you have two bosons, then you get a plus sign. There are no phases in the wave function for symmetric fermions or bosons. Here we see that if we take an anion all around another anion, we get this phase. But the spin of the particle is obtained from essentially the same thing. The spin of the particle is the square root of this phase because for fermions, we exchange them and for bosons, we exchange them. For fermions, we get minus one and for bosons, we get plus one. So the square root of this phase is essentially the spin of the particle. So the spin of the particle is essentially the square root of the phase that you would get if you took the same particle around the same particle. So let's take n' to be equal to n. We take the square root of this formula and we get the spin. So the spin is n squared over the spin of the particle anion with electric charge n is given now by n squared over 2k. So we find fractional spins. n is an integer. This is comment number one. This leads to intrinsic fractional spin. But that's allowed in two dimensions because the little group is u1 and its universal cover is r. So we are allowed to have fractional spin in two dimensions. So these classical sources became anions. If you first quantize them, you see that they behave like anions and there's interesting quantum mechanics problems associated to the many body, you know, Schrodinger equation for anions. There is a beautiful paper by Wilczek written in Wen, I think, which discusses some hydrogen atom-like problems for anions from the 90s. Okay, that's the first comment about statistics. The second comment is about how many anions are there. And this is the point where the quantization of k comes into play. One may observe that if one shifts either n or n prime by an integer proportional to k, this cancels out. So this phase becomes trivial. Okay, so we see that if we study charged particles whose charge is a multiple of k, then they have no observable a heron of bone phases. And therefore, they make no impact on the low-energy physics. So the low-energy physics, namely the topological filter, it doesn't have infinitely many anions. It only has finitely many anions given by those which have a non-trivial phase with some other anion. Therefore, both n and n prime should be defined modulo k. This is the first approximation to the answer. So we have finitely many anions, and that's essentially why k has to be quantized. There isn't a case quantized, at least in flat space. You would see some disease if k was not quantized, that there would be infinitely many anions. So if k is quantized, we have finitely many distinct anions. Well, it's not obvious that this would lead to some disease. Typically in physical systems that you see in the lab, like in quantum Hall effect, I'll explain that in a second. You'll see that there are finitely many anionic excitations, but it's not very clear why there would be a strict problem because there are infinitely many anions, but there's one more step that I could explain, which would show that, which is that if you try to compute the partition function of this topological filter on the closed manifold, to get a finite answer, you need finitely many anions. It turns out that the Hilbert space on the torus is isomorphic to the space of anions. It's like an analog of radial quantization for topological filters. So you get an infinite dimensional Hilbert space at infinity many states of energy if there are infinitely many anions, which sounds like it's a six system. Okay, now I want to make this just a tiny bit more precise. The point is that the spin was defined as the square root of the phase, and therefore the spin is intrinsically only defined module one. Indeed, you cannot know the spin of an anion more than module one. If I tell you that the spin of some anion is one over six, you can distinguish an anion of spin one over six from an anion of spin seven over six, because if you measure our own of bomb phases, all you can get is the square root of the our own of bomb phase, and that's only defined module one. Indeed, these two anions could differ by just a normal particle, which is not anionic. You could take a spin of six anion, put on top of it some proton, which has spin, let's say, well, put on top of it a Romeason, which has spin one, and together they will give seven six. So you cannot distinguish by low energy measurements these two choices. You cannot distinguish these two cases by actual experiments where you measure the spin. But by measuring long distance interference, like these kind of things, you cannot distinguish these two cases. And therefore this is defined only module one. So therefore there is a small subtlety with the number of anions. There is a small distinction between even k and odd k. That's the last thing that I'm going to say about it. For even k, we see that the anion with charge k is completely trivial. It's spin vanishes, and it's a a lot of bone phases vanish. So for this case, there are k anions. For this k, for this case, however, k two plus z plus one, if we put n equals k, this is a half integer rather than an integer, and it still has some consequences at long distances. It means that you have a fermion at long distances, and it affects various observables, especially when you put it on a spin manifold. So here we have 2k anions, because you need to go to 2k to completely trivialize everything. So the anion with n equals k in this case is very interesting. It's completely transparent. It has no aural of bone phases, as you can see from that formula, but it has a half integer spin. And you know such a particle. It's called the electron. The electron is a particle which has no aural of bone phases, but it has a half integer spin. So if you look at the quantum Hall effect, usually for odd k, this anion is called the electron. So that's how in the fractional quantum Hall effect people identify the electron. It's that transparent anion with n equals k, which looks like a transparent fermion. So if you look at Lafflin's wave function for k equals 3, that's the electron. It's a very nice fact that connects to this discussion. Yes, yes, there is also a one way to measure this braiding matrix, abstractly, is by considering Wilson loops for anions that braid each other. Like this. There is an abstract definition of this braiding matrix from the braiding of anions from the Wilson loops of anions. Okay, so I hope you got some intuition about what Sharon Simon's terms do. They seem like they completely gap the system, but they do not. They lead to some anions. Then there was a lot of work in the 90s on first quantization of anions where you just study a fixed number of anions, heavy anions, moving in a box with these finite phases, and some potential. But the modern work is about the second quantization of anions where we also allow anions to pop out from the vacuum. Yes. If k was in q you would still get finitely many anions. It might seem okay from the discussion here, but one thing that I can tell you which I won't prove is that the Hilbert space I already said that in words, but I'll write it precisely for the record. The Hilbert space on p2 if you put these anions in a periodic box the Hilbert space on p2 has k states. That's true whether k is even or odd. The Hilbert space on the torus does not see this transparent electron anion, or transparent electron. The Hilbert space on p2 has k states. That's perhaps the easiest way to see that k must be quantized. Perhaps I should just say one word on history. The fact that k has to be quantized in non-Abelian was appreciated from the get go. The reason is that if you write non-Abelian action there is a certain gate transformation called the large gate transformation which makes the action non-invariant and less case quantized. People have appreciated immediately that for the non-Abelian k has to be quantized. But if you look all the way into the 90s there are still papers discussing that in the U1 case there was no obvious disease with fractional k. In the mid 90s these papers stopped appearing. People have understood that this makes really no sense. For U1 it's much harder to see that k must be quantized. And indeed perhaps this is one way to understand it. But if you just do experiments in infinite flat space you don't see an immediate disease that's k to be fractional. As you said if it's a rational number it seems okay. For U1 this quantization business is perhaps the most subtle one among all the cases. Are there any questions about the basics of Abelian-Chern-Simons theory? Now let me just tell you about the more recent work on second quantization of that. Second quantization of enions so this is what you get if you try to make these enions pop out from the vacuum. So let's do the simplest example now. So I'll do the simplest example and I'm only going to do the case of Abelian-Anions. So I'll do only Abelian-Anions associated to Abelian-Chern-Simons theory. There are generalizations in the literature by now for far more complicated cases but for pedagogical reasons I'll just do this case. So you remember that we studied the gauged complex scalar field H. We had a long discussion about the duality between the void system and the particles in this model of the gauged complex scalar field H. Now what we want to do is to think about the H field that's creating other than ordinary charged particles. So the way we're going to write an action is that we will have the kinetic term for H as before and the gauge field is going to be called A so this is the notation for the covariant derivative it's exactly as it was before EAH plus IAH So we have a kinetic term for H then we have some potential for H which I'll specify in a second and then we have the gauge field part of the story which involves as before 1 over 2G squared d mu A nu minus d nu A mu squared and now comes the new piece which we want to add I over 4 pi A epsilon mu A mu d nu A rho and this is the action Now this is the model with k equals 1 So I'm now starting the simplest possible case where k is equal to 1 in terms of our previous notation Now this may look this may look stupid because one of the main points here was that when k is equal to 1 it's almost trivial so in k is equal to 1 there are no interesting braiding phases when k is equal to 1 all the spins are either integer or half integer and if you remember the formula for the braiding phases all of them vanish here it is so when k is equal to 1 all the braiding phases are trivial so it seems like we haven't actually introduced non-trivial enions yet to introduce non-trivial enions we would need to put a k non-trivial k which is very interesting because even for k equals 1 there is still something going on which I've emphasized here that's why I told you about it when k is 1 we don't have enions but we can have one fermion so in this you should already figure out that somewhere here there is going to be hiding a fermion even though there are no fermions in the Lagrangian this term modifies fermion even though it started its life as a boson so you should already anticipate fermions appearing from this discussion so strictly speaking this model wouldn't have non-trivial enions but it will have some surprise nonetheless because there is this fact that the charge particles with charge particle with electric charge 1 when it's very heavy when this field h is very heavy it's going to behave effectively like a fermion the person who first figured out that in the presence of a churned simons term a boson can look like a fermion which is essentially in this is Poliakov but Poliakov never discussed the second quantization what Poliakov did was to say if h was a classical field it creates effectively charge 1 particles which are heavy and non-dynamical and their statistics get transmuted a la this discussion here from the spin you can see that the spin becomes a half so as far as I know the first person who realized that the spin gets modified from 1 or 0 to a half is Poliakov but as I emphasize he never studied the second quantization he just made an observation about classical sources coupled to churned simons terms which is exactly what I discussed in the beginning of this lecture yes I just count here for k equals 1 it's already non-trivial there's a gap you're asking if k equals you're asking if u1 level 1 topological field theory is considered trivially gaped or not okay since you asked I'll just say two words about it I didn't want to get into this discussion so suppose you just have u1 level 1 churned simons action so meaning suppose there was no h but just these two pieces what is the correct terminology for this model is it trivially gaped or not the answer is the following the answer is that this model has a fermion hidden in it even though there are no fermions in Lagrangian there is a hidden fermion because the transparent line is fermionic statistics so this model can be defined only on spaces which have spin structure from the get go but on those spaces it's trivially gaped so people think about this model as the trivially gaped SPT phase that's the modern terminology but one should remember that it cannot be it must be put only on spaces where the spin structure is chosen so it's called a fermionic SPT phase if you want the precise terminology nowadays so it's basically trivially gaped to first approximation but one has to remember that there is a fermion any other questions okay so when we write Lagrangian we immediately draw this axis of m squared that's our automatic reaction to Lagrangian so where v of h is as before m twiddle squared h squared plus lambda twiddle h to the fourth so we always we analyze the weakly coupled limits what are the weakly coupled limits there is the huge positive mass squared and the huge negative mass squared for huge positive mass squared the analysis is already been done on the blackboard like what I did so far is essentially huge positive mass squared so let's do it first huge positive m squared means we can integrate out h, h makes no difference and what we get at low energies is this plus that okay so it's huge positive m squared our effective field theory at first sight has two pieces we have a kinetic term we have a kinetic term f p new squared plus 1 over 4 pi a p a with this epsilon tensor now we can this this can be further simplified because what did we find in the exercise from yesterday we found that this leads to a massive photon so we throw away the massive photon it's not part of the low energy effective theory so when we go to even lower energies all we have is the churned simons piece so this so all we have eventually is just the churned simons piece which is a d a with this epsilon tensor and as the gentleman here remarked since there are no enions there is just one world line of transparent fermion this is really trivial there are no enions so this is a trivial phase the filter in this case in this limit is really trivial because the level is 1 but I want to remark about this I want to make a remark about this so at long distances there is really nothing in this model but it's worth mentioning that the world line of the H field is a fermion so the H field which is an electrically charged electrically charged field the H field is a fermion even though it looks like a boson and there are no fermions in the original Lagrangian the world line of H particles are fermions so if you have two such things and you exchange there if you have two such particles H particles and you exchange their roles you get a minus one factor in the wave function so even though there is nothing at long distances it's worth remembering that there is a massive fermion so this model in this phase we really see just a massive fermion there is nothing at long distances but it seems like this model has a massive fermion somewhere in the spectrum now let's discuss the other limit the other limit is also very rich in physics in the other limit H wants to condense and when H condenses we have a Higgs mechanism so here the theory at long distances is most certainly trivial because A disappears it gets amassed through the Higgs mechanism and H disappears since it's given by the gate field and the radial mode of H is like the ordinary Higgs particle so everything is massive so this is of course also trivial but who can remind us in the Higgs mechanism of the U1 particle vortex duality what was the most important excitation what was the name of the excitation in this phase that we studied in some detail which excitation did we see here somebody remember we didn't call them monopoles how did we call them magnetic vortices because yeah monopoles is reserved for three plus one dimensions very good so here the most important excitation was the magnetic vortex so I want to tell you something about the magnetic vortices in this phase the magnetic vortices in this phase are very interesting and they behave slightly differently from the magnetic vortices that we encountered in particle vortex duality so remember the magnetic vortices how did it work we looked at the Stuckelberg action which is the effective action so to speak at long distances at the Higgs phase so the Lagrangian was let's say that H we wrote H as absolute value of H times e to the i psi that was our notation for H and then we wrote an action that looked like d psi a squared this was our Stuckelberg action and then I told you that you can then create vortices where psi has some vorticity around some point which is the core of the vortex and to cancel the energy at long distances we adjust the gauge A to be 1 over r and that's how we create finite energy localized magnetic vortices approximately localized magnetic vortices but here there is a small twist an interesting twist in this story because the action for A also has a Chern-Simons piece and it turns out that it makes a difference for the vortices so so there is another piece that we need to write which is this A, D, A so actually many people got interested in the properties of vortices in the presence of Chern-Simons in terms and by now there is pretty extensive literature about it I'll give you references at the end a very nice homework exercise which is not entirely trivial but easily doable if you think about it is to understand a small fact about these vortices in the presence of Chern-Simons terms a small fact about these vortices in the presence of Chern-Simons terms is that they have a fermion zero mode more precisely they turn into fermionic particles they have a spin a half I didn't mean to say literally a fermion zero mode what I meant to say is that when you compute the angular momentum of this vortex without the Chern-Simons term you find zero and with the Chern-Simons term you find that it has a spin a half because the Chern-Simons term leads to some electric fields so to speak near the vortex so this is a very nice fact that it's easy to establish by yourself you can just look for the solution compute the angular momentum and you'll find that it is exactly a half so it's not an anion but it's a fermion so the magnetic vortex becomes a fermion furthermore you can do a slightly more precise computation you can compute the spin the spin is defined module one in Chern-Simons theory but as I told you in the full microscopic theory you can compute the spin precisely because the spin does the angular value under rotations because now you have the full microscopic model so you can compute the spin exactly not just mod one so here the spin of this H-field can be computed exactly and it turns out that in some convention it's a half and in the same convention this is minus a half this is spin minus a half and this is a half so there must be a phase transition this is an interesting example because both phases were trivially gapped and if you were Lando and Ginsberg in the 50s you would say okay we have two trivially gapped phases there is no phase transition but that's incorrect because the rotation has spin a half and here it has spin minus a half and a half and minus a half are not the same this is the eigenvalue under rotations two dimensions so there must be a phase transition there is a mathematically more precise way to say that there must be a phase transition if you know about SPT phases but I'm not going to explain that here so there must be some phase transition as I said there is another way to prove that there must be a phase transition but I won't explain it here to those who have never seen it before we have a model where there is a spin a half particle on one side a spin minus a half particle on the other side and there is a phase transition does anybody know a model which has a massive fermion on one side a massive fermion on the other side and the spin goes from a half to minus a half did anybody ever see such a model? exactly so this is just a free fermion let me just show you how it works it's dual to a free fermion very good, very good guess so let me show you how it works so consider the model of a single complex fermion single complex fermion in two plus one dimensions this is the model for a single complex I think it will be called Dirac fermion in two plus one dimensions for positive mass for positive mass there is a representation in the system in the Fox space which is a single fermion and it has spin a half for negative mass it has spin minus a half there is still one state in the Fox space it has either spin a half or minus a half and at m equals to zero we have a second order phase transition second order phase transition so the idea is that here the fermion becomes massless so this is a free fermion this is a striking prediction if people can ever simulate this model on the lattice they should see the scaling exponents of free field theory namely of a free fermion theory and they should see that out of this Lagrangian which had no fermions from the get go there pops out a fermion an effective excitation which is the fermion and we see that this fermion can be viewed in two different ways we can think about this fermion either as coming from the world lines of the electrically charged particle H so we can think about H as if it became like a fermion or we can think about this fermion as a magnetic vortex that became a fermion due to the Tern-Simons piece so this is a story about second quantization not about first quantization because here we have the full Fox space of a free fermion not just one fermion yes you are asking if H is a fermion well in massive phase of course near the phase transition H is very complicated ok so we have to write the dictionary that's what I'm going to do now H strictly speaking as we discussed is not well defined H is gauge variant ok so H has to be attached to a Wilson line it's like a pro particle with charge one electrically but as I told you if you have an electric charge one it also obtains a small magnetic field the Tern-Simons piece so now this is like the H particle our H particle started its life as an electrically charged particle but now it became like a little solenoid because of the Tern-Simons piece so even though H as it appears in the Lagrangian it's a boson when you interchange these two H's you get a minus sign but H is not a well defined gauge variant operator yet it leads to a particle in the Hilbert space that looks like a fermion I'll write the dictionary now more explicitly so I'll write the dictionary here between these two models so here we have the U1 gauge theory plus H that's level one Tern-Simons piece and here we have a free fermion I want to write the dictionary akin to what we've been doing for the other case it's a vortex type duality but it's much more surprising than the previous one it's even more surprising than the previous one because it's a boson fermion duality so this is like a bosonization duality in two plus one dimensions boson fermion duality two plus one dimensions you might have heard about bosonization in one plus one dimensions and this is a generalization to two plus one so let's write the dictionary here in this model what are the global symmetries maybe some audience participation which global symmetries exist in this side of the duality it's the same as in the previous case that's a hint what was the symmetry in the gauged model in the previous case does anybody remember yes very good there is the magnetic symmetry so on this side of the duality we have the U1 magnetic symmetry which acts on monopole operators and on magnetic vertices epsilon mu new row f new row with a 2 pi on the fermion side what is the symmetry what is the symmetry of a free fermion particle number yeah fermion number so it's just psi dagger gamma mu side and you can check that it's conserved it's just the ordinary U1 symmetry it's just the ordinary U1 particle number symmetry so the symmetries match the phases match and an interesting element in the dictionary is what does h squared map to so when we change the mass on this side of the duality on the fermionic side of the duality we change this mass so therefore h squared maps to psi dagger psi that's another element in the dictionary perhaps the most interesting element in the dictionary is to ask what does psi map to somebody already asked that what does the elementary fermion operator map to on the other side of the duality can somebody guess not e to the i h e to the i h is not well defined ok so let's not confuse the magnetic vortex is a particle that is created from the vacuum by some operator so psi is an operator so not a particle it's true that the fermion particle is mapped to the magnetic vortex particle but now we're asking about the operator map so what is the operator that creates magnetic vortices from the vacuum in the Higgs phase now h times the Wilson line would not create a magnetic vortex h times the Wilson line is not a local operator psi is a local operator it's a genuine gauge invariant local operator so you're looking for some operator that would create a magnetic vortices no a Wilson loop is not a local notice a hint this operator is charged under the u1 so on this side of the duality the corresponding local operator should be charged under so which operators are charged under the magnetic u1 monopole is good so this is the monopole you see that this map is extremely non-local the fermion maps to the monopole on this side of the duality and the fermion bilinear maps to H squared and you can continue a little bit the dictionary you can say that fermion excitations in the fox space map to either H-Wilson lines or magnetic vortices depending on the phase you understand we can continue this map a little bit in more detail than what I said here so this is the simplest example of how boson fermion duality arises and this is a particularly trivial example because there are no enions here so in this system there are no enions it's just a free fermion and a free fermion doesn't have any enions it's just a massive spin one excitation it's been a half excitation so I don't want to start a new subject so I'll just say go for if there are any additional questions for the last five minutes if somebody wants to just about which limit and twiddle squared so in this story here I slightly deviated from the philosophy of these lectures the philosophy was to look always at very long distances and ignore all the massive stuff so in this model if you looked at very long distances both in the huge m twiddle squared phase and in the huge negative m twiddle squared phase there is nothing it's trivial both here and here so you would see no signal of a phase transition that's except that's the first part of the answer the second part of the answer is that even if you look at very long distances there is something more sophisticated that you can do to detect a phase transition but I did not explain that fact at first sight it seems that there is no phase transition so for this reason in this example I also considered let's say the first non-trivial state in the Hilbert space to try to understand its quantum numbers so I deviated a little bit from the philosophy so let's do it again a huge positive m twiddle squared the gauge field the gauge field is unhixed and it's massive because we have a turn Simon's mass so what is the simplest excitation in this theory there is one excitation that is just the spin one the spin one gauge field that's what we explained yesterday that A becomes massive and it's a spin one massive excitation that's one excitation in the Hilbert space I could edit here in the fermion description that would be maybe some fermion bilinear of spin one you could see it on the other side of the description an interesting excitation on this side on this part of the phase diagram is that since now the gauge field is massive you remember this logarithmic divergence due to the insertion of one particle that doesn't happen here because the gauge field is massive so this logarithmic tail goes away and therefore it makes sense to ask what happens to the H particle now the H particle even though it's not gauge invariant H is not gauge invariant the H particle is okay but like UED in 3 plus 1 there is an electron because now there is no long range tail and it's a finite energy configuration and that H particle is like an electrically charged particle but because of the transimons piece it obtains a small magnetic flux and it looks like a fermion so we can claim with some confidence at least at large m squared we can say it with full confidence that in the Hilbert space there is a spin one half particle and on this side and convince ourselves that they have spin one half so I deviated a little bit from the logic but it was in order to convince you that there is a phase transition because here in the Hilbert space there is a spin one half and here is minus one half so cannot change continuously and since there is this phase transition then we just guess that it's a free fermion sort of made sense one can scrutinize this guess much more one can compute various discrete anomalies and various things and everything works this guess can be really put on much more solid grounds by doing additional more sophisticated computations any other questions? yes so since there are trivial gaps if you took a big big manifold there will be just one vacuum there is no topological filter here at long distances but the model does require spin structure so I'll say if you wanted to compute the partition function on a big big torus you have to specify the spin structure because you see that there is a hidden fermion here and once you do that you'll find always one ground state and yeah so this is a very very deep and good question maybe I can explain the question to everybody the question is that you're asking you're saying that this model is bosonic it has no fermions so in order to put it on some non-trivial three-dimensional space spin structure well on this side of the correspondence it seems that we need to choose a spin structure because we have a manifest fermion so we need to say if it's periodic or anti-periodic over cycles right so the question is how is this contradiction settled so this contradiction is settled in two ways there is the less advanced way to understand why it's okay and there is the more advanced way so let me tell you both points of view which are very important if you ever dive into this literature that this is a very important piece of the discussion of how is this even consistent can I remind you how it works in two dimensions in two dimensions there is a famous duality between the two-dimensional Ising model and the free fermion that's on Sager how does it work in two dimensions in two dimensions this manifest does not require the choice of a spin structure and this does the way it works in two dimensions is that this is not a free fermion the duality of on Sager was not between Ising 2 and the free fermion it was a free fermion where you also sum over all the spin structures so here you are instructed to sum over all the spin structures so when you compare partition functions on one side you take some manifold and you don't care about the spin structures on the other side you sum over all of them so that's one way in which this can be resolved maybe on the fermionic side we have to sum over all the spin structures so now what happens in two plus one dimensions is kind of similar what happens in two plus one dimensions is that there are two points of view one is that this turns Simon's action since it has a transparent fermion if you look at the original papers in mathematics on this turns Simon's action you'll see that this requires a spin structure so even to write this Lagrangian on three manifolds you need to choose a spin structure because this particular object is not globally gauged environment on non-spin manifolds so to speak that's one point of view so you could say that both sides require a choice of spin structure but in fact the better point of view is like this it turns out that you can choose this gauge field to be rather than a U1 gauge field you can choose it to be a spin C gauge field you know what's a spin C gauge field okay a spin C gauge field is a gauge field which allows you to sum over the spin structures so you can choose this gauge field to include the sum over the spin structures and then perhaps I'll just stick to the first point of view which is simpler to explain because I don't want to explain spin C now so the first point of view which is simpler is that in fact in an implicit fashion both sides require a choice of a spin structure because on this side there is a level one turn Simon's turn which requires a choice of a spin structure because it's not well defined on manifolds without it and this side manifestly requires a choice of a spin structure so it's not entirely analogous to two dimensions but using spin C structures and other ghostly you can make it look more like in two dimensions in the case of free fermions the theory is like parity invariant means the one side but we see that in the left hand side the current is not parity invariance it's another excellent really deep question this model a free fermion is clearly parity invariant and also I should say that the mass term breaks parity so a positive mass are related by a parity transformation which you can see from the spectrum of the fox space spin goes a half goes to minus a half while this model does not have parity symmetry in the Lagrangian if you look at the Lagrangian it doesn't have any third of parity because the sign here is plus and not minus so another fact about this duality which I didn't even mention is that at long distances this model acquires a central parity symmetry so even though you start from a microscopic model which has no parity at long distances everything becomes parity invariant approximately so this is an instance of an emergent time reversal symmetry so this model has an emergent time reversal symmetry at long distances and this is a crucial fact about this model and it's also quite a common phenomenon that you start from a model that has no symmetry but it acquires a symmetry at long distances this also shows that this duality cannot be exact it must be only a property of the long distance theory around the phase transition because there is no exact parity symmetry in the bosonic model it's only a long distance property of the model that's why the duality that's why the duality only holds in some vicinity of the phase transition any additional questions in the fermion case if it's massive no at the quantum level you get parity breaking you get the chenzymon's term wait you're asking about the free fermion yeah the massive free fermion it doesn't have parity at the quantum level it's just a free field theory it's like this it's just free field theory this model with non-zero mass it describes the free fermion in two plus one dimensions I know but if you couple two we don't couple it to anything the claim is that the complicated bosonic model is dual to a free fermion with a mass that's it so once you have the mass it's clearly not parity invariant but at the massless point it is that means that the bosonic model near the phase transition acquires a newly found time reversal symmetry since we are out of time but this last lecture of Zora I allow for one more question if any this last opportunity was something if we consider quiver gauge theory with another u1 but another chenzymon's term but with minus one level we have an accidental z2 symmetry if we exchange two gauge the gauge fields so how can we interpret this in symmetry on the fermionic side well this would be a completely different model to analyze what happens if I have two u1 gauge groups maybe some matter that is charged under the two u1 gauge groups and I take the levels to be one and minus one so what you are saying is that since one and minus one are related by time reversal and there is some accidental time reversal maybe there will be an accidental exchange symmetry this is a set of interesting conjectures but you have to try to write down the model try to find that duality and see if it works I believe that the concrete model you are asking about hasn't been studied at least not that I remember so yeah I mean if you have this intuition you can try to work it out ok I wanted to give like one minute some references and that's it if you want to look at references about particle vortex duality it includes some part of what I said it does not include some other part of what I explained it is just a subset of the things that we discussed there are nice lecture notes by David Tong they are free online on his website and they cover a few of the things that I said and some of the computations that you may need to do to fill the gaps recently there was also a small review that came out by condensed matter people so it will be from a completely different point of view emphasizing the connections to the fractional quantum whole effect and to quantum phase transitions and de-confined criticality that was by Mclitsky et al so you can read a little bit about that in that review so these are two reviews about particle vortex duality and generalizations they are off and hopefully in the future there will be some reviews from a more high energy perspective which will also include more material but at the moment there aren't any maybe some people will write some reviews so this is about particle vortex duality now about the connection well in the first lecture I gave you a little bit of material about a young millsteering about four dimensional young millsteering I gave you a little bit of material about domain walls in young millsteering and how Cher and Simon's terms arise on domain walls and I haven't actually explained the connection to all this stuff I haven't had time but if you want to read about it you can look I mean there is no review but you can try to read some literature which you can get all the references that you need and also looking at the references to that paper you can find all that you would need to get into the subject so you could start from the paper of Gaiotto and myself so just look at all the references and all the citations to this paper and there is quite a bit of literature but there is unfortunately no review yet of that story and the connection to this particle vortex duality but you could look from that you could start from that and yeah so one may hope that in the future there will be a more comprehensive review of the subject written from a high energy physics point of view ok so this is it let's thank Zor for this beautiful set of lectures