 Okay, okay, let's get started. I have an announcement before this last talk of Nati. This afternoon is going to be a special session to felicitate the physics contributions of Kumar Narayan who is one of the main organizers, who has been one of the main organizers of the school for the last 25 years. I urge all of you to attend it. It will give you a perspective on the history of strength theory a little bit, and also the more classical period of strength theory when the world sheet was important, modular inverse was important. So you haven't heard anything about that during this school, but this session will give you kind of a new, different perspective about strength theory. So I encourage everybody to attend it, and now we'll start with the last lecture. So even though they updated it up there, they still insist that I give the same talk that I gave yesterday, but I will not. What's that? It will be what? Okay, I can do that. I have no problem. I'm about the end of lecture two out of four that I planned. So I'm not going to finish my notes anyway. I need more chalks, longer chalks. Okay, good. So yesterday, we discussed something bad with the sound, with some feedback. There was, we started discussing dualities and we discussed particle vortex duality and just by manipulating the standard particle vortex duality, we derived the new boson boson duality, which looked a lot, this theory that we studied was a gauge Wilson Fisher model that we denoted like that and I'll symbolically write it like that so that I tune the coefficient of phi square to zero and what I really mean by that is no symmetry at the point phi square equals zero. So what I really mean by tuning is that I tune it to the fixed point of the infrared theory and different people might have different regularizations and then what they mean by mu square in the Bella grandeur would be different but what is meaningful is that there is a fixed point and what I call mu equals zero is being at the fixed point. So this way, if you and I have different regularizations and you define your mu one way and I define my another way, once we found the fixed point we can both adjust what we mean by mu such that mu equals zero is the fixed point. So I did not include it. Also the coefficient of phi to the fourth will let it flow all the way to the infrared. So it's no longer a parameter in the theory so I'll denote it schematically by one but what I really mean is that we start with the Lagrangian as last time. With the phi square in with the phi to the fourth we flow to the infrared, we tune mu and we call that mu equals zero. So this is shorthand for all that and we also said that we add here i over four pi bdb. So it's not just the Wilson-Fischer fixed point coupled to a gauge field with a chance simons level one and we added to it coupling to background field two pi bda. So this is what you really mean by that is this theory in the UV that there might be a gauge coupling for b and there might be a mu term and a lambda phi to the fourth and we flow to the infrared and we look for the fixed point. We readjust what the parameters are and in the end we have this theory that doesn't have a free parameter. It has a global U1 symmetry. A is our device to probe the global U1 symmetry. It's a classical field and the claim yesterday was that this whole thing in the infrared is the same as a free fermion coupled to the same background field psi. So in the past five minutes I fulfilled the requirement of the organizers that I would give last talk and I also the requirement from here or the request that I give it again. So now we can proceed and what we'll do now is use another trick which is the same trick essentially that we use yesterday that once we have one duality we can plug it in and do the same manipulations on both sides and derive many other dualities. So again, we have this duality. Recall we derived the boson, we started from a well-established boson boson duality. We actually derived from it another boson boson duality which is essentially this theory with some signs flipped and some added term, classical terms. And then we said this looks like a fermion so that's an assumption. We are going to assume this duality and now I'm going to derive consequences. For the rest of the talk I will just derive consequences of this duality and I'll derive many other dualities and we'll present checks on the other dualities and if any of them fails it would mean that this duality is wrong. As long as they are checked then this gives us more confidence in this duality. It did, it did, it did. So there's a question of whether you mean the particle or the operator. I'm answering your question. So you could mean two different things in the question. One is what is the map of operators? Can somebody sitting in the UV in the bosonic theory identify an operator's side? That's a way to interpret your question. So let me answer that. In fact I did it yesterday. I say that on this side of the duality we can consider a monopole operator for B. It's not a soliton. It's a monopole operator and because of this term it is charged under the u1 symmetry of B. So we multiply by phi or phi dagger depending on your conventions. And we argued yesterday that this is mapped to the operator's side. So in the UV you can really see the operator's side. So even in the UV here where the theory is interacting and complicating you can see that it has an operator which has spin a half. It's a fermion. It carries charge one under the u1a symmetry. But it's not free. So in the UV I'm trying to answer your question. So there is such an operator in the UV it has spin a half and charge one and it satisfies a very complicated equation of motion. Claim is that in the infrared the equation of motion of this fermionic operator is that of a free fermion. That's a non-trivial dynamical statement. The fact that there exists an operator of spin a half and charge one that's easy to establish and that's true in the UV here. The claim of the duality is not that this theory in the UV is the same as this. It's only the infrared is the same. So we identify this. Another way, let me finish this and then you can ask your question. Another way to interpret your question is about states in the Hilbert space. But one thing at a time. This thing has been going on for some 30 years. When there was still in Russia and I had not seen him we had a very long email exchange and it took three weeks each way with long letters written long long hand and the style was exactly like this. And I was younger. I don't know whether he was younger than or not. So another way to interpret your question is about the states. So the operators are always defined in the UV. They are independent of what the parameters are and so forth. The parameters determine which phase the theory will be in. They determine the spectrum in the infrared. That's a separate question. So let's discuss the parameters and the phases. So we said that the operator psi bar psi is mapped to the operator phi absolute value square or maybe with a minus sign. Doesn't matter. So that's still in the UV. This is the map of operators. Now we discuss states. In order to discuss states we need to know which phase we are in. And we discussed yesterday three phases. One is mu equals zero. That's a transition point where this theory has a fixed point and this theory has a fixed point. In this language the fixed point is a free field theory in this language we don't know and the claim is that it is free. At that point the states are massless particles so it's very confusing what we're talking about. So we should better be in one of the other phases to formulate your question precisely. And there are two phases depending on whether mu is positive or negative. In this side of the duality when mu is positive we have a fermion with positive mass and when mu is negative we have a fermion with negative mass. And they each have spin a half and they either spin this way or that way because we are in two dimensions. And there is an anomaly that tells us that these two states are not exactly the same. But there is one free massive fermion. Now we go to this side of the duality. And now I'm getting closer to your question. Can I see this state not operated fermion that I see here? And I explained yesterday that depending on the sign of mu the answer here is different but in both cases we see a fermion. For one sign for positive sign the spectrum includes five quanta. They are five particles. The five particles are the excitations in the spectrum. Mu is positive, make it much bigger than anything else. You have five particles. But the five particles are charged under little b so we have to be a little bit careful about them. And little b is a gauge field which was not Higgs so its dynamics is important and it has churned Simon's term. A massive scalar field coupled to a churned Simon's term acquires fraction of spin and fraction of statistics. So phi itself acquires spin-a-half and therefore it's a fermion. And we can also check that it has a charge one under big A. In the other phase it's even closer to your question because in the other phase what happens is that phi condenses. Once phi condenses two things happen. First it Higgs is the gauge symmetry. From this point on all the subtleties with churned Simon's terms are not present. But what we get in return is that this is the Abelian-Higgs model and it has vortices. Which I think were mentioned in one of the talks yesterday maybe even yours or maybe Misha's. These vortices are strings in three plus one dimensions and they are particles in two plus one dimensions. So this system with negative mu has vortices. And if you go through the analysis there you see that these vortices have spin-a-half and charge one under big A. So they have precisely the right quantum numbers to be the free fermion in the other description. Now you can ask your question again if I did not answer it. Okay, thank you. Okay, so this is officially now the end of lecture three. And we are going to manipulate this duality. And the trick of manipulation will be exactly the same as we did yesterday. And this is the main trick in this game. So we are going to add terms to the two sides of the duality and convert A, the classical field to a dynamical field. And specifically the theory we would like to understand this is a free fermion coupled to a classical gauge field. We would like to study three-dimensional QED, namely a fermion coupled to a gauge field. So once we have the duality we'll be able to give a Bosonic description of this theory because wherever we see psi with a covariant derivative we'll just use the right-hand side. So let me do that. So I'm going to add to the two sides of the duality. I'm going to add I over two pi, ADB. And whenever we add a new gauge field we can add. So the idea is to promote A to be a dynamical field. So whenever we have a dynamical field we always have a classical field coming together with it that couples to it like that. And we also, nobody can stop me from adding one over four pi BDB. You will soon see why I did that. So we just add that to the two sides of the duality. So the duality is like an equation. We can add the same thing on both sides. So it's still true. And here it's trivially true because these are classical terms. They don't affect the dynamics at all. Now since the duality is true for every A I can integrate over A. So the way I do that is by replacing A by lowercase A. So on the left side of the duality I have this. That was there before. And I'm adding I over two pi. And B is now a new classical field. So I've just done that. So this is QED coupled to a classical background field that for some reason has a BDB which so far it looks totally unnecessary. On the other side of the duality we have many more terms. So let me copy. Have that. Now we have plus I over four pi BDB plus I over two pi BDA. There's something wrong with my signs I think. That's okay. Plus I over four pi sorry one over two pi terms I added there. That now A is going to be a little A. I added a DB plus I over four pi. So what have I done? I added this thing to the two sides. And wherever I saw big A I replaced it with little A which is shorthand for I'm going to integrate over A. A is going to be a dynamical field. But now we can do the same thing we did yesterday. We look at the terms in A. So the terms that A multiply A appears only linearly on this side. Since A appears linearly it's easy to integrate it out. So the general rule is that whenever we have e to the I minus I over two pi A dx. X is whatever it is. Then the functional integral over X gives us a delta function of delta function integral over A gives me a delta function of X. This is a trivial theory. You can think of it as being a Lagrange multiplier that forces X to be trivial. So if I just look at these terms they tell me that B is minus B. Right, but with the coefficient here so what does it mean? It's clearly a delta function of the derivative. It means that dx is zero. That's obvious. But I claim that it really means that X is a pure gauge. So dx equals zero is weaker than saying that X is a pure gauge. I also need to worry about the holonomies but that depends on the specific normalization. Since A is not really a number but it's a connection then A might not be single value. So that might be, I might need transition functions in A and when I integrate over these that really sets X to be zero and what I mean by that is a pure gauge. So that was a good question. And so now I can integrate out and I forget about this and I can forget about that. And I can also forget about little B because it's the same as plus B. So these two terms cancel. This becomes big B and I might have messed up a sign but oh I see my mistake. I had a minus sign here yesterday. Quality I wanted. So with this minus sign I have it here. So now B is minus B. I can cancel these two terms and what I'm left with, so now I'll do it on the blackboard, this theory with a lot of fields boils down to up to redefining what I mean by B, dB pi square plus pi to the fourth and everything else cancels out. So what do we see here? On the left hand side we see QED, a theory which is also known as U1 level a half because of the parity in number. This is a standard theory that there is huge literature on it with different number of flavors and there's really huge literature on that. And on the right hand side, we find the Wilson Fisher or the O2 Wilson Fisher model or the XY model. This result is extremely surprising. It's extremely surprising for a number of reasons. First of all, QED in three dimensions, so forget the coupling to B for that matter, just a free fermion coupled to a gauge field is a strongly coupled theory and it is controversial what this theory does in the infrared. In fact, lattice people have not yet converged on what it does. There are people who claim that there is a first order transition. There are people who claim that sidebar side gets an expectation value. We have a claim here. I don't might not be right, but the claim is very specific. This theory has a second order point. It's a non-trivial fixed point and not only is it a non-trivial fixed point, it's a well-studied, well-known fixed point. This is the XY model. That's the first reason this is surprising. There's a controversy in the literature of whether such a fixed point exists at all and we claim not only does it exist, it's a theory that everybody is familiar with and if you're not familiar with the XY model, you should be. This is a well-studied model. It describes things in nature. There are measurements in the lab measuring the critical exponents of this fixed point. This is well-established with the transition of superfluids and so forth. It's exactly the same fixed point that this controversial theory flows to. It's a very specific statement. Number two, QED, whatever it is, is claimed to be a theory that is not time reversal invariant. For classical A, there was this Tuft anomaly, which means that the free fermion is time reversal invariant for any classical, for zero A, but with non-zero A, we had this anomaly and it wasn't really time reversal invariant. When we integrate over A, we cannot avoid the question of what happens when A is non-zero because we integrate over this and that means that the theory is not really time reversal invariant. So three-dimensional QED, with one flavor, is not time reversal invariant. That's a statement. What does it do in the infrared? We claim that this is the same as the Wilson-Fischer theory, which is time reversal invariant. So the claim that this is not just a fixed point, but it's a known fixed point, is even more surprising because we start from a theory in the UV that does not have time reversal symmetry. In fact, that's the canonical example of a theory where time reversal is anomalous. So you start from a theory that classically has time reversal symmetry and you go through a long discussion arguing that it's actually not time reversal invariant and at the end of the day, it does become time reversal invariant but with a different time reversal transformation. The time reversal transformation or the time reversal symmetry in the infrared is not the same time reversal that we saw in the UV that was violated. So the theory in the infrared has a new time reversal symmetry that acts locally on the fermion side. Question. These are massless fermions. Let me clarify your question. In general, when you want to say something is massive or massless, there is a natural way to discuss it. When the particle is massless, there is an enhanced symmetry. So you don't just declare what the parameters are because what the parameters are can depend on your regularization, the way you define the theoretical distances. You use Pauli-Villas or dimensional regularization. You put it on the lattice one way or another. What you mean by mass could be different. But if there is a point with enhanced symmetry, you could say, aha, what I call m equals 0 is the point with enhanced symmetry. So you and I have different regularizations. What you call the mass is not what I call the mass. And I want to communicate with you and say, look, what I call m equals 0 is the point with the enhanced symmetry. That is known technically that this is a natural definition. Now, in this theory, in the free fermion theory, it's more tricky because the point m equals 0 does not have an enhanced symmetry. There's no enhanced symmetry at the point m equals 0. And therefore it's a little tricky to say, what do you mean by m equals 0? There's no special symmetry. So I'm going to define m equals 0 as the point where there is a transition. This theory clearly has a transition because for mass positive and large, we have a fermion in the spectrum with positive mass. And for mass negative and large, we have a fermion with negative charge. So there must be a phase transition between them. I call m equals 0 the point of the phase transition. Now we can communicate because we define what we mean by m equals 0. And then m equals 0, we claim, is a second-order point. It could have been a first-order point, but at least we know what we're talking about. So the claim here is about m equals 0. Now, for the free fermion, this parameter m, or we call it mu, breaks time reversal symmetry for the free fermion. In this theory, since we integrate over the gauge field, there isn't even such a symmetry. So there's no symmetry at this point. And indeed, if you go through the change of variables that I did here, the fermion mass from here is mapped to the boson mass here. And the point mu equals 0 in this case, so let me write it explicitly. So psi bar psi is mapped to the operator phi absolute value square. And indeed, in this theory, mu equals 0 is not any special as far as symmetry is concerned. Mu equals 0 in this side is characterized by having a transition at that point. There is no symmetry that is enhanced at that point. And similarly, on the left-hand side. So this is... If it's second-order, not still. If it's second-order, it is... No. I would call mass 0 when there's a particle. When there's a particle, I know what the mass means. This theory is not free. In lecture 3, which spilled a little bit to the beginning of this one, the mass was meaningful because the theory in the infrared was a free theory. It was a free fermion. Now, the theory in the infrared is interactive. So the notion of mass is not well-defined. There's no mass of particle. There's a parameter in the Lagrangian. No, there's a deformation from the fixed point. And it is what it is. It is irrelevant in the infrared. So it's relevant in the UV. The kinetic term or the Chern-Simons term? The kinetic term is a relevant operator in the UV. It dies in the infrared. So we get 1 over g squared times db1ad square. And as g becomes big, 1 over g becomes small. And in this shorthand notation, I send it to 0. So you see, if I wanted to write the full Lagrangian, there would be more terms here. There would be 1 over g squared and db on this side. And on this side, there would be 1 over g squared. No, sorry, a. And on this side, there would be 1 over g hat squared times db squared. And there would also be mu at mu squared and pi squared. There would also be a fermion mass here that we tuned. So there are all sorts of terms that either flow to 0 or we tuned them away. And in order not to write too many terms, I did not bother to write them. So this is shorthand notation. What we really mean is a UV theory with all possible couplings. We flow to the infrared. And after we flow to the infrared, we either do not write them or write them the way they are, or there are parameters that remain relevant in the infrared. So these are all the irrelevant parameters in the UV. I just don't bother to write them. And all the relevant parameters I tuned to a fixed point. So they're still a parameter. I need to explore the deformation. But we would still like to map the operators, because this is a very dramatic claim. So in order to make it more palatable, we would like to map the operators in more detail. Because I was asked about it. But before we discussed the bilinear, we would like to ask, where did the fermion come from in this theory? This theory has no fermions. And conversely, where is the order parameter phi, the order parameter for the U1 breaking in the fermionic language? So I'm going to answer these two questions. And let me start from the fermionic theory. In the fermionic theory, this is QED. So it has a dynamical gauge field. Whenever you see a dynamical gauge field, you should start in monopole operators. So this is a monopole operator for A. And the monopole operator for A, just by looking at this term and integrate by path, we see that the monopole operator is charged under B. So it's an operator, it's a scalar operator, carries spin zero, and it is charged under B. In this language, there is such a guy, which is phi itself. So we identify the monopole operator in B with phi itself. What about the fermion? This theory has a fermion. Should we identify it with anything in the bosonic theory? That's a question. Well, we don't have to. It's not gauge invariant. Thank you. So operators that are not gauge invariant, either we can multiply them by something else to make them gauge invariant, or we don't discuss them. So in this case, previously the fermion was gauge invariant. So we had to do a double symmetry coupled to the classical field A, so we had to discuss it. Now the fermion couples to a dynamical field, and therefore we don't need to discuss it. So what do we have here? We have a theory that has a U1 symmetry acting on the local operator. We identified it on both sides. There is one relevant operator, so this is a relevant operator and there is another relevant operator that does not carry any quantum numbers, and it's this guy. So we can relate them between the two sides. We can also consider deformations by... Oh, I'm behind in time. We can consider deformations by the relevant operators. That's somewhat less interesting to do turning this guy on because it breaks the symmetry. So let's preserve the symmetry and turn on this guy. So now we are going to have massive QED on one side, and the Wilson-Fischer fixed point on the other side. So for the case of the bosonic side, we've already analyzed it before. This theory with positive mass, the U1 symmetry is unbroken, and since it's unbroken, there are massive particles, the phi quanta, which sit there and there are massive particles. In the other phase, the global U1 symmetry is spontaneously broken, and there is a massless goldstone boson. I am not going to do it in detail here, but the same thing can be obtained here by considering the fermion and giving them a mass. So that leads to the same answer. So what we have seen is that... What we have seen is that we derived a new duality which answers a long-standing question. What does QED in three dimensions do? And the answer is something we already discussed before. So that's new. To learn about the new quantum field theory. And this identification has passed some elementary tests. There are more tests that I'm not discussing here today, but they are in the literature. So that's what I wanted to say about this duality. But in a similar way, we can easily construct more dualities. So we could start, for example, from here. Let me erase all the irrelevant stuff. We don't need all this. We don't need all that. So this is QED. So it's a fermion with a dynamical gauge field coupled to a classical gauge field in some term. And that's just a purely bosonic theory. Once we have this duality, we can easily create more dualities in the same spirit I've done before. So, for example, I can add to the two sides of the duality I k over 4 pi bdc plus sorry, plus i over 4 pi this is 2 pi because it's off diagonal, this is diagonal, i k over 4 pi bdb. So I can add this term to the two sides of the duality and then declare that big b is little b, i.e. I'm going to integrate over it. I'll manipulate the two sides and I'll find something. And that something might not be interesting. I might be able to do it a few more times and generate many more theories that I can analyze this way and generate many more dualities. So this is a lot of fun because this is really a machine that spits out dual pairs. And you have to be careful when you do that not to make mistakes and be careful about the plus sign and the minus sign, etc. And a lot of that has already been done. But this leads to a whole set of new theories. So let me give you an example. And I'm not going to do the algebra on the blackboard because I've already done it a few times in these lectures. But let me write one example so I can have one i psi bar d slash psi and you can take it as an exercise for you to work out. It's really not a lot of work. I over 4 pi a plus i over 2 pi a p plus i over 4 pi c dc and I can even add the gravitational trans-simon that I suppressed before. And that's dual to b of i where plus 5 to the 4 the same as before minus 2i over 4 pi pdb plus i over 2 pi so let me say it in plain English or in words. The theory on the left hand side is QED namely a fermion, a single fermion coupled to a gauge field and that gauge field has a u1 before we had u1 level a half. So this has a half buried in it but now we add one more from here so this is u1 level 3 halfs and it's coupled to some classical fields that we can suppress. So this is a cousin of QED this is the same QED but the gauge field also has another trans-simon term. We could be interested in this theory with an arbitrary coefficient here. It would be u1 with an arbitrary k. Yes. Which level? There's no k here. I added something like this with an appropriate k. I think you get this thing for k. If you take this thing I think with k equals 2 either 2 or minus 2 you will land on this. I think it's minus 2. Well there's a question of whether you edit or subtract it and there's some missing the minus signs in my notes so I shouldn't say but it's either 2 or minus 2 and you will land on this. I'm happy to do the algebra on the board if you want but this is really straightforward algebra and it's identical to the algebra I did for twice. Once when I did this trick of adding such terms to the two sides of the particle vortex duality and then I did it again this morning to derive QED. So it's identical to that and in general you can do it with an arbitrary coefficient here as long as you do it on both sides and you end up with this. This was changed because of k. So we end up with u1 level 3 halves so the naive QED has level half it always has to be half integer and now we add one more so we have 3 half. This is another theory we could be interested in and people have been interested in this. QED with any number of fermions with various levels for the chair and Simon's gauge fields is used literature on the subject I think even Spenta wrote on that if I'm not mistaken. Yes or? The non-Abelian that's also interesting we can discuss the non-Abelian one. So this theory is huge literature on it and it couples with some background fields because it has a global u1 symmetry so we can couple it to C and the claim is on the other side this is again the Wilson-Fischer theory but it's a gauge version so it's coupled to u1 level minus 2. So both the theory on the right and the theory on the left a theory is that there was an interest in them. The Wilson-Fischer theory can be gauged and once it's gauged it can be coupled to a trans-Simon's term and again people have discussed that in the past here the fermionic theory is QED so it's already gauged and you can add a trans-Simon's term to it this has been discussed in the past and one might think okay we have two separate theories but the duality claim is that this theory is the same as that one so we put some order in this long list of possible fixed points in three dimensions by identifying some of them some of the line operators can be mapped and some of them cannot be mapped and in order to see those that really carry meaningful information as far as global symmetries can be mapped those that do not carry such meaningful information do not have to match on the nose in general you can map even for local operators operators that are charged you have a charge operator here charge operator but then only those that are kind of the lowest dimension charge operator because the others can mix and you should also be able to map the relevant operator because they tell you where you go and the renormalization goes through the irrelevant ones are getting trickier yeah in supersymmetric theories you can do better but here we don't have supersymmetry so you cannot do better yes in the first one is it no yes in the sense yes it's a good question but the answer to the good question is no there are two kinds of things in infrared so duality is divided into two kinds first exact dualities example N equals 4 super young males another example the icing model with Kemevenia duality that's first class class number 2 are infrared dualities and in infrared dualities they are again further divided into two categories category 1 you have two different UV interacting theories that flow to the same fixed point case number 2 one UV theory flows to another and that's meaningful when this thing is infrared free the distinction between them is not sharp so for a given theory the distinction is sharp but if you start making deformations it's not sharp so imagine you take something like this and you add a relevant operator to add here a relevant operator now this one might flow to a non-trivial fixed point and this one will have to flow to the same non-trivial fixed point because of the duality but now you say that this and that flow to the same infrared fixed point so it looks more like this and you could also go the other way around with appropriate deformations so what do we have here here we have a fermion so we have a free fermion coupled to a gauge field and the relevant coupling here is the gauge coupling so the relevant coupling here is the gauge coupling and here it's the Wilson Fisher theory the relevant couplings are the 5 fourth in the gauge coupling and the non-trivial claim is that they flow to the same point I'm not going to do it here but I want to mention that all these dualities are the generate cases of a much richer structure with non-abillion gauge groups so you can repeat the same story with non-abillion groups so you have some fermions and they couple to some SUN and you have some bosons and couple to some UN and you can have transimons so there are now lots of knobs that you can turn changing the gauge group it could be an orthogonal group or a unitary group and you can change the transimons level of dualities between them and there are lots of cross checks so this is a very rich subject and one of the conclusions from this rich subject is that this particular theory which I presented two dual descriptions really has more dual descriptions with non-abillion groups so here Spenter should get the credit because he did study these theories with non-abillion groups so for example this particular example is the same as a fermion or QED or just straight fermions couple to SU2 level a half and a half is again because there is a single doublet and it's also dual to another theory which is the Wilson Fisher theory couple to SU2 level minus 2 A minus 1 so lots and lots of dualities so lots of theories that you could just be interested in bosons fermions couple to this gate group that gate group with different levels and they all can flow so these theories are totally different in the UV they have different degrees of freedom different symmetries and they all flow to the same infrared fixed point so we have already discussed the fact that this theory and this theory flow to the same infrared fixed point what's the global symmetry here what's the global symmetry of this theory we've discussed that my presentation there was a classical background field that coupled to it it's this theory it is coupled to a classical background field what letter did I use for it well that's it's not a level because it's a classical field it's U1 so this theory has a classical U1 symmetry actually to be precise it's O2 because there is also charge conjugation this theory this theory is this one also has a classical U1 symmetry magnetic symmetry so this one also has a U1 but if you write the global symmetry you write the Lagrangians here you will see that these two theories really have an SU2 global symmetry or more precisely SO3 so these theories have an SU2 global symmetry and the non-trivial dualities tell us that they flow to the same point so I haven't done that in these lectures I'm just stating the fact I'll soon make it more plausible so the interesting statement is that these two theories that we were discussing before have a global U1 symmetry and they flow to a fixed point that has an SO3 global symmetry we have seen a lot of that in two dimensions if you have a compact boson in the radius the global symmetry is U1 or U1 for the left-movers and U1 for the right-movers as you change the radius there is a quantum symmetry the global symmetry becomes SU2 and it actually appears twice once SU2 for the left-movers and one SU2 for the right-movers that's well known that's I don't know 70s or 60s even before my time there are two dimensions in supersymmetric theories there are theories that in the UV have a U1 symmetry and in the IR have an SU2 symmetry that comes under the title of three-dimensional mirror symmetry now we see the same thing in a non-super symmetric theory so let me make it a little bit more plausible and give you some more insight of where this thing comes from so the various theories we discussed are various, the Wilson Fisher the O2 Wilson Fisher fixed point and it could be coupled to a U1 gauge field and the U1 gauge field could have various levels so the simplest one which appeared yesterday it was the Wilson Fisher fixed point coupled to a gauge field no chance, Simon's term at all this theory we showed in the first lecture is the same as the Wilson Fisher fixed point without gauge fields how is this duality called? this is the duality between the 5-4 theory and the 5-4 coupled to a gauge field we gave it a name we didn't give the name, the name was given by others I repeated the name that's particle vortex duality that's known from the 70s today we'll learn that this theory is also the same as QED this theory is the same as QED and the natural order parameter is the monopolar the monopolar operator from here and that was mapped to the scalar field of phi scalar field phi of then we also discussed the Wilson Fisher theory with U1 level 1 so we take the same theory we have a gauge field without a trans-Simon's term it's the same as the Wilson Fisher theory now we add trans-Simon's level 1 this theory is also dual to a theory that we like what is it? give you a hint somewhere on this blackboard well it is on the blackboard this one, no this is QED level 1 half look at the far end of the blackboard it's the free fermion and the monopolar operator is mapped to a fermion now the monopolar operator requires spin because of the ADA type coupling or the BDB type coupling in the Lagrangian and the monopolar operator we said always has so the monopolar operator if we have K BDB 4 pi the monopolar operator has spin K over 2 that appeared in the first lecture appeared in the second lecture this is the spin of the monopolar in this general Simon's theory so for K equals 0 we got a scalar which we identify with this scalar for K equals 1 we get a fermion spin a half which we identify with the free fermion and now we are discussing the Wilson Fisher with U1 level 2 or minus 2 it doesn't matter and we found many dual theories what's the spin of the monopolar operator in this theory 1 so what can an operator of spin 1 be what other operator of spin 1 are you familiar with a master's vector field is not gauge invariant but an operator of dimension of spin 1 which operators have spin 1 the quantum field theory that has I'll give you a hint it might be conserved conserved current so the monopolar operator of this theory is mapped to a conserved current so it is an operator of spin 1 in the UV because there is such an operator in this theory but it's not conserved by the time we go to the infrared its dimension goes down it becomes conserved and it combines with the global U1 symmetry that we see with the naked eye so by looking at the UV we see a U1 global symmetry this is the magnetic symmetry and we see that the lowest dimension operator charged under that global U1 symmetry has spin 1 this doesn't mean that it's conserved and therefore it doesn't mean that there is an SU2 global symmetry but it does mean that as we go to the infrared its dimension could go down and it has the potential to become conserved and the claim of the duality is that it is indeed conserved and there is a global SU2 symmetry I've already mentioned the analogy between this and the two dimensional problem in two dimensions when we have a compact boson there is a global U1 symmetry the shifts x shifts x by a constant and there are winding modes and the winding modes carry charge under that U1 symmetry and for large radius and for large boson the winding mode has very high dimension but as we crank the radius down the dimension of this winding mode comes down until it hits dimension 1 which is the dimension of a conserved current and at that point the global symmetry is enhanced to SU2 this is well well understood it's known as the Frankl-Katz construction and it appears in string theory all over the map I haven't studied that I strongly encourage you to study this what we see here is a three dimensional cousin of the same phenomenon but it's much more interesting because it happens in an interacting theory whereas in two dimensions it happens in a free field theory and this phenomenon in interacting theories in three dimensions was also known in the supersymmetric theories in supersymmetric theories you often have a global symmetry and in the infrared these U1s become the carton sub-algebra of a non-abillion group and the extra elements that enhance the abillion group to a non-abillion group come from monopole operators so this is well understood in supersymmetric theories and now we see that the same phenomenon happens even in non-supersymmetric theories and now you can ask your question which vector boson? When you said that it was interaction with gauge field I called it vector field so now you have a partner to raise this field to non-abillion one so in this sense I'm wondering like I think you're mixing two different things think number one in this theory there's a dynamical U1 gauge field this dynamical U1 gauge field is not associated with any symmetry because whatever it couples to is coupled to a gauge field and therefore it's not a new global symmetry even that we have a U1 gauge field which is dynamical it's not true for classical gauge field is its own conserved current a quantum gauge field like this one it couples to a conserved current but there's no global symmetry instead there is a monopole operator and there is a magnetic symmetry so whenever you have a dynamical U1 gauge field there is a global U1 symmetry associated with it but it's more subtle the conserved current is f mu nu so that is the U1 that is being enhanced the U1 that is being enhanced its current is f mu nu but in terms of states do not appear remember you answered it's the same question you can either discuss states or you study operators when you ask what's the symmetry you ask about operators when you ask how it is realized in the Hilbert space you ask about states that's a different thing the question of what the operators are that's something that you see in the UV it's independent of the dynamics independent of the theory that we have phase transition there would be a superconductor we don't care when you discuss states you should analyze it separately for each value of the parameters especially when they are phase transitions different phases the symmetry might be realized differently for example it might be spontaneously broken or it might not be spontaneously broken so for any one of these dualities you can turn on various mass terms positive or negative and analyze semi-classically what the spectrum looks like and hope that you would be able to land on your feet namely if you see a massive spin zero particle here with some charges you go to the other dual description and again you see the same particle so for particles it's a separate question and in this case we can ask is the symmetry spontaneously broken etc etc in this description the new symmetry so the u1 is there the global u1 where was my diagram this global u1 is there everywhere and the monopolar operators are charged under it and the particles in the various phases are charged under it there is among other things a particle in some phases in the phase where this u1 is unbroken there are massive particles with spin 1 that's a massive particle it has spin 1 not a conserved current, no nothing there is no su2 symmetry there is no su2 global symmetry the statement about the enhanced su2 symmetry is only about the infrared not about taking the uv theory and turning on masses so when you want to discuss particles that's a separate question and here in the uv it's not going to shed any light on it having said that we can discuss these two theories in these two theories the full su2 is visible in the uv and therefore in each phase of these the su2 should act either should act on the spectrum if the su2 is unbroken or be spontaneously broken and there will be goldstone bosons so you can analyze all that and it has been analyzed and it actually works but you see there are many theories and many parameters so there is a lot of work to do and it was done okay I reached a natural point to stop and I really said a tiny fraction of what I plan to say well give about a half of what I plan to say and it is a tiny fraction of what is already known about these dualities but I hope I gave you the necessary background to read these papers things that are hard to find in the literature like the proper statement of the anomaly and some introductory comments on Chairman Simon's theories which allow you to read the papers I should just mention that we discussed boson boson dualities that is known from the 70s yesterday and today we discussed boson fermion dualities and you can iterate these and find also fermion fermion dualities so that is one direction you can take you can also change the gauge group in all sorts of different ways you can also change the potential if you have the gauge group it is sufficiently complicated you have enough scalars instead of having just one scalar you have several scalars and then you can write different terms in the potential that break the symmetry there are different global symmetries and there is a very rich structure of fixed points that you can find or might not have some dual description using totally different degrees of freedom so it is a huge zoo of examples that can be studied and some of them already have applications in condensed matter physics in fact some of these things were motivated by applications in condensed matter physics some of them were actually done by condensed matter physicists not quite in this language so I feel that there is a lot that can be done I even prepared here a lot of things that I did not cover and I hope that the young people in the audience could take it up and push it forward so stop here