 OK, good. I think we can start with the third of the series of Nantes lectures. Can you hear me now? So the first lecture, we talked a little bit about anomalies. The second lecture, although in quite a watered down version of the story, but still quite illuminating. The second, we talked about John Simon's theory, which is a very simple three-dimensional field theory. Again, I just gave you a glimpse of what John Simon's field theory is. Talked only about the Abelian theory, and only about U1. The story for non-Abelian groups like SU2 is a lot more interesting. And it can also be repeated for discrete groups, and groups that are more complicated, like we have products, and quotients, and et cetera. And then we discussed fermions in three dimensions. And we gave a slightly improved presentation of the anomaly. And we said that the partition function for the fermions is given by eta. And let me give the correct normalization. So the partition function z was the absolute value of the determinant of d slash e to the minus i by eta over 2. And eta is a function of the background gauge field. And the background metric. And then we can turn on masses, and depending on the sign of the mass. So this thing, eta e to the minus i pi over 2 eta, is plus or minus e to the i minus i over 8 pi. That's a gravitational term that I'm not writing down. And the point was that this plus or minus sign is important. And sometimes people suppress it and just write it without the plus or minus sign and say that they approximate the theory without just the functional integral just with the absolute value. And then they say it's not gauge invariant, and you have to add level 1 half. And that's what we do here. So that's what people sometimes say. This is a more precise version of it. And then we can turn on masses. And when we turn on masses, the more naive presentation that people often use actually gives the right answer. For positive mass, z has a phase, which is e to the i over 4 pi minus 4 pi a da with an interval. And for m negative, z is like a phase with a phase 0. So for positive mass, we are left with u1 level 1. And for negative mass, we have with u1 level 0. And again, highlighting the fact that time reversal is broken. For one sign of the mass, we get level 1. For the other sign, we get level 0. Although this is not a dynamical field. So this is a review of last time. And now we'll put this aside. And we're going to start a totally different topic, which will soon be connected to that. And the topic is particle vortex duality. This is a story that has a very long history going back to the 70s. And this is some kind of a duality in three dimensions. So let me describe the two sides of the duality. So the topic is, and I'll explain the term once we understand what it is. So we start with a theory that has a single complex scalar field phi. And we write a Lagrangian for it. So I'm going to write the Lagrangian like that. So the phi square and the phi to the fourth interactions are relevant in four dimensions. So we start from a free field theory, which is given by this interaction. And then we turn on these relevant interactions. Does the terminology relevant and irrelevant? Is it known? So relevant means the dimension is less than 3. Irrelevant means the dimension is larger than 3, because we are in three dimensions. And marginal is when it's exactly 3. So we exclude the marginal here. We just write the relevant operator. So these operators, the coefficients here, have dimensions of mass or mass square, positive. So as we go to lower energies, effectively, the coefficients become bigger and bigger. And if we don't tune any coefficient, the theory is massive, and that's the end of it. But we can tune the coefficient here to be 0. So if we tune the coefficient here to be 0, this is the UV theory. It flows in the infrared, in the IR, to a non-trivial fixed point of the renormalization group. It's also known as a CFT, the conformal field theory. So let me write here mu. I've not even bothered with the dimensions. And this model, since phi is a complex field, is known also, this fixed point, is known as the xy model. Because we have a complex field, so in the lattice description, we have an angle in each lattice side, which is really the argument of phi. Another name for this model is the O2 Wilson Fisher point. Wilson Fisher fixed point, because Wilson Fisher studied that model, named in their honor. And it flows to a non-trivial fixed point, which has a global U1 symmetry. Now, the spirit of these talks, whenever we have a global symmetry, we couple it to a background field. And that's the same as in the first lecture, a right here or covariant derivative with a background gate field B, which is, again, not dynamical. It's a classical background field, which allows us to keep track of the global U1 symmetry. Let's discuss the physics of this model a little bit. So the way we think about such quantum field theories, this theory is interacting. We have no Lagrangian for it. Then abstract conformal field theory, the only thing we can do is start in the UV, write the Lagrangian, deform it a little bit, make small deformations here, and check what they do here. So we tune one parameter, which we can roughly think of as mu. So with mu is tuned to 0 in some units or appropriate subtractions, we have a CFD. And lambda becomes very big, and it's effectively 1. What happens when mu is positive? When mu is positive, the field phi is a positive mass square. And when it has positive mass square, we can just forget about it. Just a massive particle, we integrate it out. The spectrum is gapped, i.e., no massless particles. And the global U1 symmetry is unbroken. And what are the massive excitations? The massive excitations are the phi particles. So we have particles, massive particles, with U1 charge. So this is the physics of the model with mu is positive. What's the physics of the model with mu is negative? When mu is negative, we have a Mexican head potential. So who can tell me what happens then? I can't hear you. There's a goldstone boson, right? Because the U1 symmetry is spontaneously broken. The spectrum is not gapped, which is roughly the argument of phi, this bar phi, script phi. So this is the massless goldstone boson, signaling the spontaneous breaking of the symmetry. So the symmetry is spontaneously broken. We can't say that there are particles that are charged under the symmetry. And as we tune mu from one side to the other, there is a second-order transition, which distinguishes between the two phases. So this is really elementary. Now let's make it a little bit more interesting. So we'll gradually add various bells and whistles to this model. So we put this aside. And now we study another model, which is very much like this one, except that the field B is dynamical. Yes, it's marginally irrelevant. Irrelevant, already in the UV. Yeah, it's positive. So it's safe to exclude it. There is some discussion in the literature whether it could change things if you add fermions or gatefields or a large N and so forth. This is an active field of research. And there is at least one person in the audience who knows about it more than I do. And I'm sure there are others sitting in the front row. But I'm not going to tell you whose name to your left. So we'll make the field B dynamical. And my rule in these lectures is that background fields are uppercase and dynamical fields are lowercase. And I kept that term in all of this notation in the first two lectures, and I will continue today. There's a scalar field phi. So in order not to confuse with that one, I'll call it phi hat. And it's very much like the previous model, square plus lambda hat phi hat to the fourth. And again, the same thing when we could add a kinetic term for B, but it will not make any difference. Once we have such a gauge field, we can add another interaction since we are in Euclidean space. It has 2 pi, which I can abbreviate using form notation. So what do we see here? This is a model. What is the symmetry of this model? The symmetry is u1, whose current, j, is db. We can see from here. So whenever you have a u1 gauge field, there is a possibility of having a u1 symmetry. It's kind of trivially conserved. Let's discuss the dynamics of this model. Mu and lambda hat, mu hat and lambda hat are relevant. Same as before. And we can tune mu hat to a critical point. And there is a CFT at the critical point. So at mu hat equals 0, there is some CFT. What happens when mu hat is positive? When mu hat is positive, phi hat is massive. So we can forget about it. It's a heavy particle. Forget about it. There is no global u1 symmetry for rotating phi hat, as we had before, because this symmetry is now gauged. Since it's gauged, it's not a global symmetry. We cannot use it to classify particles. So phi hat is a massive particle, but it does not carry any global symmetry charge. This is what happens in this phase. B, the gauge field, remains massless. This is unlike what we had before. Now, little b remains massless. And we can think of it as a scalar field. Because in three dimensions, a gauge field is dual to a scalar field. Let me write the transformation. d mu phi is like epsilon mu nu rho d nu b rho. So we have a massless scalar field. And we already mentioned that there is a global symmetry, whose current is this. And therefore, phi, the massless field, is the goldstone boson of this broken spontaneously symmetry. So phi is the massless, the goldstone boson, the spontaneously broken u1, global u1 symmetry. So that's the physics in this phase. There is a global u1 symmetry. It's spontaneously broken. And there is a goldstone boson, which is the dual of the gauge field. What happens when mu hat is positive? So that's with positive. The mu hat is negative. What happens in the potential? Can anybody tell me? Or how? Sorry? Higgs mechanism, very good. The gauge symmetry is spontaneously broken, because we have a Mexican hat potential. But unlike what we had before, before we had a goldstone boson, now, in this case, the u1 gauge symmetry is spontaneously broken. And since it's spontaneously broken, it eats the B field. And the spectrum is gapped. So the spectrum is gapped. What are the excitations of the spectrum? The Abelian-Higgs model has vortices, finite energy vortices, where the phase of the Higgs field rotates around by 2 pi. So there are particles. So they are particle-like excitations, which are the vortices. And the global u1 symmetry, whose current is db, is not spontaneously broken. So they are charged under the global u1 symmetry. So we have two phases with the second order transition between them. For one sign of mu hat, the symmetry is unbroken. The global symmetry is unbroken. The spectrum contains, the spectrum is gapped. And the massive excitations carry charge under the global symmetry. For the other sign of mu hat, the spectrum is massless. It's not gapped. And the u1 symmetry is spontaneously broken. And once the u1 symmetry is spontaneously broken, there is a goldstone boson, which is this fine. Notice that this is very similar to what we have on the left side of the line. The answers are the same. But the words are completely different. In fact, when we compare the left side to the right side, we see that mu is like minus mu hat. The phase with positive mu is mapped to the phase with negative mu hat. And all the words are the same. The details are completely the way we got to the answer and the words we used to describe them were totally different. But the outcome is exactly the same. So this is the hallmark of duality. We have the Wilson-Fischer fixed point. So we have the free phi, free phi, determined by a relevant operator. That takes us to the Wilson-Fischer fixed point. And then we have free phi hat coupled to the dynamical gauge field B. And the coupling becomes stronger. And that also takes us here. The same fixed point. So we can say that this is the Wilson-Fischer theory and this is the gauge version of the Wilson-Fischer theory. And the claim is that they are dual to each other. So I haven't proven the duality. I gave some evidence by analyzing the various phases, analyzing the symmetries, and analyzing the spectrum of excitations. But in this particular case, there is almost a completely rigorous proof of heat on the lattice. There are some assumptions, but it's almost rigorous. Excellent question. Excellent question. I'll soon explain what it does in more detail. But if you try to compare them, what you had in mind is that phi is the same as phi hat. What does it do in the field? Because the symmetries get. What I really emphasize, phi is not the same as phi hat. Phi is a nice, good operator. Phi hat is not even gauge invariant. We shouldn't discuss it. There was a discussion yesterday of whether we should or should not discuss things that are not gauge invariant. Perhaps it was even with you. Somebody asked me this question, and things that are not gauge invariant should be excluded. We don't discuss them. So phi is not the same as phi hat. The degrees of freedom here are not the same as the degrees of freedom here. For example, the phase of this phi, which is the massless goldstone boson in one phase, is dual to the gauge field B here. Showing that the map between the two sides is highly non-trivial. It's a non-local transformation. In some situations on the lattice, it can actually be worked out explicitly. Now to give you a better answer, let us discuss the operators that we should consider. What are the operators in the theory? So this is a fixed point. It has a collection of operators. And these operators can be described using the UV degrees of freedom. So in this theory, there is a natural order parameter, a natural operator, which is phi. It carries charge 1 under this u1 symmetry that couples to big B. And in one phase, phi has an expectation value. And in the other phase, it doesn't. So who can tell me whether phi has an expectation value here or here? Where does phi have an expectation value? The second one. Very good. So here, phi has an expectation value. Here, the expectation value of phi is 0. And that's the hallmark of the spontaneous symmetry breaking. In the other language, on the other side of the duality, we have this guy phi hat. But that's not gauging variant. So I'm not going to discuss it. Instead, we can do something else. I've defined monopole operators in the lecture yesterday. How do we define a monopole operator? We'll remove a point from spacetime. We specify boundary conditions of the field. There is one unit of flux around it, mB. Now we look at the Lagrangian. And if we integrate the current, which was dB, around that flux, we see that the monopole operator acquires charge under the global symmetry. So mB has charge 1 under u1B. So the map of the duality is mB is mapped to phi. So the scalar field, which is an elementary field in this description, is a monopole operator in the other description. Highly non-local phi hat would have been mapped to a vortex of the other side. But the other side, the vortex has infinite energy. So we don't want to do that. And on this side, phi hat is not gauging variant. So that's not good. In the other side of the duality, we've also map discussed the operator phi hat square. And that's not mapped to phi square, but there is a crucial minus sign between them. Who can tell me where we saw the minus sign before? Is this minus sign anywhere else on the blackboard? That's right. Thank you. So this minus sign is the same as this minus sign. So the map of operators is quite complicated. Phi, which is elementary on one side, is a monopole operator on this other side. Phi hat, phi square, is relatively simple. And the map is relatively simple, but there is a crucial minus sign between them. And that's how the duality works. OK, so there's a question. So the way I presented it here, I presented that there's two different continuum theories. They're really different, flowing to the same infrared theory. That has not been proven. What is proven on the lattice is that some lattice action that is believed to flow to the same fix, to the fixed point, which looks a lot like the right-hand side. They do some change of variables on the lattice and integrate it at degrees of freedom and integrate by path and this and that. And you find something else on the lattice that looks a lot like this model. And then the map between the two theories is exact on the lattice, but it includes somewhat complicated interactions. And the path that needs to be kind of ironed out is that they both flow to the same fixed point, et cetera, et cetera. It looks all right because they have 10 degrees of freedom, the same tuplings, and so forth. And that has been established numerically beyond any doubt. So the fact that these two different UV continuum theories flow to the same infrared fixed point is the high energy physicist's way of saying that these two lattice actions give the same physics. And for the rest of the talk, I would just assume that as a fact. There is a huge body of evidence for that. These dualities are better established than any duality that high energy physicists talk about. We take that for granted. And it was actually first invented by a high energy physicist. This is Michael Peskin. And then later by this Gupta and Halperin, or a condensed matter physicist. That's a nice example of how the two fields influence each other. So on the lattice, you can do that with a Vilan action. There's a trick to do that. You don't write it this way, but you could do this duality using Vilan tricks. And again, this is something that you know better than I do. Yeah, it makes no difference. Let me edit. Now, no, G is dimensionful, so it makes no difference. I can't hear you. There's an infinite number of what? Well, it carries no quantum numbers. Phi absolute value squared carries no quantum numbers. So each theory has an infinite number of operators. I just gave an example of something. Yeah, there are many relations between them. There's an infinite list of operators, and there are many relations between them. And you have to impose an equation of motion. And this is a very interacting, very complicated quantum field theory. The only thing we can do, the UV description is good for some deformations of it. And it's also good to give us kind of to account for the symmetries, the lowest dimension operator carrying that symmetry. So when we say phi, we really mean the lowest dimension operator carries charge, which carries charge 1 under that symmetry. That's all we mean. We don't mean literally phi. Another way of saying it, if we take the UV theory and compute correlation functions of phi, we go to very, very long distances that would project on the lowest dimension operator. We shouldn't take the operator. So for example, phi bar square, we can use the equations of motion and say that d square phi bar is the same as whatever phi bar, phi absolute value square. So there are lots of relations between the operators, and there's an infinite number of them. I just wrote the simplest one. I just discussed the simplest one that carries charge. And this very relevant operator that takes us between the two phases. That's what I wrote. The full dictionary is much more complicated. And it might depend on how you regularize it and so forth. So different people will regularize it differently. And the detailed map between the UV operators would also be more complicated. OK, so we have this nice duality. It's called particle vortex duality, because in one phase, one side, this one has particles, carries charge, ordinary particles. On the other side, we have vortices, which behave like particles, but they are really complicated collective excitations. This is, again, a common thing in duality, something that is simple in one language is complicated in the other. So now I'm going to assume these dualities right and derive new dualities. So it's like creating with equations, and except the equation will be this double arrow notation. And the double arrow notation means that the Lagrangians are not the same. The Lagrangians are not the same, but the functional integral is the same, but only at long distances. And that's what my double left-right arrow notation means. So we have these two Lagrangians. And let me write it so that it's easier. I'll mix 1 over 2 pi bdb. So I'm now going to manipulate the two sides of the duality, preserving the fact that the duality is right. So it's not a new assumption. See what we get. So we have a background field b. So nobody is going to stop me from adding i over 4 pi bdb. So I'm going to do that with the two sides of the duality. Let me restore my left-right notation. And I'm also going to add another term. Let's make sure I'm doing it with the right signs. So far, I added only classical fields. So that makes no difference. And it's the same b on both sides, the same a. So we had the duality, which was complicated quantum mechanically, but we just add terms to the two sides of the duality. But now, let's pretend, let's make this b into a dynamical field. And to highlight the distinction between the two sides, let me put the hats here. This b will now have hats, because everything on this side has hats. Another classical gauge field. Background. Notation is always uppercase is background. Classical. Lowercase is dynamical. That I'm going to integrate over. So the duality between the two sides is true for every value of b, for every classical value of b. So let's integrate over b on both sides. How do I integrate over b? I integrate over b by turning it into a lowercase, because that's my notation. My notation is lowercase is integrated over. So that's something that is easy to do on the blackboard. And I've already changed the little b here to b hat. So now I can do the same manipulation here. OK, this duality is also true. Let's manipulate it a little bit. This one, leave as it is. And notice that the b appears here only here, here, and here. Nowhere else. So for fixed values of b hat and a, the b dynamics is the theory we discussed yesterday as the almost trivial theory, u1 level 1. So this theory can integrate out b. And it's dual to another theory. And I'm copying the expressions from yesterday. So I'm copying the first term first, plus lambda hat, phi hat to the fourth. I'll soon stop writing all these things because they're always the same. And now this is a quadratic integral, because it's u1 level 1. So if I'm not mistaken, it's i over 4 pi b hat, db hat, minus i over 2 pi d hat, dA, minus i over 4 pi, adA. And then there's a gravitational term, minus 2i. This is a gravitational transimus term. And I hope I did not make a mistake. And this duality should be valued only in the IR. So let's compare this one and that one. Terms 1, 2, 3 look exactly like 1, 2, 3, except that they have x, the only difference. Here we have u1 level 1. And here we have u1 level minus 1. So let's first set a to 0 and ignore the gravitational fields. This is a Gage-Wilson-Fischer theory. So we start from the Wilson-Fischer theory. It has a global u1. We gauge the u1. We add the u1, transimus level 1, with u1 level minus 1. That's the duality we have just derived. You should be extremely surprised by it. Let us set a to 0 first. And then we'll turn on a. So imagine a and the metric at trivial, so a equals 0. And let's even forget about the metric. The Gage-Wilson-Fischer theory is time reversal invariant. We add the transimus term level 1, so it's not time reversal invariant. In fact, the theory on the right-hand side is the time reversal image of the theory on the left, and not the same. But we have proven that they are the same. So what's going on here? What's going on is that this theory, what we have just shown, is that the Gaged-Wilson-Fischer theory with u1 level 1 is the invariant, despite appearance. Time reversal acts in a highly non-trivial way. It maps phi to phi hat. It maps b to b hat. That's not the naive time reversal symmetry that exists here, because it's not even a symmetry here. So there's a naive symmetry you might want to call time reversal symmetry here. And it's violated by this u1 term sign. And there's another symmetry you might want to call time reversal symmetry here, which is also violated. But in the infrared, there is a new time reversal symmetry that acts in a non-local way. It maps phi to phi hat and b to b hat. And the statement of the duality is that the two are the same. So this symmetry is there at long distances. Now let's restore a. When we restore a, we see that this time reversal symmetry is not exact. So all the terms up to here are more or less the same. Up to flip of sides, that's the statement of time reversal. But here we have two more terms. So time reversal symmetry, it's time reversal invariant. But this symmetry is not preserved when a is non-zero. So when a is non-zero, this symmetry has an anomaly. It's shifted by that amount. So time reversal on the Lagrangian is the Lagrangian back minus i over 4 pi a dA minus 2i trans-simos gravitational trans-simos. So these are two surprising facts. Number one, this theory is time reversal invariant, which we wouldn't have expected, acts in a non-local way. And number two, when we couple it to background field A, there is an anomaly, which is precisely the anomaly of a free fermion. So this theory has a global U1 symmetry. It has time reversal. And time reversal has an anomaly, which is precisely the anomaly of the free fermion theory. And now we are going to analyze the two sides of the duality along the same lines we did before. So what are the order parameters? What are the operators in the theory? So it will be essentially the same in the two sides because the only difference is some signs here. Phi is not a good operator because it's no longer gauging invariant. So what can we do on this side? Suggestions? We can't use phi, so what are we going to measure? What would be nice operators to study? Well, we've already had before. I can attach a line, but then it will not be a local operator. So let's first ask, what can we do at the point? Well, we've already discussed the magnetic, the monopole operator. So we can attempt to consider the monopole operator. But because of this term, the monopole operator is not invariant under the gauge symmetry of B because it carries charge 1 because of this term. So what are we going to do about that? Well, multiply it by phi. So this is a nice operator to study. And it is gauging invariant. It's mapped under the duality through the same construction on the other side. Also, if you look at this term, you can integrate by parts. And you see that if you have a magnetic monopole, this carries charge 1. So this guy, this operator, carries charge 1 under the global U1 symmetry charge of the A charge 1. So this operator carries charge 1 under the global U1 symmetry that A couples to. What else do we know about this operator? It has interesting spin. We've already discussed this, so imagine we ignore everything else, we just look at this BDB. We already discussed the monopole operator in this theory yesterday, and we discussed its spin. What was its spin yesterday? U1 level 1, we had a monopole operator, and we discussed that it has spin 1 half. So the monopole operator has spin 1 half. There's another way to understand it. This is a charged particle in the background of a monopole. A charged particle in the background of a monopole also has spin 1 half. That's another way of understanding this. So this guy has charge 1 under U1A, and it has spin 1 half. That's already kind of interesting. We have a bosonic theory, and it spits out a charge 1 spin 1 half operator. Let's analyze the phases of this theory. So we really have to do it four times with mu hat, mu positive and mu negative, and mu hat positive and mu hat negative. So I'll do it fast on the backboard. Mu positive. Mu positive here, phi is a massive particle. This is before. It carries charge on the B. B is there at low energies. So at low energies, all this is gone. We have our theory now U1 level 1. We can integrate B out, and we have ADA. So its spectrum is gapped, and the low energy theory is I over 4 pi, maybe with a minus sign, ADA, and there is this gravitational trans-simon stuff. I think I need a tool. And there are five particles. Five particles. Particles. And you can show that they have charge 1. So they have charge 1 under B, and through the interactions, they have charge 1 under B and spin 1 half. What happens for mu negative? For mu negative, phi gets an expectation value. It takes us B. We forget about B. This term is absent. So it's also gapped. This interaction is 0. And we have vortices. And again, the vortices have charge 1 and spin 1 half. And if you go to the other description, mu hat negative gives you that, and mu hat positive gives us that. So what do we know about this non-trivial theory? We gave two different descriptions in the UV. The Wilson-Fischer description with the gauge field with trans-simon level 1, or the same story with level minus 1. They both flow to a fixed point. There is a global U1 symmetry. The nice operators that are charged under that symmetry, the basic one is a spinner. It has spin 1 half and charge 1. The theory has two phases, depending on whether mu is positive or negative. Both of them are gapped. Both of them have unbroken U1A global symmetry. The low energy theory either does or does not include this term, which is U1 level 1, for the U1A. And the spectrum of excitations are particles which carry charge 1 and spin 1 half. That's what we could have established so far. Have we seen a theory like this before yesterday? Which theory was it? The free fermion. The free fermion in three dimensions does exactly that. It's massless. M equals 0. It has a global U1 symmetry. It has time reversal with an anomaly, very much like this. It has two phases with massive fermions, either with a positive sign or with a negative sign. And the low energy theory is either U1 level 1 or U1 level 0. There is no level for the background A. So that suggests it works like a duck. Ducks like a duck, it is a duck. So we just declare that this theory is really dual to this theory. And it's nice operator that we discussed. The magnetic monopole with phi, which was dual to the magnetic monopole with phi hat, is dual to the free fermion operator side. So everything we said so far is consistent with this statement. That's an assumption. I cannot prove that statement. But it looks pretty good. Everything we can check about this theory, about this assertion, is correct. And it is interesting because this theory is quite complicated. It has five-fourth interaction. And it couples to a gauge field. And the gauge field have a trans-simalist term. And we float to the infrared. It's extremely strongly coupled. And the statement is that this extremely, extremely strongly coupled theory, at long distances, is the free fermion. So duality is most interesting when we have a complicated theory at short distances. And it becomes free at long distances. So at short distances, we formulate the problem. This is the theory, what does it do? And at long distances, we read off the answer. And the answer is that this theory, despite appearance, despite the fact that it looks complicated, despite the fact that it looks interactive, despite the fact that it looks like it's not time reversal invariant, despite all these facts, it's actually a free theory at long distances, a free theory of fermion, a free fermion. This is bosonization of fermionization in three dimensions. It's known in two dimensions to be possible. And here, we do it in three dimensions. But again, I emphasize that's an assumption. But for the rest of these lectures, I'll continue to make this assumption. I'll make the assumption that this duality is true. That's correct. So depending on it, that's right. So thank you for bringing this up. You could have asked this question already before we turn on the Chern-Simons term. Imagine we did only this without the Chern-Simons term. So this theory, the gauge version of the Wilson-Fischer theory, depending on how strong the interactions are, lambda versus the gauge coupling, either has a first-order transition or a second-order transition. That's not true in the other side of the duality, where we have just five. But the statement of the duality is that they flow in some range of parameters, they flow to the same fixed point. So I need a weaker statement. All I need is that there is a range of parameters such that the statement is true and that I find this infrared fixed point. That's the more precise statement. And the range is that when the gauge coupling is sufficiently small, this subtlety arises when the gauge coupling is sufficiently large relative to lambda. Forget the dimensions. Some power of g should be sufficiently big relative to lambda. But I'm going to treat the gauge couplings in this whole statement always as very simple, very small. For example, what did I do here? I turned this big B into a dynamical field. So what I really had in mind is turning it on, and I need to introduce some gauge coupling. And I take this gauge coupling to be sufficiently small. So this is some fine print that I decided to gloss over, but with such an excellent audience, I'm not. I'm caught whenever I'm trying to cut a corner. This is very special to k equals 1. In fact, if I have time tomorrow, I'll tell you what k equals 2 does, which if you thought that this is bizarre, k equals 2 is equally bizarre. And it's different. Now, there's a very rich structure here because you can write many Lagrangians with fermions, with out fermions, with gauge couplings, and with various levels. There's a rich spectrum of possibilities. And they do many interesting things. So this is one thing. Well, it happens in other cases, but not in this family. There are cousins of this family where it does happen in another fashion. This thing is full of surprises. No, no, no, no. These theories are well-defined as three-dimensional field theories. I'm glad you asked that because this is a point that needs to be clarified. I am working with strict three-dimensional field theories, no bulk. And there is an anomaly. The anomaly is only in the classical fields. This theory is perfectly sensible as it stands. There is an anomaly with the classical fields. And nobody forces me to add the classical fields. First of all, I'm never, never going to extend dynamical fields to the bulk. I will never do that. Only classical fields can be extended. Second, in this case, I don't even want to do that. I'll have to pay a price if I don't do that. And the price is that this theory is time reversal invariant with an anomaly. So for a equals 0, it's time reversal invariant. All the correlation functions are time reversal invariant. But with nonzero a, I have this shift. Where did I write it here? But if the metric is flat and a is 0, I don't care about that. So as long as I'm not interested in contact terms or complicated a's, this is totally harmless. Now, there's an option. So imagine I am interested in complicated metric and I am interested in complicated a. Then I have the option to attach a bulk. And then I can attach the same bulk to both sides and put a half of this anomaly on each side on the bulk. And then I'll have a theory which is time reversal invariant on the nose. It's exactly time reversal invariant. But it has this bulk. It is very important that when I have a duality between theories, and I want to attach them to a bulk, I have to attach them to the same bulk. Otherwise, the duality is not true. Because what is allowed to live in the bulk is only the classical fields. Only classical fields can live in the bulk. And the duality is a statement about the quantum fields. So I have this bulk with a boundary. I have something in the bulk, the same classical fields, the same action, everything is the same in the bulk. The duality is a statement that can peel this boundary and put another boundary and the physics is unchanged. That's a statement of the duality. But the bulk is unchanged. That's a very, very interesting question. So you can ask, what's the connection this duality to say ordinary S-duality in four dimensions with a boundary? This is actually a member of the same family. There's a close relation and interesting interplay between doing S-duality in four dimensions and these theories on the boundary. But I should really emphasize that S-duality in four dimensions is a statement about changing the gauge coupling in four dimensions. It takes a theory with dynamical field, not classical. Dynamical field with coupling constant E. And you write it as a theory of other dynamical fields with coupling constant 1 over E, maybe some pi's. The spirit here is different. The spirit here is these are classical fields. I don't care about the gauge coupling in four dimensions. In fact, I like the gauge coupling in four dimensions to be zero because I don't want these guys to fluctuate. So A that can live in the bulk is a U1 gauge field which can live in four dimensions or three dimensions. But even if it's in four dimensions, it does not fluctuate. It's a classical field. So addressing questions like what happens under S-duality is not a good question. You'd be surprised how confused the literature on this subject is. You never, never do S-duality on classical fields. You can do S-duality on dynamical fields. And then the coupling constant changes, et cetera. But you never do S-duality on classical fields anymore. Yep, everything matches. All the correlation functions. That's a claim. We don't know how to prove it, but that's a claim. For better or for worse, we can prove a lot. We can check a lot, but we don't have a complete proof. But the claim is very explicit. This theory, complicated as it is, well, we had a hierarchy of theories. We started with the particle vortex duality. This can almost be proven. And it's already surprising because these are two different interacting theories. So that's already something that would be surprising. But people were surprised about that in the 70s. Then we manipulated it. You shouldn't be that surprised, because we took the same statement and we manipulated it. So no wonder we derived a new duality. This is this part of the duality, this duality. Or the relation between this and that. And then we analyzed this theory that, well, I picked a particular path to derive it. If it looked at some point, why did I add this term or that term? This was designed so that I would find this one. And then we realized that it's the same as a free fermion. And therefore, we postulate with a big assumption that this is the same as the free fermion theory. There's a huge body of evidence that this is right, but not as big as for the more elementary boson-boson duality. So that's true in the infrared. It's not an exact symmetry. So these theories are interacting. So what do we say here? So we have two bosonic theories with some gauge coupling and 5 fourth and et cetera, et cetera. And we flow to the infrared and we tune mu. And they flow to the infrared. This theory, we claim, is the free fermion with all its extra symmetries. Now these theories don't have the extra symmetries, because the full description of this is the same as the free fermion, but with some irrelevant operators. So all these extra symmetries of the free theory are there only at very, very low energies or only at very, very long distances, not otherwise. So where did they arrive? So I wrote and mapped this operator. There's also one more operator that we can discuss, and we have essentially done all the work for this. We said that phi square was mapped to minus phi hat square. And who can tell me what this maps to in the fermionic theory? OK, so psi is a fermion, and phi square is a boson. So that's not so good, but you're on the right track. It has to be constructed out of psi. That's right. In fact, we've said that, because we said that when we deformed the theory by phi square, we're turning on this mu, the spectrum includes a massive fermion. What do we do in the fermionic theory? We turn on this psi bar psi. This is the master, and then the spectrum includes the massive fermion, the same massive fermion. So this is the map of the duality, and it has various checks. All these three theories have the same symmetries. We map the operators, this phasist on the deformations by these operators. Map, match, there is time reversal symmetry with an anomaly, and there are a few other checks that I have not done here. Just for completeness, we could imagine assuming this duality. Let's assume that this duality is true. This fermion theory is dual to this boson theory. We can run our argument in reverse. If we run our argument in reverse, we will derive the boson boson duality, the particle vortex duality, which is known. This is common in dualities. We have to make sure that we can run the reasoning back and forth, and we always land on our feet. So the assumption here is stronger than the assumption of particle vortex duality. But if we're the bosonic particle vortex duality, but if we're willing to assume that, we don't need the weaker assumption. It actually follows from it. So that's a good thing to know, because once we'll start having checks, that something we'll have to wait till tomorrow, we'll derive many more dualities. So we'll assume this, and we'll derive a huge set of dualities, which again will be just the tip of the iceberg of all possible dualities. And at every point, we could crash. At every point, we could run into a contradiction, which would mean that the whole logical structure is wrong. This is common in dualities, called the web of dualities. You assume duality 1, you derive duality 3, you assume 3, you derive 17, and you move around. And at every point, you have various tests. And every time the test works, you feel more confident about the whole structure. But we should always remember that it's all conjectural. Is this a good point to stop?