 So, last week we discussed, we were discussing Chen-Simon's theory, pure Chen-Simon's theory, and we discussed a very useful correspondence between 3D Chen-Simon's theory, two-dimensional chiral algebras, and essentially the point was that once we have this tool, essentially this tool with some more technology that we have not explained, allows us to understand all properties of the Chen-Simon's theory that we are interested in and to perform computations in this theory, in particular to compute all correlators that we might be interested in. Now we mentioned a small piece of this correspondence, in particular we say that if we are interested in what is the spectrum of lines, and there is pin H, we can read it off the chiral algebra because this corresponds to the irreducible representations or integral representations of this algebra, and the dimension, the conformal dimension of the primary, these were H lambda, and H as the dimension of H lambda, mod 1, because the spin is defined mod 1. So, and then this correspondence goes on, and there is more structure here, and one can use to compute everything we want in the Chen-Simon's theory, in particular the Chen-Simon's theory is a solvable theory, one can use these to compute everything we want, and so this allows us to show and prove, we will not prove it but it can be proven, the first duality that I want to present which is 11 rank duality, and so in our definition of Chen-Simon's theory we start, so we have a Lagrangian definition, so we start with some gauge group G and some level K, but what it might happen is that, so we start with two descriptions that look different, so different gauge group, different level, but nevertheless the quantum theory is the same, okay, and 11 rank duality is one example of this phenomenon, in particular it turns out that SUN at level K, let us use this symbol now for a duality, is the same theory, the same quantum theory as UK level minus N, okay, so these of course look different, the group is different, the level is different, but the quantum theory is the same, in particular the spectrum of lines is the same, and all the correlators between the lines if used the dictionary is the same, and this duality as I said this can be proven because both sides are solvable and you can solve them and you can see they are the same, in particular what you can do is to see that both descriptions lead to exactly the same carol algebra, and so if the carol algebra is the same everything that you compute with it is going to be the same, okay, we will not go through the derivation of this, the reservation essentially starts with the two dimensional bosonization, so the fact that free fermions can be described with a WZW model and then one has to do some manipulation and one can prove this, now just to give you some flavor of this duality we can have a look at what is the spectrum of lines, so we analyze this case because one example that we gave of this was the example in which the group G was simple, compact, connected and simply connected, so SON is in that class and we said in that case two dimensional carol algebra is a fine, the algebra at level K and we said what are the integral representations of this, in particular where representations where if you take the weight, the highest weight of the representation and you take the product with the highest root, this has to be smaller or equal to K, now if we translate this in the case of SON what we find is the following, to remember for SON we can represent the representations or if you want these sets of dinky labels in terms of young diagrams, in particular for SON the young diagrams will have two, if I have some young diagram here, so the young diagram should have at most N minus one both, so the height of these columns should be at most N minus one, I'm trying but it looks the same, the young diagrams, so first of all young diagrams that representations of SON should have at most N minus one boxes in the columns and then this condition means that the number of columns should be at most K, so essentially the lines in SON level K are all the young diagrams that you can draw inside the rectangle with size M minus one and K, now we have not discussed this case here but one can, because this group is not for instance simply connected but one can do a similar thing here and it turns out that the lines here up to a small subtlety that I will mention in a second can also be classified integral representations by young diagrams but this time the number of boxes in the columns should be at most K and the number of boxes, the maximum length of these rows should be at most N minus one and so the map of the lines is that you have to take this young diagram and flip it along the diagonal and then it will fit in here, so it's the same rectangle but it is flipped along the diagonal, is this clear? I would like to make two comments about this duality, the first comment is a technical remark, it is the fact that really this duality is not a duality of standard Chen-Simons theories but it's a duality of spin theories or spin Chen-Simons theories, so I don't have time to explain this point, I can give you some more details if you ask in the discussion session but okay, just to mention that one has to do a small operation on this theory if we really want to be precise and really have two theories which are really dual, okay but this is a small remark and then a comment is that there is a special case here, you see if you take N equal to one, okay because if you take N equal to one here, well SU1 is the trivial group with just the identity, so this theory becomes trivial and so what this duality tells us is that the theories, let me write them as UN level one are a bit special, in fact these are examples of what we can call trivial topological field theories, so these are theories for which there is a single gap vacuum on any special manifold, now remember one of the, so topological theories in general describe gap systems, okay, yeah it's not really important because okay I wrote these, so well thank you for the question, I wrote this duality but you can take a party transformation, so what party does is to invert the sign of the transimus level because the transimus actually for writing components, the Lagrangian is a mu d nu a rho plus two thirds a mu a nu a rho and then there is a epsilon mu nu rho, so if you change, if you do a party transformation this changes sign, the transimus level gets a minus, so you can perform, so first of all this theory is, transimus theory is not party invariant, okay if you apply a party transformation it changes the level goes into minus itself and what is a duality, you can apply party on both sides and then you have another duality in which you have SUn level minus k which is due to UK level n, okay, so I will not write the dualities twice, it's always meant that we take positive numbers for n and k and then there are the corresponding ones and so in particular these also are trivial, so topological theories and transimus theories in particular, they discover gapped systems where there are no dynamical degrees of freedom but still as you see they are not just trivially gapped back because depending on the topology of space we can have a degeneracy in the Hilbert space, so multiple degenerate states at zero energy, now if you wish in a trivial topological theory there is one single gapped state for any topology and so really there is nothing non-trivial going on here, up to the fact that still there is a framing anomaly and in fact the fact that the Hilbert space contains a single state on any spatial topology means that the partition function should be just a phase, however this phase should capture the framing anomaly and so all the content if you wish in these theories is the framing anomaly and since there is no dynamical content it turns out that the partition function is fixed classically, okay, there's no dynamics and in fact this partition function is just a local integral of the background fields, so in particular just to be concrete in these theories the partition function turns out to be equal to, so it's a phase and now it's a local integral of something that so let me write in this way minus 2 n gravitational transform term, okay, so this is a classical integral, this is a classical functional of the metric and write it here, so we can write it, so we could write in three-dimensional terms, let me write it in four-dimensional terms, so taking some four manifold whose boundary is the three manifold we are interested in, this is the transformation term constructed with the spin connection, okay, so this object is purely classical, okay, it is just a background field, we are not doing gravity, so G is not a dynamical field that we are integrating over in the particle, it's just classical, but this captures precisely the framing anomaly of the theory, in particular these two tensimons graph is corresponds to C equal to 1, if you work out the numbers, D corresponds to C equal to 1 which is essentially the framing anomaly of a system whose boundary is a real scalar or a complex fermion, okay, other questions? Okay, so now I would like to leave the topological word and start talking about theories with the dynamical gapless degrees of freedom and so the first example that I would like to discuss is particle vortex duality, is not, as I say, so we are discussing just field theory and so this term, this is a function, I don't know if you can see it, it's a function of the metric, it's a purely classical object because for us the metric is just a background field, it's not part integrated over and it's consistent with the fact that this theory is not dynamics, if you wish, there is a phase which captures the framing anomaly but it's completely classical, so the full partition function is a classical integral, a local function of the background fields, yes, it's constructed with the spin connection, yes, sure, in fact, I mean we know that this is a topological invariant up to the framing anomaly because, so we said that if you change this framing, so if you change the way, so if you want, the reason is that there is this huge denominator here, so this one is not properly normalized to be well defined in three dimensions, you remember we have this discussion that we complete with a four manifold and the result is unambiguous, completely unambiguous if when you have evaluated this object on a closed four manifold, you get a multiple of two pi and from this one you don't because this corresponds to C equal to one, in our convention C equal 24 is an object which is, I mean give you phase one, so it doesn't give you anything in three dimensions, but this is one 24th of that, if you take U24 then you don't get any phase at all, other questions? Okay, so I want to discuss particle vortex duality, this was proposed by Peskin, Dasgupta and Halperin, so in order to understand this duality, so let's start discussing the Abelian X model in two plus one dimensions of course, so what is this? So we have a Abelian gauge field, so U10, there is no transformous interaction, it's just young mills, and then we have a complex scalar, we charge one under this U1 and then we take a potential which is a mass term and a quartic interaction which is relevant in three dimensions, but we take M squared negative, so this potential is the Mexican hot potential, okay, we take this mass large, so we can do a classical analysis, and so since this is the Mexican hot potential, there is symmetry breaking because phi condenses, once this symmetry is caged, this is the X mechanism, and so the system is gapped. Now it turns out that in this theory there are vortices, these are some solitons, some solitonic configurations such that, so if we take, so this is our space, it's R2, and the space of vacua is a circle, okay, so you can construct, you can try to construct solutions of the equations of motion which are time independent and such that as you go around a circle very far from the origin, you also wind once in the space of vacua in field space, okay, so you construct a solution where phi and radial coordinates of R and theta is some radial profile e to the i theta, so when you wind once in space, you also wind once in field space, and so you construct a radial rotationally symmetric configuration, and in general you also turn on, you also have to turn on some field strength, and in particular since there will be some field strength to construct these solutions, the ones that wind once in field space, they have one unit of magnetic flux, so essentially these solutions have some bump of flux around the origin, and they differ from the vacuum around the origin, and if you go far away they essentially go to the vacuum, now you can write down, so this is a very simple theory, you can write down the equations of motion, you can try to solve these equations, they cannot be solved analytically, but you can easily solve them numerically, and so you can construct these solutions, and then you can ask, okay, what is the mass of these solutions, you compute the energy of these solutions, and what you find is the following, so if you are just considering the theory with the scalar without the gauge field, you would find that in fact these solitons do not have finite mass or finite energy, because you would find that the energy is logarithmically divergent with the size of your, of your, of your, you construct your solutions, and you see that the solution here for instance for phi, well just for phi because we are considering which there is no gauge field now, approaches the vacuum with a power law, okay, and then you plug into the energy and you find that this energy is, is logarithmically divergent with some infrared cutoff, however if you consider now the theory with the gauge, and so now you have also to do this, and you are once again studying your solutions, you find that now the vacuum is approached exponentially, and so now the energy is finite, abelian x-model, so very different behavior, and so this theory has finite mass solitons, and so the idea is that if we go at very low energies and we turn on a mass which is, which is small compared to the scale set by the, by the gauge coupling, we can describe the theory as a weakly interacting model of, of, of these solitons, and in particular we can write down a theory in which the solitons are described by a fundamental field, and so we describe essentially these solitons as weakly interacting particles, and we introduce some scalar field phi tilde, and we write a weakly interacting theory for some massive scalar field, because these, these solitons are, are massive, and so the theory that we will write is just a theory of a scalar field with some potential with a certain mass, and then we can include the first interaction, which is a quartic interaction, so this phi tilde, this, this field phi tilde represents the vortices. If you want is the creation operator for, for vortices, for these solitons. Now notice, so here m tilde squared is positive, and notice that here up to the fact that we have this positive mass, this is precisely the O2 vector model that Igor was talking about yesterday, okay? This is a complex scalar, you can write it in terms of two real fields, there is O2 symmetry, corresponds to rotating this field, and taking the charge conjugate, this precisely the O2 vector model. Now, what happens if we try to invert, are there questions so far? So what happens if we try to invert the sign of this mass, okay? So let's see what happens if we take now positive mass here, and then negative mass here. Now if we take positive mass here, now this field phi is massive, so okay, if the mass is large, we integrate it out, we are left with u1 level 0, Young-Mills theory, but in fact this is a free theory, okay, this is a free photon, and a free photon in two plus one dimensions can be dualized into a free scalar, but importantly a free compact scalar, we can call sigma, and this dualization is a standard dualization that for instance you do in four dimensions when you want to do electric magnetic duality of an abelian, free abelian gauge field, so essentially you are right, here you have your Young-Mills action written in terms of f, f by star f, you are writing star f as some d sigma, okay, and that is because if you wish the Maxwell equation of motion say that this is closed, and so you write it as the differential of something, okay. Now it is important that this sigma is compact and one way, one simple way to see that this sigma should be compact, by the way compact I mean that sigma should be identified with sigma plus two pi, or if I want it properly normalized, here I should put the correct mass scale, which is probably g, and one way to see that this should be the case is because for instance, so suppose that we are looking at some field configuration on S2 times S1, and then in the pat integral we would be summing over fluxes on this S2, because one of the two pi integral of f should be quantized, and in the pat integral we should sum over all these quantized fluxes, but now if you use this map, you see that in this configuration the sigma is constant, and so if you want to reproduce these field configurations, you need configurations where this sigma is constant on the S1, and so if you integrate it, sigma cannot go back to itself since the derivative is constant, it has to shift by two pi, and okay here I'm not careful with all the coefficients, okay. So this sigma lives in S1. So here I'm doing pure classical analysis, I'm taking the mass large, and the mass is very large, and I do a semi-classical analysis when the mass is large, phi is massive, I can integrate it out. Again, I take large negative mass, and there is condensation. I'm not trying to understand here what I meant for small mass, which is hard, I'm just doing very large mass compared to the mass set by the gauge coupling. Yes, yes. Yeah, I'm starting that, I'm saying what happens in that case? Yes, I'm saying I'm taking some particles with the large mass. This is purely classical analysis. This one here? Yes, we will come to this in a second, but yes. So in this theory we say when the mass is positive, now we integrate out the scalar field, and we arrive with this free photon, which is a free compact scalar, S1. So now what happens in this theory? So now in this theory we take, if you wish, a large negative mass squared, and well, also here there are no mysteries, because if the mass is large, now again we have the same phenomenon, we break the symmetry, but this time we break a global symmetry, and so we have the Goldstone mechanism of the Higgs, and so we have a Goldstone boson, which is S1, because the symmetry that we break is a U1 that rotates this phytid, and so from here we also get some S1 free scalar, okay? And so surprisingly, we started with the Higgs, and attempt to describe these vortices with some dual theory that was working for some value of the mass, but in fact, this duality works also for the other value of the mass, okay? We have a match. Now in this theory, which is the O2 vector model, so in both theories we find for large values of the mass, we find two phases. Oh, yeah, so, I mean, of course, when you do this, if I take sigma to be canonically normalized, now sigma is dimension one-half, so I want to put here something which is dimension one-half, which is G, because G squared is dimension of mass, G is dimension of mass, square root of mass. So if you wish, it is true that this theory is free, and you might ask, okay, so what about the gauge coupling? If it is free, there is no, I mean, this doesn't appear anywhere, but still it looks like it's setting a scale for us, and in fact this scale also appears here, because this is a free scalar, but the radius of the manifold, of the target manifold needs a scale, which is this very same scale. So we found these phases, so let me plot m squared, and so what do we say? So for positive m squared, here there was an s1, a linear sigma model, okay, this would like to be a nonlinear sigma model, but in this case just one free scalar, so it's actually linear, and here the system is gapped. Now in the case the O2 vector model, we know that in fact, so first of all we know that there must be at least one phase transition in the middle, because these two phases are different, but now we know that in the O2 vector model there is in fact just one phase transition, and this is a second-order phase transition, it's a CFT, and so the claim of the particle vortex duality is that in fact, also the gauge theory, it is dual to that, and so it has precisely one phase transition, and this phase transition is second-order, okay. This is not obvious, but this is the claim of the duality. Now as I say, this is not obvious, but this has been extensively checked, in particular with the lattice Monte Carlo simulation, and as far as we know, everything's consistent, so we do believe that this duality is correct. I think that there are clashing claims, for instance, this is the very case that some people would like to say that there is SO5 and unsymmetry, but it's not clear if this is true. I mean, there are no definite results, right, from the numerics. Yes, I mean, I don't have much to say on the case you won with two scalars, that will be somehow outside of the domain of the claims of these dualities. For one scalar, for one... As far as I can see, yes, I mean, okay, I don't know what you mean by a lot, but I think there is consensus with one complex scalar, the lattice Monte Carlo simulation. That is also some theoretical work that has been done, so one can write explicit non-linear map on the lattice fields, but it's also numerical lattice Monte Carlo simulations. Yeah, yeah, I mean, I can give. So, in any case, I mean, in this, indeed, for this that the following dualities, as I said, we cannot prove them, and so I will state what the conjectures are. We can make checks, various type of checks, but I will never make a claim that some duality is correct. So eventually, we'll have to do numerical simulations and see whether these proposals are correct. But the proposals are sharp, so either they are correct or wrong. So in this case, for this duality, I will use the following notation. So u1 level 0 with phi dual to just phi tilde, which is the O2 vector model. Okay, so in particular, in this notation, I will just write the fields that we have every time there is a complex, sorry, every time there is a scalar, I will implicitly assume that there are a quartic interaction for this guy. We'll not write them, but they're always implicitly there. Okay, so what checks can we do of this duality? So okay, we already checked that the phases match, at least when the mass is large and one can do a semi-classical analysis. We can try to compare the symmetries. So in this, for this theory, the global symmetry is O2, which is a u1 that rotates the field phi tilde times charge conjugation, which is a z2. There is also parity and time reversal. Now what about this theory? So clearly here the symmetry that rotates phi is not there because it's gauge, it's not a global symmetry. However, there is another global symmetry, which is called the magnetic or topological. This is a symmetry whose current is constructed out of the field strength. So this is epsilon mu nu rho, f nu rho, and this is conserved just because of the Bianchi identity. Okay, in terms of form, we'll write this in the following way. In fact, what we can do with this does, well, if you go in the case, so if you consider the case in which this was massive and we're less with the free photon, this current, which is precisely equal to the sigma, what it does is to shift sigma. If you go in this phase, what this current, what this symmetry does is to shift that free scalar, and so in fact, this free scalar is itself a Goldstone boson, and so for this reason we are sure that it cannot receive, there cannot be any potential from some additive corrections because this is actually a Goldstone boson for the broken magnetic symmetry. So this is my next U1, which is mapped to this U1, and then there is also charge conjugation as well as parity and the entire reversal. So the symmetries match. So what about operators? So in general it's very hard if you have a duality to draw a complete dictionary between the operators. In principle, there should be one, but in general it's very hard to write it. However, we can map certain simple operators, especially if we consider the simplest operators with a certain charge. Now here there is an operator which is charged under this U1, it's just the complex scalar phi. So what is the operator here? The corresponding operator which is charged under this magnetic symmetry. So first of all you take the complex conjugate of the scalar field, and then it maps the gauge field to minus itself. This is the general action of charge conjugation, and it turns out that, well you see it from here, since charge conjugation maps a mu to minus a mu, it also changes the charge, so it acts on this current, and that is the reason why I'm using this strange symbol which is not just a product, the semi-direct product, because charge conjugation really acts on this symmetry. So everything matches nicely. Yes, so the operator in this theory that is mapped to the complex scalar is the monopole operator. So let me explain what does this mean. Monopole operators, so monopole operators are local operators, they are called sometimes disorder operators, because they are not defined as the most standard local operators we are used to as some polynomial function of the fundamental fields in the Lagrangian, but rather they are defined as some singular boundary conditions at some location in space. So what does it mean? So suppose that I want to compute a correlation function of these operators, so if I want to define these operators essentially I have to tell you how to compute correlation functions. So suppose that I want to compute a correlation function of some monopole operators at x times something else, and now the way to do that is to set up a path integral, so let me call phi all the fields in the theory. So in general here we insert whatever standard operator we have, however we specify that when we have point x, we cut a small ball around x, and then we impose a boundary condition, some non-trivial boundary condition at x, and this condition, for instance in the Abelian case, so for an Abelian gauge theory, is that there is some non-trivial flux on these two sphere. Let's say that I want to consider the basic monopole operator, so there is one unit of flux on this sphere, and this is some singular boundary condition because since the flux is constant, if I shrink this sphere it means that the connection is, or if one of the field strength is becoming very large, is diverging. So the prescription is that correlation function monopole operators are computed in the standard way, however I have to integrate over field configurations which are not smooth at x, but they have some prescribed singularity at x. So this is what monopole operators are. More generally, if we have some non-trivial, sorry if you have some non-Abelian theory, we should specify that on this S2 there is some non-trivial gauge bundle, and since these gauge bundles are parameterized by pi1 of g, where g is the gauge group, in fact this is the group of magnetic charges that monopole operators can have. These are conserved charges. And in this case here it's in fact a u1, because the magnetic charges are all the integers, so the corresponding group is u1. Now, in the presence of the transimus interaction, what happens is that these, let me call them bare monopole operators are not gauge invariant. They have some, well maybe. So if we have transimus interaction means that the bare monopole has gauge charge. One way to see that is that if we are on a manifold, not three manifold with a boundary, and you do a gauge transformation of the transimus action, you get a boundary term. And so in particular if you try, I'm not doing this computation because I don't have much time, you can try to do it. If you do a gauge transformation, you will get a boundary term on this sphere. When this sphere is very small, essentially you pick up a phase that depends on the gauge transformation and x, and we interpret this as the fact that the operator is not gauge invariant transformed under gauge transformations. And so in order to construct gauge invariant operators, we have to dress this bare monopole with some of the fundamental fields in the theory. So we have to construct some composite object. And because of this fact, monopole operators in transimus theories can also get flavor charges or spin, essentially because you dress them with the fundamental fields, which can be charged under some other global symmetry. Well, I'm jumping between Lorentzian and Euclidean. But no, I mean this, so this definition is both, I mean you can do it in Lorentzian, this is also Lorentzian signature. I mean it's a local operator, so you can do it in Lorentzian, you can still cut a ball around the point and you impose until you have boundary conditions around that point. You can do it directly in Lorentzian. Okay, however, okay, for the particular case that we are looking at, this is not an issue yet because the level is zero. So in fact the bare monopole operator is gauge invariant. And this bare monopole operator in fact is, as we said, is precisely the operator which is charged under this magnetic current. And so it's precisely the operator which is due to these fields, phi tilde. The other piece of the operator map is that if we take phi squared, should be mapped to minus phi tilde squared because this was precisely the operator that we were using to do the relevant deformation to move into the two phases, okay, with the minus sign. Okay, now we can write the duality in a slightly more precise way in the sense that we can include more, a little bit more data. And this will be useful later or tomorrow. So in particular, since we have global symmetries, we can couple this global symmetry to background fields. This is extremely useful in general in quantum field theory. And tomorrow we spend a lot of time talking about this procedure. This is useful because essentially once we couple to background fields, we can construct observables which are essentially the partitions function of the theory on various backgrounds. And of course, I mean it's also useful because if you compute functional derivatives, you compute all the correlators. And so in particular, we can rewrite this duality coupling to the background fields. Okay, let me just introduce a background field for this U1. Okay, in principle, we could do better, but this would be complicated. So we introduce some b mu o. And let me just, as a notation, use capital letters for background fields and small letters for dynamical fields which are part integrated. So I can write the duality in the following way. And this is very schematic. So I'm not careful about coefficients here. The only thing I'm careful about are the coefficients of St. Simon's terms because they should be quantized. Okay? No, so the claim is that this is a duality of, so this is an infrared duality of around the second order phase transition. So if you want they flow, so the second order phase transition is the same CFT. The claim is that if you do small deformations, you get the same. And now if you take larger deformations, I mean, the duality is still correct if you are at small enough energies. In particular, when we said, okay, let's take some mass, for instance, the billion X model. There are these vortices. Of course, if you go at energies which are larger than the mass of the vortices and then you reach the scale, the mass scale set by the photon, of course, you see that the theory, I mean, contains a photon and so on. But if you remain at small enough energies, you just keep... So these are the CFT. Yes, so that is a claim about the CFT. Okay, so here I'm writing the duality. So what I've introduced is the field B, the background field. So here it's obvious, okay, because how I coupled this theory to a background field. It goes in the covariant derivative. Here maybe it's a little bit less obvious. Let me also say that I'm not writing the Emile's term. It's implicit, but it's there. So the coupling is through this term here, and what this term here is nothing else than the minimal coupling between the magnetic current and B. And if you... Okay, this J is star F. If you are writing in terms of forms and you do some partial integration, it becomes this mixed-transiment term. Okay, so this mixed-transiment term is just a standard coupling, linear coupling of B to the current. Then I would like to discuss another example. It's the example of flux attachment, how it's called in the... In the... Or bosonization. So flux attachment is some simple idea, some smart simple idea that I think it goes back to... We'll check. And that is the following. So if we have some particle... So for instance, suppose we have some bosonic particle, and we manage to attach to this particle one quantized unit of magnetic flux, then this particle becomes a fermion. And one way to understand that is that... So suppose they have two of these objects, which is a particle attached to one unit of magnetic flux. Now we like to exchange these two particles. So I can do that by taking one and moving it around the other one by half rotation. Now when I do that, I pick up Aron of Bohm phase. But since this is just half of the rotation, this phase is minus one. Okay? So when I rotate this, I get a minus one. And so this is telling me that these are fermions. Okay? So this is the basic idea. We can realize this idea in the relativistic context of quantum field theory, again with Chen-Simon's theory. And so in particular, we can consider the following theory, which is u1 level one with phi. So very similar to the theory that we had before. But now I have a Chen-Simon's theory and level one. Okay? So why this model realizes this flux attachment? Well, because... So a unit of flux is a particle with a unit of flux is essentially something which is created by the monopole operator. So we look at the monopole operator here. However, for what we said, there is one unit of Chen-Simon's level. And so because of that bare monopole has gauge charge one. It's not gauge invariant. How do we make it gauge invariant? Well, we have to dress it with one mode from phi star. If you want the compressed conjugate, which has a gauge charge minus one. Okay? So we have to dress it. What does it mean that we dress it? So for instance, we can define this operator in radial quantization. And if we do that, we would go to S2 times R. We would put one unit of magnetic flux because this is the basic monopole operator. And then we would put one zero mode of this scalar field phi star. But now, if we look at this lambda problem, so we ask, okay, what is the spectrum of a scalar in this magnetic flux? If one of these problems is solved, so the spectrum of the Laplacian for this field here. And this problem is solved by the magnetic spherical harmonics. And the result is that, in fact, there are two zero modes. And these two zero modes form a doublet under a rotation. So this is a spin one-half. And so when you dress this monopole with one of these zero modes, the gauge invariant operator becomes a spin one-half operator. So it becomes a fermionic operator. So if you want, this was the bare one. And the gauge invariant one has a spin one-half. The claim, which again we cannot prove, of bosonization is that, in fact, at low energy, if we, maybe this is a point that I have not emphasized before. So if we tune value of the mass, even in the previous case, there was a mass term. And if we want to hit the CFT, we have to do a tuning of it. So here as well. So if you tune the mass appropriately, the low energy tier, in fact, is a free fermion. So this spin one-half operator, in fact, this kind of a free fermion. So you want one high. It's just a free fermion. And so in particular, as you said, the monopole operator, the gauge invariant monopole operator is mapped to the fermionic operator side here. And for instance, the mass term, and this we are going to check in a second, is mapped to a mass term here. So in fact, if we propose this, we should check if this makes sense. And so, okay, first of all, we should check the symmetries. That works exactly the same way as before, because this side is, yes. Yes. So I will, I only want to discuss gauge invariant operators. Yes. Otherwise, I will write bear if it's not. I'm almost done. Yeah. So the analysis of the symmetries is the same as before. It doesn't change. So there is u1 times charge conjugation here. There is magnetic symmetry and charge conjugation here. What about the phases? So what if we deform? Clearly, the phases of this theory is quite different than before, because this is a free fermion. If we give a mass to the fermion, we have a gap system on both sides. We don't have a Goldstone boson. If we do the analysis here, well, when the mass is negative, it's the same as before, because this field condenses and there is x mechanism and this is gap. It's the same as before. The difference is when the mass is positive, because now we are left with u1 level 1. It's not u1 level 0. And this is one of the theories that we discussed before. This is a trivial topological quantum field theory. So it has a single gap vacuum on any spatial topology and so perfectly matches with a fermion. Okay, let's stop here.