 One thing we discussed was the phase diagram of the O2 model. And then we focused a little bit more on the Goldstone phase. And in the Goldstone phase there were this rather rich physics, we had this vortex, and we saw some hints of gauge theory. And then we formalized this correspondence that the free compact scalar, which is the low energy effective field theory in the Goldstone phase, is in fact entirely equivalent to a free gauge field. So we see that in the long distance limit, where the effective field theory on this side of the phase diagram is well approximated by a single Nambu Goldstone particle. And the physics can be also captured by a free gauge field. And there is a dictionary between different elements. Many people ask me lots of questions over the break, and also before the lecture. And some of these questions I think are kind of pertinent to everybody. So please step up and ask more questions during the talk, because some of these questions are, I think, would be of interest to many people. And I don't have really time to repeat all of them. So it would be good to just ask them in flight. Okay, so now we're going to make a more dramatic claim about... So the idea is that we have some kind of hints of duality on this side of the phase diagram. But the idea would be to try to extend it to the whole phase diagram, which is a genuine and non-super symmetric duality. And the idea is that intuitively, here we saw this vortices, which reminded us of charged particles. But the vortices here were dynamical objects. You could create a vortex and an anti-vortex. It would cost you some energy, but you could create vortex, anti-vortex pairs from the vacuum, and they could move. So the idea is to add charged particles, because charges look like vortices. And so the idea is to try to add charged gauge particles to the gauge theory side of the correspondence, and to try to see if we can lend on our feet. So we start from the simplest thing, which is to add a single charged particle of charge one. So the Lagrangian is going to be 1 over 2g squared, d mu a nu minus d nu a mu squared. That's the gauge kinetic term. But now we have a new scalar field. And since I already used the notation phi for the scalar field in the O2 model, and since I used the notation phi hat for the operator, the tax on the vacuum creates stable particles. I guess now I need a new notation for a scalar field. So I wonder what would that be? H. Okay. So let's do dh squared, and some potential for H. Now H. H is a complex. H is a charged one scalar field. So now you'll see a non-supersymmetric duality appearing sort of out of this mess. And it has really amazing properties that I'll explain. So it's a charged one scalar field, meaning that the derivative acting on H is nothing but the ordinary derivative minus IAH. It's the usual covariant derivative. And now we have to say something about the potential for H. And as before, well, it has to be gauge invariant. But as before, I'm going to truncate it at the quarter to quarter. So we have M squared, but now it's a new M, let's call it M twiddle squared. H squared plus some new lambda, H to the 4. Okay. So that's the model. So let's study the physics of this model. We already know a lot about the free gauge theory. And we have some intuition about, you know, charged particles coupled to gauge fields from the books, from graduate courses. So let's try to analyze this model, making the same sort of a self-consistent assumption that can be verified at the end that this is an irrelevant coupling and everything depends only on this coupling. So we'll vary M twiddle squared and try to draw a phase diagram and see if we land on our feet. So that's going to be here. So we draw the same line, M twiddle squared. And we have to analyze the model in limits where it becomes a weakly coupled. Now, here there is one conceptual issue that I will get to at the end, which is that unlike the O2 model, which had one quartic coupling lambda, here there is G squared and lambda twiddle. So this is an interaction parameter because if you normalize the gauge field canonically, this G pops out here. And so it leads to some interactions between the Higgs field and the gauge field. And likewise, this is an interaction. So you see that these models are clearly not the same. They even have different number of parameters. They have a different number of degrees of freedom. They are not equivalent in any exact sense. But the duality that we'll discuss soon will be a long-distance duality. I'll discuss this conceptual point soon. But for now, just I wanted to say that the model becomes weakly coupled when M twiddle squared is much, much bigger than both lambda squared G to the 4. Oh, yeah, lambda twiddle squared. Thank you. So this is where the model is weakly coupled. When the scale is much, much bigger than on the interactions. Remember that we learned that when there is an interaction, then it leads to an expansion at long distances that's badly divergent. So we need to take this limit for the expansion to make sense. Okay, so what happens when M twiddle squared is huge and let's start from positive. M twiddle squared is huge and positive. So somebody from this section, you have nobody present, no? No, here, okay. I'll compromise. Yeah, it's the same theory as before. So namely it's a free photon, okay? So here we have a free photon. That's our effective field theory because we just get rid of the Higgs. It's like people didn't know that there is a Higgs, you know, when they discovered the electrodynamics because it's heavy. You integrate it out, nobody cares, yeah? So at long distances you don't care. So there is a free photon and we have a dictionary here saying that free photon is exactly the same as spontaneously broken magnetic symmetry. So this model has a global symmetry which is u1 magnetic which is generated by the current 1 over 2 pi epsilon mu nu rho f nu rho. So this is the magnetic symmetry. This is an exact symmetry of this model, an exact u1 symmetry. Now, I just want to make one remark, which I won't say again, this is O2 symmetry. So you might want the symmetries to be the same just for starters. And here there is just a u1 symmetry at this point. So there is another Z2 which is called charge conjugation. So there is also charge conjugation which takes the gauge field to minus itself and the Higgs particle to h dagger. And that's how we get O2 because these two generators combine to O2. Anyway, so in this phase we get a free photon which is exactly the same as Nambu Goldstone. And so it can be written as an effective filter that looks like f pi twiddle squared d theta squared. So we see something that we already saw here but it appeared on the other side. It appears on the wrong side of the phase diagram somehow. Anyway, now let's do the other limit of large negative m squared. At large negative m squared this is also familiar from textbooks because the Higgs field then wants to condense and we have a Higgs mechanism. So when the Higgs field condenses and there is a Higgs mechanism the gauge field disappears, right? Like the W bosons, they become heavy. And so what remains at low energies? Nothing. So this is trivially gap. However, I want to tell you something nice about it. Well, in this side of the duality the vortices for theta are not good and they correspond to charge particles. Let me just write it. Vortices for theta, vortices of theta they are charge particles. But here I want to say something more a bit about this phase. Remember that here I made a little note that in this trivially gap phase there was a massive charged particle phi and it was a stable resonance. It's the lightest particle in this theory. So what is the analog here? I want to discuss that a bit. It's an interesting problem. So it's useful to try to write the Higgs field in terms of a phase and the radial mode. And so let's write the Higgs field in terms of a phase. Let's call it psi and the radial mode. So if you go to the deep into the Higgs phase where the scalar field condenses then the only thing that remains is this guy and the action is like a Stuckelberg action. So there is this thing that you saw in Peskin and Schroder, I'm sure. There is this Stuckelberg action which looks like a deep psi minus iA squared. So that's the effective filter at very long distances. I mean it's a trivial theory. So saying that this is the effective filter at long distances is kind of void of content and this theory is completely gap and trivial as we said here. But this is a useful thing to write because it allows you to exhibit some vortices. So we can ask, remember that vortices for phi did not exist, they had infinite energy. But what about vortices for psi which is the phase of the Higgs field? When a complex scalar field condenses we already convince ourselves that there are no vortices. But the Higgs field is coupled to a gauge field so maybe when there is a scalar field that's coupled to a gauge field there are no vortices. And indeed the point here is that this previous computation that led to a logarithmic divergence is now fixed. We can cancel the logarithmic divergence by picking an appropriate configuration for the gauge field. So if we go very far, we have some vortex core and psi goes around and picks a 2 pi phase. But now we can pick a gauge field little a that goes like one over the distance. From the core. So let's say that this is R. If we pick a gauge field which goes like one over R with a component in the polar angle this will cancel this logarithmic divergence that we've previously encountered and in fact it will render the vortex into a genuine, now a genuine finite energy particle. So now there is a particle which is called a magnetic vortex. This particle is called in the literature a magnetic vortex. The reason that it's called a magnetic vortex is that because you've activated a gauge field to cancel the logarithmic divergence now this gauge field carries some flags and you can check that this flux leads to a magnetic field. So the way it looks is like there is basically a magnetic field through some circle here. So the particle looks like a little tiny bit of magnetic flux that's moving around. So it's like a moving solenoid. So that's why it's called a magnetic vortex. And this particle carries the charges of the U1 magnetic symmetry. So this was a symmetry in the problem and you can check that this if you integrate J0 you integrate J0 you get one in some units. Where J0 is defined as the zero's component of the current you compute the total charge since this particle carries magnetic field it gives one because J0 is the magnetic field in space. So the total magnetic field is one in some units. So we have a perfect candidate to mirror the stable particle created by an ordinary complex scalar field that's the magnetic vortex. So we see essentially the same thing a trivially gapped phase that maps to a Nambo-Golston phase a U1 this is U1M charged this is charged under U1 magnetic. In a trivially gapped phase we see a stable particle which is charged under the unbroken symmetry on both sides. Here it looks like a magnetic vortex and there it looks like an ordinary particle a 5 to the 4 particle. That's why this is called particle vortex duality. It's very confusing because there are two kinds of vortices in this theory. There are bad vortices and good vortices bad particles and good particles The bad particles are the electrically charged ones and the bad vortices are the vortices of the ordinary scalar field. The good particles are the 5 particles and the good vortices are these vortices. So it's very confusing it's particle vortex duality but it's really particle vortex duality for the good vortex and the good particle not for the bad vortex and the bad particle I'll write it more we'll write the dictionary more but in any case you see the same phase transition so you're compelled to say that this is also ought to because why not? It's the same phase transition so people were compelled to make this conjecture and it's correct it can be checked on the lattice but this conjecture is not an exact duality now I want to come to two points one is the dictionary and the other is the nature what does it mathematically mean for these models to be dual so now we'll dedicate some time so I can erase that and now we can write a more sophisticated dictionary so unlike the dictionary between the free photon and the compact scalar this duality cannot be proven and nobody has proven it yet it's something that's observed by doing theoretical tests and consistency checks so one side of the duality is the gauge theory which is the kinetic term 1 over 2g squared f mu squared plus dh squared plus m twiddle squared plus lambda twiddle h to the 4 m twiddle squared h squared plus lambda twiddle h to the 4 that's one side and the other side is d5 squared plus m squared h squared plus lambda 5 to the 4 and now there is a dictionary now nobody can prove this duality but we understand how various objects on one side map to various objects on the other side from these basic considerations but nobody has a full dictionary it's an open problem and it might be unsolvable at this point so let's start with the dictionary first it's always the symmetries it's the simplest so the symmetry here is u1 magnetic and the conserved current is 1 over 2 pi epsilon mu nu rho f nu rho so the symmetries this is the u1 symmetry of that side I ignore charge conjugation on that side we have SO2 which is the same as u1 and the current is phi d mu phi star that's the usual expression the usual expression for the usual expression for the particle number for a complex scalar field so that's entry number 1 in the dictionary entry number 2 the operators so we see we see that both have a known fixed point but somehow it's reversed the trivially gapped phase appears here on the right and on the left here so it seems like the m squared parameter and m twiddle squared are not quite the same m squared couples to phi squared and m twiddle squared couples to h squared so the next element in the dictionary is the relation between h squared and phi squared and in this relation there is a funny minus sign this minus sign is super crucial that's the thing that tells you that the phase diagram is reversed in the duality this may look disturbing because h squared and phi squared are positive so how can it be that there is a minus sign and this is okay because these are composite operators so the square of an operator in quantum filter is not a positive definite operator I missed it, yeah yes about what? yeah that's what I'm going to discuss the meaning of this duality I'm going to write a little bit the map we'll discuss the physical meaning of this duality a little bit so the next thing here is that there is a monopole operator that we defined before this monopole operator creates these magnetic vertices from the vacuum so when the monopole operator acts on the vacuum in the trivially gapped phase it creates magnetic vertices on this side of the duality it's clear what creates the stable particle in the trivially gapped phase it's phi so we have a mapping for phi so you see phi squared maps to h squared but you cannot take the square root of this formula because h doesn't exist h is gauge variant phi the thing that maps to the monopole operator and phi squared maps to h squared not to h sorry so phi doesn't map to h the gauge doesn't exist so this is the last element in the dictionary but this is a questionable element so this guy is not really rigorous it's just a qualitative thing this is the bed vortex you can also say that phi vertices map to electric charges electric charges are h it's not gauge invariant so it's really kind of I'm shaking just from writing it but here we have the phi vortex so both objects don't exist they have infinite energy and they're not gauge invariant on this side of the duality but they kind of are similar so we're still right I can still mention it okay so we have many elements that agree on the two sides and this is the central reason why it's called particle vortex because this is a particle and this is a magnetic vortex in the trivially gapped phase in the spontaneously broken phase they both create goldstone bosons so it's trivial but in the massive phase it's a non-trivial mapping now about the meaning of this duality that's what you asked about right? can you just pick up because I can't hear anything that's right so the fixed point here appears at some point which is given by m twiddle squared in some units of g and lambda twiddle and there the fixed point appears at some m squared which is some number times lambda squared but these numbers don't have to agree it's a non-trivial reversal because this comes to my next point which is the meaning of this duality so there is some kind of mapping and the phase transition agrees and so this is the simplest example of non-supersymmetric duality for an interacting filter this is supposed to be the simplest non-interacting simplest non-trivial this is the simplest non-trivial duality it's non-supersymmetric that's the great thing about it you might maybe some of you are experts on supersymmetry and you might have thought that this concept is limited to supersymmetry but as you see it's not there is a non-trivial non-supersymmetric duality in two plus one dimensions it's called particle vortex duality and this is actually proven on the lattice and it passes a very large amount of consistency checks theoretically but I just wanted to explain what it means so clearly nobody says that these two models are the same in fact these two models don't even have the same number of degrees of freedom for instance in this model if you go to very high energies you see a gauge field and also a complex scalar field well in this model you see only a complex scalar field so the number of degrees of freedom doesn't match that's point number one point number two the number of coupling constants doesn't match here there is one interaction sorry here there is one interaction here and another interaction in the covariant derivative but here there is only one interaction the Feynman rules are not the same these models are different nobody says that these models are equivalent however these models become the same only when you go to long distances and the long distance theory is still non-trivial because there is a fixed point the second order phase transition so the statement is not that these models are exactly the same and magnetic vertices have exactly the same properties as the stable particles the statement is only in some vicinity of the fixed point the two models can be mapped to each other so there is some vicinity of this fixed point where the physics of these models is very similar or identical but it does not extend to the full phase diagram these models are really different in general okay this is what's called an infrared duality yes the question is what is the motivation for introducing a scalar field a propagating scalar field well on the other side of the duality in this effective field theory the bed vertices these ones they were dynamical objects so you could create a vortex, anti-vortex and they would move so if you just had a free gauge field like no H then there won't be dynamical particles, dynamical charged particles there will be some charged particles there will be classical though to have dynamical charged particles you need to introduce a propagating charged particle to reproduce the physics in this side of the duality now of course you have to play you could say oh how did I know that there is only one scalar field, not seven so you just check with one and if it works you declare victory you have to check, you have to guess nobody can prove it, it's a guess but it passes many consistency checks H is not composite, H is elementary any other questions? yes the question is if it's important that the phase transition order is second well to have non-trivial long distance suppose you have this kind of suppose you have some other model in which the phase transition is first order so you have something like this then this and then this some kind of picture like that and then you go to the long distance limit what do you see? well not much because in this all the correlation functions are exponentially small so they vanish in the ordinary sense of the long distance limit so there is a little bit here there are two super selection sectors and maybe a domain wall so if you can match the number of super selection sectors and the properties of the domain wall there is some kind of context to the infrared duality but it's very little in the normal filter it's much more because it's like an infinite set of scaling exponents and OP coefficients the very good question so there is a very good question here that sometimes in physics you hear the word universality and sometimes you hear the word duality and the question is what's the difference or what's the connection so I'll explain that point it's an important terminological point so universality is what Mr. Wilson invented universality is something that was invented in the 70s mostly by Wilson who got the Nobel Prize for that and universality is the following statement so let's say that we are in the scalar model I'm just writing again the scalar model Lagrangian so this has the following phase transition between the you know Nambo-Goldstone mode and trivially gapped phase and there is an O2 Wilson Fisher fixed point so Wilson invented the whole scaling hypothesis that operators that are irrelevant do not make an impact on the long distance physics so therefore Wilson would have said that you know if you added phi to the 6 or phi to the 8 or any number of other terms it would have not made an impact on the phase diagram because near the fixed point they all decay the irrelevant operators decay near the fixed point because their coefficients go to 0 so universality means universality means that if you have a complicated lattice model or a filter with many relevant operators the phase diagram is controlled by very few of those the concept of duality is entirely different it means that two completely different physical systems with different classical limits made out of different degrees of freedom can have the same long distance limit they are not connected by a series of operators that decay to 0 they are made out of completely different fields the Wilson hypothesis is about adding irrelevant operators to a given filter duality is about starting out from completely different classical limits and arriving at the same quantum fixed point well that's called duality I would say universality is what Wilson yeah universality is the statement that the higher terms don't matter and duality is the following statement so you know the universe is quantum so basically what what happens is that there is some quantum stuff but then one quantum system can have different classical limits so there might be a classical limit here there might be a classical limit here so we have one quantum model which is the old two Wilson Fisher interacting conformal filtering but we found two different classical limits by adding various things one is the one is the 5 to the 4 model okay so one is the the xy model xy model which is that and the other classical limit that we found was the Higgs field coupled to a gauge field so like a gauge field plus a Higgs field these are two different classical limits because they both have an h bar going to 0 there is some I mean there is h bar here is not defined because it's a quantum model there is no h bar there is nothing that appears when you start from a classical field theory and you quantize it so here there is h bar 1 and here there is h bar 2 but the quantum theory doesn't have h bar quantum theories don't know what is h bar it's only classical limits that are quantized introduce a small parameter right so we have one notion of h bar in the xy model which is you just start by writing Feynman diagrams for the old two model that defines h bar that defined in this model where you just start from a gauge theory coupled to some scalars and you quantize it so you have two kinds of expansions in h bar and the miracle is that they converge on the same quantum object the quantum object is universal but there may be many classical limits to the same quantum object this is called duality this is called universality there is yet something else which is that some people believe that there are very few quantum theories constrained by consistency so typically you expect that quantum theories have lots of lots of classical limits maybe we did discover many of them what Wilson discovered is that each of those classical limits still it is kind of you don't need to specify much about it that all these things won't matter that's what Wilson discovered that each of the classical limits is kind of specified by just few parameters but there may be lots of lots of classical limits to a given quantum system the hypothesis that there are very few quantum systems is the underlying philosophy of the bootstrap and so in that perspective duality is not surprising because there are very few quantum systems any other questions say again I can barely hear you the dictionary yes what the gauge theory is a particular classical limit that has a well defined phase diagram this is an infrared duality so nobody claims that the physics here and the physics here is the same the claim is that the physics here is the same that's near the phase transition where there is non-trivial quantum models the fixed point so the duality is only valid near the transition now in this light the question that Matteo asked about what our first-order transitions also can lead to dualities is especially kind of annoying because here there is nothing almost except for maybe a domain one so it's really a question of semantics at that point it's an excellent question so the question is what are the possible applications well first of all there are constructions like physical constructions where the gauge theory description is a much much better approximation so let me give you an example there is a phenomenon that's called superconductivity in which the wave function condenses so the wave function has a non-zero norm in the vacuum and in condensed matter this is described basically by a Higgs mechanism the wave function is like a charge particle in second quantization and the phase of the wave function is a gauge field and the condensation is achieved due to a potential for the Higgs field very much like what appeared here at some point this view of H and you write a model and then you ask what does this model do you have a complicated gauge theory coupled to a Higgs field and this is the first quantization of the Schrodinger wave function for the Cooper pairs and then this is where the duality comes in because you can say oh this model I've already seen it before near the phase transition it's the O2 model and then you say okay so there must be a phase transition between the superconducting phase this would be called sometimes the superconducting phase to some other phase or the phase transition because you know the scaling exponents from all this work on Wilson Fisher models in the 80s and the 70s the Epsilon expansion and the Buttra so suddenly you have great predictions for the insulator superconductor transition that's one type of application another type of application is that in some of these constructions especially started from a superconductor this U1 magnetic symmetry would not be maintained because many lattice models break this symmetry so in fact the Lagrangian in many of these condensed matter applications will have additional operators which break the magnetic symmetry the operators that are charged under the magnetic symmetry are the monopole operators so condensed matter theories would often add monopole operators to the action to mimic what they see in the lattice because the symmetry is not present in many lattice constructions and now this is extremely hard to analyze because as I told you the monopole operator cannot be even expressed locally it's a complicated boundary condition but using the duality you can now like in one minute say everything because on the other side of the duality this is like scalar field theory with some symmetry breaking term let's say you could add x squared where x is this x plus i y so x is the real part of phi without like the real part of phi squared now this is trivial to analyze I mean it's an O2 model you know, silly quadratic piece it's classical field theory and you can easily analyze what happens to the phase transition and for this particular deformation it's like literally 5 minutes to convince yourself that this will turn into the Isink filter the O2 will turn into the Isink Wilson Fisher fixed point and also about that everything is known so this type of duality is useful because in some applications in condensed matter and in particle physics one description is much more kind of natural from the other but then once you have the duality you have other descriptions that provide you with the answer for the quantum phase in a much more natural fashion that's one type of applications but other than that it's just an interesting general question in physics of you know to try to classify the possible quantum theories and which classical limits they've got it's like a broader question that is essentially also very important like string theory is a very similar picture but it's really the same it's one quantum theory with many classical limits any other questions before I I forgot when I started unfortunately 1145 I guess ok so what I want to do now so if there are no more questions about particle vortex duality what are the criteria for distinguishing good from bad the good question in ethics any other questions I mean maybe you have some context or completely in some can you say the context ah in this sense I thought like in this context no this it doesn't this if you act on the vacuum with the gauge invariant gauge non-invariant operator create like a charged particle and as I told you a charged particle in two plus one dimensions has infinite energy and it's not a good state in the Hilbert space and the same happens on the other side of the duality you have a logarithmic divergence in the energy so this is like an analogy it's not really a part of the dictionary so there are approximate configurations that could mimic that like you could have a box and then a charged particle would be ok and also a vortex would be ok yeah no much more yeah yeah so here you see so I mean in this analysis one makes a lot of implicit assumptions if you think about this picture deeply there is no way to make consistency checks in the classical regime because the classical limit is very different it's only this that agrees but what I'm doing here is that I go to the classical limit and I check the phases and I do the same here and then I just cross my fingers that the phases transform to each other by one phase transition it could be that there are seven phase transitions here in principle and the other model has eight phase transitions and none of them agrees but one hopes that at least classically the phases that you see are those phases between which there will be a phase transition and then you're compelled to make a guess there is no way to directly compare these theories at strong h-bar we can compute here at small h-bar 2 and here at small h-bar 1 but we're interested in h-bar 1 equals h-bar 2 equals 1 and there is no way to make a computation at least not with currently existing technologies so one does what one can and then one makes a conjecture and makes consistency checks now in supersymmetry there isn't a supersymmetry allowed to discover so many dualities it's because there are many quantities that can be computed directly here bps quantities that are independent of h-bar that's how people made so much progress in the 90s in superstrength theory and in supersymmetric duality because there were a bunch of quantities that you could just compute here but they didn't change and you could just compare so if you start from here and you start from here and you get the same answer you say voila here we cannot do that there are some things we can compute which are independent of h-bar like topological invariance or some what's called the SPT phases which I'm not teaching you in supersymmetric context one could compute continuous functions not just like three or four integers so there are very few things that you can like Francesco is here I saw him before so like if you look at Francesco's papers on these non-supersymmetric dualities he computes some integers there are like three integers you can compute which do not change as a function of h-bar they are called SPT phases I do the same when I write papers on the subject but it's just very few integers and in the checks that you can do in supersymmetric gauge theories you can compute partition functions so we can do less but the duality is not less true in fact this duality is more is better verified than any supersymmetric duality because people put it on the lattice and check any other questions? yes, yes, yes most certainly they put the Abelian Higgs model on the lattice they vary the mass and they find the phase transition and they measure the critical exponents that are the same one that Wilson and Fischer computed for the O2 model people were able to compute the critical exponent of h squared and they saw that it's like 1.512 to three digits exactly the same as the O2 model it's very impressive this case is better verified than any supersymmetric duality any other questions? ok, so next what I wanted to teach was a little bit a more modern thing which is essentially the following so here the main point was that there were these stable particles here in this language and there were some vortices in this language the magnetic vortices and physically these magnetic vortices are just like particles with a little bit of magnetic flux and they behave like ordinary particles so if you take them around each other there are no interesting phases they behave like ordinary bosons so what I want to teach now is generalizations of this duality for situations where these vortex particles behave like anions ok, so that's the next topic generalizations of this whole story for a situations where there are anions and churned simons terms so for that what I'm going to do now I'll start teaching churned simons theory for the next 15 minutes I'll finish it tomorrow and give you the simplest generalization of this duality with churned simons terms and if I have 5 minutes tomorrow at the very end I'll make contact with Yang-Mills theory the domain rule that we discussed at the beginning now we'll just discuss a billion churned simons so this is going to be free field theory we'll just try to analyze that free field theory and then do something more interesting with it this is the action churned simons captain is here and it's a particular it's a particular term that you can add only in 2 plus 1 dimensions now one thing that's annoying about it is that the gauge field appears with no derivative so you might be suspicious about gauge invariance so let's check the first sanity check is to verify that this is gauge invariant so you have to take a mu to a mu plus d mu of some omega and this is clearly gauge invariant because this is an anti-symmetrize thing but this this piece gives you d mu omega epsilon mu mu rho and then d mu a rho now of course you just put the derivative on the other side and it vanishes so this is a total derivative as long as you're in infinite space with no boundaries you don't have to worry very very very very much but as long as you're in infinite space you don't worry so this is gauge invariant and the question is about the physics of this model it's a gauge invariant a field theory for a gauge field remember that without the churned simons term this model was equivalent to a free compact scalar but now it's not there is no way that I know of to modify the compact scalar theory to include this k it cannot be included in any way that I know in the compact scalar theory so we cannot use that language anymore as soon as k is non-zero so the remark number one was the gauge invariant remark number two is that k has to be quantized it must be an integer this is not so easy to see but we'll do some computations soon and you'll see that if this k is not an integer you get some disease in the theory so I'll derive that k is an integer from some computations that we'll do down the road for now it may look mysterious why k has to be an integer but it's gotta be an integer for non-integer k this model makes no sense you'll see why so the question is about the physics of this model so remember that at k equals zero the model had one massless particle at k equals zero we had one massless degree of freedom just one mass one object that's moving at the speed of light one quantum moving at the speed of light now for non-zero k I'll explain it soon but the point is that there's one, still one degree of freedom we haven't changed the number of degrees of freedom but now it's massive now when you have a massive degree of freedom in two plus one dimensions like in any other number of dimensions you can just go to the rest frame we can go to the rest frame and ask what is the spin we have a particle and we can ask what is the spin of the particle we just rotate around it and ask what is the eigenvalue of the little group so if k is positive the spin of the particle is one if k is negative it's minus one the spin in two plus one dimensions is just a real number and in fact it can be even a fraction as we'll see for anions but this is not an anion this particle is not an anion it's a genuine propagating particle with integer spin if k is positive, its spin is one if k is negative, its spin is minus one your most important homework exercise perhaps if you don't know this fact the most important homework exercise for this course is to compute the propagator compute the propagator in your favorite gauge like C gauge C gauge is convenient and try to understand this fact now there is a hint I want to give you a small hint when you do this computation in arc C gauge you'll find also a pole at zero momentum so you might think that there's still a mathless particle but that pole is porous and you can prove that that pole is porous by computing the correlation function of f mu f rho sigma and you'll see that that pole goes away and in this Green's function all the poles are physical and you can read out the quantum numbers very conveniently and you can also read out the spin so the effect of the Chern-Simons term is that it makes the photon massive which is very weird without a Higgs mechanism so it's a Higgs mechanism without a Higgs field in the Higgs mechanism however I want to compare this with the Higgs mechanism it's a source of many many confusions so I want to compare this mechanism that leads to a mathless particle with the Higgs mechanism just that you don't get confused because this is something that lots of people asked me about various contexts so if you study the same action same action that we had before you're in the Higgs phase let's say you're in the Higgs phase so we have to say how many degrees of freedom there are so before we included the Higgs field say the Higgs field is gone there is one mathless degree of freedom if the Higgs field does not condense but now as we know from particle physics when we are in the Higgs phase there is another degree of freedom the longitudinal mode so in this model in the Higgs phase two degrees of freedom which are massive not one one is spin one spin minus one it's very different in the Higgs mechanism we take a mathless gate field and turn it into a massive thing but now there are two degrees of freedom spin one and spin minus one and here there isn't an additional degree of freedom it's the same old mathless photon that becomes massive and it has either spin one or spin minus one depending on the sign of K now this is of course consistent because parity reverses angular momentum this model is invariant under parity so these two things are related by parity because parity reverses angular momentum but this is not invariant under parity this breaks parity so it's okay that there is only one excitation and there is no partner under parity okay in fact parity takes K to minus K so it's good that for K positive the spin is one and for K negative the spin is minus one it's very different people often call it a Higgs mechanism without a Higgs field but it's not very different okay the next thing about it I have five more minutes is the physics of this model long distance physics the central question is what is the long distance physics what is the long distance physics of the model with non-zero K about zero K we already know everything so the first naive answer to this question is that the long distance physics is trivial because there is only a massive particle there is nothing massless so the naive answer is that it's trivial because there are no massless particles naive answer in fact this answer is incorrect this model has non-trivial long distance limit even though there are no massless particles you remember that in the first course in the first session I gave you a classification of phases and there was one element there that was non-trivially gaped it's a gap model but it has non-trivial physics in the infrared in the form of anions probe anions so this is the naive answer this is incorrect the correct answer is that it's non-trivially gaped so even though there are no massless particles even though there are no massless particles let me just write it down it's perhaps the most important thing here if you don't know that even though there are no massless particles there are probe particles which are very heavy that lead to our own of bone phases which are measurable even at long distances there are probe particles so the probe particles themselves are very heavy they are like classical sources like this j mu so these probe particles they come from coupling a mu to j mu classical we'll discuss it now in a lot of detail so there are classical sources for the gauge field which behave like anions behave like anions and they lead to our own of bone phases AB phases so even though the model has no actual fluctuating fields at long distances some probe objects behave like anions and of course you can anticipate that in the future we will want to make these anions dynamical we will want to second quantize these anions so far there will be just first quantized or even classical but then we want to second quantize them so they could fluctuate so it's important to understand the classical limit first before we second quantize it ok so what we have to understand is the response of this theory to classical sources let me just write the equation and then we adjourn so that's what we're going to study tomorrow so taking this model and adding classical sources we lead to some Euler Lagrange equations which are like generalizations of the Maxwell equations Euler Lagrange equations that you will get for this model would look like d mu I'm just writing them qualitatively for a second d mu f mu nu plus epsilon nu rho sigma f rho sigma j nu and here there will be a k and here there might be a g squared this is the qualitative form of the equations of motion that you'll get when you write the Euler Lagrange equations for the system and this is just a classical function it's not quantized so somebody gives you a distribution of charges and you find the electromagnetic fields like you did in school these equations are a little bit too complicated to analyze and in fact we don't need the first term that's what I wanted to argue we don't need the first term to understand the physics of dissonance because it has two derivatives well this has one derivative so this is more important at long distances so if you want to understand macroscopic phases you can throw away the first term so in fact we will simplify these equations to just the first to just k over 4 pi so this is actually the equation that we're going to solve epsilon mu nu rho f nu rho equals j mu classical so this is the equation that we're going to study it's a highly non-standard equation you're not used to electromagnetic fields with a linear response you're used to electromagnetic fields with a second derivative response so there are waves but of course it's not surprising that it's the first derivative equation that it's the first order equation but as I told you there are no waves there are no mathless particles this model is completely gapped it's non-trivial gapped so there are no waves and so the equation is the first order and so the plan is to just take a distribution of charges solve this equation and compute our own of bomb phases then we will establish the existence of anions in this model then we will second quantize the anions and get generalizations of particle vortex duality this is the tionionic