 Day three of the school, so we start with Yifan's third lecture, right? About non-invertible symmetries. Fourth? Oh, sorry, please. All right. Good morning. Recording in progress. Well, back to everyone. So today will be the last lecture from me, non-invertible symmetries. So in the previous lectures, we have been focusing on the big picture, the general structure of non-invertible symmetries, and especially we spelled out a lot of this general structure in detail in two-dimensional CFTs. And in this lecture, we'll apply the knowledge we have gained to identifying non-invertible symmetry in the very simple CFT, just to, you know, get our hands dirty. That is in the case of this one of the standard first non-trivial CFT that you probably encounter in any CFT class, the Ising CFT in two-dimension, okay? As I explained, this very simple CFT already gives an interesting instance of a non-invertible symmetry generated by a topological defect line, okay? So let us start with a brief reminder about, well, a brief review of the CFT, okay? So first of all, the CFT has a center charge equal to one-half, okay? So it's one of the simplest CFT, and the operator spectrum equivalently, the hubris based on S1, contain these primaries, okay? We subscript, you know, in the scaling dimensions, identity operator, the spring operator of dimension 1, 16, 1, 16, and the energy operator of dimension one-half, one-half, okay? And together with, because we are talking about two-dimensional CFT, the operator spectrum organizing two-dimensional primaries and descendants, here I'm listing the primaries, and then the rest of the states in the hubris space are generated by the verisoral descendants. This CFT has a famous symmetry, a very simple symmetry, that is the Z2 splint flip symmetry. The symmetry act on the CFT, this symmetry acts on the CFT in a very simple way. All of the operators in this hubris, corresponding states in the hubris space are even under the symmetry except for the sigma operator and its verisoral descendants. The sigma operator and their verisoral, its verisoral descendants are odd under the Z2 symmetry. And in terms of the notation that we have introduced, let's call this symmetry generated by a topological defect line, eta, okay? And in terms of the picture we have been drawing, this means that with eta acts on the operator sigma, okay? By enclosing it, whenever we encounter this kind of graph in your correlation function, we are free to shrink this graph and obtain just the sigma operator by the way it's opposite sign at the same location, okay? As I've told you in the very first lecture, a hallmark of symmetry is that it gives rise to selection rules. In particular, in this case, because sigma operator is odd, this implies immediately any correlation function that involves odd number of sigmas. In particular, the case with three sigmas, in terms of arbitrary locations, better be zero. But this CFT is very simple. It's so simple that you can actually compute all the correlation functions of these operators. I will not explain the technology, but you have to trust me that you can compute all the correlation functions. And after you compute them, you will discover that in fact there's some other kind of hidden selection rule that says any correlation functions that involve an odd number of epsilon operators inserted at arbitrary points is again zero, okay? So if you try to extrapolate between from this picture down to here, you want to say that there may be some hidden selection rule that could be explained by some hidden symmetry, okay? And that is the symmetry that will be identified. It turns out that will be a non-invertebrate symmetry. Just a historical remark. So there is a non-invertebrate symmetry. The kind of the signal there, this non-invertebrate symmetry, goes all the way back to Cranbers-Wanier. There's some already suggestion of such a non-invertebrate symmetry in this simple model. Since the work of Cranbers and Wanier, I believe in the 1930s. So back then people studied a lattice version of the CFT, Icing lattice model. You can think about as the statistical model in two space dimensions, and it's described by a Hamiltonian normalized by the temperature in terms of Nier's neighbor coupling of the sphinx on the square lattice. Imagine you have some square lattice, and on each side you have this sphinx, know by Si, and it can take two values, plus minus one. And this notation means that you introduce this kind of Nier's neighbor coupling for each pair of Nier's neighbors, okay? And K is the dimensionless coupling constant. As I said, I was briefly saying, the same model, there's a very similar model related to this. That's described by one-dimensional transfer sizing model. That's the quantum version of this statistical model. This model is so simple that you can actually study its entire phase diagram as a function of K. And because we have observed the temperature over here, you see that this is equivalent to, to tune K is equivalent to tuning the temperature. So in particular, high temperature will correspond to the small K part of the phase diagram. And low temperature will correspond to large K. And in this limits, it's very simple to solve the model, okay? I will now solve it here. But you find that high temperature, you have some non-degenerative vacuums, okay? And on the other hand, you have a low temperature, you have a doubly degenerative vacuums. They correspond to spontaneous breaking of the Z2 splint flip, okay? As a consequence, you expect there to be a phase transition between these two different phases. This is usually referred to as the disorder phase. This is related to, referred to as order phase because the spontaneously symmetry breaking, okay? There's some phase transition over here, which happens at a special value of K, which is denoted by Kc. That's a critical C, a critical value of K. And over here, the description is given by, there's a second order phase transition. And the phase transition is described by the Ising CFT, which is the CFT we reviewed over here, okay? In particular, the sigma operator, the spin operator, is the CFT version of this individual spins that live on individual lattice sites, okay? What is the statement of the Kramer's Wernher duality? The Kramer's Wernher found, when they studied this Ising model, okay? As a function of K, there's some mysterious duality between the high-temperature and the low-temperature phase of the theory. They find, what is the duality? They find the following equivalence relation, okay? That under the identification between a dual coupling, which I call K-dual, there is relation, okay? Ising, and you already see that if one is small, the other is large. And correspondingly, the duality relates to the low-temperature and high-temperature limit of this phase diagram. And the more detail, it was found out that the precise statement is that Ising at a low-temperature is equivalent, or is dual, to Ising at high-temperature in the infinite volume. But to be precise, you also need to introduce another D2 gauge field. The reason being that if you don't worry about the global issue, sorry, if you care about the global issue in particular, like the number of the ground states, obviously, these two phases will not match, okay? But introducing this additional D2 gauge field, which gauges the Ising's D2 symmetry over here, fixes the problem, so that it's actually an exact duality. And because this duality relates the low-temperature and high-temperature Ising model, and I should also say that in terms of the CFT, this phase diagram can be captured by the CFT, the Ising CFT, couples to a relevant deformation, well, deformed by a relevant deformation, figured by precisely this energy operator, okay? Which essentially dual to the Hamiltonian over here, where m squared is the coupling. And translated into this phase diagram, this point corresponds to m squared equal to zero, so we are at the fixed point, and the high-temperature phase corresponds to m squared bigger than zero, and the low-temperature phase corresponds to m squared smaller than zero in my convention. So you may have different conventions for your definition of the action, and there could be an opposite, these two phases could be flipped, okay? But this is my definition. In any case, this duality can be written more explicitly as a transformation on this couplings from m squared going to minus m squared, okay? Because it goes from low-temperature to high-temperature and the vice versa. And correspondingly, because how it couples to the operator epsilon, your Ising CFT, this is equivalent to sending epsilon to minus epsilon, okay? So this is already some hint that the Kramer's-Wende duality may be related to this hidden symmetry that explains this lecture rule at the fixed point. While the duality relation doesn't end here, there is also the famous duality relation that relates the sigma operator to the mu operator, okay? So we call sigma is the outer operator, okay? It corresponds to this spin-degree freedom on the lattice. When it takes a non-trivial expectation value, that corresponds to the spontaneous symmetry breaking phase of the Ising model, okay? On the other side, this is the desoldered spin operator, okay? This is the operator that takes VEV on the desoldered phase or high-temperature phase of the Ising model, okay? So when I draw this diagram, okay, this is in terms of this other parameter, sigma. All right. Then at the critical point, which in this case coincides with the self-dual point of this relation, meaning that sinh 2Kc is equal to 1, this duality first implies, sorry, this duality implies that the Ising CFT is equivalent to this Z2 over fault. This operation of coupling quantum field theory to Z2 gauge field is essentially implementing the Z2 over fault, okay? And as is common for dualities, duality typically relates two descriptions of the same system, and if there are some parameters you can tune, then typically what happens is at a special point at the parameter where the duality becomes a self-duality, the self-duality can be described by a symmetry. For example, you see this happens a lot on the modular space, conform manifold of two-dimensional CFTs. One typical example is the duality of a compact boson. Two duality maps the compact boson radius R to 2 over R, in certain units. That is not a symmetry in general, but at the self-dual radius, which is described as Z2 level 1, the duality becomes a very special Z2 symmetry in the Z2 level 1 WSW model. So we're expecting something similar here, and you will discover that it's indeed the case, but the difference is that in the case of Z2 level 1, from the compact boson special radius, that symmetry is invertible. Here we'll discover the corresponding symmetry would be non-invertible, which is correlated with this very special feature. So this is a question we want to ask. So there's many ways to pin down what this symmetry is. Okay? So let me give you one argument that uses this very simple picture that we draw many times. That is the modular invariance. For different perspectives, you can, two different perspectives, you can adopt to view the torus pattern function twisted by insertion of some putative topological defect line. So this is again the object we were considering is simply the torus pattern function of this CFT twisted by this topological line. So right now we are not assuming what this topological line is. We're agnostic about the details of a line, but we'll deduce constraints from this very simple equation. And once again, this is the same object, but with the topological line oriented in a different way. Sorry, the other way around. And as we said before, this equality leads to the following relation between a pattern function that's twisted in the time direction by the line defect insertion weighted by the Hamiltonian associated with the CFT generated by L0 and L0 bar. And on the other hand, we have the trace with no other temporal insertion of the hubris space twisted by the insertion of the topological line. And the astral version of the torus modulus enter into this expression. So I'm writing this for general C, but here C is equal to one-half in this specific example. And the fact that the topological line preserves the stress sensor means that the right hand side and the right hand side will have decompositions into the basic building block of 2D CFT, representation theory building block of 2D CFT, namely characters of the erasoral algebra at center chart equal to one-half. That is very constraining because unitary representations of erasoral algebra at center chart equal to one-half only comes in three families. There are one-to-one correspondence with the operators that we are talking about in the Ising model for the chiral representation. So that means the left-hand side let me call it is a linear combination of the character associated with the identity operator, okay? Let me call it alpha one, okay? Alpha zero, sorry. And the coefficient multiplying the character associated with the 116, 16 representation, okay? And the character associated with the h equal to one-half representation, okay? And they correspond to physically the three operators, identity, sigma, and epsilon, as well as their descendants. And these coefficients, which we are being agnostic about, determines how this operator acts on, this putative line operators acts on those operators, okay? The line operator acts on those local operators, okay? There are too many words about operators. Hopefully there's no confusion. And we can do the same thing for the right-hand side, okay? But as we said before, the right-hand side in some sense is more constrained. So even though we are, because we are agnostic about the detail of the line, we don't know much about this defective cube space. But just from the verisoral symmetry, we know whatever this defective cube space is, it must be built from the basic building blocks, which involve these chiral representations with weight zero, one-half, and one-sixteen. Okay? So you have a sum over two numbers. Each of these numbers can take value in a set of three possible representations. Okay? And the summand involves this coefficient, which is a positive integer, a non-negative integer, that comes for the degeneracy of the representation in the given two-state hubris space. And they are multiplied by these characters. And this very simple relation, so we have this equality, this very simple relation weighs the constraint that this coefficient of positive integer is surprisingly strong. Okay? So I'll leave that as a homework to show that there are three independent solutions to this equation. Okay? Meaning that all the other solutions have the same positive integer combination of the solution I'll write down. Okay? The first solution is such that three solutions, the first solution is when they are both equal to one. The second solution is one minus one and one. Okay? And the third solution is for two zero minus square root of two. Okay? So this corresponds to the identity sigma and epsilon means that this is nothing but the trivial line, the identity line. The second one, X non-triviality, triviality on everything except for the sigma operator as well as the sentence, identifies this operation generally by this punitive line as the Z2 flip symmetry, Z2 spin flip symmetry. So this is the object that will give rise to this duality defect. Okay? So this is the only other possibility and as you'll see, this is indeed a consistent solution, a consistent symmetry operation. Okay? In the sense that we will give an alternative description, a deterioration of this result that does not rely on any assumption. Okay? So here it's like we assume the line exists, this is the most general form it can take. The line exists. Okay? Is it okay that some coefficient is negative? So this coefficients appear on the left hand side. Okay? What's important, what's not true about the equation is that these coefficients are always non-negative. So it's your homework to see that having certain coefficients being negative is fine but for example you cannot have this coefficient being negative. The coefficient function is a... That's right. It's twisted in a temporal direction. Okay? So this left hand side keeps track of how symmetry acts on the hyperspace with no twist. Sorry, and what are the corresponding value of the coefficient n? You are keeping it arbitrary. Sorry? Come again? The coefficient n i j are arbitrary. Totally arbitrary. So to solve this equation all you need to know is that this coefficient has this property but you leave it in general and you will find the solutions will be in general a positive integer linear combination. Sorry, non-negative integer combination of these guys. But once you fix the alpha the n is going to be fixed? Yeah, so once you fix alpha the n is obviously fixed but if you are forming alpha you don't need to make assumptions apart from what I have said. Sorry, just one other clarification why in the left hand side you are assuming that this is diagonal? Right, so this is what we know from something we just erased. Right, so we know that the way this to what a defect line acts on the operators will be such that it cannot change its scaling dimension in particular it maps a varisoral multiplet to a varisoral multiplet. In the IC model there is no degeneracy in the hubris space without twist. So for a given varisoral representation there is only one operator so there is no other possibility than having just an overall number. Alright. Okay. And we can learn a bit more from just this very simple table. From this how it acts on the operators in the untwisted hubris space you can already infer the fusion rule. In particular the fusion rule obviously of eta squares identity that's what we expect for z2 symmetry. Moreover when you fuse the duality defect with eta in either way you will recover the duality defect. That is because this entry is 0. Okay. Furthermore if you square the duality defect you get 1 plus eta. So these rules are uniquely determined just by postulating the most general fusion rule with non-negative fusion coefficients and consistency with this table. This defines what's known as the icing fusion rule which is not so surprising. Okay. And as I said before once you have a set of duality defect lines and you have specified the fusion rules there's the consistency condition just coming from locality of the theory says that they should furnish to this whole fusion category structure. Meaning that it should have consistency solutions to that symbols. So there's a consequence of locality essentially the isotope invariance. Okay. In other words the f symbols associated with junctions formed by this topological defect lines would have to satisfy the Pentagon equation. And that has been solved. Okay. So here I'm listing the solutions. This was actually this problem was solved in generality that generalized this particular fusion rule in the case when the Z2 is replaced by a Zn by the work of Tambara and Yamagami Okay. So in general we call it Tambara and Yamagami fusion category associated with the general abelian symmetry. They generalized the case with just eta. Okay. But here I'm listing the full data of the fusion category in the case of this particular icing fusion rule. Okay. So in particular your simple objects are these three topological defect lines that satisfy this fusion rule which I showed before. And there's only one non-trivial junction where none of the three lines involve the identity. So whenever three of the external legs involve the identity that correspond to a trivial junction as we said before that comes from just bringing the identity operator in the bulk to the defect line. This is only non-trivial junction in this game. Okay. And then non-trivial junctions enter into various non-trivial F symbol associated with this fusion category which I've listed here. So the important thing is about this sign. Okay. Which is similar to what appears in the complex symmetry case. Okay. The fusion symbols, the F symbols are faces in general. But the characteristic of this non-invertible symmetry case is that F symbols are in general not just faces. It will involve these matrix elements. Okay. Which are not not faces. Okay. And we'll use there's a question. Sorry. But we know that Tambara Yamagami's fusion categories are not uniquely fixed by diffusion rules. Very good. Very good. So here I jumped over one small subtlety is that if you just use this particular fusion rule for the Z2 case, there are two solutions up to gauge freedom to the F symbols. Here what I'm writing down is the F symbol that will be relevant for the Ising CFT. Okay. And there is some physical meaning to the choice of... Sorry. There is some physical meaning for the choice of one of the two solutions. Ah, it's because the CFT actually realized just one of them. I think CFT realized one of them. The other is realized by the SU2 WZW model at the level 2. And there is a way to see that. That's right. But I'm afraid I don't have time to explain that detail here. Okay. But thanks for the question. Okay. And just from how this solution looked like, okay, and it tells you how the duality defect acts onto the hubris space, we have these diagrams, okay, which we have been drawing before. So acting on identity, which is equivalent to inserting nothing, this gives you square root of 2. We're just copying down the table over there into the diagram we have been drawing. This means that whenever you have this duality defect encircling nothing, you can just shrink it and the cost is to include additional factor of square root of 2. If you have instead the epsilon operator inserted in the interior of this loop generated by the duality defect, you have the cost of introducing extra minus sign. Okay. And similarly, instead if you have the sigma operator inserted inside the loop, you will just have zero. Okay. It's completely annihilated. And this is another signature of this this duality defect line being non-invertible. So it annihilates certain operators. Okay. But there will be in a sense where this operator will be recovered. Okay. But it will recover in a way that will be consistent with this duality relation. And this is what we will see next. Okay. So instead of as we said before a feature of this non-invertible duality defect is that it doesn't just give between the, you know, the hubr space without twist to itself. Okay. It also gives maps between defect hubr space twisted by a given line to defect hubr space twisted by a different line. Okay. In particular, gives maps from defect hubr space without twist to defect hubr space with twist. Okay. And such linear operations are represented by again this kind of diagram. So this is a diagram that we call lasso diagram. Okay. When we discuss the generality, but with potentially with a non-trivial junction so I'll be using the red to represent eta and using the, okay. This is not very red. The red line to represent this eta symmetry defect and the white circle to represent the duality defect. Oh, I should also have said that here the defects that's involved in the icing fusion category they're all self-dual. So I do not need to keep track of the arrow. Okay. Just in case you worry about this. Question. Again about the f-symbols it's not true that you always have the solution with trivial f-symbols. Come again? Is it not true that you always have a solution with trivial f-symbols? No. No, okay. So as I said before f-symbol what f-symbol is doing is giving you the change of basis between different representations of the junctions associated with four external legs. Because of the change of basis it will always have to be invertible so it cannot be trivial. But I... Yeah, I don't mean zero. I mean that essentially you have no anomaly. Yeah, so well, trivial so trivial could be zero, trivial could be one, identity matrix. Yeah, one, let's say one. So that would not be consistent because you have multiple fusion channels. That would be okay for that's the case for non-anonymous group like symmetry. But because you have multiple channels you cannot put one here and here to be consistent. So for some fusion rules you necessarily have anomalies. Come again? What you're saying is that for given the fusion rules there are fusion rules that imply automatically that you have tough anomalies. I'm not sure I get your question. Maybe it's better to ask that again in the discussion. Okay, good. So what I was saying here is just that there's additional actions linear actions of the symbolic defects, the duality defect represented by these white circles acting operators and sending operators not to the hyperspace without twist but to a hyperspace twist. In this particular case twisted specifically by the eta line. So once again because every junction is topological so you can shrink this diagram. So this produced for you an operator that's attached to this eta line. And this is precisely the operator mu that's relevant here. How do we see that? So once again because the theory is simple enough you can actually specify the entire hyperspace. So every state that lives in here is a state in the hyperspace twisted by eta. So similarly to the IC model without twist here the hyperspace is completely fixed. It consists of the operator that corresponds to the remnant of the fermion before you bolonize the size of dimension one half on either side. And just one other operator of dimension one sixteen one sixteen. And for the same reason as before this topological operation of shrinking this diagram encircling this operator does not change its scaling dimension and because there is a single operator of dimension one sixteen one sixteen it has to be proportional to this guy and the coefficient can be fixed by asking this operator to have normalized two point functions. And what you find is that the coefficient with that particular choice of normalization is square root of two. And similarly just to be consistent with such a duality operation another thing that you may wonder if it is this kind of diagram, this again a lasso diagram but encircling instead the mu operator so like this. So for this to be consistent this better give you something similar to this. So it should be zero. Why is this zero? This is why I draw this over here. This minus sign is crucial. So look at this region apply that as move because this operator scalar you can freely rotate this red spoke all the way around and because in the process of moving all the way around it crosses this point once at which you use this operation it give up a minus sign. It will say that would mean that this diagram is equivalent to minus times itself that implies this diagram vanishes. And furthermore a similar exercise what we did over here tell you that if you have this diagram again shrink it so what this diagram means is that it's a map induced by the top of defect line on the hubris space twisted by eta into the hubris space without twist. So because when you shrink it there's no dangling line defect. So this better produce the state in the hubris space without twist and what you find is that indeed you recover this operator the spring operator. So what we see from this diagrams is the following if we just focus on how the duality defect acts restricted to the hubris space without twist it looks invertible it looks non-invertible because the spring operator. But somehow this non-invertibility can be recovered once you realize there are other non-trivial maps that max one defect hubris space to another in particular the hubris space without twist to the hubris space with twist. So there's a sense in this particular non-invertible symmetry that once you include once you take into account how the duality defect acts on all the defect hubris spaces it is really invertible and not losing information. And that is why it makes sense to talk about it as a you know as a remnant of the duality of the duality. So there's another picture that is helpful to represent the way the duality defect acts on local operators is to instead imagine that you have the duality defect over here and some local operator inserted around it. This diagram we draw when we discuss the topological feature of these defect lines. And as we know as we have drawn this many times you can deform the duality defect line as a consequence of topology invariance these diagrams produce the same observable when inserting correlation functions. But because because of this diagram this implies and you apply the fusion rule over here so the fusion rule here is important so let me So you apply the fusion rule for these two corners sorry you apply the f-mool for that two corners using that rule what you find is 1 over the square root of 2 times this diagram plus attached to the eta line and then further attached to the duality line. And then you use what we have already got over here so there's one diagram I didn't draw which is the case when you have this line attached to the epsilon I'll leave that exercise to convince yourself that this is 0 from a similar argument and this implies combining with this fact that this is equivalent to flipping the sign of epsilon so the cost to move across this energy operator to move the duality defect across the epsilon operator to flip the sign you can do a similar analysis and this is okay and this explains this selection rule so we have achieved in explaining the selection rule using this symmetry and the way you do that is to put a duality defect line at one end and move it all the way you pick out all the faces and you get a contradiction if this is down to 0 and that's how you argue the selection rule a similar exercise in view of time let me again leave as homework by doing the same procedure again using the f-symbols we wrote over here what you will find is precisely the expected relation that says that when you move the duality defect across the sigma operator which is charged on the Z2 you will get instead all the operator which lives in the hyperspace twisted by the Z2 line this explains this duality mapping between sigma and i mu and you can do it also inversely move the mu across and you bring the sigma so combining this picture this explains the Kramer's winding duality at a self-dual point with this non-trivial transformation rule that sends this basic operator in Icing CFT which naturally are local operators in the Icing CFT after the dual operator so the mu operator is not a good operator it's not a good local operator before you do the dual operator because it is attached to a non-local line but after you do the dual operator this becomes a local operator and this operation explains in what sense this is the precise symmetry so let's not discuss a kind of explicit example of how non-invertible symmetry works in motion let's now given this example let's not deduce some consequences so the main point here is that this symmetry just like a Z2 symmetry we discussed it in the very simple CFT like Icing CFT but the structure of the symmetry it shows up in a very interesting system and it can be used to deduce non-trivial consequences for example on RG flows so this is what we will discuss now the dynamical consequence for RG flows so what is the general picture for RG flow so you have some we focus on the case when you have some quantum field theory which we will call TUV which is the UV description for some potential RG flow you can trigger some RG flow for example by some relevant deformation or by gauging this will end up with a non-trivial IR face in general if the theory is strongly coupled and there are several possibilities for what the IR face may look like so possibility number one is trivially gapped meaning that there is a unique vacuum that is gapped only massive excitations about the vacuum possibility number two is vacuum degeneracy or in other words a multiple discrete because it is gapped this corresponds to the case for example when you have some discrete symmetry that is spontaneously broken in two dimension context RG is gapless in other words described by a non-trivial CFT so these are the three general possibilities for RG flow so let's focus on the two dimension case from now on and for example consider RG flow trigger by a CFT in two dimension perturbed by a relevant operator that's the meaning of being relevant and also to preserve the Lorentz symmetry okay so symmetry that's preserved by RG flow generally leads to constraints on what the IR face diagram may look like and here we want to consider this non-invertebrate symmetry so we first need to introduce in what sense is non-invertebrate symmetry preserved by RG flow of this form a non-invertebrate symmetry generated by the Vodger defect line called L is conserved and L is transparent to the deformation operator we can move it across the operator without introducing any face factor that's the meaning of transparent just like how stress sensor is always transparent to the Vodger defect this ensures the Vodger defect remains topological under this deformation okay and then there's a very simple theorem once we have defined what it means for a symmetry generated by a non-invertebrate Vodger defect to be preserved on RG there's a very simple theorem we can state the theorem the very simple theorem states the following if a quantum field theory admits a Vodger defect line L such that its quantum dimension is there on the cylinder is not integer positive integer but preserved then IR theory IR phase cannot be trivially gapped okay so the only possibility will be a non-trivial TQFT which is described in multiple vacuums or a gapless CFT and this is a very simple theorem there are refined statements that constrain how precisely the symmetry is spontaneously broken let me just quickly give you the argument for this theorem which is very simple to prove so the proof proceeds by contradiction essentially the same steps that you will go through in solving that homework problem will lead to the proof of this statement and the proof actually only uses something even simpler okay so assume because we are proving my contradiction we assume the theory close to a TQFT or a gapped phase with one vacuum there is only one operator of dimension 0, 0 in this TQFT okay and while it preserves the topology of defect L so we can simply consider this again the corresponding function twisted by the topology of defect in the time direction in temporal direction and which is related by a modular S transformation to the configuration with the topology of defect L inserted in the twisting the spatial direction okay but in this TQFT we can compute the differential function very simply on the left hand side we have a trace of this line weighted by twisted by this insertion because of the TQFT the Hamiltonian trivial you literally just get a trace there is no non-trivial Q dependence and because there is only one state in the super space there is only one state in the D by R this just give you the number and that's nothing but the depth of this line operator but using the right hand side on the right hand side instead you have a trace over the defect that's punctured through by this defect line and there is nothing inserted because there is no more twist and it just corresponds to one and just because you have a super space this has to be a non-negative integer so we arrive at the contradiction to the assumption in the theorem thus we have proved the theorem so if the number of states in the twisted sector is bigger than one would you still call it a trivially gapped phase so it could be that in the twisted sector there are more than one state as long as there is only a trivial vacuum in a single state in the untreated sector I would still call it a trivial gapped phase according to this definition that's the circumstance where the theorem applies but it can be refined does this also apply to higher dimensions then the very same argument applies in higher dimension and indeed that is how the higher dimensional version of this dynamical constraint is taking this form having derived in higher dimension you have more choices for what you call a torus depending on the choice of the space d-1 dimensional slice so now use the other side the other table that I wrote before to apply this dynamical constraint to another less trivial example what is the simplest simplest next non-trivial CFT is the tri-curializing CFT so let me just write over here for example the application of this theorem so here I've written down the operator content of the tri-curializing CFT this is the the next minimum level of 7 over 16 it has 6 primaries and because the theory is so simple you can actually solve the entire you can actually identify all the topology factor line is theory and there are 6 of them the detail is really important there are two distinguished sub-categories this is one called the Fibonacci category it is also called the Lyon category it is generated by a single topology defect line W and it has this very interesting fusion rule it squirts to not quite itself but with an additional identity line also appears and this line is also same as its dual okay and the Ising category appears also in this tri-curializing CFT and this is correlated with the fact that similar to the Ising CFT the tri-curializing CFT also have the Cremers-Wanier type duality its self-stool under Z2 gauging okay but what's important about this particular sub-categories generated by the topology factor line W and N is that this W and N they have the common feature that their VEVs are non-integral in particular the quantum dimension of this duality defect line is square root of 2 on the other hand the quantum dimension associated with this W line because they have to solve the same polynomial equation given by the fusion rule you can find out is a golden ratio okay and so if you have any RG flow that preserve the symmetries and from the general theorem we just discussed then it's guaranteed to land on a non-trivial IRF space so we just have to go through the table to find out which operators if there are preserving these lines okay and as we said preserving means that you have this line over here and you can move it across without introducing any factor okay so this is the equivalent to to that this diagram is equal to the VEV of the topology defect line multiplying this operator so in term of this diagram this will be the equivalent relation so we just have to go through this table and look for operators phi when L is either the W line or the duality line that this equation is satisfied and it's easy to spot that for the duality line the relevant operator so you're looking for h because h is equal to h bar you're looking for operators which are smaller than 1 the relevant operator will be this one so this is the operator that preserve the duality preserve the duality defect and if you look for operator that preserve this W line then the only option is the sigma prime operator that is relevant so what this means is that the tracheonol we have the immediate prediction that the tracheonol ising deformed by sigma prime okay and deformed by epsilon prime okay in this case we preserve the Fibonacci category okay in this case we preserve the ising category okay so they cannot be trivially gapped so if gapped there has to be non-trivial vacuum degeneracy okay and you can actually derive a stronger result which I did not explain that you can actually show the minimal number of degeneracy is 2 and here the minimal degeneracy from a similar argument is equal to 3 okay but there's no other symmetry for this deformation there's no other symmetry if we don't know about this non-inverbal symmetry having some degeneracy typically want to interpret as something being spontaneously broken that leads to this degeneracy just like the d2 degeneracy in the ising case in the low temperature phase by here there's no if you don't have this non-inverbal symmetry there's no other symmetry that's responsible for these degeneracies okay instead these degeneracies are enforced by non-inverdible symmetries in particular this is a prediction and one can go ahead and check if this is actually the case given the tracheo-dual ising model okay this rg flow is integrable and you can check this statement so if it's gapped indeed which corresponds to one particular sign of deformation indeed you have 3d degenerative vacuum this side is not integrable but you can check numerically and you will find out that indeed has a 2d degenerative vacuum it turns out that for this deformation there's another possibility as I said the general theorem would solve the possibility of having a single vacuum a single gapped vacuum there's another possibility of being a non-trivial CFD saturating this symmetry and that is nothing but the ising CFD with the opposite sign of this deformation okay so this is just a very simple application of the general theorem we derived and this is a similar theorem can be applied in higher dimensions to deduce similar statements I think I'm running out of time so let me just summarize these lectures by listing some other aspects which I did not discuss but you are free to ask me afterwards so let me just summarize so in these lectures I've spent a lot on the basics of diagonal symmetries and hopefully through these lectures I'll convince you that they are as good as the usual symmetries in particular they give rise to selection rules and leads to constraints on RG flows IR phase diagram of some UV description what I didn't discuss is that in these lectures I focus on the bosonic theories and to watch with the effects in the bosonic theories there are extensions to the fermionic case just like the extension of bosonic group-like symmetry the fermionic case including the fermion parity and that story can be tied together with what I discussed using the bosonization duality in two-dimension and higher and as I just discussed this non-inverbal symmetry similarly to usual symmetries can undergo symmetry breakdown and it can be used to explain the generacy of the vacuum and something I also didn't discuss is that this non-inverbal symmetry if they are not anomalous they can be gauged so there is a precise way to gauge some sense non-anomalous non-inverbal symmetries and this is the way to produce other CFTs starting from CFTs with non-inverbal symmetries and the list goes on essentially the list contains all the nice things we like about usual symmetries and it's a more general it's kind of a more general richer framework and lastly let me just try to connect to the other lecturers the very nice lecturers by the other lecturers at the school just posing some questions for you to think about the obvious question in relation to Laura's lectures the non-inverbal symmetries in the context of celestial CFTs and the potential applications of these non-inverbal symmetries on constraining the asymmetrics of massless particles and in relation to Matias and Kevin's lectures a puzzle, current puzzle is to understand these non-inverbal symmetries in the context of ABS CFTs and in relation to the notion of all global symmetry in quantum gravity so in particular in the context of this non-inverbal symmetry in the context of the ABS3 tensionless strength which is a very nice explicit pre-glound to discuss the non-inverbal symmetry in the quantum gravity in that case describe some explicit string theory which we will hear more from Matias and similarly in higher dimensions perhaps in the twisted horary the string theory in the bulk is much more complicated perhaps one can make some progress using the twisted horary so that is the end of my lectures but hopefully this will not be the end of your journey in the world of non-inverbal symmetries so sorry for going over time I'll take questions