 we'll continue telling us about non-invertible symmetries. Recording in progress. OK, well back everyone. So let's continue our journey into the realm of non-invertible symmetries, in particular, in two-dimension couple of filters. So here I summarize some of the things that we talked about in the previous lecture, and also some of the things that I'll briefly explain now. In particular, in the previous lecture, we introduced this notion, a mathematical framework known as fusion category, which is a natural generalization of group symmetries, taking into account their anomalies into some more general object, and which we use later to describe symmetries in conformal field theory that are associated with non-invertible topological defects. So as we said before, objects in this fusion category correspond to topological defect lines. And in particular, in the CFT, there are associated with this twisted defect keyword space. This is the hubris space on the cylinder. When you quantize, this is the time. And you quantize on the circle, which is punctured through by the defect line. And then there's a notion of direct sum in the fusion category. And that boils down to the fact that on the CFT side, a hubris space associated with the direct sum of lines splits into the direct sum of the hubris space associated with the individual constituent lines. And then there's a tesser product operation in the fusion category side. And that allows you to decompose a product of lines into a direct sum. And that boils down to this picturesque fusion product on the CFT side, which is a special case of OPE between line operators in one plus one dimensions. And then there's this notion of a dual object, which Paul pointed out last time, and I forgot to introduce. And that, on the CFT side, corresponds to taking the CPT conjugate of a given line. So in any observable involving this line defect, you are free to replace this diagram with this particular orientation by our bar, but in the opposite orientation. Question? Here? For the CFT. The CFT just means that it's a composite line. So meaning that if you compute a correction function where we insert a direct sum of the line, that correction function becomes a sum of the correction functions in which, in each summand, you have the individual line inserted. And then there's a notion of the morphism in the fusion category. And on the CFT side, they simply correspond to topological junctions. These individual junction vectors are denoted by v. And they live in some juncture vector space. And that has a natural interpretation in terms of the hubris space over here, a generalized de facto hubris space, a simple generalization of the both, where you have more than one lines puncturing through the circle on which you quantize the theory. So you have l1 bar here, l2 bar here, and l3 here. And this picture is related to the picture over here just by doing radial quantization on this circle. Doing radial quantization, you map this point-like operator to a state on this circle punctured through by these defect lines. And the requirement of this object being topological boils down to the condition that you want to require the corresponding states in the subspace to have h and h bar equal to 0. Just like I did in this operator, so it can move it around, costing no energy. Question? In this radial quantization picture, no, the choice of the morphism, how does it reflect on the cylinder? So it boils down to a particular vector in this junction vector space. So the junction vector space is a subspace of the superspace restricted to the eigenvalues of l0, l0, bar equal to 0. In general, that could be a dimension bigger than 1 vector space. In general, it's a vector space. So this v, if it exists, is a vector in that space. OK, so you choose a basis. That's right. So if that space is empty, that tells you that this topological junction does not exist. Thanks. Yes. Are you allowing also for non-trivial junction on the trivial line? For example, if you infuse the trivial line with itself, and this can be. When you say trivial line, you mean identity line? Yeah. OK. Are you allowing for non-trivial junction? Let's say local operators in bulk. So here, and I will get to that. But just to answer your question quickly, so I think you are imagining a case when, for example, one of the three external lengths is identity. Also all the three. If all three are identity, then the implicit assumption I'm making in the CFT is that there's a unique vacuum. And that will answer for you that there's a unique, just one-dimensional junction vector space there, and they are proportional to the identity operator in the bulk. And more generally, if you take one of these lines to be trivial from the argument I'll give shortly, if L1, L2 are simple, then the corresponding junction vector is again coming from identity operator taken to the line. Identity operator in the bulk taken to the line. OK. Thank you. Any other questions? OK. So these are the basic defining data. And there's one more ingredient. These are this F symbol. I will not redraw that diagram again. But essentially, like a matrix that keeps track of the change of basis in the junction vector space associated with four external lengths, L1, L2, L3, L4. There are two different ways to represent this junction vector space by factorizing a fourfold junction into a pair of threefold junctions. And that change of basis matrix is captured by this F symbol. And that symbol is subject to this Pentagon relations. And that physically just corresponds to a consensus on this change of basis operation. OK. So let's use this kind of the general dictionary to deduce some simple consequences, in particular relating to the fusion rule, that specific features of this fusion rule. So so far, we have not been assuming any special property of this line. For example, this line can be decomposable. But it turns out that if this line are taken to be indeconsible or simple, there's a nice property of the fusion coefficient, which can be interpreted from the point of view of this topology of junctions. And this will come to that next. So let's first introduce the notion of simple topology of defect line. OK. A simple topology of defect line is defined by having a one-dimensional junction vector space between itself and its conjugate. So it's easy to see that this condition implies that this defect line is decomposable. OK. Let's see why that's the case. First of all, for any tdl, for any defect line l, it's easy to see that the dimension of the junction vector space between itself and its dual has to be bigger or equal to 1. Just because the junction will be something like this, OK? And you can include identity here. But you can forget about it as well, because it's identity. And you can always bring the identity operator in the bulk to the line. And this give rise to the non-trivial junction vector, associated with l and l bar. OK. So if I draw l bar, OK? And identity. So if l is decomposable equal to l1 plus l2, OK? This implies that the dimension of the junction vector space must be bigger than 2. This is because this factorization, I mean, this kind of distributive property, associated with the hubris space, defect hubris space will translate to a distributive property for the junction vector space. And there's at least two contributions coming from the fact that for each l1 and l2, there's a non-trivial topological vertex with identity, OK? So just from this, you conclude from this definition that simple tdl must be decomposable, as we have expected. It turns out, strictly speaking, being decomposable doesn't necessarily imply simple in this definition. But this will actually lead to a generalization of the fusion category I'm talking about here. So I'm happy to talk to you afterwards about that. But let's, for the moment, focus on the case where actually in decomposable also implies this condition, OK? Now, once we have introduced the notion of simple tdl, we can talk about the fusion product of these simple objects. So writing the similar relation over here, over there. Now, in general, there's some coefficient that appears in the decomposition of the tensor product of the two simple tdls into all the simple tdls that you have in your theory, that generate all the non-inverbal symmetries. I claim this coefficient is equivalent, it's equal to the dimension of the junction vector space. So we can write it explicitly, associated with these three lines, i, j, k, defined over here. So how do we prove this? So let's focus on the simple case where l, i, and l, j are taken to be l and it's conjugate. And l is a simple line. And we want to show that this is equal to 1 plus other non-identity simple lines up to some degeneracy. We want to show. So how do we show this? Let's consider the torus function. So I imagine the square represents the torus in the coordinates which have two pi p resistive, both in this direction and in this direction. Think about this as time and this as space. So the opposite side is identified. And we insert this topology of d5 lines as follows. And because of the definition of the dual coming from the conjugation, I can replace this also by the original outline but re-invert its orientation. The torus has a modulus tau, as usual, that controls the shape of the torus. Now there's two different ways to compute this observable, the torus function in the presence of these two lines. The first way is to take directly the limit where tau goes to the infinity, which means the time cycle becomes infinitely long. In that case, the torus function projects what essentially everything propagating in this time direction gets suppressed and the only dominant contribution will come in from the vacuum, states of dimension 0. So h equal to h bar 0 states dominate. Quantizing on the circle. And this picks out precisely the contribution from the junction vector space, multiplied by this divergent vector that depends on the center charge. And q is equal to e to the 2 pi tau, as usual. So it's a different way to represent the modulus of the torus. So we see this quantity shows up, which is what we want to relate to the fission coefficient here and here in the special case. Sorry? Sorry. Sorry. You're all right. Good. Good. Thanks for pointing that out. Tau is going to i infinity. Very good. We can proceed in a different way. Because we're on the torus and we have this parallel defect lines, we can apply the fission of these lines, the fission that's over here. And we use this postulated expression. In general, you have multiple terms. But suppose this term is n, n times 1 with some coefficient. You have n times the contribution on the same torus function with no line inserted, a corresponding identity line. And then you have contributions coming from all the rest of this right hand side, which all involve non-trivial defect lines inserted in this time direction. Now you take the limit that tau goes to i infinity again, just like before. This parting function dominates over this one. Why? Because we have the Tadpole condition that tells you only identity operator can end on a dimension 0 operator. So this is computing the thermal function in the hubris space punctured by a single Tdl. There's a non-trivial ground state energy that is above minus C over 24. And it's in this case, the ground state energy is saturated at minus C over 24. So if you take the tau goes to i infinity limit, this guy dominates and you get precisely this expression. And you equate these two. And you conclude that n is equal to 1. Now, just to generalize, we can already use this result. So what we want to show is over here. And we can write this equation in the form similar to this by fusing both the left hand side and the right hand side with orientation reversal of k, of lk. We consider the fusion of three objects in a similar configuration. And this will contain a piece like this, precisely this coefficient, plus sub-leading, sorry, so this is times the identity line plus non-trivial lines. And once again, you can go through the same argument. So go to the torus. And now you have three topology defect lines inserted along the time cycle. And in the limit where tau goes to i infinity, playing the same game, you find the dimension of the junction vector space goes up, multiplying the divergences, the exponential divergences due to the cascading energy. And doing the same comparison, this tells you that indeed, this fusion coefficient is the dimension, the complex dimension, of the junction vector space. So from time to time, just to save some writing, I will drop the l when I label the defect lines, just to save some time. So this is a very simple formula. But the implication is actually quite deep, depending on how you think about it. But there's a perspective that was emphasized in the recent paper from Troy Fordova, Seng Lan, and Xiao. I think it just appeared some weeks ago. They give a very nice interpretation for this formula and this generalization to higher dimension, which I want to say a few things about here. So you can think about this as the planning function, or precisely, the planning function as one of a topological quantum mechanics. In other words, it's just one-dimensional TQFT. So one-dimensional TQFT is nothing but a bunch of ground states with energy 0. And the fusion coefficient is the planning function of this topological quantum mechanics over the manifold, which is precisely the manifold on which you wrap the line when deriving the fusion rule. This generalizes, so this is in d equal to 2, for d higher than 2. If you again consider this kind of non-inverteral symmetries, say, of cold dimension 1, this fusion coefficient will be replaced by a planning function of d minus 1 dimensional TQFT. As you can imagine, TQFTs in higher dimensions are much more interesting in the one dimension. And that will lead to interesting structures for different properties for this kind of fusion product. In particular, the planning function, this planning function is no longer required to be integral on general three-dimensional manifold. So that leads to various exotic but now understood properties of the fusion rule in higher dimension. Yes? Is this like a proven statement or is it based on? Yeah. This is, I think, as far as it's not a proven statement. It's a statement. It's a kind of a conjecture. And it holds for all the examples that we know at the moment. It will be good to understand to what kind of generality this is the statement. Because in higher dimensions, I wouldn't even know to define a simple object very well. So a lot of things. That's right, that's right, that's right, yes. So this nice junction vector space, I'm talking about in higher dimensions, the junctions. I mean, if you talk about generic junction, it would be point like you can do. But if there is a non-generic configuration of this defects, the junction is typically also not zero-dimension. So the junction vector space will involve some topological lines and so on. So it's more complicated. It's a very nice idea. Yes, it's a very nice idea. And I think I agree with you that it should be more developed, should be developed further. Sorry, can I ask, is there some intuition why these TQFT lives in one dimension? I mean, I would expect that the TQFT lives on the junction, so zero-dimensional. The TQFT lives on the, so because on this fusion picture, the two manifolds are wrapping one, sorry, the two defect lines are wrapping homologous manifolds. And you can take it to be, I mean, essentially identified as far as the boundary concern. And the TQFT lives on that one-dimensional manifold. It is true that when you look at the junction, then the TQFT, only operators in the TQFT shows up in the junction because there's no way for it to propagate anymore. Yeah, because this is related to a question I did yesterday that in higher dimension, I would expect that if you look at the picture with the junction, the 3D effects can, in principle, live on different manifolds. That's right, that's right. So where does the TQFT lives? Yes, I mean, it can come out, for example. Yes, right. And so where does the TQFT lives on? Which of the three manifolds? Yeah, so the TQFT here, so this is a very restricted statement. It's a statement about doing this particular fusion parallel manifolds that are homologous. And the TQFT lives on that manifold. Okay, okay, thank you. So I agree with you that there should be generalization of this statement. We have more complicated configuration of this two-ball defect lines, but it's still to be developed. That's a very interesting question, but that's beyond what I'm talking about here. Let me just point out the similarity to this. I think some of you are familiar about similar kind of fusion product that appears in supersymmetric gauge theories. For example, study Wilson lines in Boyden-Ruth 2 theories, and you encounter similar fusion rule for the BPS lines, okay? In that case, there are also a similar interpretation of these coefficients, okay? In that case, you are talking about the fusion product supersymmetric lines. As I mentioned, generically, if you're just taking the fusion product of line operators, the fusion product singular is not well defined, but for supersymmetric lines, there's a way to regularize the divergence supersymmetrically, and that gives a well-defined fusion product, and in that case, the coefficient also has a very nice interpretation. It has interpretation as the supersymmetric index of quantum mechanics, okay? It's not a topical quantum mechanics anymore, but the pattern function is replaced by the supersymmetric index, okay? So this is just a comment for people who studies these line operators, okay? This NIJK correspond to Suzy index of certain quantum mechanics, okay? So a useful reference is the paper from Gautam Morineski. So the next thing I want to talk about is it's a symmetry action of this topical defect lines potentially twisted, or in other words, defect humorous space, okay? Because this is the one place that really distinguish the non-invertebrate symmetries from invertible symmetries, okay? So what is the kind of action we are looking for? So as I already said before, the defect humorous space comes from quantizing on the cylinder, okay? We're quantizing on the spatial cycle of the cylinder that's punctured through by a defect line. Let me call it L1, okay? And say there's some state inside this hubris space which I have been calling the defect humorous space labeled by L1. I want to define, without the defect line, as I already said before, the action of the topological defects on the hubris space on S1, they're correspond to encircling the corresponding operator on the plane, right? So as before, we have this picture, this enclosing picture after you shrink the defect line because it's topological, you can shrink it and the local operator you produce after shrinking enclose how this defect act on local operators. These operators are in one-to-one correspondence with the states. So this is equivalent to some operation on the state in the hubris space with no twist, okay? Okay? Now we want to generalize this notion to the case in the presence of a twist in a spatial direction by this non-verbal symmetry. So the kind of intuitive picture you want to draw is something like you want to include another topological line, okay? So L2 in the along the spatial direction, okay? Wrapping this cycle, okay? But for this to define a symmetry, as we know symmetry is equivalent to topological property of this various configurations in its lectures, you want to make sure that this junction formed by this line is to be topological. We already know what these junctions are, okay? So first of all, let me write down the representation of the same operation on the plane using the usual mapping between the cylinder and the plane and that corresponds to this kind of diagrams. So this is the operator corresponding to this operator in the defect hubris space and then drawing the same diagram just mapped to the plane, okay? So to specify this symmetry action, we need to specify the choice of junction. As we said before, a convenient way to specify a junction involving multiple external legs more than three is to resolve this junction into three fold junctions, okay, for which we treat basic building blocks for higher fold junctions. So there are two different ways to resolve as kind of the same case as for how the F moves arise, okay? And give rise to these two different configurations, okay? And in general, here could be some intermediate topological line that appears in the fusion product of L1, L2. And similarly, another diagram you can draw which corresponds to different way to resolve this four fold junction, essentially you're resolving in two different ways like this way and this way. And in general, this could be some other topological line if the fusion product is not commutative. Well, L4 is inside, is a potential simple line appear in the fusion product of L1, L2, okay? If L1, L2 have a commutative fusion product, this can be taken to be the same set but in general, it could be different. And we call these diagrams lasso diagrams, okay? Just because how they look like. And these diagrams define different actions of topological defect lines on the twisted hubris space in this case twisted by L1, okay? They're different actions. They correspond to in this picture, different choice of the junction vector, okay? And even more generally, you can even have a diagram where the internal line that is immediately attached to a local operator that defines the twisted sector can be different from the outgoing line to the end, okay? By considering more general junctions over here. For example, a stain to the case when this internal line is still labeled by L1, okay? This outgoing line could be some other line L4, not different from this L4, okay? And what this diagram means is that after you shrink, okay, I do not need to draw the other arrow. I mean the lines goes to infinity. After you shrink the topological defect network, okay? That's encircling this operator with these various junctions. What this gives you is an operator attached just to the L4 line, okay? Some other operator, let me call it, coming from all this transformation, let me call it fight crime, okay? The location doesn't change. But it becomes starting from operator that's attached to the topological defined line labeled by L, you end up with operator attached to the topological defined line L4. And what this means is that this defines operation, a linear map from the defect cubar space twisted by L1, okay? Which correspond to these operators to the defect cubar space twisted by L4, okay? And the point of non-invertibility is that these maps, each individual maps are in general not invertible. Let me call this map F, okay? And I should also say that in the invertible case, you will not have this non-invertible maps if L4 is different from L1, okay? So in the non-invertible case, the first non-invertible thing is that when you restrict the case when L1, L4 are the same, this map is in general not invertible, okay? And furthermore, there exists non-invertible maps between the cubar space twisted by L1 and L4, even when they are not the same, okay? So that's the two features that distinguish the action of symmetries on the twisted cubar space in the case of non-invertible symmetries versus that of the group-like symmetries. Do you get any consistency conditions on these maps because of the pentagon, no? Yes, so yes, these maps are not independent, okay? These maps are closely related. For example, these two are related in a way by the F move, okay? So what I'm trying to say here is that if you just want to have a most general action of this kind of a TDL in circling an operator attached to another TDL, the most general action is specified by the choice of junction vector over here. You can represent this junction vector purely in these spaces by including all the L3s over here and all the choice of the three-fold junctions over here. You can also do the same thing over here, okay? And these two different bases are related by that move. No, I was meaning, okay, maybe if you take like two operators inside, no? Because you can just draw the diagram of the pentagon then. Here you do the diagram of the symbol, no? Sorry, maybe I'm not completely sure what you're talking about. Here, the F move just concerns us. No, I know, I know, I understand. No, I will, maybe in the discussion, sorry. Okay, sure, sure, yeah, yeah. Okay. And I don't have time to go into that detail, but the action of symmetry on the twisted sector is in general a very important object, okay? For example, in the case of group-like symmetries, the anomaly associated with asymmetry is equally encoded. So it is encoded by the natural phase up here in F move. But it also equally encoded in the symmetry action in this twisted hubris spaces, okay? In that case, L1, L4 are the same. And in particular, you observe this phenomenon that the symmetry, the group-like symmetry, acting in the twisted sector can develop projective representations, okay? Even though acting on the untwisted hubris space, it's faithful, okay, the linear representation, okay? And there's some generalized notion for this non-inverteral symmetry, but I will not get into that general detail now, okay? Instead, let's move to another physical object. Yes. Just to make sure I'm understanding, if the symmetry was invertible, Yes. this junction could still be non-trivial. I mean, L1 could be different from L4. And this means that this symmetry is giving an automorphism of the set of lines in the case of invertible. I mean, you're picking L2 and it's acting on the... So if all the lines are invertible, Yes. then for this junction to be topological, the L4 has to be same as L1. Just from the junction, I mean, as we said before, the topological junctions are only if this condition are satisfied. So if this is, I mean, if these are all invertible, L3 has to be the product of L1, L2. And because this is already L2, this has to be L1. Of course, I mean, you can choose to insert something that's not topological over here. But then you cannot do the procedure to shrink the diagram down to a local operator, like what I'm doing here. So you will have some two-point function and there's an attribute OPE, which you can still study, but it will not give rise to this nice linear map between local operator and local operator. In that case, the OPE coefficient in particular of this non-trivial local operators would enter into this map, okay? Okay. The next important physical quantity that, you know, encode information about fusion category in the CFT is the notion of the VEV of a topological defect line, okay? Which is also known as the quantum dimension of L in the fusion category. This is a very simple notion, okay? So the VEV is defined as the expectation value of the topological defect line on the cylinder, okay? So you have a cylinder and you have this topological defect line inserted in this orientation, okay? So that's the definition. They're a nice property of this topological defect line, okay? Infinite long cylinder. You can consider infinite long cylinder and on which you insert, you know, a composite, Li and Lj, so call this Li and insert another Lj. The fact that you have this fusion rule, so for now I'm always taking the lines to be simple unless I say otherwise, okay? Because on the infinite long cylinder, everything is projected down to the vacuum. The state of propagating this direction is projected down to the vacuum. It means that the VEV will satisfy the same polynomial equation. So you can think about this as some kind of cluster decomposition on the cylinder for these lines, okay? That's right, that's right, yes, yes. But here I'm just using the fact that down the cylinder it projects to the vacuum, okay? So that all in the intermediate. And because they're topological, can make them as arbitrary far apart lines, okay? So this means that the VEVs are highly constrained by this polynomial equation, so this very particular property that the coefficients are positive integers. So these are gonna be very special algebraic numbers associated with these VEVs. Now there are additional constraints on these VEVs coming from uniterity. Uniterity would imply in particular that these VEVs are actually bounded below by one. Okay, this is a slightly, whoops, I did it again, sorry. Slightly untreated to see, so let me go through that argument, okay? The argument actually is a very simple argument involving the consideration of the torus which will come back, so it's worth about to go through it in a little detail, okay? But let me first say explicitly, let me first say the consequence of this, okay? So in the case when LI, these lines have expectation equal to one, that's the case when it's invertible, okay? Because this condition, going back to this equation, implies immediately that only one fusion coefficient can be non-vendishing and has to be one. Okay, so there can be at most one term on the right-hand side of the fusion product and so they're invertible, okay? They generate some group like symmetry as usual. And if the VEV is bigger than one, it's non-invertible. So in other words, if you have some aligned defects, which is topological, a quick way to diagnose whether you have a non-invertible symmetry or not is just to compute this cylinder expectation value. And if you find it to be bigger than one, then you know right away that you have a non-invertible defect, okay? Emphasize this is the case in one plus one dimension. This statement does not hold in higher dimension precisely because the coefficient will depend on this and attribute to KFT, okay? Which is the function no longer integer in general. Yes, question? Yes, sorry, but if you normalize differently the lines, yes, you get different quantum dimensions. Good, so okay, welcome to that. Well, the next time I should make that clear. Okay, if that's not, I'll ask you again. So how do we derive this statement? And this derivation should also address this question. So we'll be using locality, okay? In particular, locality of the quantum field theory, in particular, in this case of 2D CFT, define a torus with the insertion of a line. So the object we'll be studying is the torus function, okay? Where's the defect line inserted? Let me call this L, okay? The locality of the Euclidean CFT, or quantum field theory in general, implies that we are free to compute this function. So let me label this function. So we're free to compute this function in different ways. We're free to quantize the theory in different ways. So we can quantize on this spatial slice and this defines naturally this function, okay? Which is R per L, just to denote this particular choice of quantization, which can be interpreted as a trace over the hubris space on this spatial cycle, which is not punctured by any defect line. So it's ordinary hubris space, weighted by the action of the line, denoted by L hat, okay? The same L hat that appeared over there, and weighted by the usual Hamiltonian evolution factor, okay? Into DCFT are determined by the L0 and L0 bar. Now, there's a different way to compute the same pattern function, okay? So instead of quantizing, treating this as direction of time, we can treat this direction of time. That's a consequence of locality, okay? That you can quantize the theory in different way. And in this case, so this is equivalent to S transform, okay? So if I still think about this direction as time, I want to rotate my picture. And as a consequence of that rotation, the line now, so I'm going to do this direction actually. It's just a combination for the S transformation. It's a 90 degree rotation in this way, okay? And this defines naturally the punting function. Now, for the, it's a punting function, it's a thermal punting function that's represented. So let me write in terms of the inverse, or the, sorry, the S transformed values of the shape modulus of a torus. So this is T2 tau, okay? And this is interpreted, okay? Now as the trace over the hubris space, the defect hubris space, because now the defect punctures through the circle on which you quantize the theory. But because there's no other defect insertion, this leads to this simple thermal punting function with no additional insertion, okay? And q tilde is the analog of q when, so q is as before, q is e to the two pi i tau, and q tilde is just e to the minus two pi i over tau, okay? Now, the fact of our quantity equates these two sides. All right, are you assuming c equals c bar? Very good, okay. So let me forget about subtlety with the gravitational anomaly under the multirutor transformation, and for simplicity to focus on the case when c is equal to c bar, okay? And in that case, the right hand side has a property because of the trace over hubris space, okay? Hubris space is some vector space, in particular some vector space. It has this decomposition, okay? Into characters, okay? And because we have a topology defect line that preserves the real sorrow symmetry, this would decompose into characters as I'm trying to see of the following form, okay? But for now, actually, we will not need to this specific form. All we need is the fact that this is positive if we restrict to the case when tau is equal to IT and tau bar is equal to minus IT, okay? Because these coefficients are positive integers, okay? And each of the terms are positive, okay? So I mean, okay, you don't really need this. You just stare at this with this assumption is already positive, okay? Because each weight is positive and you're summing over all the states. And what this implies, okay, the reason I write this is to address the question, okay? This is not required for the derivation over here, but I write this to address the question is because if you rescale your, what do you mean by L here? It will be intention with this being a, you know, a quantized integer. So you can take a direct sum of this line. That is fine because it's gonna retain its integrality, but you cannot apply it by arbitrary number, okay? And that's what, in what sense locality is crucial, you know, in specifying these lines. Without that, you can modify it by arbitrary phase, for example. There's still some subtle kind of counter term you can introduce for the line, but that's not relevant on the cylinder with a flat geometry, okay? So there are subtleties of additional phase factors that you can introduce for the line by introducing one-dimensional counter term. If you're interested, you can ask me afterwards, but I will not show up in this context. So what does this equality imply? This equality implies, so if we take, so specify to this case, and take the limit that T goes to infinity on both sides. Once again, on the left-hand side, because T is going to infinity, Q is going to zero, the term that dominates the left-hand side will be the contribution from the L hat acting on the vacuum. So here I should write L hat, okay? So this means that the width of this line, which is the expectation value of L hat on the cylinder, okay? Is equal to the limit of something as manifestly positive, which exponential before? Here? No, I've moved, I'm gonna start from here. Move it to the right-hand side, good? Okay. So we're halfway there, okay? We prove it as bigger or equal to zero. We have not shown that it's bigger or equal to one, okay? For that, we use the first defect, okay? That if we take the fusion product of the defect line L and it's conjugate, taking them to be simple, we have on the right-hand side, one plus other non-trivial lines, okay? And from this property, we already know this is bigger or equal to one, okay? And then we use the CPT invariance of a general polynomial theory that says that these webs are the same. And as a consequence, we have shown that the web of L is bigger or equal to one. So I went through this kind of argument in detail because this is actually how various non-perturbed constraints from this fusion category symmetry have been derived in two-dimension and also in higher-dimension. It's generalization of this game using the locality of the particle, whichever direction you call time can lead to different expressions and that can lead to constraints on what kind of phases can appear under a symmetric RG flow. Okay. So now I just want to introduce the general data. After this basic building block, let me just introduce the general data for CFT enriched by non-invertible symmetries. So whenever you have a symmetry, apart from trying to study constraint from the symmetry on the original CFT observables, okay? And another thing one can study is the additional observables that are brought to life because of this additional symmetry. In particular, those leaving the twisted hubris space from inserting this vertical duality, I mean vertical non-invertible defects, okay? So CFT, we call that CFT without defects, in two-dimension. The basic data is captured by the hubris space on S1, okay? Which is related to the local operators by the radial concentration. And together with the OP coefficients between these local operators. And this data are subject to bullstrap constraints, which are again consequences of locality of the Euclidean observables. For example, the four-point crossing equation can be thought of as consistency in cutting and gluing the four-point observable on a sphere with two punctures. You can cut along this cycle, and you can also cut along the other cycle, okay? They give you two different OP channels and the equivalence between the composition in the two OP channels coming from inserting composite of states on the circles, at least to the bullstrap constraint that constraint this data, okay? And there's similar relation for the torus one-point function. And it's an entrepreneurial fact that a set of data that satisfy the two kinds of consistent conditions, there are infinite set of consistent conditions because you need to consider arbitrary insertions labeled by states in here, okay? And similarly here for the one-point function. But the entrepreneurial fact is that once this set of conditions are satisfied, this defines the consistency of T, okay? So all the cutting and gluing of the specific conditions arbitrary general remand surfaces are automatically satisfied once these two basic moves fulfills. This generalizes with DDLs in a natural way. So in the moment you have to watch your defects in the CFT, as I said, there's this additional structure that appear, okay, just from the locality of the conformal field theory. You have this defect here where it's based, okay? And you also have this additional OP coefficient, sorry, so let me write it here. You also have this additional OP coefficient that involve operators not in the defective birth space. So they can generally be represented on the plane by some diagram like this, okay? Where the external legs are some topological lines and there's some topological junction leaving here. But the external operators correspond to operators in the corresponding twisted hubr space. So these are the additional data you bring to the game. And to fully specify non-invertible symmetry in the CFT after having some evidence for such objects is to solve for these quantities, okay? And they are subjected to similar blue-strap equations. For example, the four-point function, once again, but now attached to a network of this to what would defect lines, okay? All these lines are not just the mnemonic for OP exchange, but are extra defect lines. And depends on whether you cut in this direction, okay? I don't want to draw the circle again, just to mess up the diagram. Or this direction, okay? Inserting complete set of states and that leads to constraint on this data. So this provides, there's a similar generation for a torus one-point function which I'll not draw. This provides an axiomatic approach to identify invertible symmetries into DCFT. Okay? So having non-invertible symmetry into DCFT, because we discussed, if we assume the existence of this topical defect, it will imply all these structures that satisfy cutting and cooling as a consequence of locality. So if you are given just abstractly safety data like the local Hebrew space and this OP coefficient, the thing you need to do abstractly to identify a non-invertible symmetry in this CFT is to solve for this additional data, okay? If you find a solution, then you are sure that you have this non-invertible symmetry, okay? But as you can see, this is gonna be a formidable task for a general CFT because there are many apple strap. Yeah, so I actually wrap up. But as you all see, there are additional physical argument that allow you to infer the existence of topical defects without solving this equation explicitly. But this equation can be solved explicitly in rational conformal field theories, okay? Let me not get into the detail, but let me just say that in the next lecture, we'll discuss explicit examples that realize the symmetries and produce solutions to these equations without actually solving these equations explicitly. I'm going slower than I expected, but that's fine. Let me stop here and take questions.