 učeniti trenu, inf центрov, i bolo spokovnit surprised, da, after this lecture and before the discussion, at the end of the one hour lecture, there is going to be a group picture. So please don't leave. I am not sure exactly the details about this is going to happen. Also for online participants, so somehow together we take a picture. Is there anything else to say? I think we are ready. We will continue with the second lecture about non-invertible symmetries. I should start the recording. Recording in progress. Thanks for coming back. We will continue the journey on invertible symmetries. This morning, I reviewed for you the general notion of symmetry as topological operators. We saw briefly how the ordinary symmetries of social waste continuous groups and discrete groups fit into this framework. Now we are going to generalize that using this, based on this picture of symmetries as topological operators, in the morning, we discussed the case when the symmetry of social waste is a group, and this is defined by the topological operator on the co-dimension 1 manifold of the space time. We are ready to generalize. The first generalization that retains the feature of being topological as well as some structure of a group is the notion of p-form symmetry, a high-form symmetry, general p-form symmetry. This is realized by a higher co-dimension. In other words, p, co-dimension, actually in my notation, will be the p plus 1 co-dimension topological defect. So instead of defining a topological defect defined on the co-dimension 1 manifold, in general, you can have a topological defect defined on a higher co-dimension manifold, and they generate the p-form symmetry and the p enters into co-dimension in this way. The particular example is the one-form symmetry in four-dimensional gauge theories. Historically, they are known as center symmetry, because the corresponding group coincides with the center of the gauge group. In this case, p is equal to 1, and the generators are two-dimensional topological operators, and they are generators that charge the unitary operators corresponding to the one-form symmetry. And the operators in the theory that are charged under this one-form symmetry are both in lines. This is the kind of characterizing feature of higher-form symmetries is that the charged objects are no longer local operators or particles, but rather I extended objects such as both in lines. And if you want to think about dynamical objects in your theory, and think about, say, strings, they will be charged under this one-form symmetry. This is generation number one. Another generation, as you probably already guessed from the title of the lecture, is to consider non-invertible fusion, or non-invertible, which come from general fusion. So previously, the topological defects, as I said, the symmetries are characterized by the topological defects, their fusion and how they act on local operators. In this case, it's generalized in the case of how they act on these extended operators. And the fusion previously always takes this form. They follow some group modification law. Instead of generalizing on the co-dimension, we can generalize this notion of a fusion product to something more general. So I'll denote two such topological defects by L, I, and L, J, because I'm staying in two-dimension. And the corresponding topological defects are one-dimensional, represented by these lines. We know these two topological lines, because they're topological, looked from far away. They look like a single topological object. And in general, it could be reducible. And it could be decomposed into some of individual topological lines, without assuming there's underlying group associated with these topological defects. And the point is that on the right-hand side of this fusion product involves multiple topological defects rather than a single one is a signature of non-invertibility. So when you talk about group modification law, of course, you always only have a single term on the right-hand side. It's meaningful decomposition. And as you will see... Sorry. Can this be viewed as a generalization of the OPE expansion? That's a very good question. Indeed, I'll make a comment related to the OPE of more general defect in a moment. Yes. So this is a special case of OPE expansion, involving extended objects here, line defects. Question? Why are there no signatures of the indices i and j on the right-hand side? Right. So what k appears here depends on i and j. So this step implicitly depends on the choice of i and j. But then being kind of a schematic here, later we'll write explicit equation. In particular, that's a good point. So some of the lines here could be degenerate, meaning that I'm summing over... I'm just doing arbitrary decomposition here. So I'm not assuming each one appearing here is irreducible, for example. So if you write the right-hand side in terms of incomposible lines, the coefficient may also carry the degeneracy of how many times it appears. We'll be precise about that later. Thank you. Here, just a schematic. The moment you have this non-trivial sum, having more than one term appear on the right-hand side, it's already a signature of non-invertibility. Sorry. Question? The definition of having more than one term needs the definition of some building block. I think a sum of lines is a line. That's right. So that's a very good question. As we will get to that, from what I've told you here so far, there seems to be no physical meaning of how I split the right-hand side. For example, when I have a single line as before, why can't I split a single unitary as a half? Here is obvious, because if you take a half of a unitary, it's not a unitary anymore. Here, we'll also see that there's a constraint precisely how to decompose this, coming from the locality of the corner field there. So you cannot arbitrarily scale your line defect. That's constrained by locality. OK. All right. So let me now come to this question in relation to a notion of OPE between these extended operators in a more general context. So indeed, this can be thought of as a special case, OPE between defects. In this case, I'm focusing on line defects, saying D equal to 2. But this is a general notion you can consider in hard dimension. But in general, the OPE of the line defects is much more complicated. For example, just to make connection to existing knowledge about defects, let's consider an example of conformal defects. So a large class of non-trivial defects are these conformal defects, the conformal defect lines, and D equal to 2. So these defect lines are again kind of denoted by this in the same picture, let me emphasize, this is the conformal defect in general. OK. And it is defined such that the stress sensor, so in general, it actually can be an interface, is defined such that the stress sensor on this side satisfy this relation. No, let me think about this direction. So this is the z, this is the z plane, and for convenience, I'm calling this direction and imagine the z direction, which is the location of the interface. In general, there's a discontinuity in the stress sensor for conformal defect, even in the same theory. The discontinuity in the stress sensor is captured by what's known as a displacement operator, which is the operator on the conformal interface, let me call it I. The special case of topological defect corresponds to the case when D is trivial. So the stress sensor is fully continuous across the defect. Now, for the general class of conformal defects, there's this notion of OPE that you can discuss. So I take I1, I2, there are two conformal defects. And that's kind of the picture that you would want to have also for conformal defects, a similar picture as that. But there are some important differences. For example, even if you take these two to be the same defect with opposite orientation, and you expect the identity defect to show up, the trivial line to show up here, but this coefficient would involve divergences in particular from the cashmere energy. So considering OPE will be corresponding to the limit where you take the two defects close to one another, and you will have this cashmere energy divergence that's proportional to a cosmological constant term integrated over the defect, over the line. And moreover, you have subleading divergences due to relevant operators on the line from this fusion product. So it's physically very distinct from what happens here in the moment when you have this discontinuity. The discontinuity in the stress sensor is what give rise to this cashmere energy which is essentially measuring some kind of imbalance between the stress sensor inside of this region and outside. Because in a non-tobodral case you have this divergence piece and there's the structure of this fusion although you can write it down, but it's much messier. And in some special cases, for example, if your system has supersymmetry, in that case because of the contribution from Bosnian degrees freedom and for supersymmetry line you can achieve a special case where this cashmere energy piece is a divergence vanishes. And this is why in supersymmetry theories it's meaningful to talk about fusions of supersymmetry both lines and so on and this kind of fusion product is well defined. Sorry. And you may be able to derive similar relations for example from localization. There is no way to regulate this divergence then. Sorry? There is no way to regulate the divergence. You can, but for kind of a generic theory there are many divergences over here for each of these relevant operators. If there is only a single divergence coming from the cosmological constant term then that's more kind of there are ways to deal with but facing the entire tower of suppleading divergences from relevant operator means dimension smaller than one in this context on the line there is no obvious way to achieve meaningful answer. Question? Sorry, this might be a knife question but is this divergence related to both the UV and IS structure of the theory because in one case you are taking delta to zero which is the short distance structure but you are also integrating over some non-compact direction presumably. Right, so here you can make it compact but the divergence stays it's because of the limit. So the divergence is mainly UV divergence. That's right, that's right. Thank you. That's my image. Okay. So that's just a quick comment. All right. Well, let's fulfill this promise to provide some tower example before we jump into general structure some tower example for non-invertible symmetries. Okay. The tower example is not going to be the tower example of this inverse symmetry will take place in a very simple theory which is the discrete gauge theory with gauge group a discrete group g. Someone of you already asked me about this during the discussion in formal discussion. So in this case there are interesting topological operators which are extended objects. These are Wilson lines. And these are the Wilson lines that you would have. They are just analog of the Wilson lines you would have for continuous gauge theory. Okay. So they are labeled by representations of the corresponding group. In particular the in-decomposible Wilson lines are labeled by ERAPs of the group. Okay. So explicitly you can write them in terms of the exponential of your gauge field over a curve as follows. Now in general continuous gauge theory like in Yamile's theory this Wilson line would not be a topological object. Okay. That's because the connection in general is not flat. But here it's topological because discrete gauge fields are always flat. It doesn't support discrete gauge field cannot support any curvature. As a consequence this provides possible realizations of the topological defect operators that we may interpret as symmetries. Okay. And they claim they are non-invertible. They give us non-invertible symmetries. If g is non-abidient. So what is the simplest question? Yes. So what happens when the connection is not flat? Ah. So when the connection is not flat then this Wilson loop is not going to be topological defect. But they will fall it can fall into this realm of conformal defects. And as I said before their fusion in general is much more complicated. There are these divergences and from these divergences there is the regularization ambiguities. But in special cases like with supersymmetry you can still have this well-defined fusion products when all the lines preserve the same superchargers. And how can you assure that in this case you have a flat connection? Oh. It's actually by construction. If you're talking about the discrete gauge field there's just no there's no coverage. Okay. Discrete gauge field backgrounds are classified just by how many. Sorry. Just by the transition functions. Thank you. Sorry. What are the connections for a non-abidient discrete group? Ah. So this is the same thing that I drew before. The to specify a gauge field equivalent to you know, specify a gauge connection for a discrete gauge field is equivalent to decorating your protein function with a network of symmetry defects. Each of them will be labeled by, you know, this ug where g is element of the group. Okay. And they're joined by this topogru junctions which you introduced in the last lecture. So this specify tiling of your space time by networks like this specify a particular gauge connection background. And the discrete gauge theory means that you are summing over all these configurations. Right. And if you if you don't like this notation. Well, okay. So this is I think this is the best I can do for general non-abidient groups. For discrete discrete abidient group sometimes you can introduce some kind of Lagrangian multiplier field and write everything in terms of u1 gauge field. But this is the this will be the what we need to do for this general non-abidient discrete gauge groups. Is there a question? As you also draw in this picture and as you presented this morning there are maybe two different point of view on fusion both as picking defect and stacking on the same manifold or as junction as you draw here. Is it obvious that this is exactly the same thing and I mean maybe in for lines it is almost obvious because you can put lines only on 3-year manifolds but in for higher dimensional defects you might put you are you are similar on different two-dimensional manifold for example and I don't I don't know how to see that it is the same. Yes. You caught me. That was correct. So when I draw this picture this represents the discrete gauging in two-dimension. There is a similar picture but it is harder to draw for the three-dimensional gauging in this case in that case the basic junction will not be of this shape but it is similar basic building blocks and similarly in higher dimension. OK, but the two point of view are always the same. Both as stacking defects and with junction are always the same. No, so as I said before you have the topological defects and they satisfy some fusion rule and on top of that there are these junctions the existence of junctions can be inferred just from the topological property of the lines. OK, but there is some extra data which is the you know the specifying each junction. OK and that goes into defining this gauge theory background. The key point is that the freedom that you have in defining these junctions will not affect physical quantities like anomalies as opposed to the corresponding symmetry. So in this case here for example I'm talking about the discrete gauge theory in which case I have dynamically gauge the group so I'm implicit already assuming there's no anomaly associated with the symmetry in this setup. So like I back to this claim so the simplest perhaps the simplest example of a non-impedient group is the case when it's the permutation group of three elements. OK In this case we have three irreducible representations the trivial representation right as one for the moment for the moment it will become clear why I call it one and there's an additional one-dimensional non-trivial representation as the sign representation OK and then there's a vector representation which is the which is the two-dimensional representation for S3 let me call it V OK and as is true for Boston loops in general OK you can decompose at the level of the representation that associated with the gauge group we can decompose the fusion product between two Wilson loops into Wilson loops labeled by the representation that appear in the tensor product OK So in particular the Wilson loop associated with the sign representation when you fuse it with itself you just get the identity that's because the tensor brought out the sign representation with itself it's the identity representation similarly identity when you fuse with the sign representation it does not change representation and the same is true for the vector representation OK So far I'm writing fusion rules where only one term appear on the right-hand side and that's the signature of so far we're just uncovering some invertible symmetries in the theory Now invertible symmetry will be generated by the Wilson loop associated with the vector representation Now the key is that the tensor product of the two vector representations of S3 decompose into three representations all these three representations appear in the tensor product as a consequence on the right-hand side you have this combination of Wilson loops appearing OK and this is the non-invertible symmetry that we promised in this very simple toy theory If you are still confused about something that I said before let me just focus on the case for d equal 2 and then this is the literary picture that it means for this is the literary kind of a network you sum over to define this discrete gauge theory OK but this construction what I was trying to emphasize before is that this construction works in general dimensions and you always for any discrete gauge theory in any dimension you will have this non-invertible symmetries generated by this Wilson lines as long as the gauge group is non-belian when it's belian all the irreducible representations are one-dimensional so and you will not have this non-trivial tensor product that involve more than one term in the decomposition on the right-hand side OK and this is a very simple model that I realized this symmetry and it turns out that this symmetry is usually denoted by S3 wrap S3 OK meaning that the individual generators of the symmetries are one-to-one correspondence with representation irreducible representation of S3 OK and later we'll see this kind of trivial symmetry will show up in the interesting two-dimensional CFT namely the three-state POTS model so here I discussed the kind of trivial example a trivial theory where some non-invertible symmetry shows up but the general lesson is that given non-invertible symmetry can show up in many theories in particular we'll see that show up in some more interesting theory the three-state POTS once we have settled this very simple toy example let's start with some general generality of non-invertible symmetries to spell out the general structures in more detail and once again I'll be focusing on dimension 2 OK and as I've alluded to before the mathematical structure here we'll see the mathematical structure here generalize the groups to what's known as the fusion category and we'll unpack the ingredients that go into this definition step by step so the basic starting point of defining a non-invertible symmetry in two-dimension in general quantum field theory is to first specify the set of topological defect lines which we will abbreviate as TBL and we'll denote by Li individually and analogous to the fusion rule that we wrote before and we generalize to the case when there are more term appearing on the right-hand side this topological defect lines generally satisfy some fusion algebra or fusion ring which tells you how to decompose products of these generators in the fusion product and when I write the equation like this it should always be interpreted in this way OK so you are taking two you are taking this topological defect lines to lie on manifolds which are homologous and you are taking a limit where they approach one another and you decompose this configuration into the individual topological defect lines and in particular among this set is this distinguished element which we'll call one that correspond to the identity or correspond to the case there is no insertion at all OK and you are free to apply this to any fuse this with any of the line over here and it will return to you the line that you started with another notion is something that is already asked in the context of discrete symmetry that generalize beyond the fusion ring is the notion of topological junctions the existence of topological junctions can be inferred from the topological property of these lines but to specify these junctions individually requires additional data so this topological junctions are defined as follows for any line I call it L3 that appear in the fusion product between L1 and L2 so whenever I write the product of a line it is interpreted as in the sense a fusion product for any line that appear in the fusion product you have a topological junction of this type generalizing the topological junction that we saw when these are lines that generate group-like symmetries the junction relies on a specific specification specification of this point as we can put out of a point operator which is denoted by V in general when you specify external topological defect line there may be multiple choice of V and this V in general is a vector in the vector space which we will call the junction vector space topological junction vector space to be precise and this topological junction vector space because these are point-like operators you can equivalently think about this as kind of a TDL or topological defect line changing topological operators in particular in a special case when L2 is identity V will be the topological changing operator between the defect line L1 and L3 and in general V will be the topological defect changing operator between the fusion product L1, L2 and L3 and later we will give a more precise kind of picture for what this topological junction vector space is when we discuss this topological defect line so defect lines in two-dimensional CFD the last ingredient to fully specify this notion of fusion category is the notion of F symbols and this captures possible general possible factors you get from a topological change a topology change in a network of topological defect lines similar to the phase that we encountered when we changed the network of part of the background describing part of the background for discrete symmetry so this topological change is commonly referred to as an F move which is the basic move to change the topology of this kind of diagram describing a two-dimensional background for the discrete symmetry and also generalize to this background of TDR network so I'm just redrawing the diagram we draw before but in a different way that's conventional to in convention to the literature so 1, 2, 3, 4 denotes 4 topological defect lines and A labels this internal topological defect line and all these junctions are topological junctions so they are specific implicitly by some specific vector in this vector space the F move corresponds to a topology change in this network by reconnecting the second topological line to another leg and in general some other intermediate topological defect line may appear over here and this change of basis is captured by some numerical coefficient known as the F symbol so here this thing can be thought of as a matrix the 1, 2, 3, 4 are external labels that specify the external lines that enter into this diagram and this indices specify the change of basis between these two configurations so why do I say this is a change of basis what I mean precisely this is a change of basis in the junction vector space between four external lines L1, L2, L3 and L4 so you think about this pair of trivalent vertices as a basis for the topological junction between four external lines because there are two ways to resolve this four point junction there are so there are two that give you two preferred basis for this space and this F matrix captures the change of basis can you comment a bit about operational change of this arrow because in the first picture you draw the two in go in the vertex you have two in going and one out going but in the other diagram you have two out going and one in going sorry which picture for this picture how important is the two in going and one out going so it's just because I want to make sense of this notation so I can draw in going and then it will be the L3 bar I'll come back to that you will define what you mean by bar thanks for the question can you explain again why there is a whole vector space of junctions right so ok so it's not I think that notion is not completely clear at the moment but you can think about this way so the point is that this is like a point like operator so having one junction you can multiply this point like operator by arbitrary complex number that's still topological it's just a different way to define this junction so if you have two independent such junctions in the two dimensional complex vector space and that's one way to think about it later we'll give a physical interpretation of how this junction vector space comes about in the two dimensional CFT it will be related to the hubris space of the two dimensional CFT on a circle with defect points and this will be characterized by specific states in that hubris space it's a subspace we'll make that precise but in order to define this f symbol why don't you need to specify the junction very good so I cheated a bit I said it but I didn't write it because it just makes the equation more complicated so on the left hand side there's a choice of v1 and v2 over here and on the right hand side is also the choice of junction vectors here v3 and v4 and this matrix will involve this additional index v1 v2 and v3 v4 and you'll sum over all these choices for simplicity I'm just drawing the case of a one dimensional so that the information is completely captured by this coefficient otherwise there is a further indices is there some interesting case in which the junction is always one dimensional or yes of course for example the duality defect which shows up in two dimensional ICM model as well as in this three dimensional and four dimensional gauge theories they all fall into this category in having one dimensional junction vector space as we'll explain the dimensionality of this junction vector space will be tied to the coefficient in the fusion ring when you write to the right hand side in terms of indekomposible objects and whenever the fusion product such that only coefficient one shows up in the decomposition as we'll explain that corresponds to the case when this is one dimensional as for the questions we'll try to jump ahead so if you didn't understand the last comment don't worry we'll come to that alright so these symbols are not arbitrary so they satisfy some consistency condition this is something that you can already kind of inferred from the fact that when you specialize the case when the defects are invertible this symbol is nothing but the phase that we introduced before that captures anomaly associated with the corresponding discrete group like symmetry ok and as we said before the anomalies phase is subject to this consistency condition that essentially tells you that this is a cycle and that's where the group cohomology classification of anomalies comes in and here we can be more explicit and recall where the consistency condition comes from the consistency will come from looking at a 5-fold junction so it's a topological junction 5 external legs so for simplicity I'll not keep track of the arrow ok and then there are 2 internal legs labeled by A and B so this specified if you once specify the junction vector over here this specified particular junction in the space between 5 external lines 1, 2, 3 and 4 1, 2, 3, 4 and 5 ok now there's a consecutive change of phases you can do ok I think I was too ambitious with my space management what I'm going to draw will not fit in the region so let me move over here let me start with this diagram once again 1, 2, 3, 4 and 5 ok A and B and as I said this specifies a particular junction vector in this junction vector space with 5 external pdl specify 1, 2, 3, 4, 5 and of course there is a various ways to resolve so you can think about this as specifying the choice of junctions leaving here ok so this is the general junction, topological junction with 5 external legs and for any vector here there will be elements inside this vector space now there are different ways to represent vectors in this space using the topological nature of this of this pdl so in other words you can try to resolve this very singular looking junction into like a triplet of trivalent junction in this picture and each different resolution will give you a different basis for this junction vector space so in the next over here is an arbitrary choice of junction vectors of these trival junctions ok and there is a consistency coming from comparing this to the following so let me draw it over here so this is the famous pentagon equation we can perform this change of basis given by this f move from this basis to this basis by using this f matrix and we can do it multiple times now do the f move focusing on this part and you get to this picture and we can do it again focusing on this part and apply the same f move and we get to this picture of course there will be different changes is entering into each step correspond to a different labeling of external labels as well as the choice of internal lines that appear in that channel but then the point is that going through this round we established the change of basis between this basis for the 5-4 junction vector space and this basis for the 5-4 junction vector space but there is a different way to reach the same change of basis matrix is to do first to this picture which correspond to doing f move on this block so you reconnect 3 with this other side of this diagram and then do f move on this region and you get to this basis again and in this process another different the same f matrix with different external indices shows up and by consistency in the basis between 2 special basis in 2 different ways you must get the same result that schematically look like a product of 3 f matrices with certain indices that's been contracted is equal to the product of 2 f matrices going this way and this is the famous pentagon equation because this look like a pentagon which is the generalization the cosycle condition for g symmetry for invertible symmetry where g is discrete and in which case the pentagon equation is the same as the cosycle condition that defines the g anomaly so now we have introduced all the basic ingredients that define what is known as the fusion category so it's not something mysterious it's just some objects that generalize groups and knows about the anomaly of such ways of groups to the case where the building topozor defects are non-invertible oops I'll pick it up later together the topozor defect lines their fusion junctions so these are individual building blocks for this underlying structure and f symbols they lead to the fusion category the structure of the fusion category so this is just some references in case you have heard about the category before and you want to make connection to the notion of category in this language in this case the underlying objects for the fusion category are the topozor defect lines the fusion will describe the tensor product between these objects in this fusion category the fusion category has in particular a structure of a tensor product and these junctions leads to morphisms between objects in the fusion category in particular between objects and this is why the trivalent junction defines a morphism and these f symbols are known as associators which specifies isomorphisms between the tensor product of three objects that actually has six external labels and that's why it's also called a 6j symbol and this fusion category structure as we already saw from various parts of this structure it generalizes discrete group symmetry in particular in a special case when all the topozor defect lines are invertible in corresponding some group the fusion category is nothing but the usual discrete group symmetry but at the same time taking track of the anomaly through the f symbol it's in a sense more data than the group but the data contains is precisely what we care about in physics and the second comment is that the structure of the fusion category is extremely constraining of course if you don't introduce this non-year variable defects then it's just as constraining as it is for constraining anomalies associated with the given group symmetry but here the constraints manifest in the following way, let me just give you an example if you start from a set of the water defect lines which you postulate in some fusion category symmetry and you postulate the fusion rule for example if you just postulate a single topozor defect and you postulate the fusion rule of the form L squared is equal to 1 which is a trivial defect plus N which is some degeneracy times L itself this is only gives a fusion category N equal to 0 or 1 so this is very surprising because why can't we have a single non-year variable defect that satisfies such a fusion rule and the constraint is coming from the F symbols and in particular the pentagon equation that constraint F symbol and the punchline is that when the N is so the story is that when N is large enough there is just no solution to the pentagon equations ok and as well later we'll see all the structure of fusion category symmetry would necessarily comes out from the locality of a quantum field theory so that tells you that this symmetry is just simply not possible in the quantum field theory sorry about question does this constraint come from unitarity or it's independent of unitarity this term is independent of unitarity it's just let me start it depends on how unitary you want the theory to be so to be absolutely sure I think we're implicit assuming some notion of unitarity because if we forget about unitarity then for example the fusion products may have funny coefficients and that would ruin this story but here the fact that we're talking about a fusion category actually relies on some notion of unitarity that's a very good question otherwise there is this general notion of some pseudo fusion category very good question sorry when you look at the solution of the benton equation in which range do you look for f should be a phase there should be there is some condition very good question so as I mentioned briefly over here for each junction vector there is ambiguity if the junction vector space is one-dimensional you have ambiguity coming from the complex number and if you keep track of ambiguity in this equation you see that depending on your choice of normalization of this vertex here and here that will change the f matrix so in general this f symbol is not completely unambiguous but a statement here is that there is absolutely no solution whatever choice of normalization you take for the vertices but in general you will have family of solutions for the f symbols but we only study them up to this equivalence relation that is the analog of the exact co-cycles in the context of group cohomology so the phase coming from the f move for the group symmetry case is abstract ambiguities coming from the exact co-cycles and that is the physical origin of that is coming from the redefinition of this point like operators and another comment I want to make over here is that just like in the case of groups there are ways to classify groups and that is known so you may ask if there is a classification of this fusion category symmetries and the answer is that and there is no classification as of yet except for low rank cases rank means the number of indy composable objects which we will come to shortly but on the other hand from physics as we will see we will be able to produce tons of examples for the fusion categories so it is really down to the mathematicians to actually pursue a classification that will include all the examples that we will cook up and lastly it is related to the question is there is some structure associated with the pentagon pentagon equation that is up to this gauge freedom that we have associated with redefining each point like operator that live at the junction this topological point like operators the pentagon solutions the pentagon equations have only discrete solutions ok so again this is up to the gauge freedom that we talked about redefining these junctions and this is very nice mathematically this is known as the oknienu rigidity this is something that the mathematician has proved so it says that the pentagon equation does not allow continuous family of solutions when you remove potential continuous ambiguities coming from these redefinitions and this is very nice because this is a similar feature that we have for anomalies anomalies are quantized and they cannot change under continuous deformations and because of this feature they are bound to make non trivial constraints on quantum field theories in particular for RG flows which is a continuous process that is the mathematical framework underlies this non invertible symmetry in two dimensions ok we have not really put in any physics as we will see studying these symmetries in the context of two dimensional quantum field theories would make many of the structures more transparent and it will also lead to additional physical information which is not obvious at all from this basic building blocks ifan just in two minutes we should move to the discussion but maybe you got a lot of questions so you can take a little bit more if you want how much this is already ok if I can have 10 minutes ok so there are some extra structures let me explain coming from studying the symmetry in two dimensional QFT with this invertible symmetry so to be explicit we will restrict to two DCFTs and this is not really a loss of generality because as we alluded to over here the structure of the student category is rigid so to the QFTs as obtained from CFTs by RG flows is naturally captured by the statement we will be making here but the CFT or TQFT ok but the notion of a 2D CFT will help me to make some explicit statements using so this is convenient because we will be able to use radial quantization and as we will see shortly they will help me to specify what these operators are in radial quantization because in this case in the CFT I have radial quantization on top of that I have what defined for the conformal symmetry and there are h bar and h this is really where I am using the CFT structure and the key as already mentioned that leads to this extra structure in CFT for the student category symmetry is the locality and this is something true for general quantum field theory which can be thought of as consistency in cutting and gluing or quantum field theory pattern function with observables so you can cut and glue your observable pattern function with observable in different ways and glue them back together it will lead to the same results that's one notion of locality one way to phrase the notion of locality and as you will see this locality will lead to this fusion category structure starting from the basic objects the topological defects all the others will just follow the topological defects which we postulate as a symmetry in the theory the topological defects acts on local operators and we can be quite precise in the context of two dimensional CFT so imagine you have a local operator over here as I said before the way a unitary operator acts on a local operator is by enclosing it and because this operator is the logical it means that in particular it commutes with stress sensor so it will preserve the conformal way that social wizard is a local operator and it will produce another operator which you already know by L hat acting on operator phi is operator with the same same h bar as the original operator before you act by the topological defects and moreover this also implies that the topological defects will map the soror primaries in the two dimensional CFT to the soror primaries in the entire multiplets and equivalently from the radial quantitation you can think about this picture as meaning on the cylinder where on the bottom of the cylinder insert to the corresponding state correspond to this local operator so it's a primary operator and it's a state inside the hubris space on S1 and this is the line insertion that acts on this hubris space that comes from this radial mapping the coordinate change that we use in the radial quantization and this is the equivalent meaning that tells you how this is why we use the L hat because L hat can be also thought of as the operation of this symmetry defect acting on the states so this is the if you wish, this is the generalization of what Wigner wrote for unitary but now for this more general operation linear operation L hat and the last another ingredient and there is questions the other important ingredient coming from the quantum field theory in the presence of this topogel defects is the defect hubris space and this is the consequence of the locality of this topogel defects and this is the generalized notion of the twisted sector of a two dimensional CFT that you associate with discrete symmetries the idea is that you study again the theory on the cylinder but you can choose to align the topogel defect now in the time direction in each case the hubris space gets deformed and the general state inside the hubris space denoted by psi the hubris space gets deformed into this twisted hubris space in a special case when L corresponds to group like symmetry defect this is nothing but the twisted the hubris space with the twisted boundary condition imposed on the spatial circle but in the case of a non-year-old defect because of locality you can still define this defect hubris space and by the same the same conformal map that relates the virtual concentration to the cylinder this is equivalent to specifying this operator is equivalent to having a local operator that sits at the end of topogel defect line and once again because of the topogel nature of these lines this again fall into representations of the two copies of your soror symmetry and in particular will assume the following condition which is a physical assumption let me call it the typal condition that is the identity line can end topogelically and this is equivalent to having a faithful representation of the symmetry in this case the non-invertible symmetry generated by this topogel defect on the hubris space on S1 the reason is that if other topogel defects and topogelically operators so in this case psi in general is an operator with h and h bar that's not equal to 0,0 so in the case of a topogel junction will be the special case where h bar is equal to 0 but in general this is not possible and indeed for CFTs we want to impose the condition that this is always not possible so having h and h bar equal to 0 for psi it's always impossible when the topogelically defect line is not an entity and that's the equivalent to requiring the symmetry that you are studying to be represented fistfully on this hubris space I think I'm running out of time so maybe let me stop here and answer questions and we'll catch up next lecture okay so let's thank the speaker recording stopped