 Next will be on this topic of non-recording-in-progress invertible symmetries. I should say that during my talk, if you have any questions, please just speak up. Let me start with some motivations of this subject, in case you're not familiar with this recent mini-revolution. So we'll start with some motivations. As you probably all have learned from physics courses, that symmetry is perhaps arguably the most important guiding principle in dealing with physical problems. First of all, symmetry is ubiquitous, meaning that even messy system can exhibit symmetry properties at long distance. Second of all, symmetry gives rise to optimization principles for the physical observables. And examples of that take the form of selection rules, or the usual laws you can derive from charge conservation. And thirdly, symmetry can often give rise to non-trivial constraints on dynamics of the quantum systems, such as constraints on the randomized group flows in quantum field theory, as well as constraints on the possible IR phase diagram of a given UV description. And given these nice things about symmetry, it's a very natural step that given a physical system to first identify, therefore, given any physical system, a natural first step to solving that system is to first identify the symmetries of the problem, and then deduce the dynamical consequences. So now I have said some nice things about symmetry and how we use symmetry to constrain physical systems. Let's review a brief history of symmetry, or the success of symmetry. And besides, the recent history of symmetries in quantum systems, quantum mechanical models. So the first list in this historical list is not so recent. It's from the 1930s. That's when the foundations of symmetry in quantum mechanics was laid down by pioneers like Wigner. In those cases, the symmetry will be associated with the group, which are denoted by G. And then the power of symmetry was greatly used in the study of quantum field theories. In that context, the central subject will be this continuous symmetries, such as the space time Lorentz symmetries, as well as the say the extra magnetic magnetic symmetry, as well as the gauge symmetries in the context of a center model. And their power have been used in the context of studying the quantum field theory in the 1980s. It produced many amazing results. And then shortly after, perhaps not well appreciated at that time, is the discussion of discrete symmetries, which is the generalization of these continuous symmetries, which are associated with continuous group, to the case when the group is generally discrete. Started from the paper and written around this time. This was not as much appreciated in the high energy community when it was first put out. But it has spurred activity development in condensed matter community, in particular, condensed matter theory community, because these discrete symmetries naturally show up as symmetries of lattice systems. And people in the condensed matter community have used the results from Tiger Whitten to understand phases of these systems that are described by lattices at some cutoff scale. And this was around 2010s. And then there's an interesting interaction between the high energy community, coming from the first description of this discrete symmetry anomalies from this paper, to the condensed matter community doesn't go one way. There's also a feedback from the condensed matter development for discrete symmetry back into the high energy community. And one prime example is the work Gautel Popolsten-Weiberg in 2014. Now, in this paper, they discuss generalizations of this discrete symmetry, which have received active development development in condensed matter theory, and further generalize it one step further. And in particular, a key result of this paper is this generalized notion of symmetries viewed as topological effect operators. This is a notion that encapsulates or captures all the previous notions of symmetries and allows further generalizations. And the further generalization claims in two branches, roughly speaking. So there are many further generalizations since around 2015. One type of generalization is to still insist on having some group associated with the symmetries. This leads to the higher form symmetries, which is already discussed in the original paper. And this is generalization of the old well-known center symmetry in Yamio's theory. And another generalization is to still having some notion of group, but consider a higher version or generalization. And this give rise to the higher group symmetries. And all these guys have the feature that they're associated to some underlying invertible structure that is characterized by a higher group generalization. And their steady progress today to date to understanding the symmetries in various physical systems and anomalies associated with symmetries, which can be used to deduce dynamical consequences. But that is not the focus of this lecture series. Meanwhile, around the same time, another branch of development in generalizing symmetries come from generalizing the notion of group to something more drastic. So this also happens around 2017. This gave rise to the notion of non-invertible symmetries. If you look into the literature, these symmetries come with many names. But in this lecture, to avoid any confusion, I'll stick to the name of non-invertible symmetry. But in the literature, you may also see people referring to these symmetries as categorical symmetries, fusion categorical symmetries, and so on, or higher category symmetry. The definition of symmetry, which we will discuss in detail in the lecture series, relies on this perspective that treats symmetries as topological defect operators. So the non-invertible symmetry will be associated with topological defect operators, but it will be associated with this general topological defect operators, which are not associated with group. In other words, they are not invertible. All these notions will explain in detail in the main parts of lecture. And why do I say this is a drastic departure from the previous notion of symmetries? That is because from the foundations of symmetry that was laid down by Wigner in the 1930s, a crucial assumption made for symmetries is that it should define a unitary operator on your hubris space. But that is closely related to the notion why symmetries discussed in the past are always associated with the group. The consequence of non-invertibility is that the corresponding symmetry operator, the topological defect operator, would not become a unitary operator acting on a hubris space. So that's a very drastic departure. So you may have some immediate questions on your mind since this is something so drastic. The first question is that even though this is some objects as allowed by this perspective, just insist on topological defect operators but without associating with two groups, do they really exist in interesting quantum field theories? Or is it just a mathematical fantasy, a generalization that has no physical origin or any, and furthermore, no physical interpretation? Yeah, yeah, so I'll come back to that. Yes, yes, yes. So when I say, when I discuss this, I do not mean that the symmetries are new. The symmetries always exist. It was just not put into this universal framework. And indeed, there are examples which I'll mention shortly. As is true for any symmetry in quantum field theory, they always exist. They're just waiting for us to discover them. And the second question is if you can generate, we can identify some examples. Do these examples produce new dynamical constraints? In other words, if they only produce constraints, which can be re-derived using ordinary symmetries, then it's not very interesting. So why do we need to develop this new machinery? The key question is, do they produce new dynamic constraints? OK. And the answer is yes, as you may have imagined. Otherwise, I wouldn't be giving this set of lectures. So let me now get to some examples of these non-invertible symmetries. And along the way, I'll give some references for which the lectures will be based on and some additional references for further reading on this subject. We can think about what I'm going to say as a guide to literature on non-invertible symmetries. So the first development for non-invertible symmetries comes in two dimensions, a spacetime dimension equal to 2. As was already pointed out in the audience, there are already examples of these non-invertible symmetries in rational conformal field theories date all the way back to the work of Valente, 1988, and furthermore, the work of Petkova and Zuber in 2000. In which case, they discuss non-invertible symmetries in rational conformal field theories, such like the Liouville theory, well, such like the rational minimodels, and there are also generalizations in the Liouville theory, which is not really RCFT, but it's for certain line operators, a certain topological defect, in that case, they can be described, and thanks to the work of Valente. And there are huge generalizations of this original work by Frolic. Fox, Ronco Schwieger, in the sequence of papers starting around 2006, which essentially fully specified the set of topological defects in RCFT that preserve a chiral algebra, some symmetry that's beyond the universal symmetry. We'll get to that notion later, and this will be typically referred to as some kind of Valente lines or some suitable generalizations when RCFT is defined by a non-diagonal model invariant. And shortly after this work of Frolic, Fox, Ronco Schwieger, there are also realizations of this non-ievertible symmetries or this non-ievertible topological defects in lattice model in two-dimension. So classical lattice model, a statistical lattice model in two-dimension by the work of Asen and Fentley, the sequence of two papers, which spells out concretely at the level of a lattice model this non-ievertible symmetries for the special cases like the Ising lattice model. And they also discuss some generalizations to I think paraphernal theories. So before, there were always examples in this RCFTs and lattice model related to RCFTs. And what I was saying here was the recent development that put this previous development into a more general framework that started with the work of this gentleman and also around 2017. And this puts this discussion of topological defects or non-ievertible symmetries in two-dimensional quantum field theories into a very general axiomatic framework and consequences of such symmetries for RG flows in two-dimensions. What is discussed in many other papers, including this one, as well as this one. I'm drawing work with Ryan Thorgren, as well as this nice paper from Komagowski and Company. And that's the development in two-dimensions so far. As you can see, this is very recent, very rapid developments from 2017 all the way to 2021 and today. Thoughts are unfortunately running out of space, but there are further developments in discussion of two-dimensional invertible symmetries even this year. And going beyond, now I have discussed the two-dimension. The natural question is to ask if such non-ievertible symmetry also exists in two-dimension. And the answer is also yes. And there are many exciting work in this case, starting from last year, in particular this work from Troy Cordova, Shin Shao, Lam and Shao, sorry, which will discuss a version of topological defects which generalize those discussed in these papers to the case of four-dimensional gauge theories. In particular, the Yang-Mills theory with orthogonal gauge group. And similarly, for super Yang-Mills theory and for super Yang-Mills theory with unitary gauge group. Turns out that such non-ievertible symmetries show up in those contexts as well. And there was also a related paper that appeared around the same time discussing similar and also different topological symmetries, non-ievertible symmetries from topological defects in four-dimensional theories, as well as in three-dimensional theories from the paper from Kaidi, Omori, and Zhen. In particular, in these papers, they also generalize the results obtained from these papers on constraints from these non-ievertible symmetries, RG flows, to the case of four-dimensional theories. In particular, they can prove statements like that in the four-dimensional gauge theory with orthogonal group, for example, where these non-ievertible symmetries are present, you can argue that any RG flow preserving this non-ievertible symmetry has to end at a non-trivial IR phase. Saying, we're now in trivial constraints RG from these instances of non-ievertible symmetries. And in some papers that just appeared, I think, a week or two ago, there are further generalizations of all the framework that I've already reviewed over here. So there's a question you can ask here, already here, is that there are two branches of generalizations, which I mentioned. One generalization from these topological operators is to consider higher form symmetries and higher group symmetries, who are all, who are still invertible, okay? And another generalization is to talk about this non-ievertible symmetries. And you can ask if there's some kind of more general structure that put this together, okay? And this is some working, that's work still being developed, but there's already a nice paper on the subject that's started to discuss in this general framework from these authors, appeared a few weeks ago, which discuss this kind of larger framework that's known as higher category version of the non-ievertible symmetries, okay? So that's a brief historical walkthrough for the developments in the field of non-ievertible symmetries over the past decade. What is the general lesson that we learned from all these developments? The first lesson is that quite on the contrary, before you may expect, because this is such a drastic departure from the usual group symmetries, which is ubiquitous in quantum systems, you may think that these symmetries are exotic and rare, but quite on the contrary, this non-ievertible symmetries are also ubiquitous. So let me give you two kind of intuition for why they're ubiquitous. The first intuition is that almost all known CFTs in D equal to two have them, okay? In fact, the only two DCFT, which I know that does not have a non-ievertible symmetry is the SU2 WS model at the level one. Second, there's a very nice recent work from Tombatakis and Devnachary, which explains that in dimension bigger than two, the non-ievertible symmetries are as common as non-anomalous higher form symmetries in these theories. So in other words, if you already accept that the higher form symmetries are interesting objects of study, then we are behooved to also study this non-ievertible symmetries because there's universal constructions as shown by these authors that gave rise to non-ievertible symmetries by the mechanism of condensation, okay? As this authors put it, higher gauging, which is a way to gauge this non-anomalous higher form symmetry on some hierarchical dimension, some manifold space time to generate non-ievertible in the same theory without changing the theory, okay? So higher gauging is different from ordinary gauging in the precise sense that it does not change the theory, okay? But they allow you to recover, to discover that you have this non-ievertible symmetries in the same theory, okay? Once you have this non-anomalous higher form symmetries, okay? And because of that, it has been active field of research, okay? And in many efforts, again, identifying non-ievertible symmetries in quantum field theories, okay? Particularly in non-quantum field theories, and deducing the dynamical implications, some of which we will review in this lectures. And then over this kind of games, it's what we played in the context of symmetries, even before we talk about, for the ordinary symmetry, even before we talk about non-invertible symmetries. So it's really running the old game of constraints from symmetries using this new tool, this non-invertible symmetries, okay? But more crucially, from these exercises, we want to find some universal patterns, such as a notion of anomaly or this non-invertible symmetry. So anomaly is a very powerful universal pattern that was observed in studying constraints from group-like symmetries on phase diagram of quantum field theories and RG flows, okay? Having an attribute anomalies greatly constraints where the RG flow can end. So you want to find similar patterns in the context of non-invertible symmetries that you give us some general notion of anomalies and they are, as we'll see in this lecture series. So that is the motivation for these lectures. Let me now give the outline for the rest of these lectures. First, we'll review this notion of symmetries as this general notion of symmetries as topological defect operators, okay? Secondly, we'll graduate from this general framework and focus on the case of non-invertible topological defects which define non-invertible symmetries and we'll see that the mathematical structure that replaces group in the context of non-invertible symmetries will be a fusion category. This is precise in the case of one-to-one dimension and in the higher dimension, it will be some higher category. Don't worry about this word if you have not heard about the category before. We'll make all these words explicit and if you want to refer to this as this mathematical object, you can just refer to them as non-invertible topological defects. That's equally good. And thirdly, we'll discuss or I'll provide for you a toolbox for identifying topological symmetries, to identify non-invertible topological defects which give rise to these non-invertible symmetries, okay? Which is essentially the tools I will use in these papers to discover many of the non-invertible symmetries in well-known theories, well-known CFTs and was further generalized in these papers to identify similar symmetries in higher dimensional theories. Okay, so the tools that I will discuss will apply both the two-dimensional case and the suitable generalization will apply to the three and four-dimensional case as well, okay? Lastly, along the way of these discussions from time to time, I'll discuss dynamical consequences of the non-invertible symmetry that we identify. Okay? And I should say that given this list of recent development, I think most logical way for me to proceed is to focus on equal to for the generalities because this is the case where the axiomatic framework for this non-invertible symmetry is mostly well-developed, okay? So the axiomatic framework for these symmetries are still developed, still to be developed higher dimensional case, although there are already hints for some structures, okay? But to introduce these symmetries in a more concrete setting, I'll focus on the equal to two, okay? And I use these examples in the equal to two-dimensional theories, non-invertible symmetry in two-dimensional theories to illustrate the main ideas, the general structures. And hopefully this will provide you with the basic knowledge to follow the recent literature that discuss the higher dimensional cases, okay? And I think this is the moment that I should also emphasize the references I'll be using, using mostly in these lectures, I'll be this paper and this paper. But the other paper that's included in this historical outline provide other references for further reading into the subject, okay? And most importantly, what I hope through these lectures is to kind of motivate you to study these symmetries and hopefully the knowledge that you gain from these lectures would be sufficient for you to start your own venture in the realm of non-invertible symmetries. With that, let's jump into the first part that is to review this notion of symmetries as topological defect operators. So let's start by, let's start slow and recall what conventional symmetries look like in quantum mechanics from the eyes of Wigner, okay? Conventional symmetries in quantum mechanics. Symmetries in quantum mechanics are defined as transformations of states in a hubris space and similarly of operators, which define your observables, okay? Such that transition amplitudes or probability amplitudes are invariant, meaning that things like this from two states and operating insertion is invariant under this transformation. Question? Why should it not be invariant up to an overall phase up to some- Oh, because I'm talking about the amplitudes, okay? So if you're just acting on the states, then in general the state is defined physically up to overall phase and that will be related to something I'll mention next. Okay, thank you. But the key is that this phase is universal for the hubris space. So when you look at this bracket, it will cancel, okay? That phase is known as the protractive phase space. I think we're, as a refinement of the original discussion of Wigner, okay? This is what I'm gonna say next. So this is the definition of the symmetries but Wigner in 1930s axiomalizes in terms of axiomatized the notion of symmetry in terms of certain operators in quantum mechanics. That is, this symmetries, okay? They're equivalent to two notions or two kinds of operators. It could be either associated with a unitary, linear operator acting on hubris space, a anti-unitary and also anti-linear operator on the same hubris space, okay? And the latter case is not important for these lectures. It will be relevant if you include say time reversal symmetries, okay? And because symmetries, thanks to Wigner is associated with this unitary operators, okay? Focus on the case without time reversal activated, okay? It's associated with unitary operators. Naturally, there's a notion of group, okay? In particular, this unitary operator, let me call them UG, okay? Where G is element of the group, okay? And the group law is realized by the product of this unitary operators acting on the hubris space. Is this part visible to, for G and G prime inside the group? And this is just the group modification law for this group elements. And this unitary is acting on the hubris space, okay? Or it states in the hubris space. In general, as follows, okay? And acting on operators by conjugation. And the, related to the question that was asked before, the representation of the symmetry operators acting on, sorry, the symmetry operators acting on the hubris space in general does not form a representation for this group representation, linear representation for this group law, but instead a project representation that is related to the phase ambiguity associated with this state in the hubris space, okay? So far, we're just talking about general quantum mechanics. We haven't used any fact about locality, something that's crucial to the quantum field theory, okay? Let's move on to quantum field theory. In this case, we have the Northo theorem. The constraints, while the structure of the symmetry is more stringent, in particular, the Northo theorem, Northo theorem, implies that if you have a continuous symmetry, a U1 symmetry, which is a charge symmetry associated with the electron, for example, it will imply the existence of a special operator in your quantum field theory, namely, a Northo current, which is conserved. And the Unitary operator that Wigner introduced that applies for general quantum mechanics can be reconstructed from this conserved current. In fact, there's something even more elementary you can define that is the notion of a charge. So, for example, for the U1 case, a U1 symmetry of an electron, and this will be measuring the charge associated with the electron, and it can be defined as follows. It's an integral over the spatial direction at a constant time of the zero's component of this current, okay? So it measures the charge of a state that is specified in your quantum field theory by quantizing the theory on the constant time. And this object is T independent as a consequence as the conservation law. And in terms of this charge operator, Wigner's Unitary is just given by the exponential of these charges. The group element, in this case, is determined by the symmetry transformation parameter, alpha, in this way. So that alpha can be thought of as a Lie algebra, in this case, trivially, for the U1 associated with the symmetry group. Wigner's Unitary acts on operators. So similarly in the quantum field theory, the symmetry action on operators is also captured by these charges and the unitary's. In particular acting on local operators, the symmetry variation acting on local operators at insertion O, sorry, insertion X is determined by the equal time computation of the symmetry charge Q with this upward insertion. Okay? So we saw that in the quantum field theory framework, at least for community symmetry, we can recover all the ingredients that Wigner introduced for quantum mechanics, but we'll have something more. We have this local object, this is conserved current, and then we can define these charges. We can go one step further by focusing on the case when the quantum field theory is relativistic, which is equivalent of the recrotation to Euclidean quantum field theory with in particular a rotation symmetry, in which case you are free to orient your choice of constant time slice. Okay, you're free to choose your direction of time, and consequently there's a generalized notion of the charge operator associated with a general, not necessarily a constant time slice, but a general called dimension one manifold in your space time, denote by sigma operator D minus one of the hodge dual of the current one form. So here the current one form is defined as follows, okay? And the hodge dual satisfy the hodge dual, give you a D minus one form, that's why you can integrate over this D minus one dimensional sub manifold, and the current conservation is equivalent to the closeness condition for this hodge dual, for this D minus one form, okay? This is from the conservation. Similarly, the unitary is given by the exponential of this charge operator as before. And the way this operators, this charge and unitary operators act on your local operators is by enclosing it, okay? So for example, if you have a local operator over here with subscript Q, which will define its charges under this U1 symmetry, for example case, to be explicit for case when the symmetry is a U1, okay? In closing it by the charge operator, okay? For example, taken to be the sphere of D minus one, intuitively this will measure the charge that's associated with this operator, okay? Source by this operator. If you encircle the same operator by the corresponding unitary, okay? This will implement the symmetry transformation on the corresponding operator, okay? So in other words, you can shrink this ball and replace the left-hand side by the symmetry transformed operator. Now a crucial property of these integrals, the charge operator defined this way and the unitary operator defined this way is that they are topological. Meaning that even though in these pictures I've drawn such that the encircling called measurement one and four is a sphere, it doesn't really depend on the metric on the sphere or the shape of the sphere, it only depend on the topology. And that is a consequence of this being a closed D minus one form. And consequently, that this kind of integral only depend on the topology of the manifold integrating over as a consequence of the Stokes theorem. So what's the implication of this topological property? So these objects, okay? These objects, this unitary defines the prototypical example of a topological defect operator. The notion of topological is what I have just explained. Why it's called a defect is because it involves, it has a dependence even though topologically on the higher dimensional manifold. It's not inserted at a point, but it's wrapping some higher dimensional sub-manifold as opposed to a point. That's why it's called a defect. The fact that this topological is closely related to constraints and correlation functions in the following way. So imagine some general correlation function that you are studying in the quantum field theory with various insertions. And potentially symmetry topological defect operator which I'll call UG inserted somewhere in this correlation function, okay? And suppose that in the neighborhood of this topological defect, we have an operator insertion which we make it explicit and call it five. This, the fact that this object's topological means that if we focus on this region, go over here, save some space, we are free to deform the locus of the symmetry defect or the topological defect operator, okay? Keeping the location of insertion fixed. For example, we can deform it all the way to something like this, okay? And this equality means that we are free to replace this configuration locally by this configuration without changing the correlation function called isotopic invariance. Furthermore, if you have more than one topological defect, more than one topological defect operating inserted in your correlation function, there's the fusion, okay? Which is the characterization of the group modification associated with the unitary operator that Wigner defined in the context of the quantum field theory, okay? In this case, just to be concrete, you can think about the diagrams in drawing is the projection of the higher dimensional diagram down to two-dimensions. And if you're not comfortable with that, just focus on the case when space time is two-dimensional, okay? So the co-dimension one manifold, the sigma's always just lines for circles. In the presence of two such topological symmetry defects, topological defect operators, they're all mean the same thing. If there's nothing inserted in between, because they're topological, you can zoom far away, okay? And they look like a single defect insertion, which is labeled by the product of the individual group element enters here. Sorry, the same sigma, okay? So I'm taking the limit where they collide and in which case the same sigma, and given by this, okay? So, all right, so the orientation is related to how I label the defect by group elements. In particular, if they are not oriented, sorry, if they are oriented the opposite way, like this, this will be equal to the trivial line, meaning that there's no insertion. The reason is that, so this is the, is the, it's a bar, okay? You're noting the opposite orientation. The reason is that you can connect the two ends of the line. In each case, you're just a form of circle with no insertion, and that would be the case that you're detecting the charge of the identity or the charge of the vacuum, and that's trivial, okay? So this is how the two orientations are related. Now, once we understand the fusion, we can go one step further in this picture. So the, recall the usual way you determine water entities is to reduce the insertion of the, reduce the insertion, like a, sorry, reduce the current function with the insertion of this unitaries to something that depends on the charges of the insertions, okay? And then you can deduce consequence such that the charges of the insertions have to add up to zero for this current function to be non-trivial. And to do that step, you want to be able to move across this module symmetry defect away from this insertion. And there's one step involved which comes from doing a fusion of the two ends of this module line, okay? And that's equivalent to doing this, okay, locally. And as a consequence, what you find is something like this, okay? And this generally is equal for a general group, for a symmetry defect or social with a general group as the transform operator in the presence of a line which have been moved across the operator insertion, okay? And this feature, the fact that there's a simple way to move the symmetry line across the local operator insertion with no other line attached is actually a feature of this invertible symmetries and you will see how this generalize in the case of non-invertible symmetries. How many minutes do I have? We started late, right? Okay, ready to summarize this part of the discussion, this review on symmetries as topological defects, okay? So some comments, okay? The first comment is that the isotope invariance, okay? Meaning that the deformation of the observable in the presence of this topological defect operators is invariant under small deformation of the line as long as they do not cross other operating insertions together with fusion which tells you how you can further move the line across the local operator, okay? And with the information of how defects act on local operators, okay? There are three ingredients. The three ingredients completely characterize the conventional symmetries as topological defects. The topological defect operators, I'll use them interchangeably. For continuous symmetry, because the presence of the current which is something more fundamental, this looks like a complicated machine to produce something that's well-known, okay? So why is this useful? This viewpoint of topological symmetries as topological defect is useful already in the context of invertible symmetries when the corresponding group is discrete. In which case there's no north or current and this topological operators, the topological defects should be treated rather than as the fundamental objects. In particular, you can just like in ordinary continuous group symmetries, you can gauge this symmetry by coupling theory to an attribute gauge background by turning out gauge connections. In this case, the topological defects give you a natural way to couple the theory to a discrete symmetry background. Let's focus on the case of d equal to two just so that I can draw explicit pictures. In which case the topological defects labeled by g to specify the transition function, the discrete gauge field, a discrete g background, is essentially just determined by specifying transition functions from patch to patch. So you can triangulate your space time and the background is specified by transition functions from different patch, like from the left to right. Particularly the transition function, for example in this picture, will be given by this g element labeling the topological defect. And furthermore, if you take two topological defects, instead of doing this fusion, complete together because they're topological, you can imagine fusing just one ends of the defect. And this gave rise to, and you do not break the topological property through the way. And this will define for you some topological junction. Okay, everything is topological. And together they allow you to form a general network, and this is equivalent to a generally discrete symmetry background. Okay, so you can think about this as a patch of the full network that you tile on your space time. So here, this is topological because as all these junctions, you are using this basic building blocks. And a very nice thing about coupling a theory with certain symmetry to the gauge field background is because it's a, sorry, it's a very nice thing because this is a useful way to detect anomalies associated with this symmetry. We're precisely to hold on the associated symmetry. Okay, so it's not a sickness of the original theory with this global symmetry, but it's a feature and it can be detected by gauge non-invariance once you couple the theory to gauge background gauge fields. In this case, in the C3 case, this topological defects. Detected by change in the network topology. So the basic ingredients I was talking about here isotope invariance and stuff like that ensures that topological network is invariant under small deformations of all these vertices as long as they don't cross each other. Okay, but if you focus on a patch of this diagram, for example over here, okay, there's a way to change the connectivity of this diagram by performing a gauge transformation. For example, in this region, by doing a gauge transformation associated with say G1. And by doing that gauge transformation, you go to another diagram that with a different connectivity, doing this way. Okay, so all these internal legs are unambiguous because of the topological vertex condition. The third leg is always determined by the labeling of the other two legs. Doing this gauge transformation essentially removes this line by reconnects G1 with G2. Reconnects the topological defects labeled by G1 to G2, giving rise to this diagram. And if there's no phase associated with this gauge transformation, that's equivalent to saying there's no anomaly. So once again, anomaly is diagnosed by coupling the theory to the symmetry background, meaning a gauge field background for that symmetry, in this case, a network of topological defects. And if there's a phase coming from doing a gauge transformation, then that signals the anomaly. In this case, it's equivalent to changing the topology of the network. And this phase is generally denoted as this, and this, as we're saying, encodes anomalies. And this phase is not arbitrary, just like usual anomalies are subject to vestuminal consistency conditions. Similarly, this anomalous phase are also subject to a finite version of the vestuminal consistency condition. And this leads to the constraint that this phase is something classified by the group cohomology, H3G, with U1 coefficient. The co-cycle condition is coming from the consistency condition. And the fact that you are quotiently out by exact co-cycles is related to that, the fact that at this vertices, there's some choice of gauge, there's some gauge freedom. And that gauge freedom can allow you to remove part of the phases. But the part of the phase which cannot be removed by this ambiguities over here, which is related to local counter terms in the context of continuous symmetries, is precisely this anomaly, okay? And that's captured by this non-trivial cohomology classes. And you will see generalizations of this for non-invertible symmetries in the later lectures. Let me end here. Okay, let's thank Yifan for his lecture. Recording stopped. And now we can move to our discussions.