 Okay, we'll let us start. So for the first lecture of the second day, we are happy to have Laura Dane and she will continue to talk about Celeste's one please. Thank you very much. Good morning, everybody. Thanks for coming. So yeah, we will keep exploring this topic of the infrastructure of gravity in flat spacetime. Let me start by summarizing the main point that we saw yesterday in the first lecture. So yesterday I tried to present the rich fact that the symmetries of asymptotically flat spacetime are way bigger than the ones of Minkowski asymptotically flat spacetimes. And we saw a precise ansatz for the metric, which is asymptotically flat as one approach is the boundary, the null boundary, a future null infinity. So the symmetry of exactly flat spacetime are given by the Poincare group, which consists of four translations and six rounds transformations. Now the symmetry group of asymptotically flat spacetime is an infinite dimensional enhancements of Poincare given by an infinite amount of what people have been calling super translations, which are spanned by this arbitrary function of the angles t, which comes in this new components of the infinitesimal vector field. And similarly, there is a sense in which we can enhance the Lorentz part of the Poincare group to an infinite amount of super rotations spanned by conformal killing vectors. So we have two copies of them in this complex coordinates, y, which is depends only on z and y bar, which depends only on z bar. So these are conformal local conformal killing vector on the sphere. So this is what is known as the extended BMS group, extended because in the first place boundary master and sex only found these super translations. But now people thought it was a good idea to extend the Lorentz part to also an infinite amount of symmetries, because in conformal theory, we are very used to having this zero type of symmetries. So this is what we saw yesterday. And today I want to tell you basically on the importance of these symmetries for the scattering problem in flat spacetime by showing you that these symmetries provide an infinite amount of conservation loss for the S matrix. And so this is the relation we will start looking at the BMS and the S matrix. So this is a result due to Strominger is now already 10, a bit less than 10 years ago. So the result can be stated as follows. BMS symmetries imply an infinite amount of conservation laws, which constrain strongly scattering amplitudes of massless particles in flat spacetime. So the original references for these papers and to show that, of course, I will not have time to go into the details of all the steps required for establishing such a correspondent, which is a neat mathematical statement. But I want to tell you what are two main ingredients you need in order to achieve this. So the first ingredient is given by BMS charges. So far, I've been telling you about symmetries. But as you know, from Nutter's theorem, when you have a symmetry, there is a conserved quantity associated to it and the BMS charges are nothing but the Nutter charges associated to these asymptotic symmetries. To the BMS, of course, asymptotic symmetries. So you might, I mean, actually, this topic of building charges associated to asymptotic symmetries is a whole topic on its own in GR. So it could be a lecture just about that. There are many techniques involved, which trace back to the work of Wald, and Wald and Zupas, and other people in the 90s. It's known as the cover-and-face-space formalism. If you are interested in knowing how building these charges, I would refer you to this PhD thesis, which, which I think contains the most accurate and state-of-the-art prescription for BMS charges. But let me just give you one result, what this charge is for super translation. So I will denote by qt, the charge associated to super translation symmetry. And this is just taking this simple form. So where it's basically a pairing between this function of super translations and the mass. You remember this M was the bandimass aspect that enters into this one over our expansion around flat space. And this integral is over the sphere located at the past of the future of null infinity. So I will have to define this because I don't think I did it last time. So you remember we had this null boundary, future null infinity, square plus, which was parametrized by a retarded time u and coordinates. Now if I take u goes to minus infinity, I arrive here at the past of future null infinity. And similarly we had, I mean, I didn't present the things in advanced coordinates. But as Francesco was asking, there is all these stories also valid for incoming particles. And in this case, everything is labelled in terms of this advanced time v which is t plus r. And taking v goes to plus infinity. You land on this location which people denote by prime minus plus. There is the future of the past boundary. So v goes to plus infinity. Okay. So, yesterday I've been writing everything in terms of just one components of this boundary in terms of as cry plus. So strictly speaking, I have defined for you a one copy of the BMS group living on the future. And I will denote by with a plus. Yes, coming the charge living on, meaning on future null infinity. Hi, one question. What is the qualitative difference between the past of the future null infinity or future of the past null infinity, and the special infinity I zero. And so we'll come to that but so I zero is here. And on this diagram, it looks like they are the same thing. But actually, it's just an artifact of this pen rose diagram. Actually, these locations are definitely far away. So, and actually, more precisely, special infinity is not included into the conformal compactified space time. And this is actually a very important point and I will come back to it right now. About how do you basically, what happens when you cross is spatial infinity, but yes, patient infinity is not neither sky plus minus neither. Not at the limit point of this. Yeah, it's really speaking isn't I could not even draw it in this diagram. Thanks a lot. Thank you. I have told you about BMS here on the future, but there is another copy of BMS symmetries on the past. And this charge will denote the BMS charge living on the future. But there is also, there exists also a charge living now on scribe minus plus. So the first ingredient is the actually to be able to construct these charges. And now you, you probably see where I'm getting at. What I'm getting at is that in order to define the scattering problem in space time. So we need to know how, how the quantities that are defined at the past are related to the one living in the future. So in order to make the statement true and actually to uncover that BMS symmetries provide you a symmetry of this matrix, a fundamental ingredient, and this is, I think the lack of observation of the statement was because nobody really realized what we needed to do to match these two to disconnected pieces. And some major propose that there is a precise way to map the quantity of the space space time from the past to the future through a so called anti-podal matching at anti-podal matching at I zero. It's just calling anti-podal matching. So this anti-podal matching will give us a junction condition or gravitational scattering problem. So I'm focusing on massless scattering, which is needed to relate the in incoming states to the outgoing states. So without such a matching, you cannot talk about scattering basically. So this anti-podal condition. Well anti-podal in the just in the sense that we will relate quantities on the celestial sphere as cry minus plus with the one at the anti-podal point of the sphere as cry plus minus. So to do that. So if you can consider you remember I use this Z bar coordinates. So a convenient choice for discounting coordinates is to say that a point Z and Z bar, the coordinates at that bar will represent a point theta five on scribe plus, or the anti-podal point pi minus theta pi plus pi on scribe minus. So this is just a way to choose my coordinates here in such a way that when I will give you a Z bar, it will either represent a point on scribe plus or the anti-podal related point on the past. So it's just a way, it's just a choice for these coordinates that will make this anti-podal matching way easier to write down. Because now basically this coordinate system implements naturally this anti-podal relationship. So what it means is that the condition that we will impose is that the boundary mass aspect, so it also depends on you, evaluated at scribe plus minus will be equated to, well okay, you go to minus infinity so I will not write it down because it's not a function of you anymore. So that's why I'm clicking on this location will be equated to the sphere to the corner at scribe minus plus. So that's the anti-podal matching condition we will impose. And it's really crucial. Because not only without that, you cannot make this statement about this metric, you cannot make any sense out of that. But this condition is actually the one needed and very natural to a certain amount of extent. For instance, you can see that this CPD and Lorentz invariance, and they have been a long literature in general relativity about this sort of matching condition. So it's a very difficult problem to see in general how the field will behave when you take you goes to minus infinity. It's a it's a delicate problem in GR and how they evolved from the past to the future. So, yes, yes. So, in general, M is different before the limit so you have M use easy bar on scribe plus and some M to the V is easy bar and then on the limit you do this or the the function of the function as a function of you and me is supposed to be somehow related. So why you use the same symbols. It's very good. So, indeed, in principle, these M, the M here is this one of our our term is in the GBV components of an expansion of the MS type that's cry minus. Because these are two unrelated quantities. But now, indeed, I am making the claim that if you want, yes, this in principle would be a different function. But now I'm identifying this function at this, at these corners. So is there a physical intuition between why you you want this condition like understand maybe the math gets better but Yes, I mean you can you can see that if you have if you are a non interactive particle and you you enter the space time from the past infinity you will cross. You will cross the square the sphere at scribe plus at the antipodal relation. That's somehow natural in this sense. And it's a really non trivial statement about the structure of of the space time, which is, so I want to emphasize this thing is, yeah, has not been fully proven but it's, it's proven in a posteriori when we will see that it's equivalent to the Weinberg stuff theorems so it gives a proof posteriori that this matching should hold. So maybe you also mentioned that this is a, it seems that it is an artifact of the Benros diagram that these two places are the same now, but they are not the same. No, they are not. Okay, so for other quantities. This matching is not true. Other quantities what you have in mind like other terms in the expansion, for example, yes, yes. So I will actually I will also impose this matching. This matching on the other function. Here I'm doing everything for super translation so this will be enough, but it's, it also matches. Remember this angular momentum aspect we saw yesterday this will also be matched and also the the shear, this gravitational shear will also be required to obey a similar antipodal condition. Okay, but so if you match all the quantities there, it is not like if you are imposing that these two places are the same if you can analytically continue all the, all the function in zero. It seems like you are identifying this to. The quantities are not strictly speaking defined are zero this. There was an expansion over around the null hyper surface. And if you want to give me a data on like, there are people do on some kind of time like Cauchy slice. So you have to do a total and you want to describe an assembly expansion around a special infinity, you will have to resort to a totally different gauge and, and stuff so these are a lot of work by Friedrich and other people where you have to resolve a space like either you can put a system of coordinate and foliated by hyperbola, or you can use another gauge where you, where you foliated by what you describe as your as a cylinder, there are many ways to do that but they are really different. Expansion around I zero. And since I would be interested in mass scattering anyway, no masses particle goes there so it's, we will not need to do that for the purpose of this lecture, but it's an interesting story and people have been looking at BMS symmetries around I zero. Okay. So now that we have this matching on the face space. I'm just, I will not enter into much details but I'm actually restricting myself to a certain class of space times which are the so called crystal do look like our mind kind of space times which obey certain conditions. I don't want to go into too much technicalities about that. But if you're interested about the assumptions behind this. Let me, and the other break during the discussion. So, the, this antipodal condition will break the combined BMS class. For example, we have two copies of the symmetry one leaving the future one at the past, the BMS class and BMS minus action, which act in principle independently on the future and the past. It will break it down to a diagonal subgroup that preserves this antipodal condition. But basically what he will amount to do is it will amount now to identify the super translation parameter at the, at the past of the future with the one of the future of the past. So again in principle as Marco was making a point this T function is in principle a very different function but now I'm I, if I want to preserve this matching I will, it will induce this, this choice, which basically fixes one frame in terms of one other. And so we have this symmetry matching now, as we are going through around I zero. And now we can put the two ingredients together to show immediately that what will this matching will amount to do is, it will amount to equate the BMS charge at the future and at the past. And this is the statement that the energy is conserved this we know, but now the super translation enhancement is telling us that the energy is conserved at every angle. And the quantum version of this equality is a wide identity. associated to super translation symmetry is the statement so if it's a symmetry it should commute with the S matrix. So this will be the form of the world entity that is implied by a super translation symmetry. And then, and I will tell you briefly. How you can see that this word identity is actually nothing but a very well known theorem and quantum theory, which is Weinberg, soft, gravity. Soft gravity is a gravity whose energy is taken to zero. So, I will present the main things we need to, in order to show this sort of statement. Getting at a more defined, more affinated definition of massless scattering in flat space where I will introduce the thing that we are more used to now in quantum theory with three fields creation and relation operators and, and most of the work to show these identities actually translate these two languages the language of general activity, and something symmetry and charges with the one of the quantum theory that we are more used to. There any question on this conservation. Yes. The reason that we don't need this condition for simple flat spaces scattering is this that me zero or is there something else. Because I think there's a discussion in Weinberg spoke about matching the generators of Lawrence group in the for the in a state and out the states. Is this related to this. So the first part of your question. No, don't keep the microphone because I'm not sure understood the question. You asked about what is zero if this thing doesn't hold what the quantity and the body mass. Yes, so what's the question is whether this holds. This condition is trivial for simple flat spaces scattering. I mean why why we don't bother to talk about. Yeah, no, because we are we're interested into, you know, these are something that space phase space with. And how this bond the mass I mean one bird doesn't care I guess the bond the mass aspect and the angular momentum. This is just a different point of view. It's just really two communities talking different languages and working in a different setup. And so I came into the realization that they were actually describing the same physics in different ways. So, let's, let's go to a mass scattering in flat space if there is another question just interrupt me. So, I will sometimes okay I will write down very fast. Now I will be using. I mean usually we we are using not bond the coordinates but Cartesian coordinates. And I just want to write for you how these two things are related to each other. Just a simple change of coordinates but again most of the work has to do with translating things into this particular. So this is just a change of coordinates from Cartesian to this bond the coordinate I used yesterday, and this maps the to the line element. We had yesterday was gamma zero the wrong string metric. So a mass less particle of energy Omega. Cross the celestial sphere. Yes. At the point. So now I will use a different notation. I will use a different notation I will use different letters to the to to distinguish the, the, the coordinates that and that bar, and the, the one the coordinates that labored the moment of the particle, but this w w bar, not to be confused. Omega with the w. So I hope this will not lead to confusion. I have, you can parametrize the for a moment of the particle in you on shell mass less particle, like so, where omega time q mu where q mu is a null vector, which can be parametrized like so. Because of these angle w and w bar so this parametrization is not unique, of course, but this is very convenient. When we want to match things with bond the because you can see, when you see how you go from Cartesian to bond the coordinates. This parametrization looks not so not so crazy. The particles. So I will be talking about spin one and also spin two particles, mostly about gravity tones, but they have, they can be have a polarization vector epsilon mu with the plus denoting positive electricity, which in terms of this null vector parametrization can be taken to take this form. And similarly, for negative electricity, epsilon minus can write it like so. So you can check that these guys satisfy these relations. So they are suitable for realization vectors normalized. And, okay, very good. So this is just mostly introducing notation. So what is important is that we have much less particles, have an on shell momenta, and I can write, instead of writing p mu I can write everything in terms of three numbers. Omega the energy and the point, the w and w bar, at which the particle will cross the celestial sphere. So this is the most natural thing to do. So we will want to talk about. Let's take mass scattering of gravitational field so at late times, the gravitation, the gravity tone will become free and can be approximated by the mode expansion that I guess you're all familiar with. So I'm considering outgoing gravitation, which is a very, very late time is free. And this is the usual textbook formula that you can find. But I have to write down just to introduce the dimensions and notations, where you have the usual creation and alienation operators. Exponential at each, I to the p dot x where x are the Cartesian coordinates. And you have a sum over the two electricity is pleasant minus level by this alpha a dagger e to the minus I p dot. And now we have a polarization tensor. It can be plus or minus electricity, which is, we're choosing a game so that it can be written as absolute new times epsilon new. And a out and have dagger satisfy the usual. Commutation relations. Write it fast. Prime dagger is delta alpha a b times to omega, mega is P zero to pi cube. And delta three of P minus P prime. Okay. So I'm guess, I'm guessing you you have seen this expression before and now the thing we will. The main thing we have to do is we have to express this expression. You have to write down this expression in bone decoordinates, you are as a bar, take a large radius expansion. If we want to match it with the BMS asymptotic expansion that I wrote yesterday. And then what you can show is that it will take the following form. But remember that what I introduced yesterday, this shear function. So this that this encodes the two polarization modes of the graviton was the piece in the Z bar components of the metric. So to extract it I need to take the perturbation age divide by our. The limit as our goes to infinity of this quantity. So this is just a definition now it's it's an operator you see I have operator representation for for this object in terms of creation and elation of gravitons fields. And so this is a definition if you want but it will match it will be identified with the shear in the BMS expansion. And now you need to do this business right down H in Bandy and take the large R limits. And there is a little computation to do that. I will not review here. I'm just writing the result. Okay, minus. Okay, minus out. This depends on omega. That's bar, you have a dagger and you have exponential e omega u. So to obtain this. There is. You see you need to write this in coordinate in Bandy coordinates and take the large R so they will be a rapidly oscillating phase. In the exponential you need to take stationary phase phase approximation that will localize this exponential. And it's done in detail in the exercise in the book of Andy storminger that I've written the reference yesterday is a stronger lecture notes, you can see the detail in exercises. But the important thing I want to emphasize on is that doing this larger expansion will localize the point, the direction in momentum at on the celestial sphere so you see before I had. I made a distinction between the Z bar coordinates of this of a point in space time and the point w w bar with the particle is pointing towards the celestial sphere. Now I'm doing this large or expansion. These two angles will be identified so I have only now something that will depend so omega omega bar have been identified to Z bar. So this little computation that that does this is just saying that we are identifying the point on the celestial sphere with the particle courses as quite, quite pleasant quite well here I'm talking about outgoing states as quite fast. So I can pose here takes there is some question on the notations or on what we are doing. Sorry Laura a very elementary confusion so in typically when we have a particle states, we take say gravity on with the definite momentum. So here, we are integrating over this energy so what, what is the relation with the usual particle states, can you repeat again. So here I'm integrating over. Yes, if you want this is the expressing the state in a direct opposition space, which is related to the momentum space by a Fourier transform. It's because you see we're. So basically this is basically a Fourier transform which inverse the energy with the time because I need to map precisely I need to express this in this basis to map it, map it with the bond the bond the story. The state is going to be say I can state of a boost, or what the associated states will be in a state of boost, or. So the associated state. So usually, let's see, yes, so usually they would be in if I were in the position in the usual. The momentum space, it will be energy. Again stays. Now if you want they are the Fourier transform of that. So they will, they will. The transformation will be a bit. The thing that they will diagonalize will be a little bit different. So then, eventually I will go to another basis I will not stay very long in this you basis if you want. I will go to the celestial basis where I think will transform nicely under boost and Lawrence transformations. So yeah, it might be a bit and not not familiar to write to write the the gravity on mode, like so, but precisely we, most of the things to establish the connection between these, these two topics is precisely to sometimes go to some basis which are not some from one point for the other. And then I will go to a different, totally different basis. Hope it clarifies the things. So this is for a gravity ton and you have an analogous expression for a spin one particle. Let me. Yes, I mean it's it's very similar but you, you prefer that. So, well okay I will, I will, I think I'm running out of time so I will not, I will not write that but basically you have a similar expression for gauge field. Okay, so there's that components of that. And I'm selecting the leading piece in the large or expansion, which now normal follow for each field is to start with R to the power zero. So I just have that here instead of absolute apps, the, you will have epsilon. Z, instead of ZZ. Yes, maybe I should say what this this hat here is one of our square times this epsilon. I have defined here. Okay, so this. So what, what is this expression means again to to reiterate. So, let me call this expression one. So here you will really have the same expression, you can find it into and this book, epsilon hat is epsilon Z divided by R. The powers are a bit different than from gravity but okay so let me write some words. So these are, these will define for a boundary operator, leaving on on scribe. So these two operators, one and two. What they do is they annihilate a positive electricity, grab it on or photon, grab it on for expression one button for expression two, and create a negative outgoing, grab it on or photon. So we have retarded time you, and at the point that the bar on the celestial sphere. Okay, so here you have CZZ so we describe one electricity. And a similar expression for CZ bar Z bar with some, some details change about the polarization and so on and so forth. So that's this makes sense. I hope. So this this we have a sort of a boundary operator that lives in in this null. Hyper hyper surface. And now what I will tell you is that the zero mode of the field strength associated to these boundary operators once inserted into the S matrix actually leads to this universal formula, known as Weinberg's theorem. So I will not have time to review in details. So theorems, I will just sketch very schematic way, what these are. So this trace back to the sixties also so it's funny to notice that it was at the same time that bondi master and Zack is there. They work one better than this side was was studying the infrared structure of scattering elements and other people like low burn net Gellman and others. So, a soft theorems is a statement about S matrix, which involves a soft particle. So, let me just sketch what these are, and you can go into the Weinberg's literature if you want to know more on that. He's basically saying that you have a scattering process involving a certain amount of particles, which all carry a momenta, and you add to this process, a soft gravity or soft photon. But let me focus on gravity, which will, with momentum will be written like this just I'm calling just came you. And where the energy of the, of this soft guy is taking to zero. These things will, this amplitude will factorize and will be given by the same amplitude but now without the soft one times a number, and this number is called a soft factor. So, gravity speaking, you have an amplitude with n particles. So this particle which are not soft or sometimes called hard. This amplitude will factorize as omega goes to zero. So, where as zero is the, let me write down what this is factor is a gravity as zero. So it's just, just a number, which only involves the moment of the hard particle. A pole, Weinberg's pole. So importantly, these poles goes like omega to the minus one. And this it does here is just plus or minus just plus one if the particle is outgoing and minus one incoming. So this theorem is meant is called universal because the form of the of the of this amplitude factorization doesn't depend on the nature of the other hard particle. Similarly, there is a sort of universality into this formula. It holds for gravity but also for scaring of photons and also gluons. There is a soft photon theorem a soft glue on theorem. The form of the factors are of course different, but very similar. I can, in the last five minutes, I can basically tell you the main conclusion that I wanted to get at is that the BMS symmetries is what identity that I've written before. Once we do this dictionary between Bondi and one Weinberg. They are exactly the same. The same statement. And just this will serve as a sort of a review and just it will amount. It will help me to also extend what I've been presenting to other cases. Just a summary table. We have this wild identity. If you remember this was the statement that the S matrix commutes with the super translation charge. What am I trying to say, but we have a gravity tone. We take is this. That's the bar component. We read of the shared of that. Yesterday we saw that the asymptotic symmetry of a flat space include super translations by T by the U. There are super rotations. And then Y, Y, Z, D, Z, plus Y, Z bar, D, Z bar. And the thing I'm claiming and that stronger and others proved in a rigorous way, is that this wire identity as it to symmetries gives us Weinberg sub theorem. So the leading is the one I have written here, which contains a pole. I will come back to the sub leading story in a moment. But before that, let me just say that you have a photon, you make a similar expansion. You have now the gauge version. This symmetries, which what people call a large gauges symmetry, which basically changes the field. Like so where the epsilon is an arbitrary function of the angle, and the word entity associated to that is now not surprisingly the leading of photon, which also goes like. So you get to the minus one. And how you do that, what is actually the realization that this is implementing implemented by inserting inserting in this matrix, some currents for super translation. There is this so called super translation current, which gives the following. So I'm just rewriting here, the word identity. So if you insert this current into this matrix. It will take the following form. There is an integral story here, let me just write everything and then I will explain it more. I'm just trying to collect a lot of statesman. And then we will see later that in the celestial holography problem program, these currents will be implemented in a very natural way. Let me just mention this sub leading story. So what I remarkably what people found is that they knew that there was the super rotation symmetry. And then they thought okay it seems that there is a identity between why there is a relation between symmetries and sub theorem so why don't we look for the extra term into this expansion. There's an expansion in, in omega, where now this sub leading sub factor doesn't go like one over omega but now like omega to the zero. And also in strong in your phone, that indeed there is a theorem a sub theorem associated to super rotations, which now is sub leading compared to the one with the Weinberg pole but that exists. And remarkably, these, these statements can be implemented to insert through an insertion of an object, which looks like a stress tensor. And the realization that there was so basically that this while an entity was equivalent to inserting a current on the celestial sphere, which has the exactly the right dimension to be a stress and a conformal field theory was what I think really kicked kicked off the celestial holography program when people realize that there was something transforming like a stress tensor and they thought okay maybe maybe we, we can dig better into the conformal field theory structure of, of, of flat space, and we will find some constraints on the holographic nature of flat space times by identifying more of these currents. So I will, I know I was quite fast on that so I will stop here and take any question you might have. Thanks. Okay, let us start today's second lecture. So Yifan will continue telling us about non invertible symmetries. Well back everyone. So let's continue our journey into the realm of nine invertible symmetries, particularly in two dimension couple field theories. So here I summarize of 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 notions, a mathematical framework known as fusion category, which is a natural generation of group group symmetries, taking into account their anomalies into some more general object, and which we use the later to describe symmetries in conformal field that are associated with non-invertible topological defects. So the, 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 hubris space. Okay, this is the hubris space on the cylinder, we have quantized, this is the time, and you quantize on the circle, okay, which is punctured through by the defect line. And 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 tether product operation in the fusion category side, and that allow 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 I pointed out last time, and I forgot to introduce, and that on the CFT side corresponds to taking the CPD conjugate of a given line. Okay, so any observable involving this line defects, you are free to replace this diagram with this particular orientation by our bar, but in opposite orientation. Question? Here? What is CFT? CFT just means that it's a composite line. So meaning that if you compute a correction function where inserted direct sum of the line, that correction function become a sum of the correct 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, the 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. So we generalize defective birth space, a simple generalization of the both, where you have more than one lines puncture ring through the circle on which you quantize the theory. Okay, so you have L1 bar here, L2 bar here, and L3 here. Okay, and this picture is related to the picture over here, just by doing radial quantization on this circle. In the radial quantization, you map this point like operator to a state on this circle punctured through by this defect lines. And the requirement of this object being topological follows down to the condition that you want to require the corresponding states in the subspace to have h and h bar equal to zero. Just like I did in this operator, so it can move it around, costing no energy. In this radial quantization picture, the choice of the morphism, how does it reflect on the cylinder? Right, so it follows down to a particular vector in this junction vector space. So the junction vector space is a subspace of the super space, restrict to the eigenvalues of L0, L0 bar equal to zero. In general, that could be a dimension bigger than one vector space. In general, it's a vector space, so this V, if it exists, it's a vector in that space. Okay, 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. Sorry, when you say trivial line, you mean identity line? Yeah. Okay. Are you allowing for non-trivial junction? Let's say local operators. Ah, very good. So here, and I will get to that, but just to answer your question quickly, so I think you are imagining the case when, for example, one of the three external axes have an entity. Also all the three. If all three are an entity, then the implicit assumption I'm making in the CFT is that there's a unique vacuum. Okay, 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 book. So 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 an entity operator taken to the line. Okay, like an entity operator in the book taken to the line. Okay, thank you. Any other questions? Okay. So these are the basic defining data, and there's one more ingredient. These are this F symbol. Okay, I will not redraw that diagram again. But essentially, like a matrix that keep track of the change of basis of the change of basis in the junction vector space associated with four external X, L1, L2, L3, L4. There are two different ways to represent this junction vector space by factorizing a four-fold junction into a pair of three-fold junctions, and that change of basis matrix is captured by this F symbol. And F symbol is subject to this Pentagon relations, and that physically just correspond to a consistency on this change of basis operation. Okay. So let's use this kind of the general dictionary to do some simple consequences. Okay, in particular relating to the fusion rule that the specific features of this future. Okay. 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 into constable or simple. And if this property of the of the fusion coefficient, we can be which can be interpreted from the point of view of this vodka junctions, and this will come to that next. So let's first introduce the notion of simple. Okay. A simple to body defect line is defined by having a one-dimensional junction vector space between itself and this conjugate. So it's easy to see that this condition implies that this this this defect line is decomposable. Okay, let's let's see what's why that's the case. First of all, for any, for any TTL for any defect line. Okay, it's easy to see the dimension of the junction vector space between itself and this dual has to be bigger or equal to one, just because the junction be something like this. Okay. 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 book to the line. And this give rise to the natural junction vector associated with L. And L bar. Okay, so you have to draw the L bar. Okay. And I didn't see. So, if L is decomposable equal to L one plus L two. Okay, this implies the dimension of the junction vector space, possibly bigger than two. This factorization, I mean this kind of distributed distributive property associated with the hubris space, which trans defect hubris space will translate to distributive property for the junction vector space. And there's at least two contributions coming from the fact that for each L one and they are two. There's a non trivial to multiple vertex with identity. Okay. So just from this. To conclude, from this definition that simple TTL must be in 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 fission category while I'm talking about here so I'm happy to talk to you afterwards about that but let's put a moment, focus on the in decomposable also implies this condition. Okay. Now, once we have introduced the notion of simple TTL, we can talk about the fusion product of this simple objects. So, writing the similar relation over here over there. Now, there's some coefficient that appears in the decomposition of the tensor product of two simple TTLs into all the simple TTLs that you have in your theory that generate all the nonverbal symmetries. I claim this coefficient is equivalent it's equal to the junction vector space dimension of the junction vector space. And I have explicitly associated with this three lines, IJK, okay, defined over here. Okay, so how do we prove this. So let's focus on the simple case where I and LJ 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 one plus other non identity simple lines. So let's go back to some degeneracy, want to show. Okay, so how do we show this. Let's consider the torus function function. So imagine the square represents the torus in the coordinates which have two pipe resistive both in the this direction and in this direction, think about this as time and space. So let's go to the opposite side and inside. And we insert this to module D5 lines as follows. Okay, and because of the definition of a dual coming from the conjugation, I can replace this also by the original outline but revert its orientation. Okay, the torus has a module that's tau, as usual, the control the shape of the torus. Now there's two different ways to compute this 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 infinity, which means the time cycle becomes infinitely long. In that case, the torus function projects one essentially everything propagating this in this time direction gets suppressed, and the only dominant contribution of coming from the vacuum. Okay, states of dimension zero. H equal to h bar zero states. Dominate. Okay, quantizing on the circle. And this picks out precisely the contribution from the junction vector space. It's multiplied by this divergent factor that depend on a center charge and q is equal to e to the two pie tau as usual. Okay. The way to represent the modulus of the torus. Okay, so we see this quality shows up, which is what we want to relate to the efficient coefficient here and here in the special case. Sorry. Sorry, sorry. Good. Good. Thanks. Thanks for pointing that out. Yeah, I was going to I infinity. Very good. We can proceed in a different way. Okay. Because we're on the torus. And we have this parallel defect lines, we can apply the fusion of their office lines. Okay, the fusion is over here. And we use this postulated expression. We have multiple terms. But suppose this term is n, okay n times one with some coefficient. You have n times the contribution on the same corresponding function but with no line asserted corresponding identity line. Okay. And then you have contributions coming from all the rest. And then you have some of this some of this right hand side, which all involve non trivial defect lines inserted in this time direction. Okay. Now, you take the limit. And then you have identity again, just like before this, this party function dominates over this one. Why because we have the type of condition that tells you only identity operator can end non truly can end on the dimensions your operator. Okay, so this is completing the part the thermal function in the hubris space punctured by a single DDL. Okay, there's a non trivial ground state energy that is above minus C over 24. So if you make this case, the currency energy is saturated saturated at minus C over 24. So if you take the toggles the infinite limit this this guy dominates and you get precisely this expression. Okay, and equate this to you conclude that is equal to one. Now, just to generalize, we can already use this result. Okay, so what I want to show is over here. Okay, and we can pick out. We can write this equation in the form similar to this. By, by fusing both the left hand side and 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. Okay, precisely this coefficient plus sub leading. So this is the times that the line plus non trivial lines. Okay. And once again, you can go through the same argument. So go to the tourists. And now you have three to bother defect lines inserted along the time cycle. And in the limit where toggles my infinity playing the same game, you find the dimension of the junction vector space. So multiplying the divergences, the exponent divergence is due to the cast me 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 say some writing, I will drop the, the L when I label the defect lines just to to save some time. So this is a very simple formula. Okay. But the implication is actually quite deep. Okay, depending on how you think about it, but think about it, but there's a, there's a perspective that was kind of emphasizing the recent paper on Troy. I think it appears some weeks ago. Okay. They give the give a very nice interpretation for this formula and this transition to higher dimension, which I want to want to say a few things about here. Okay. So if you think about this as the planning function were precisely the function as one. Okay, 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 zero. Okay. And the fusion coefficient is the planning function of this to body quantum mechanics over the manifold, which is precisely the manifold on which you wrap the line when deriving the future. Okay. This generalizes. So this is the equal to two or the higher than two. Okay. So consider this kind of non-inverter symmetries, say of cold dimension one. This fusion coefficient will be replaced by a planning function by putting function of the minus one dimensional, you can have tea. Okay. So you can imagine, you can have these in higher dimensions are much more interesting and then one dimension, and that will lead to interesting structures for our different properties for this kind of fusion product. In particular, the planning function. This planning function is longer required to be integral. Okay, on general three dimensional manifold. Okay. So that will lead to kind of various exotic but now understood properties of the fusion rule in higher dimension. Is this like a proven statement or is 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 contracture and this holds for all the examples that we know at the moment. And 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, you know, a lot of things. That's right. That's right. That's right. Yes. So that this nice junction vector space I'm talking about in hard dimension the junctions. I mean, if you talk about generic junction will be point like you can do but if there is a non-generic configuration of this, the effects, the junction is typically also not zero dimension. So the junction vector space will involve some double 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 should be kind of more developed should be developed further. Sorry, can I ask, is there some intuition why these TFT lives in one dimension. I mean, I, I really would expect that the TFT lives on the junction. So zero dimensional. The TQFT, the TQFT lives on the, so because on the, on this fusion picture, the two manuals are wrapping women, sorry, sorry. The two defect lines are wrapping the wrapping homologous manifolds. Okay, and you can take it to be, I mean, essentially identified as far as the project concern, and the TQFT lives on that one dimensional manifold. It is true that when you look at the junction. Then the TQFT, you know, only operators in the TQFT shows up in the junction because there are, there's no way for it to propagate anymore. This is related to a question I did yesterday that the entire dimension I would expect that if you look at the picture with the junk 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, you mean it can come out for example. Yes. Right. And so where does the TFT lives on which of the three manifold. So the TQFT here. So this is a very restrictive statement is a statement about doing this particular fusion on parallel manifolds that are homologous. Okay, and the TQFT lives on that manifold. Okay, thank you. Yeah, so I agree with you that there should be generalization of the statement. We have more kind of complicated configuration of this double 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 case theories for example study Wilson lines importing the two theories, and you encounter similar fusion rule for the BPS lines. In that case, they're, they're also a similar interpretation of this coefficients. Okay, in that case you're 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 in super super symmetrically, and that gives a well defined fusion product. In that case, the coefficient also has a very nice interpretation. It has interpretation as the super, super symmetric index of quantum mechanics. Okay, it's not a total quantum mechanics anymore, but the pattern function is replaced by the supersymmetric index. Okay. So this is just a comment for people will study this line operators. And I jk correspond to Susie index of certain quantum mechanics. Okay. So useful reference is the paper from Galdor Moore and that's the art. Okay. So, the, the next thing I want to talk about is. The definition of this topology defect lines on potentially twisted, or other in other words defect your space. Okay, because this is the one place we that really distinguish the nine vertical symmetries from invertible symmetries. Okay. So what is the kind of action we're looking for. As I already said before, the defect your space come from quantizing on the cylinder. Okay, we're quantizing on the on the special cycle of the cylinder that's punctured through by a defect line. Let me call it out one. Okay. And say there's some state inside the super space, which I've been calling the defect your space. I want to define. Without the defect line, or as I already said before, the action of the topological defects on the hubris based on as one. They're correspond to in circling the, the corresponding operator on the point, right so as before, we have this picture. After you shrink the defect line, because it's the logical you can shrink it and operate local operator you produce after shrinking in calls, how this defect defect act on local operators, these operators and one to one correspondence with the states. So this is, this is equivalent to some operation on the state in the hubris space with no twist. Okay. So we want to generalize this notion to the case in the presence of a twist. Okay, in a spatial direction by this non invertible symmetry. So the, the kind of intuitive picture you want to draw is something like you want to include another topological line. Okay, so out to in the along the spatial direction. Okay, wrapping this cycle. But for this to define a symmetry, as we know symmetry is equivalent to topological property of this various configurations in this lectures, you want to make sure that this junction formed by this lines to be topological. Okay, we already know what this 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 now correspond to this kind of diagrams. So this is the operator corresponding to this operator in the defective space, and then drawing the same diagram, but just the map to the plant. Okay. So to specify the symmetry action we need to specify the choice of junction. As we said before convenient way to specify a junction involving multiple external legs more than more than three is to resolve this junction into three four junctions. Okay, for which we treat as basic building blocks for higher full junctions. So there are two different way to resolve as kind of as as the same case as for how the F moves arise. Okay, and give rise to this to different configurations. Okay. So in general here could be some intermediate to logical line that appears in the fusion product of L one L two. And similarly another diagram you can draw, which correspond to different way to resolve this full full junction essentially are resolving two different ways like this way, and this way. And in general, this could be some other to what if I client if the fusion product is not committed to this. So for our four. It's inside is a is a is a, you know, a potential simple line appear in the fusion problem L one L two. Okay, if L one L two have a committed to fusion product. This can be taken to be the same set, but in general they could be different. So we call this diagrams. Lasso diagrams. Okay, just because how they look like. And these diagrams define different actions of topology defect lines on a piece of the hubris space in this case by L one. Okay, there are 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. And the, the, the, the, the internal line that that is immediately attached to local operator that defines the two sector can be different from outgoing line to the end. Okay, by considering more general junctions over here. For example, staying to the case when this internal line is still labeled by all one. Okay. This outgoing line could be say some other line out for not different from this out for. Okay. And what this diagram means is that after you shrink. Okay, I need to draw the other arrow. I mean the lines goes to infinity. After you shrink the topology defect network. Okay, that's encircling this operator with this various junctions. What this gives you is an operator attached just to the L four line. Okay, some other operator. Let me call it coming from all this transformation. Let me call it five crime. The location doesn't change, but it becomes starting from operator that's twisted. That's attached to the topology defined line level by L you end up with operator attached to the topology defect one L four. And what this means is that this defines operation, a linear map from the defect hubris space twisted by L one. Okay. Which correspond to this operators to the defect hubris space twisted by L four. Okay. The point of noninvertibility is that these maps, each individual maps are in general not invertible. Let me call this map. And I should also say that in the invertible case you will, you will not have this non trivial maps if L four is different from L one. Okay, so in the noninvertible case the first natural thing is that we restrict the case when L one L four are the same. So that is in general not invertible. Okay. And furthermore, there exists natural mass between L between the hubris space space by L one and L four even when they're not the same. Okay, so that's the two features that distinguish the action of symmetries on the twisted hubris space in the case of noninvertible symmetries versus that of the group like symmetries. Yes. Do you get any like 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 bad estimate. Okay, so, so what I'm trying to say here is that you can if you just want to have a most general action. Okay, of this kind of a TVL encircling an operator attached to another TVL. The most general action is specified by the choice of junction vector over here. You can represent this junction vector purely in this basis by including all the L three over here and all the choice of the three fold junctions over here. You can also do the same thing over here. Okay, and it's two different places are related by that move. Okay. Maybe if you take like two operators inside, no, because you can just throw the diagram of the pentagon there. 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. I know, I know, I understand. No, I will maybe in the discussion. Okay, sure. Yeah. Okay. And I don't have time to go into that detail, but the action of symmetry on the justice 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 by the, you know, by the natural face up here in F moves. Okay, but they also equally included in in the symmetry action in this twisted hubris basis. Okay, in that case L one L four are the same. And in particular use, you will, you observe this phenomenon that the symmetry, the group like symmetry acting in a twisted sector can develop projective representations. Okay, even though acting on untwisted hubris space. It's a faithful, okay, the linear representation. Okay. And there's some generous notion for this 90 virtual symmetry, but I will not get into that general detail now. Instead, let's move to another physical object. Just to make sure I'm understanding if this imagery was invertible. Yes, this junction could still be non trivial I mean L one could be different from at four. And this means that this imagery is giving an automorphism of the set of lines in the case of invertible. So if you're picking L two, and he's acting on the. So if, if, if, if all the lines are invertible. Yes. Okay, then for this junction to be topological, the L four has to be same as L one, just from the junction, I mean the, as we said before, the total junctions are only if this condition are satisfied. Okay. Well, so if this is, I mean, if these are all invertible L three has to be the product of L one L two. And because this is already out to us. Of course, I mean you can choose, you can choose to insert something that's not topological over here. Okay. And then you cannot do the procedure to shrink the diagram down to a local operator, like what I'm doing here. So we have some two point function, and there's an entry OP, which you can still study, but they will not give rise to this nice linear math between local operator and local operator. In that case the OP coefficient in particular of this non trivial local operators would enter into this map. Okay. Okay. The next important physical quantity. In code information of a fusion category in the CFT is the notion of the VEV of topology defect line. Okay, which is also known as the quantum dimension of L in the fusion category. Okay, this is very simple notion. Okay, so the VEV is defined as the expectation value of the topology defect line on the cylinder. So you have a cylinder. And you have this budget defect line inserted in this, in this orientation. Okay, so that's the definition. There are nice property of this body defect line. Okay. You can consider an infinite long cylinder. You can consider an infinite long cylinder, and on which you insert, you know, a composite L i and L j, so call this L i, and insert another L j. The fact that you have this fusion rule. Well, 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 misdirection projected onto the vacuum. It means that the VEV will satisfy the same polynomial equation. So you can think about this as some kind of closet composition on the cylinder for this lines. Okay. Sorry. That's right. That's right. Yes. Yes. But here I'm just using the fact that down the cylinder. Projects to the vacuum. Okay. So that all in the intermediate and because they're topological can make them as arbitrary for our part of the lines. Okay. All right. So this means that the VEVs are highly constrained by this polynomial equation is very particular property that the coefficients are positive integers. So these are going to be very special algebraic numbers associated with this VEVs. Okay. So this is actually on this VEVs coming from unitarity. Unitarity will imply in particular that this VEVs are actually funded below by one. Okay, this is a slightly. Oops, I did it again. Sorry. So let me go through that argument. Okay. The argument actually it's a very simple argument that involving the consideration of a tourist which come back, 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 I, this lines have expectation equal to one. So 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 vanishing 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 are invertible, because it generates some group like symmetry as usual. And if the VEV is bigger than one is non invertible. So in other words, if you have some aligned defects which is logical, a quick way to diagnose whether you have a non invertible symmetry or not, it's 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. And this statement does not hold in hard dimensions, precisely because the coefficient will depend on this and not to be kept. Okay, which is the function no longer in general. Yes. Question. Yes, sorry. But if you normalize differently the lines. Yes, you get different quantum dimensions. Good. So, so, okay, we'll come to that. Well, the next time I should make that clear. Okay, if it does not ask me again. So, so how do we derive this state. Okay, and this derivation should also address this question. So, so we'll be using locality. Okay, in particular locality of the column field theory, particularly this case of to the CFP defined a torus with the insertion of a line. Okay, so the object where we will be studying is this, the torus function. Where's the defect line inserted. Let me call this L. The locality of the Euclidean CFP or quantum field thing in general implies that we are free to compute this function. So let me come. 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 special slice. And this defines naturally this function. Okay, with upper L just to denote this particular choice of quantization, which can be interpreted as a trace over the hubris space on this on this special cycle with which is not punctured by any defect line so it's ordinary hubris weighted by the action of the line, you know by our hat. Okay, the same L hat open that appeared over there and weighted by the usual Hamiltonian evolution factor. In 2d CFP are determined by the L zero and L zero far. 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 quantify the theory in different way. And in this case, so this 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 is a convention for that transformation. 90 degree rotation in this way. Okay. So this defines naturally the function. Now, for the, it's a pattern function is a thermal 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 the course. So this is T to town. Okay. And this is interpreted. Now it's the trace over the hubris space, the defect hubris space, because now the defect punctures through the punctures through the, the circle on which you quantize the theory. But because there's no other defect insertion, this leads to this simple thermal function with no additional insertion. Okay, and kill Tilda is the analog of Q when so Q is as before Q is e to the two pi I town and killed Tilda is just e to the minus two pi over town. Okay. Now, the, the fact of our quality equates these two sides. Assuming sequel C bar. Very good. Okay, so let me, let me, let me forget about subtlety with the gravitation along the under the under the modular transformation and for simplicity to focus on the case when C is equal to C bar. Okay. And in that case, a, the right hand side as a property because of the trace over hubris space, okay, hubris space is some vector space in particular some vector space. It is, it has this decomposition. Okay. Into characters. And because we have a topology defect line that preserve the resource symmetry, this will decompose decompose into characters as I'm trying to see other falling form. Okay. But for now actually we will not we will not need to this specific form. All we need is the fact that this is positive. And in the case when tau is equal to it and tau bar is equal to minus it. Okay, because this 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, each, each weight is positive and you're summing over some determine it. 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 it's right right this to address the question is because if you reskill your what do you mean by L here. It will be intention with this being being a you know a contest integer. So, you can take a direct sum of this line. Okay, because it's going to retain it into into reality, but it cannot apply by by arbitrary number. Okay, and that what that's what in what sense locality is crucial. You know in specifying this lines. Without that you can modify the arbitrary place for them. There's still some subtle kind of counter term you can use for the line, but that's not relevant on the cylinder with a flat geometry. Okay. So you can also notice of additional face factors that you can introduce for the line by introducing one dimensional counter term. If you're interested you can ask me afterwards, but it 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. Okay, I'm both sides. The left hand side because T is an 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. Here I should write L hat. Okay, so this means that the best of this line which is the expectation value of L hat on the cylinder. It's equal to the limit of something as manifestly positive. So, which, which excellent show before here. No, I've moved. I'm going to start from here, move it to the right hand side. Good. Okay. So we are, we're halfway there. Okay, the proof is bigger or equal to zero didn't already we did we have not shown that is bigger or equal to one. Okay, we use the first effect. 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. And then we use the CPT invariance of a general quantum theory that says that these webs are the same. And as a consequence, we have shown that a level of L is bigger or equal to one. So I went to this kind of argument in detail, because this is how this is actually how various non non-productive constraints from this region category symmetry have been derived into dimension, and also in higher levels. So this is the generalization of this game using the locality of the particle, which 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. Now I just want to introduce the general data. After this basic building block. Let me just introduce the general data for for CFD enriched by non invertible symmetries. Okay, so whenever you have a symmetry, apart from trying to study constraint from the symmetry on the original CFD 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 living the twisted hubris space from inserting this vertical duality, I mean vertical non invertible defects. Okay. So CFD, we call that CFD without defects into dimension. The basic data is captured by the hubris based on S1, okay, which is related to the local operators by the radio, radio composition. And together with the OP coefficients between these local operators. Okay. And, and this data are subject to bullstrap constraints, which are again consequences of locality of the 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. And there's similar relation for the Taurus one point function. Okay. And it's an entrepreneurial fact that a set of data that satisfy the two kinds of consistent conditions, they're 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 natural fact is that once this set of condition are satisfied. This defines the consistency of tea. Okay. So all the cutting and gluing capacity conditions opportunity general remand surfaces are automatically satisfied once this two basic moves for bills. This generalizes with TV else in a natural way. So you have to watch your defects in the CFT. As I said, there's this additional structures that appear. Okay, just from a locality of the conformal field theory, you have this defect here where space. Okay. And you also have this additional OP coefficient, sorry. Let me write it here. You also have this additional OP coefficient that involve operators now in the defective space. So they can generally be represented on the plane by some diagram like this. So where the external legs are some topological lines, and there's some topological junction leaving here, but external operators correspond to operators in the corresponding to super space. So these are 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 this quantities. Okay. So we are subjected to similar blue strap equations. For example, the full point function once again, but now attached to a network of this to watch with defect lines. Okay, so all these lines are not just the mnemonic for OP exchange, but our extra defect lines. And that 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 inserting complete set of states, and last leads to constraint on this data. So this provides, there's a similar generation for torus one function, which are not drawn. This provides axiomatic approach to identify non-invertible symmetries into the CFT. Okay, so having non-invertible symmetry into the CFT, because we discussed, if we assume there is enough of this to body with defects, it will imply all the structures that satisfy cutting and cooling as a consequence of locality. So if you are given just abstractly safety data like the local hubris space and this OP coefficient, the thing you need to do abstractly to identify a non-invertible symmetry in the 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 going to be a formidable task for general CFT. Yeah, so I should write pop. Okay. But as you will see, there are additional physical arguments that allow you to infer the existence of the body with defects without solving this equation explicitly. But this equation can be solved explicitly in rational conformable field theories. Okay, let me not go into the detail, but let me just say that in the next lecture, we'll discuss explicit examples that they realize the symmetries and produce solutions to these equations without actually solving this equation as explicitly. Okay, I think I'm going slower than I expected. That's fine. Let me stop here and take questions.