 with a course about non-indulgable symmetries. Okay, thank you. Let me just check. You can hear me in the back and you can read what's written here also, all right. So here's a very quick recap of the first lecture. The first lesson that you were sort of told and then we substantiated it, it makes sense. Is it global symmetries? We should think of as really topological operators in your quantum field theory. And that insight then taught us to think about standard symmetries, what we would now call zero form symmetries, in a slightly more general context in terms of p-form symmetries. And p-form symmetries act on p-dimensional operators and they're generated by some topological operators which have to link non-trivial with p-dimensional operators and those are precisely d minus p minus one-dimensional topological operators. And we were so far assuming that they all have very nice group-like structures so these operators would compose or fuse together into a group. Now the first extension from, if you put p equals to zero, you just go back to a standard symmetry. Mine then, this is just a co-dimension one topological operator. But if you actually go to p equals to one, then the charged objects are line operators and the symmetries are generated by co-dimension two. So for example, two-dimensional topological operators in four dimensions or line operators in three dimensions which compose in terms of a group that we call GP. So in this case GP is one. And then I cheated here a little bit because someone asked the question in the question session that allowed me to say well actually you can really think of local operators as objects that could screen once you have certain types of local operators that could screen the one-form symmetry. So in this case you could have a Wilson line in a representation R and that could end on a local operator. If you have this local operator in your theory, then that topological defect does never non-trivial linking. There's something wrong with the audio, right? Anyway, let's see whether this is better. Okay, so if you have this linking here, you can just trip it off, collapse it and the charge would be trivial. So this is sort of the way you can then determine these higher-form symmetries, or in this case one-form symmetries, engage theories. So pure gauge theories with simply connected groups is how they have one-form symmetries that are the center. Okay, so we are not done yet with invertible symmetries. There's some sort of backlog and I'll just try to explain some things which we'll need to then actually discuss the extension to non-invertibles. So one thing that I wanna talk about today is background fields for higher-form symmetries. So if you have a standard zero-form symmetry, you know how to couple this to a background field. And so for P equals to zero, some group G zero, you would introduce a gauge field, a background gauge field. So A1 is a background gauge field. Once you add some term to your action, J times A or something. And then you could sum over A or integrate of all the connections that you would like to consider and that would gauge that symmetry. And so for P larger than zero, so for GP-form symmetry, we equally have to introduce so the concept of coupling this to a background. And the background fields here for P-form symmetry are P plus one forms. And I'll sort of always assume this, I didn't emphasize this again, but when P is larger than zero, these are billion. And I'll very often think of these as actually discreet, so finite, actually finite symmetry groups. So these P plus one fields are P-forms, a P plus one forms on your space time taking values in this group GP. So for example, if you have your SU two pure young mills in four dimensions, we said this has a Z two, one form symmetry, then this would introduce a two form background field. And that would take values in H two of M four with Z two coefficients. Okay, so we also discussed yesterday how we go from SU two, so there's basically a sense there's a global form SU two, which has a Z two center, one form symmetry. There's also sort of a gauge form. So we also looked at the PSU three, or as it's more commonly known, the SO three theory, pure young mill theory. And this has actually sort of magnetic Z two magnetic. It's a PSU two. It's absolutely, thank you, thank you, thank you. PSU two, more commonly known as SO three. And you could also just say this is spin three, right? And this has a magnetic one form symmetry. And the question is how do you go from this theory to that theory? And actually what you have to do is you have to gauge this Z two one form symmetry. So actually when you have these background fields, normally when you couple them to a standard zero form background, you would then integrate over this here. These are actually, for finite group backgrounds, actually what we have is so discrete values of this. So in fact, gauging the Z two one form, maybe I should call this here the one form symmetry electric. And this is the magnetic. The electric one in SU two becomes the SO three theory with Z two one magnetic. And so what does this mean in terms of the partition functions? So I have our theory, so T, so T is the theory with Z two one form symmetry. And I just wanna gauge the Z two electric. I have coupled this now to another background field. This is again a one form symmetry, but it actually is a one form symmetry. No, not the same background field because this is sort of here the electric. You want to be pedantic. This should be so the superscript. There's another two form field. And this is obtained in terms of just summing over all the B twos in H two and four one electric. And now we have to notation it. I choose each of the two pi I B hat where T here is this theory, the SU two theory. And there's some normalization factor which is basically the order of the groups on this case to the half. So this gives, if you look at the partition function for the SU two theory, couple it to this B two background and I'll sum over all the allowed configurations that gives us in fact the, with this weight factor here. I'll explain what this is in a second. This now depends on B hat. And this is then the partition function for the SO three theory. So what is this B hat? Is the B hat is an element in H two and four C two one magnetic. And now the question is what do I mean by this bracket? So I had this bracket here, B B hat. This actually will be appearing. This should become a phase. So this better be some well-defined sort of phase that I can shove in here. So the question is what is this peering between B and B hat? So this is defined or actually let's put it like this. This is a map from H two and four Z two times H two magnetic into you can call it U one or R mod Z. So it's a peering. Now how do I define this peering? Well H two is Poincare dual to H lower two. And these two actually had a peering already from the New York pairings. Basically this is a well-defined thing that gives us precisely a U one valued object or R mod Z valued object. I should probably write R mod Z. Okay, so more generally if we do this for a P form this has a BP plus one background and we gauge it in D dimensions. And then I need to do some of the calculation what is the degree of this form. Then you get a dual B hat. And the degree of that is D minus P minus one and that is for a D minus P minus two form symmetry. So normally there's a question. Yeah, sorry, yeah, so that's why I actually took them off. So here this whole thing you could say is B. So for P equals to one and D equals to four, right the one form symmetry is meant to one form. So B two went into B two hat. But generically you'll be mapping a P form symmetry to a Q form symmetry of this degree. So you're not mapping within when you gauge from one to the other. So yesterday a lot of people asked, oh, how do you actually write down these generators because it started off, yes, there's a question. Excuse me, I didn't hear the question so I don't know what happened. Why did you just question it? So he was just asking about, is shouldn't this be if this is a phase like you can leave it as two pi I and then you just have here R mod Z, this is fine. He was just saying you could, it's a phase. It went into U1, you can define the map either with the log or without. Today's the day of the logs. I should have had a log missing, I had a log missing, so. Sorry, just for clarifying me, H2 is a group of co-homology or- A co-homology group. Ah, okay, okay, right. Yes, sorry, absolutely, okay. All right, H2, where is the first time this appears? So HP plus one is a co-homology with these coefficients. So as I think of it as forms, and in fact maybe for people who are not familiar with co-homology with discrete coefficients, give me one second and maybe I think this might be useful and a lot of people do holography here. So I will tell you how you can think of these form fields from a slightly more holographic perspective. And this goes actually back to exactly this kind of gauging. So basically, one way of thinking about these B2s is either as elements in this sort of co-homology where you have these discrete values, or you can also think of them as coming from a theory which is actually where everything is alternative, description where these are actually U1 valued and they are actually then sitting, so this is a U1 valued description, where the fields, I should maybe call them now, little b2, have an action in the following way. So let's say we are looking at a p-form symmetry. So let's actually put, let's leave it as z2 by, we can, okay, we can put it as, let's leave it as in. So these are U1 fields, b2 and b2 hat. And they have an action here in five dimensions. This is something you will have seen before, it's like a B after theory. But now you have an action where there's an n out front. So in fact, if you look at very crudely at the equations of motion, you would get something like that n db2 is zero, or n db2 hat is zero. And so you would normally, this n weren't there, just say db2 is closed, but actually here is the freedom of n. So in fact, this also is a way you could characterize fields that have in that case now a zn valued homology class in terms of U1 fields. So this clarifying useful or there's a way of, so if you don't wanna use sort of this formulation of discrete coefficients, you can think about it like that. So yesterday people were asking me how what are actually these, the generators, right? So now we looked at these background fields and when I started off talking about these symmetries, I had the current and I constructed the symmetry generators from the north of the current. And the question is, how do you actually write down these type of things? And so now once we've introduced these background fields, we can actually do that. And so one way of sort of doing this is in fact the generator. So let's stick with this SU2 example and the SO3 example. So again, let's look at SU2 and SO3 and 40. So we have these two fields B2 and B2 hat. They're z2 valued forms. And we know in this theory, we have Wilson lines. And they basically are z2 many Wilson lines here. We have Toft lines. There's z2 many Toft lines. And well, we can sort of think of these as actually coming from the dual one. There's also one form symmetry, the z2 electric and z2 magnetic. So the actual generators of the z2 one form electric, these objects here are in fact, besides these sort of lines where you have so there's some W and some representation, right? Also the trace and the representation e to the integral gauge field. So in fact, what we actually can say is these actually become charged under the following operators. These are actually the DPG operators on two manifolds, which are now e to the integral for B2 hat. And this may be, before I actually finish this course, it'll become clear why this is true, but this is actually the operator that will link non-terratively with this line operator here. And B2 hat is the dual gauge field, the one form symmetry gauge field to the one form symmetry that's the background for this symmetry over here. And vice versa. So the one with B2 is actually the one that generates the magnetic z2 one form symmetry. And one way to see that is in terms of this sort of BF type action, and I'll explain them tomorrow in more detail. But that's answering sort of the question that people have asked yesterday. So is this, are there questions about this? So the reason why I actually have to introduce these background fields is because one of the important things that global symmetries give us are tough anomalies. And we usually write them in terms of the background fields for the symmetries. And there's a very close connection between the tough anomalies and non-invertible symmetries. And so we actually have to talk about tough anomalies before we go to the non-invertible symmetries. So the next sort of thing is tough anomalies. So these are anomalies of quantum field theory for global symmetries. So these are anomalies for global symmetries. So they are not a bug of the theory. They are a feature of the theory. There's a feature. And so normally you would, for example, if you have some G00 form symmetry, you would calculate them by some triangle diagram on quantum field theory where these are all global symmetry currents. I wanna take a slightly different perspective. I wanna say sort of the tough anomaly is essentially an inconsistency or a non-invariant of the theory under background gauge transformations of the background fields of a global symmetry. And then we can sort of apply it in this case already directly. So, but okay, before we get there, of course, one key utility of these is these are RG invariant. So you can match tough anomalies. You can start saying in the UV theory, calculate them and go to the IR. And you can match the tough anomalies from the UV into the IR and they need to agree. That's really why they're so important. And they will also be important for these types of symmetries. And whereas here you had sort of the 0000 form symmetries, they'll also be now mixed anomalies between zero form, one form, two form symmetries. And so the whole structure of tough anomalies will become much richer in the setting of these higher-form symmetries. So I wanna think of a tough anomaly as, yes, and so it corresponds to the situation when your partition function of a theory, say now for example, for one form symmetry, but you can do this for any P-form symmetry. So one form symmetry has a B2 background field. I put the theory in the background for this one form symmetry and now I do gauge transformations. So ZB2 is not invariant under background transformations. B2 goes to B2 plus delta lambda one. And I want to basically, this transformation makes the partition function pick up a sign or face and that's essentially gonna determine the tough anomaly. What I'm saying is we have the theory, we have B2 plus delta lambda one. And that's now some phase phi of B2 lambda one. So this is, you could also have put here, let me put it in color. The familiar case is you have some B1. You do a B1, goes to B1 plus delta lambda zero gauge transformation and then you pick up a phase. So that's sort of a background transformation. And so this is sort of the non invariant. Another way of saying it is the partition function doesn't depend actually on the cohomology class of B2. It depends on the representation. Okay, so how do you remedy this or how do you sort of now define the actual anomaly? So we can fix this by considering coupling the theory to another theory, a D plus one dimensional so-called anomaly theory. So if you know what anomaly info is, this will be very familiar to you. So this theory I'm gonna call a D plus one and it has the following properties. An anomaly theory D plus one. If I put it on a manifold M, let's call it W D plus one, where the boundary of that is empty, then basically a D plus one is well defined as a function of B2 and also invariant number B2 plus delta lambda one. So it's actually defined as a function of the cohomology class B2, B2 dependent. If I put it on a manifold with a boundary and the boundary is D dimensional, then it picks up a phase and that's then exactly a phase such that it canceled this phase five that we had over there. So but if you put AD plus one on a W D plus one with the boundary is MD is not empty, then I want that theory AD plus one to restrict to precisely on MD plus one to be exactly five inverse of lambda and B2. Okay, so that's essentially, I'm just engineering a theory that in D plus one dimensions, I can do back, so it depends on the background field B2, of course, but then as I put it on a manifold with boundary, it actually restricts to exactly the inverse of this phase that we had in our initial theory, this phase, the inverse of that. And that's good because now what we can do is define Z tilde as Z times A and this is now, of course, AD plus one and this is invariant under B2 goes to B2 plus delta lambda one. Okay, but what we're really interested in is this anomaly theory. So if I have a quantum field theory and it has a symmetry, a global symmetry, I do background transformations, then I can determine from this essentially the anomaly theory and that's sort of the thing that is gonna be robust under the RG flows and we'll see lots of anomalies popping up throughout these lectures. I'll give you some examples of anomalies in QFTs that are now related to higher forms in the trees. So the first example is if you take a three-dimensional theory with a one-form symmetry, some G1 and this has a background field B2 and H2, then this can have an anomaly and some E to the two pi I integral B2 B2. All right, so the anomaly theory lives now in four dimensions and the way I should really draw a picture, you should really think of this as, and this is sort of a picture that will hopefully become even clearer than tomorrow and more quantified so we have a theory T and then we essentially attach to it, this is A. So this lives in D dimensions and this is now D plus one dimensions and this restricts on here is exactly this sort of anomaly thing, in this case this is A. Okay, all right. So this could be not here a three-dimensional theory and if not this four-dimensional theory attached to it, that's this anomaly. Another example, which is also one that's quite important is in four dimensions and now it becomes more interesting because we could now write down so really anomalies for theories that we might care for for example, four-dimensional SUN N equals to one super young mills, this happens to have a ZN one form symmetry that's the center of the SUN but it also has from the N equals to one supersymmetry it has the U1R symmetry that gets by an ABJ anomaly broken to a Z2N and that then in the UV gets all the way broken to Z2. This zero form symmetry has a non-chirvy anomaly, mixed tough anomaly with the one form symmetry and so this anomaly basically is equal to E to two pi I integral A1, B2 squared where this is the background for the, so here A1 is the background for this the chiral symmetry is the zero and this is the background for the one form symmetry. And this is actually an anomaly which will be very, very closely related to constructions of non-invaluable symmetries. So this is again, there's a five-dimensional anomaly theory that lives on some M5 where the boundary of it is the 40 N equals to one super angle theory. So there were lots of things I was gonna tell you about invertible things but then everyone is here because they want to learn about non-invertible things. So I think I need to move on to the non-invertible things. You can ask me about the physics of one form symmetries and so on in the question session if you would like. There are certain interesting things to be said about that but I think I would like to at this point maybe open up for questions about anything invertible and then we will actually migrate to the constructions of non-invertible symmetries. Question about this B2, B2, so you're choosing some quadratic pairing on G1. Yes, right, yes, absolutely. So here really I should have Pontiagin squared B2 and I specify a pairing and I take a refinement of that, quadratic refinement of it. So here for example this would be, the actual normally is just with the N inside here and so then actually I need to, when N is odd or even, I need to reduce the refinement suitably. So I'm just putting that on the card. Why do you assume that it's in the comology? I might have missed why you said it, the forms. Because I want them to be flat background fields. I could also turn on something some non-trivial background fields but that's right here, these are global symmetries. I would just like to have them to be, have essentially like Wilson, like discrete Wilson lines and I have two-dimensional there for example. You could think of it like that. It's just background fields for global symmetry. Okay, so the chapter two, the sort of constructions of non-invertibles and the example I gave you yesterday were these verlander lines and two dimensions and in fact we will focus on two dimensions also at the beginning of this section but before we go there, I want to give you sort of a very quick flavor of why such non-invertible structures could exist in higher dimensions. And there's really like a very simple example that we can now understand given what we've done so far. So let's look at four dimensions. We're looking at a U1 gauge theory and so we now know this has topological defects, D2, I think I called them alpha electric and there's a D2 alpha magnetic and these were like either the alpha integral star F and DF and zero is the star F. So these are topological defects and they generate a U1 electric one form symmetry. So that's group like and it's invertible and it forms a U1 times U1 group. What we would like to consider first is actually gauging of a charge conjugation. And so this actually first appeared in a paper on the swampland by a bunch of people who I will try to recollect. I didn't know I should deal here as well in Sweden, I'm on tarot and magma, thank you. And this came out of a swampland, completeness of spectrum kind of discussion and when you gauge charge conjugation, what actually happens? So charge conjugation you can think of as a Z2 zero form symmetry, it's a global symmetry that tends A to minus A. And so what does it do to our symmetries here? And first of all, there are also Wilson lines and top lines here, but actually what does that do to the symmetry generator? So in fact, it will act also on this alpha E and it'll just map F to minus F so that you can think of as star F to minus star F so as alpha goes to minus alpha. And so here alpha is in zero to two pi. So this is how Z2 zero act. Now if I want to gauge that, I can't really just sort of stick to these guys because they're not invariant. So the only invariant ones, so the invariant ones, topological operators, they're the ones for alpha is equal to zero and then also pi, because pi goes to minus pi which is the same as pi under this, it's alpha is identified not to pi. So alpha equals to zero and pi and those are sort of operators I can write down so it's zero and D2 pi. But for the others, I have to take combinations. So I need to take a sum of these two operators. As you do that, so the invariant let's call these D2 alpha plus, they're defined as just D2 alpha. Okay, I'm carrying this electric with me. You can do the same with the magnetic. It's gonna be not very different. They're the sum of these two guys and that's an invariant. So that's also an invariant. So the D2 acts like this. But the crucial difference is that this is now composite, the sum of two of these operators and now if I actually start looking at the fusion, so if I comp, so fuse, I compose them all these ones will just have nice fusion. This is basically the identity and this one will basically square to the identity. But as I fuse now the D2 alpha plus, for example, with D2 beta plus with alpha and equal to beta. And here I now need a restrict alpha to be between zero and pi excluding these two values because they're invariant and then actually what happens is we can just multiply this all out and it'll just give us a D2 alpha plus beta plus plus D2 minus alpha minus beta plus. But these are again defined in terms of the, sorry, this is the class here. So these are just defined in terms of this thing here plus minus this thing. So but in this sort of invariant formulation you get two operators now fusing into the sum of two other operators. So this is essentially the first way you can see this is actually non-invertible, right? So this is, now after we have gauge this should be the fusion. So this actually should be the fusion, the one-form symmetry generators in the U1 modulo Z20 which is actually also called the O2 gauge theory. Now it didn't stop with this in that paper and it came out sort of early I think 2021 maybe. There were also fusions of when alpha is actually equal to beta you get something slightly more disturbing on the right-hand side. You actually get surfaces and lines and surfaces with lines and then that sort of I think really so people were stunned what on earth is going on with these types of symmetries. I don't want to actually write down what the proposed fusion was because it actually was causing more confusion than anything but I would like to actually take note that here the key is the fusion of these composite things which are these invariant combinations automatically give you a sense of where the non-invertible comes from, right? You just multiply this out in the U1 theory you repackage it and you get the sum. And we'll see this very often in the following that gauging auto-automorphisms C2 zero charge conjugation or we'll also see if you take an SO type theory there are auto-automorphisms on the Dinkin diagram or you have an AN gauge symmetry and you gauge this reflection so these are like literally a gauge algebra auto-automorphisms. All of these things will yield non-invertible symmetries, the gauged QFTs, right? So if you take for example this here you take the spin theory and you gauge this auto-automorphism you get pin plus here you get these so-called SU and tilde theories in this case charge conjugation for the U1 theory gave us O2. So these will all have non-invertible symmetries and once you realize this there are actually a lot of non-invertible symmetries in our lives. Yes, please? For what? Yeah, so you can also do it of course for the case whenever you have a symmetric auto-automorphism for E6 has this, right? You can do this here but that's it, no? E7 doesn't have any. E7 doesn't have any auto-automorphism. You need a symmetry of the Dinkin diagram. Oh, sorry, I got it wrong. Yeah, but here you can still, any auto-automorphism you can gauge and you will get these sort of things because essentially you will then get this sort of very simple observation that you have an invariant sort of combination of defects. That's the sum of two other defects and that fusion gives them non-invertible. Okay, but this is all very, very bird's eye vision because we haven't really understood what's going on here. This is not the true final fusion. I didn't give you the fusion for all the values here for alpha is equal to beta and that's basically where we'll get to eventually. But I hope this gives you a sense of the ubiquity of these, right? This is not an exotic, strongly coupled five-dimensional quantum field theory that nobody has heard about. This is four-dimensional mill theory, okay? Yeah? Does anything special happen with SO8 because of its tryality? Yeah, so there you can gauge for if you have a spin eight, you actually can gauge the full S3 and that's a very interesting thing to do because it's not a billion. Okay, but after this motivation, what I want to do now is go back to very firm ground and talk about something extremely basic and then try to build up again starting with two dimensions. Really everything is extremely under controlling two dimensions and building up the structure from 2D to 3D to 4D. So unless there are any other questions about this sort of example, I want to now actually discuss two-dimensional non-invertible symmetries and a very specific construction. So this is sort of about non-invertibles in 2D from gauging. And you'll see there's a lot of non-invertible symmetries arise from gauging things we know very well and that's why we need to learn about gauging in quite some detail. And I want to start with something extremely trivial in two dimensions. So in two dimensions, we have, well in two dimensions we can have a theory T and let's say it has a zero form symmetry which is group-like, okay? It's a group, it's a zero form global symmetry. So we now know why I was talking about these lines, these valenda lines at the very beginning of the first lecture. So these are now labeled by g elements and they have fusions h is d1 gh, i for all g and h and g, this is, and now so how do I think about this? Well I have a line, so label these lines d1g. So what is actually this here? I can very often write this sort of as, I have a g here, I have an h here and I call this here g times h. So this is essentially almost all the structure that we have here. And if I sort of have a line that corresponds to say the generator g here, right? I also in principle could ask, are there any topological operators on this line? And actually there are none as long as this line actually is essentially one of these generators. So the really interesting structure is just there's a fusion on such topological lines. There's also a little bit more structure. So if I have such a thing, I also of course, I don't necessarily only want to be able to compose two of these. I also have to ask if I have three lines, how can I actually have a g, h, and k? There are different ways I could compose them. I could first fuse g and h and then I bring in k and that should give us g, h, k. And that should be the same as if I first take k, fuse it with h and then bring in g. So this is the sort of structure and this is all group-like. Now one can refine this a little bit. One can add a little bit more structure and a so-called co-cycle omega from g times g times g into u1 that I can just attach to one of these diagrams. And one thing you could think of is basically saying this is equal to that times omega of g, h, and k. And so of course, not any omega of this type is allowed. If I now take four of them, it needs to be again compatible with that. So associativity then implies that actually omega is in what's called h3 of g, u1. So the closeness of this is precisely the condition that if I take all these four and move these around in different orders that these omega satisfy that this diagram sort of is consistent throughout. Okay, so this structure that I just defined here, really there's nothing interesting going on. I have a group-like symmetry. If you use them, there could be some additional phases. You can even ignore them if you like. And this is a structure that a mathematician and you know, we've already let string filter in the room. We will now let the categories in as well. We'll call VEC g. So this is just what you call VEC g. There's nothing interesting going on here. It's group-like topological lines and that's how you should think about this category. So what we want to do is gauge this group. And because this group actually, I didn't say anything because it's zero form symmetry. I should maybe emphasize this. A zero form symmetry can be non-Abelian. It's a group. It's finite but not necessarily Abelian. The zero form symmetry can be non-Abelian. And in fact, the interesting things happen when g is non-Abelian. So what does it mean to gauge the g zero? So in a standard sort of setting where you do two-dimensional conform field theories, you have a symmetry, you want to gauge that, you call this orbit folding. And we all probably have done examples of that in 2D CFTs. I want to do it in a slightly more categorical way that actually talks to these symmetries, but this is nothing other than sort of doing this orbit fold construction. And there'll be a couple, and I'll actually say omega equal to zero because omega is actually like a top anomaly for the zero form symmetry. So if omega is actually non-zero, then we have some issues with actually gauging this. And there'll be various takes on gauging and there'll be various levels of familiarity and they are disproportionate to the generalizability to higher dimensions. So the first take is, well, what does it mean to gauge? We introduce a background gauge field. H is an H1 of our two-dimensional space with g coefficients. So actually to do this, here we really should have assumed that this is a billion, so this is why this is a little bit easy, but you can also, you can generalize this, but let's not get, here we can assume for simplicity it's a billion. So what it becomes here, this becomes a dynamical field and in this gauge theory, so this is when I do the mod g zero, what actually is happening is I can now actually construct operators, which are just in some representation R, the gene representation, there's the trace, R e to the i integral b1, right? So these are Wilson lines I can define in that gauge theory and in fact, because b1 is closed, I can infer that these are actually topological. So db1 is zero and so in fact these WRs are topological. And now I wanna know, okay, so these WRs, if I take one and I bring it close together to another, again R1 with some other representation, what actually happens is that this just becomes the sum over some coefficients R1, R2, R3, WR3, and there's something here about three, where these are just sort of the Clebsch-Gordon coefficients when you decompose the tensor product of R1 and R2 into irreps, R3. So I really, really should call this here something like d1, R1, et cetera and so actually what happens is in that theory before, we had something that was completely group-like, we gauge it, we get a gauge, so we can construct these Wilson lines and then I'll have a fusion, which is given in terms of the structure WR1 with these coefficients here. And generically, if I have a tensor product of representations, this isn't just like a single one, but actually this will be a sum over a bunch of, yes, yes? In order to have this structure, then you need to have non-Abelian group. I know. I will give the full argument of non-Abelian also. So what you can do is... A billion there. Okay, so basically you can also, you don't need to, we can generalize what it means to put this background. Basically what it means is we take the two-dimensional theory and now we include a mesh of lines, of these topological lines and then we can ask what are the lines that are loud and are blind to these other lines? Because we're gauging this, the lines in the gauge theory should not actually feel the mesh of lines. So that's the more proper way of doing that. It's true here I'm doing Abelian and indeed when it's Abelian, this all becomes extremely trivial. But so in this case indeed, this is sort of for Abelian is a good point. That's why I'm saying this is take one, take two will be, there's a better way of doing this and take three if I have time, we'll basically do what I just explained here. So Abelian, it means that essentially these are just, the reps are just characters to determine by the characters. Then in fact we can see now that the dual symmetry is also just gonna be an Abelian symmetry. Okay, so let me do this. So the representations, right? So the character for an e to the i theta, say e to the two pi i over n even, right if I take this to the n, it needs to satisfy that this is equal to actually that entity, but then also if this is g here, this is also the same as equal to character of g to the n, which is equal to the character of the identity is equal to one. So then actually you find that character of g to the n is equal to one. So in fact these are also just the n dual lines. These are representations. So this is just saying that the dual in this case for g zero Abelian is just, it's a morphic to g zero again. But if for a moment, with disregard this fact that it came from here, this will be the general structure. More generally you will get something that actually is completely non-invertible in this case. And since you brought it up, I think I will, so what time did I start? When do I need to finish? More relevant question. In 15 minutes, including questions. Okay, then you will have to ask your question again. So the take two is, so this is actually I was a little cheating here, but now what I'll tell you is really, we can think of this in the following way. And this is really what also then generalizes to higher dimensions and incorporates essentially almost all constructions of non-invertible symmetries. And this is sort of work we've done with. And at this point, this is I posed on lecture, Baradwaj and Jingxian Wu. And then also Baradwaj and Tiwari in a more general setting. And also applications with my student, Leah was there. So Baradwaj, Tiwari and Botini who fits in here. If you have questions, Leah is right there. So anyway, so what is actually the idea of this paper here is, so let's for a moment just think about the following problem. This two-dimensional theory T and it has a global symmetry G zero acting on it. And I want to gauge this and get to a theory that I call T-Moggi. So when I do this, I can do something before I gauge or just take a product with TQFTs. So TQFTs will be an ever-recurring theme here. I can take a product of a one-dimensional topological which has with G symmetry. And I'll tell you what these are in a moment. So just take on this side, the product T times this TQFT with G symmetry. And then I gauge whilst that this theory has a G symmetry, this has a G symmetry. I'm gauging now the diagonal G symmetry. And what happens after we do this is that here, this is now not a product of two theories, the TQFT and the theory, but actually this now becomes a topological defect in this gauged theory. So this is now a D1, whatever, some label alpha. So in this, all right, so this is the theory which is now actually not T-Moggi, but T times the TQFT, G-Moggi. And the important thing is it's really, I'm gauging this diagonal G and this has become, this is now topological defect in the theory. So you might say, why on earth are you doing this? And I will say, well, you've been doing this all the time. And so in four dimensions, if you, for example, look at a U1 symmetry, a global symmetry, then you know very well the concept of a theta angle, all right, the theta F by Jeff. You can think of this as exactly a U1 preserving topological field theory that it's basically trivial and I just start with a trivial theory with trivially acting U1 symmetry. I add this topological term and then I gauge the U1, all right, and then what I get is basically in the integral F which star F, one of G squared plus theta F by Jeff. And this is essentially, you know, the theta angle, you can think of it as what's called a symmetry protected topological phase. I'll talk about those in more detail when we generalize this to higher dimensions. In this case, it's a U1 SPT phase and what you've done is you start with a trivial theory, you add this term, you gauge the diagonal and you get your favorite theory with the theta angle. So in this sense, these type of defects, we can call them also theta defects because they're really nothing other than a sort of implementation of the same idea now in these two dimensional theories and later in higher dimensions. So why is this now a useful thing to do? We can now look at this problem that we're discussing here. We had a global G symmetry. Now G can be non-Abelian, no problem at all. Now what actually is, what are TQFTs with G symmetry? It's a one dimensional. TQFTs and one dimensions are really quite trivial. They'll be characterized by a set of vacuar. There's no dynamics, it's really literally just a set of vacuar V1 until the end. And then the fact that they have a G symmetry, it means that G acts on it. So G acts on the vacuar in a consistent way which means these vacuar, so these collections of vacuar V, V forms a G representation. And so once I am at this point, I can say, well, what I'm getting here is I'm attaching these type of TQFTs here and I'm gauging these and so basically what's happening is these topological defects are given essentially by G representations. And now I can ask how it does the fusion work? Well, you can sort of see if you have V1 to Vn and you have W1 to Wm, and G is acting on these, right? And there's some decomposition into E reps of these and of the Ws as well, that their fusion is precisely, now G will act on the collection of vacuar V1, VI, WJ precisely as in terms of the tensor product representation. So if I now label these things here, but basically D1R, so this is given in terms of vacuum one to vacuum N, D1R1, let's call these here V, W, W1, Wm. All right, now the question is, what is W1R, tensor W1, D1RW? So I have to now take the tensor product of these things and decompose it again back into G because G acts on this full collection of things. So this will exactly be what I wrote down over there. So that's a much more stupid annotation unfortunately, but I don't need you, right? Because the vacuar of these guys are essentially, on this side we have just VI, WJ. That's the collection of vacuar for that TQFT. And now I have a G action on this and I just decompose it into E reps. So in this way, I generically, in particular when this is a non-Abelian group, these are now non-invertible topological lines and this is perfect sort of board usage because we had here back G, what is this set? This is actually what people call the representations of G fusion category. And that's for G non-Abelian, the fusion is non-invertible. And now I'm probably at five minutes. So I can now say, well, this was a two-dimensional theory and I had a one-dimensional TQFT and it gave me these one-dimensional defects. If I have a D-dimensional theory, then I can in fact here take TQFTs, G of dimension D minus one, stack them on top and then I would get a non-invertible D minus one defect generically. So this is the idea that we get non-invertibles by gauging these zero form symmetries and the way we actually determine the symmetries in the dual theory is by determining these GT QFTs. And I'll go through this explicitly in three dimensions in the first lecture tomorrow and then we'll see examples of quantum field theories where these non-invertible symmetries are happening and also some of you will know about condensation defects and they're sort of exactly these types of topological defects. So something else I wanted to say, right? So maybe I should write this down what I just said in words because that's the important punchline. So what we've seen is in two dimensions, if I take this vector G, so this is literally just math code word for G zero, zero form symmetry and I gauge that it doesn't have any of these anomalies, I can gauge it, then I will get the symmetry category rep G which is for G zero, just the representations. The fusion is just a tensor product. It's basically the tensor product decomposition and for a billion, this is non-invertible and these defects here, these D one, R are essentially coming from one-dimensional GT QFTs and you'll see tomorrow that in three dimensions there's also a way to have a complete invertible group like zero form symmetry and we actually here would get what's called a two category two vector G. Again, this is just code for invertible group like zero form symmetry, we gauge this and we get what's called a two representation of G and what these are essentially compensation defects for this symmetry. Okay, so are there any questions? So I understand that these are some non-invertible operations, so in what sense are the symmetries, do they preserve some correlation? Yes, so in fact, constructed it basically very abstract, it didn't tell you what is the theory T two, right? But it could have been any two-dimensional CFT or two-dimensional quantum field theory that has a zero form symmetry. Whichever, so for example, you could have, I guess you could do an SO3 WZW and act with the S3 and gauge that you will get these types of things and they would act on the CFT, on the gauge theory. They would, yeah, you can, these are genuinely zero forms of the global symmetries of that theory. So you can act, for example, in this case, you can really insert them into torus partition from right. You can insert them into D1G, for example, D1R. Let's put an A index here, Q to the L zero, for example. So you insert them into like this and you can compute correlation functions or partition functions with those lines inserted. So for verlanda lines, this is very well studied because you know exactly what their fusion is from the verlanda formula and so on. So these are, in this case, they're really acting still on local operators, right? The interesting thing here is that these now become also one form symmetries and higher form symmetries that are non-invertible that act on line operators, for example. So those will preserve the correlation functions of the line operators? Yes, so then you can, for example, you could ask, for example, just in the invertible one form symmetry case, you can ask what is actually the action on the Wilson lines on the top lines and you will see a non-invertible one form symmetries and then there's a question, how do you actually act on the line operators? Because now it's a representation of this object. It's not a group representation question anymore, but it becomes a question of how do you actually represent this sort of structure? So here we assume that G is finite group. Yeah. Yeah, if it's, I... I'm just wondering if there is some sort of minimal kind of physically sensible example where this representation category is not semi-simple. Yeah, so maybe some topologically twisted theory. More you have... Well, if I take some G not finite group, one can have non-simple category of representation, so you have, you want... Right. Okay. You wouldn't have this unique decompositions and there will be some other issues. You would have what? You wouldn't have this unique decomposition in particular in the direct set. Right, but I mean, so for example, like your Q mod Z story, or what do you mean? Yeah, I mean, but is it... Well, it's an infinite, it's an example there, it's an example of an infinite. No, I mean, there can be some sort of group, which is, I don't know, well, the simplest example is something like you get a continuous group of... Oh yeah, right, right, right, right. ...partrangle or matrices like this. Whoops, oh, oh gosh, all right. We'll sort out the... Yeah, we can clean it up a little bit. Yeah, it can't really look like this, all right. So there's one, it's like a domino thing. Chalk goes down and this goes down. Anyway, yeah, that's actually an interesting way. So actually, what is the diffusion of the result? Because you can obviously do the same thing. You can ask what are the TQFTs that are symmetric under such a thing. I don't know whether that's known, but then you can stack them. This construction is also interesting, and I don't think that has been sort of studied from this perspective. You could also think of constructing line operators in quantum filters in this way. By now not taking TQFTs, but say one-dimensional theories or maybe two-dimensional surface operators and engage in this way construct non-trivial operator, defect operators. I just wanna check again why we wanted a one-dimensional topological theory. Could you just do it? Yeah, so here it was because I wanted to, well, okay, so I was gauging a zero-form symmetry and I just could have generalized entire dimensions. In fact, you can generalize it to a P-form symmetry and then you could have TQFTs that are protected by P-form symmetries or even higher group symmetries. I mean, what are the options? I could have done a local operator, a topological local operator. There's not a lot of structure there, but one-dimensional or I just stack a full T, two-dimensional theorem top, right? That would sort of be something like generating a minus one-form symmetry. So here, the reason why I'm doing this is because I actually am constructing now topological lines that generate a zero-form symmetry. That should be the dual symmetry. What this is is really a way of constructing the dual symmetry from this stacking with TQFTs. The dual symmetry is nothing other than whatever, if I start with this and I gauge it, the B-head I had at the very beginning, but now I'm not restricting myself to any kind of a billion G or whatever. But you're not restricting. This is sort of the simplest example of the other more complicated situations. We can proceed, postpone the first questions to the discussion session. Let's thank Saco again. If you want to hear a few remarks about the pizza tonight, please stay around. So let me just make a few remarks. So yeah, you probably, in principle, you received this information. So there'll be two buses.