 So we are ready to start our morning session of this second day of this of the school so our first Lecturer this morning will be Sakura Shafferna Meki from the University of Oxford and her lectures will be about non-invertible symmetries and Of course, you are very much encouraged to ask questions. So please Sakura take it away great. Thanks very much. So first of all, thanks very much to the organizers That's Francesco Agnese Pavel and Bobby and for organizing the school I'm also inviting me to this wonderful place. It's very nice to be back here. Can you hear me and? Okay, we'll see whether you can read my handwriting so my lectures are on non-invertible symmetries and Before we get to Actually understanding what are non-invertible symmetries? I will have to serve at least give you some rundown on why do we care about symmetries in the first place? I think It's it's very clear that symmetries are at the key Sort of inside a key toolbox within theoretical physics sort of for Ages starting with, you know classical physics how you solve rotation of symmetric problems Also quantum mechanics again, how do you solve a rotation of symmetric problem and and then In quantum field theory, of course or classical field there first and in quantum field theory and Where you know, I think our main sort of focus will be When we talk about symmetries in quantum field theories, we can actually so distinguish between gauge symmetries and global symmetries and They essentially organize everything that we write down and you write down a Lagrangian for a U1 gauge theory The U1 actually tells us how to do that for a non-nuclear gauge theory and it organizes What representations of matter we can add to a gauge theory. It's a pure gauge theory and also of course it'll tell us What interactions are allowed? so symmetries are really at the core of physics So examples here, of course gauge symmetries are the say the gauge group of the standard model global symmetries can be flavor symmetries Like if you have a bunch of flavors and you can rotate them into each other They can be continuous, but it can also be discreet and so But the the key mantra sort of in physics has been symmetries our groups Or are realized by groups, right? So for example, if you take a gauge symmetry like SU3 times SU2 times U1 Maybe more Z6 or something if you want to be fancy or an SUNF or Z2 charge conjugation, etc. These are all groups. They could be league groups. They could be discreet groups, but for sure they should be sort of Things that have group properties. So actually since We're here. We might as well say what is a group? so usually I will actually think of these as Either continuous groups like elite groups or finite ones. So I have some sort of generators G and G and If I have G1 and G2 Then I have a map That says the G1 G2 should also be in G. So it's really I should think of this as a map G times G into G a multiplication map. There is a unique element E such that G times E is always G for all G and Absolutely crucially for all G There exists a G inverse Which means if I take G G inverse This is E. So there's always an element that brings you back So group like symmetries are invertible and that's also in the sense that the things they will be looking At will be non-invertible because we will relax the existence of such In inverse the existence of an inverse Okay, so in these lectures We will relax this Invertibility and we will say a Symmetry and I'll be very precise about this in a moment for now just Think of this as sort of like a symmetry It's just I'll specify what I mean with what they actually do on the on the theory etc But a symmetry will be an object more like you know has some elements and generators ABC. These are generators and There'll be a product So there'll be again. So let's say they are in some Structure, let's call it S for symmetry for lack of a better name and We have a again something that goes from S to S which will call some sort of fusion Takes a comma B Into a times B But we will not require this invertibility in fact this a comma B will generally have the structure of a sum The C is in And these can be some coefficients and so what we have is We have some kind of product structure But also a sum So this is not a normal group anymore Well, it's not just this actually more like it's like more an algebra. We've replaced essentially a group by an algebra and We'll see however Hopefully at the end of these lectures. I'll convince you that even algebras won't do the job and actually secretly these are categories and Since most people Now leave the room. I Will refrain from using the word category And I'll try to motivate everything that I do From a very physics sort of perspective So do I will not hit you with Infusion categories. I will actually try to explain to you Where these symmetries come from and how they're realized and in quantum field theories and Then I will tell you also by the way mathematicians call this so-and-so category. All right So I hope you will Stay Even if you don't like categories Okay, so you might ask and Is this Something I should care about Groups have done a pretty good job since about you know, no systems like 1918 so Since then it's been quite a successful story representations of group organize everything in quantum field theories Why do we need this and I want to give you I want to start off with a real world example That gives you a bit of a flavor. I think real world may put it into quotes and It gives you an idea of what sort of things will be looking at and this is in the context of two-dimensional Theories so a lot of you have done string theory a lot of you will know what two-dimensional conformal field theories are and then two-dimensional CFTs There are very special CFTs or CFTs have a very sorrow Times not a very sorrow symmetry and their special CFTs are finitely many primaries cow primaries with respect to the their sorrow and so if you have sort of H H bar I finitely many so H I H I bar top primaries then These are called rational CFTs and they're particularly beautiful structures One particularly nice one is the so-called easing model Which you may already know from the context of our G flow like chairs, etc So the easing model from this perspective is a is a is an RCFT Which has three such H is So we have H H bar is Actually, let's let's write the field lambda H H bar the sort of the identity Which is zero zero epsilon, which is one half one half Sigma is one sixteen sixteen and this is a c equals to one half So it's central charges one half And if you want to have a good sense of what sort of theory it is you can realize it as a Coset model as you to level K over you one Or K goes to two if you don't know what this is just think of it as the it's a two-dimensional CFT and it has a Spectrum of these type of operators and I'm using here by the left and right Verso I'm using so diagonal module invariant Okay, so these types of theories in particular the easing model have Particular types of one-dimensional line operators These are called the verlanda line operators. So RCFTs line operators and They're labeled precisely by these one epsilon and sigma. So in fact, I don't want to call them They're labeled by a lambda. So there's an L One there's an L sigma and an epsilon and how do they act and They're sort of We'll get we'll return to this example I will now not tell you exactly how you actually construct them But what they do is they act on sort of a state which has some weight lambda in terms of this operator mu It'll act as s mu lambda over s lambda What is here is the module s matrix, right? So this is the thing if you have a character lambda of minus one over tau that's Some as lambda mu chi of tau. So these are some bunch of lines and These actually have an interesting Compositions if you insert them into partition functions like for example into Torus partition function then Given how they act on these five lambdas. In fact, you will find that if you take two and insert them then the Composition to insert or mu and then new They're precisely Mu lambda We're now These here so we had here the s matrix and now And I'll conform field theory. There's also the fusion Matrixes or the s matrix here the s matrices. I also have fusion matrices and in a rational conform field theory I Can write them in terms of and lambda mu in terms of the s matrix as the sum s Alpha And this is the famous well in the formula and what happens is in these Alliance when I insert them these are precisely the fusion coefficients So they're written in terms of the s matrix and that comes from the fact that these L's act on the files in terms of this ratio of matrices Okay, so these are the types of general and this you can do for any rational conform field theory for the easing Model this simplifies and in fact with the L it access of the identity and The L epsilon is like a Z2 so L epsilon with L epsilon is L it and then the L epsilon with sigma is L sigma and it's the same as all sigma L epsilon and then the interesting one comes from L sigma Sigma is L it Plus L epsilon. All right, so ready. So this is just evaluating these ends with the death matrices for the specific easing model and then you get this type of Rule so if this were group like symmetry like for example the epsilon group like When I put two of them together the square to the identity so that it's its own inverse If you try this for sigma Well, it's one it's the identity plus the L epsilon So there's no way you could invert it and so in the end. This is sort of the hallmark of this non-invertible structure Okay, so here we have non-invertible. Okay, so you might say alright maybe That's not too spectacular. What are what do these things actually do when I look at Say the action on in this case, it's a two-dimensional theory. I can act on some local operator Vertical operator inserted and the interesting thing is the this non-invertible guy this L sigma Does something rather peculiar? And we'll see this also in higher dimensions Later in this lecture Yeah So this example is very simple, but besides being non-invertible It's a commute it's commutative this product. That's true. Yes. It's in general No, they're also non-invertible examples I Will probably not get to discussing some but we can discuss in discussion sessions later in the week when in higher dimensions They can be non-commuting cases as well. So how do these? So Essentially what we have is sort of the way you should think about this L sigma is you have an L sigma and another L sigma And you bring them together and you get sort of not just one other thing. You get a sum L it plus Epsilon So if you for example look at The sigma operator. It's a sigma was a card primary so I can actually look at if I had sort of sigma And I had this line L sigma and now I try to actually so now I want to put So I want to move this through this line So what I can do is So these lines actually and this will be something absolutely crucial. You know topological so I can deform them and So what you actually get is something like You can move this in here Sigma and here's still this L sigma and now you can hear you have an L sigma and L sigma line What you actually can do is it can use the fusion and what now happens is If this happened just the identity We could just use it off and move the sigma to the other side And I'm using here fuse these two lines But now L sigma Sigma is L it plus L epsilon so in fact I Have here L epsilon line attached. So this is what Condens matter person would call this year's sort of an ordered phase So this is you have some Local operator in here this sigma actually we shouldn't really call it sigma. It's not another field. It still has the same H is equal to 116 116, but It is attached to a line to an L epsilon line So this is what you would call a non Genuine Operator what is this here sigma here was a genuine operator exists on its own at the point operator Genuine and this would be the disordered phase So this is the example now we have started with a two-dimensional theory It has these lines. I didn't really explain to you where these lines came from But using the structure of how to act we can determine the fusion the fusion is non invertible And then we get from these non invertible fusions these type of actions on local operators Which are sort of mapping Things that exist on their own they're genuine in this case point-like operators to things that attach to lines Okay, so this is essentially the store in two dimensions and what we'll see in the following is There's a very similar story in higher Can I just ask you? Yeah, so now besides having the local operators and the line operators They also have introduced new kinds of objects. Is that right? I have one Subobjects these are another local operators or line operators, right? There are local operators with lines attached And so I call them non genuine because they're not there on their own I do need to attach a line to it for this object actually This is what the object is defined as the operator with this line attachment, right? So it's true. It's now that's why I called it now mu Because it has the same H value as sigma, but now it comes with this line attached And you'll see a similar sort of effect in when we go to gauge theories Then so genuine line operators become non genuine line operators as we go through these non invertible defects Okay, so the point is indeed in these lectures we will construct such non invertible symmetries in These and these always for me space-time just in case there's a Condense metal theorist that's gonna mess in the audience the d space-time dimension d greater than 2 So in 3 and 4 dimensions so 3d 4d All key examples and that's not obvious that such a thing would exist because so far Right as I said at the beginning everything is group-like that we have encountered until a lot of recently Okay, so this is To give you an idea of where these lectures are heading. We're trying to construct these types of Non-invertible symmetries and by the way this particular one I should say that is called And the promise money is the sell sigma and so a lot of these high-dimensional versions are called promise money defects and Yeah, please go ahead where this L epsilon line Meets this L sigma line. Yeah, can one interpret that some local operator living at that point Yeah, this is a local operator No, at the end of the slide look at the end of the L epsilon line on the other end, which meets this Here you mean here the L sigma line. Yeah. Yeah, so there's a junction Between this line and that line. I see and that's essentially You can think of it as this equation What L sigma L epsilon so this object here you can think of here There's an L sigma coming in and L epsilon going in and there's another L sigma. I see okay. Thank you Any other questions? Yeah, I'm only in the sort of generalized symmetries picture when lines and topological topological operators have a boundary You interpret there being some sort of halonomy acting around the boundary So at the end what boundary as in you you split the space into different and there's junctions between different topological Operators and the the old sort of guy off on a towel paper with the original generalized symmetry stuff And you can you know wrap closed topological operators around stuff? Oh, I will get down. We haven't even talked about any of that. That's we will we will discuss all this now This is just this is something to wet your appetite to say well, this is kind of weird I've not seen this before and Well, this exists also in four-dimensional young milk series and so that's an interesting thing But first let's learn how we actually construct symmetries. We will discuss. Yeah, please. There's another question I have a question. Where do these operators act on in which space? Right, that's a very good question. So I was that's another way of saying you didn't explain to us exactly what What these operators are, right? So here these are actually in in in two dimensions what they do generate are actually zero. They're just standard symmetries So they're like what we would call zero form symmetries and they act on these operators as you can think of it as Just like a generalized type of flavor symmetry if you want But these operators act on the Hilbert space Yeah, so we will get to exactly all these questions I'll explain in a moment when I actually say by the way, we should think of symmetries as topological operators I'll get to that Next page. Thank you Okay, okay, so these lectures will so Essentially be based on Doing as I said that the goal will be this and how do we do this? Well the key sort of Observation that underlies everything is that actually and this comes from this 2014 I'll not be giving any references except to neuter and this paper here and I am actually Writing some lecture notes which would be finished if I didn't chop off part of my finger last week But essentially there will be all the references in those notes, but the guy Otto Kapustin We're what? aside back Paper from 2014 on generalized symmetries really key observation and that is to say a symmetry a topological operator in your theory and this identification is sort of at the heart of All future generalizations that will be discussing and so this is the thing if you're taking away one thing today That symmetry is a topological operators and usually we all say a topological operator and this notation You should also take away is there is I'll call topological operators always D some subindex P. This is the dimension of Of the operator and then it'll be labeled by something else. This could be for example a group element Or other labels and sometimes I will also specify a manifold But sometimes this will be also something that I'll denote by DPA of MP Or this is the M manifold on which DP is localized But since it's topological it will actually not depend on a lot of interesting like metric structure and this just on the topology Okay, so the first step in in constructing interesting things coming from this observation was of course done in that paper and the The first termization is to invertible generalized global symmetries So you all know what an invertible standard zero form or standard symmetry is I will have to explain to you a little bit what in non invert what an invertible is before we go to non invertible And so the first lecture will be mostly about this whatever is left of it and Then lecture two This is lecture one lecture two will be studying and non invertibles symmetries 2d and I'll give you a universal construction lecture three and look at 40 3d and 40 gates theory examples and Applications and lecture four Can I put it? You are here. You can see it I had to do that and L4 it will be sort of more Generals, it will be generalized charges and what's called the sim tft But you won't know what that is anyway at the moment and if you do then you probably shouldn't be in these lectures So I won't write it down Okay, so we'll start today and You had a boot camp on What was it yesterday some boot camp yesterday? You had a boot camp today. We have a boot camp on invertible higher form symmetries So lecture one is basically Invertible if to start First understanding what are these generalized symmetries at all now? Unfortunately, there's not really a very good review at the moment. So we'll have to I'll put some other things on the web page later But the starting point is again symmetries of topological operators and let's understand that statement in terms of the standard sort of Neusser theorems or Neusser charges So what does Neusser in a slightly more modern perspective tell us is? Consider a symmetry. So G is some group. So here we have a theory. It's a symmetry symmetry We'll imply a conserved and so then I'm not sure what this tells us and so Usually we'll have first a conserved current. So it's a J mu be a current With D mu J mu is equal to zero And then Neusser basically tells us on this we can construct the charge and I'll sort of As I mean D dimensions irrespective at D should be quite larger than we go to So the Neusser charges obtained by now integrating the component of the current D minus one X J zero over some MD minus one Now a little bit more Compactly, we can just say this D star J or J Is D X mu J mu and then this just becomes an integral now over in D minus one over star J and Of course the point is that now Q dot is zero So this is We're good and fine and now this is so classical when we go to quantum field theory what we usually want to do is in In fact go from a symmetry to watered entities so how do we implement these things in QFTs so in QFT We want to implement this in terms of watered entities So for example, we can say this current conservation so D mu Insert this into some correlation function J at some point X On some operator at P Then there should be something like a D dimensional delta X minus P of the symmetry acting on So here this delta is basically here the symmetry just means the delta of the action Okay, so this equation I can sort of integrate over A volume and the D dimensional volume and now I what I actually get is The following sort of nice pictures. I can rewrite all this in terms of integral D DX And D and here's a delta function X minus P I take out in space and there's this operator inserted at at the point P and N is essentially think of it as some volume that's filling this end-dimensional Space and of course what I want to do is rewrite this now in terms of the boundary of this Using something like Gauss's theorem. So here there's some MD Which is the boundary at D minus one and So Obviously can do that by using The D star J is in fact Just our J of P and This is now nothing other than by this equality over there the charge Q of M D minus one with this operator of P So what we have is this integral with the delta function is now equal to the charge Nicolay where we have this charge and D minus one when D minus one is the boundary of the space and D is sort of this inside is the volume And this is the boundary Okay So one question you could ask is what actually is this object here this integral where I'm integrating this now over This and D well this will only be non-zero if the point P is actually inside this volume So this actually is only non-zero if P is inside and D Yeah, and this is like so the the crucial bit It doesn't actually matter so much How we integrate this because in the end this will also this actually just depends on the boundary But again, it's important that inside this MD minus one P is in is sort of contained So mathematically what this is is just a linking in fact This integral is just a linking between this point and This surface MD minus one Okay, so this expression is unequal to zero if MD minus one Contains P or mathematically this means the linking between this co-dimension one surface and the point is Non-zero So at this point is a bit stupid because it's just a point and a surface But you'll see that this is what then arises to higher dimensions So one of the important observation is that it really wasn't important so much Where this MD my what kind of shape this MD minus one is the important note here is This is sort of a topological property this linking They actually do not right to the the the specific the MD minus one geometry immaterial and So this is sort of The key pointer in fact, right so what we can write and I if you put this all together that this charge Q and D minus one OP is the linking MD minus one with this point And this is literally just a sort of topological object And so this actually is what it motivates to say, okay We don't need to do all this business with the currents and then the charge we can just think of this as there's some sort of operator that measures the charge of this operator OP at P and so this is essentially what then motivates introducing these type of things that the symmetries are really topological operators and those are exactly Realizing these sort of you know Implementing much manifestly the fact that these linking is actually a topological quantity, so We think I'm going to write the same thing that's written here so this motivates replace Some topological operator in this case a D minus one dimension operator on this manifold logical operator That implements What it measures? The charge right so if this is still a little bit confusing right to think of it is there's some in a classical setting There's a charged object like an electron. You're just measuring now the flux Through a sphere the sphere doesn't have to be a sphere could be also a cube or something a dented sphere As long as you're surrounding the electron, you'll be measuring the correct charge Okay, so this is essentially what you do and now for example for you one Symmetry if your group G is a you won Then in fact, we could write down this by explicitly so a DD minus one for say G and you won with me something e to the i alpha and then this D e to the i alpha of md minus one It's just given in terms of discharge e to the i alpha q of m D minus one And also from this, it's completely obvious that if we now stick in two of these guys We'll just get now All right, if you have DD minus one Some G and another minus one H then we get the same as We can also first collect these two things and get G times H So in this example, it's completely obvious for more complicated groups So I want to think about this for a second, but you just get DD minus one G H and D minus one and So this is what you would call these topological operators compose like groups Yes, absolutely, please one and just sort of basically just Is it difficult to put this on a discreet space or it's Yeah, I mean people do this all the time in condensed metal Yes, you can do that for sure But then I think you would want to do something slightly more like what we'll get to in a moment where these groups are maybe also Finite groups you don't have these currents So I think that's a very It's a bit tangential to what but yeah, you can In fact One place to look at is Paul Fendley's work Can you please explain again? What is this linking or where does it come from? So here it's a little Okay, take an apple right and now Basically say inside this the surface of the full apple is Andy It's a full ball and inside there's a core and I'd say the core is actually Apple pits and they're charged say under some electric charge for some reason And now you're trying to measure that charge Right, how he would you do this you would you could either integrate a charge density and just measure that That would be the correct charge or you could just say let's look at the flux Through the surface of the apple I'd electric flux that goes through it and that basically is what this topological operator would measure now It's not quite the linking is basically saying as long as these these pips are inside the apple It doesn't matter where they're located or whether the apple is like a sphere or it's not dented or whatever That's sort of the statement. It's a linking is just saying there this this point is in fact Okay, the filling Andy Has a non-chive intersection with this P. That's what the statement is In this example, MD is a boundary of some other manifold. Yes, it's just a surface of the Indian I mean this definition is more general for any manifold or Does it have to be here? so it's Right, so here this definition just depends on MD minus one Right, but it actually says it doesn't depend on right sir the really the example is You could also have put this thing into a box right as long as P is inside here It doesn't depend on the specifics of you know, is it a cube? Is it a sphere or whatever it is as long as P is inside this volume in that if you fill this That actually intersects with P then this is non-trivial It's a little bit more intuitive. I think when you go to higher dimensions when These are not just one thing is contained in the other But there's some non-trivial linking then becomes clearer why it's called linking perhaps My question is if MD is not a boundary of I mean you can imagine Different kinds of topological in these which are not equivalent to each other for example Yes, then I think you have to be more careful So here I have not okay I put on the carpet so normally would say MD minus one is sort of a spatial slice and it's non-compact So here I'm sort of anticipating that I will generalize this and these actually sort of compact spaces so yes, that's sort of By design so with this in place. I want to now make the definition what we would call now a standard ordinary global symmetry so a global so called zero form symmetry is Defined D minus one dimensional topological operators and so these are precisely these D minus one some a and there's the finance on MD minus ones and if that group like and so if Symmetry s Zero if s zero Is a group G zero Then this label a is equal to G is in G and Then they will have precisely this composition I wrote down over there. It's good like And once you have this definition Then you can start saying well, why did this stop here? Why did we not? Consider other than types of topological operators and so here the only key point was this linking between the surfaces and the generalized Global symmetry so the P form symmetries are precisely those where we now have we replace this picture in a second with some Not point like operators, but extended operators So higher form symmetry P form symmetry One more thing and this of course It acts acts on points like operators. So zero dimensional Operators and then the charge is calculated by the linking now for the P form symmetry is Generated by Okay, let me let me first code. Yeah, okay people by D It'll be clear in a moment why it has to be this D minus P minus one so So these are code I mentioned P plus one topological operators and They act on P dimensional Extended operators. Let's call them a sub P. So here we would call these Oh, it's up zero So why is this the correct thing? It's essentially in D dimensions P and co-dimension P plus one will have their exactly one extra direction where they can still link with each other So it might be like that. This is one example This is when P is equal to zero when P equals to one and say we're in D is equal to four actions with three So then this OP is now one-dimensional object D P minus D minus P minus one is also one-dimensional. So we have these D one Operators and they can exactly Link around these guys So in three dimensions And so again here, there's a non-trivial way that these topological things can link So there's a linking between the manifold M D minus P minus one and OP That calculates this charge In four dimensions right these these topological guys will become two-dimensionals But more difficult to observe this by in three-dimension can still see there's this one-dimensional line This is the charge. This is what we place this object here And then there are these topological defects of the lines as well and they link with them in three dimensions Okay, so this is There's sort of natural generalization Here there was when there's a group this group could be non-abelian, of course But here I want to think about this in fact you've already in three dimensions if you had D1 D1 And you fuse them That has to be sort of a billion because you cannot actually you can sort of move them Out and then around each other and you would basically get sort of an inconsistency. So in fact these are for P bigger than zero These are billion would usually when they group like Symmetry is generated by this and again, there's a comment when they group like I will call them P form symmetry groups and that's what we'll focus on today Can one think of the non-abelian objects in terms of germs? Which not a billion object. No, I mean some non-abelian higher from symmetries Can they be I think you would have to attach them to something else right they need to if there's a non-generate and then maybe you can have the non-commute But it's you can't like if they're just genuine one-dimensional things you can always shift them out I don't know what you mean is germs in this context or we can talk about it later okay, so When these form groups so if they again you can insert into correlation functions play the same game if they're sort of they can if they're labeled by groups D minus P minus one Then I will often not write this manifold because this already is that I mentioned and I assume that they will Be defined as a manifold so I can so compose them and I'll use this sort of terminology that this is again a Composition between these two guys D minus P minus one G H and G and H are now in GP and this is the P form symmetry group Okay, let's do an example So you know this abstract the example and this is sort of Really the thing that essentially Maxwell could have already thought about but he didn't We had to write it for much longer. So basically we do a U1 gauge theory and he'd be focused on P equals to 1 So we have line operators, but we're in four dimensions And so we have basically a gauge field f is equal to da and now we can write down the following and f of course satisfies Df is zero, but also the star f is zero And that gives us two conserved currents and now we can essentially do Neusser But just now with this two-form object and we can define these types of guys the electric one and some manifold and two It's just e to the i alpha integral star f and there's another one Magnetic on a two manifold into the i alpha integral f And again because of this d star f and df is zero. These are again topological and alpha here Should be constrained to zero to two pi. So is it important that this is gauge theory or not? Right. So one one one really really important thing is in this whole lectures Which we build up to to the very end is we can really extract out the symmetries before we Start talking about symmetry acting on a theory. So here. I'm just giving you an example Why construct these topological defects from the gauge fields? But I could also have just written down a u1 one-form symmetry generator And they compose like you once and just study it to have its properties So here it depends on these specific properties. Yes, I mean what I was thinking is that you know Basically, you're writing down extended objects and those are basically Wilson lines and no, no, I haven't written down any yet These are not there wasn't these are the topological generator. Okay, but I also will You preempting I will write down the Wilson the top lines. Okay, because what kind of you know strikes me here Is that you're working with gauge invariance objects already so? kind of Like f is already gauge invariant right so um I mean, can you differentiate between a Global symmetry like a let's say standard global symmetry not gauge symmetry and the gauge symmetry in this case If you're already working with gauging variant degrees of freedom But this is a topological operator. So right because the df is zero again Might I can put them to M2 goes to M2 with a little dent that just cancels out by cows, right? So It's topological, so this is You know But but I think you the question you're asking is a good one He's just saying how much of that is actually property of the quantum They the gauge theory here and how much is it property of an abstract like there's a symmetry structure And the message I want to actually get to eventually is it's really there's a symmetry structure That's much more fundamental and in fact it's like okay. You're saying you gave me a Basically a representation of a group in a particular gauge theory, right? It's sort of the higher form symmetry analog of I gave you as you too with a fundamental representation Now could I have also studied that in terms of just groups and its representation theory and my answer is yes I think this was your question and so essentially where we'll get to is There were there are these things they will act on line operators. What are these general charges? Just like you would say there's a group like zero form symmetry and it stay a flavor symmetry and it has representations How do I study that? So these types of questions will get to I'm not just making an example. There was another question up there. I think Okay Right and as you just said already and the charged operators are Precisely The Wilson lines and the top lines Of this you want gauge theory the one dimensional the O ones the line are line operators and they're just the Wilson line which is an e to the i Q integral a and then there's a top line which I sort of by abusive some notation Magnetic charge a dual which I can do it for this where a duo is just the a dual is just Starf of course in a more not complicated case these top lines. I need to be careful So here we would say the theory has a you one One form symmetry so put the superscript one Electric and another you one one form symmetry magnetic And these are the charged operators on of these So more interestingly if we go to so this was example one example two if we have a non-op you engage theory So we have some gauge Algebra mass fact G Say you have some a DE type for simplicity Then the line operators on factors the top lines and the Wilson lines again and What are they actually characterized by? They sent you all the line operators They're given in terms of the labeled by so the Wilson lines are labeled by some weights Of this league Alba G module or roots and the top lines are basically in the co-way like this Not the other roots. This is sort of like what kind of See Wilson lines will be in some representations of representations will be essentially given in terms of some weight Yes, I just was it doesn't have to be Just in case people preferred them. It's absolutely not important But let's stick with it because okay, we even will be simplifying it to just For simplicity, I don't have to be so in fact if you go to your favorite Lee algebra league group This will say well this quotient. There's nothing other ZG with it, which is the center G or G is Okay, let's and now what is the center of G for example for SU 2 and for SU n And then ZG is just Zn and this is just sort of saying there are Wilson lines labeled from 1 to n and the basic degenerates the fundamental Wilson line and So this is what the What the line operators are But now we were saying all we define actually a symmetry. So, you know What actually is the one form symmetry here? so now that was slightly more complicated matter Because if you take your favorite gauge theory And you just stick in an arbitrary set Let's call this set of line operators L and you call you you you put in an arbitrary Set of line operators like alpha beta inside L, but they're labeled by two elements in the centers And look at correlation function a lot of file beta with some other stuff inside Then in fact you find if you move them around each other they can pick up phases So then I die to pie I Alpha comma beta. I'll tell you in a moment what this is L beta so there's some ordering ambiguity and This alpha beta is basically the Dirac pairing and which gives you an ordering ambiguity and what this means is so this is essentially alpha the electric beta magnetic where these are the weight and the co weights respectively minus The electric Magnetic So in order to actually have a mutually a local set of line operators in your theory Which would be preferable. So these type of correlation function are well-defined. You actually need to pick a Subset so need to pick mutually local and so we want maybe to pick a subset and That's also maximally So maximal subset of this type and This is what usually is called a polarization So the theory with all these line operators isn't actually a good theory yet. It's not what's called an absolute theory only one to pick Mutually local subset of lines. Do you actually get an absolute theory and this is this choice of sort of polarization Which I call lambda in fact makes that happen And so of course if we can now have SU n Finish off with this and then you'll see where we are. And so for example for SU n For mass frag SU n the le algebra We had that L was essentially Zn plus the n And in this case, these were the worst line the top lines There's of course a simple choice that lambda is just what's called as the electric magnetic It's just Zn electric So here we have Wilson lines, but no tough lines. This is what you usually call the SU big n gauge group and then if we pick lambda is equal to Zn magnetic, right? So each one of these choices would be okay because this is zero Then this is what's called the PSUN or SU n Mod Zn Gauge theory and that has only the top lines But these are just examples. You can also pick different subsets and find different solutions to this actually being a Trivial phase Okay Okay You told me I should finish Five minutes before we have four minutes are there questions of any urgency because if not Then I'll actually talk about also briefly screening of charges And now we're almost finished and then can go next lecture to Non-invertable So can you just explain so these are which form symmetries are you considering these are what what form symmetries are these are one form symmetries So then this okay, so I haven't told you yet What does the one from say excellent question because I only told you this is now a good theory This is I picked a mutuals local set of line operators now indeed. What type of symmetry do I get from this? Well, we hope it's a one-form symmetry because we start with line operators and indeed This is sort of the the Pontiagin dual group so the Homs from Lambda into you one That's actually exactly the one form symmetry. So in this case, these are all the end. So the end actually lambda hat for the end It's just equal to the end isomorphic to that because it's a billion. These are just the characters So this is just all a billion. So it's just maps back to the end. So in this case, this is the one-form symmetry So this case you would have an electric one-form symmetry and in this case the magnetic one So this would come from pi one whereas this here comes from the center so maybe that's a good thing to summarize if you have a Say a four in fact any dimensional gauge group so you have the one-form symmetry It's essentially generated here by topological operators now G on some m2's or now G is in The end and so we have now D to G D to H is D to G H Module N so this is sort of The one-form symmetry in this particular example More generally if you gauge group G the gauge group And so here this is the gauge group gauge group If the gauge group is simply connected like for example the SUN or spin then the one-form symmetry is always the center So this case the N and this case depending on what N is Z2 Z2 times Z2 or Z4 And also the E6, E7 have exactly the center. So these are exactly the charges that line operators So now we have a line operator One and now there's a two-dimensional. They can only draw a projection D2 G that now measures this charge of These these Wilson lines or top lines And any other questions So is there a construction of this D2 as some integral? I mean like No, so okay in the U1 case we were lucky we could write down a current and some Here I'm just saying there is not a topological operator which will measure this So you can of course there are these gucaviton operators for gauge theories and you can write them down But in general, I will just have this sort of as an abstract notion I will construct these operators as symmetries from for For certain theories, so I haven't constructed to you the actual e to the integral of the current because there's no work current here Just to make sure because you have not directly mentioned it The linking stuff that you mentioned earlier. This is the linking from not theory, right? Or is this something else? It's essentially the same right if I have two in three dimensions Why if this were in three this would be a one, right? You have a String and indeed it's exactly that link. There's a linking between these two One is an infinite extended circle. The other one is a circle So you can fill one and then the other one will intersect that which is exactly that definition No more questions. We're at 31 Okay, so we'll finish then and next time we'll start with a little bit more background on these higher form symmetries because We haven't really talked about Screening and about also background fields and anomalies Thank you. We will now have a break until 11 for the second lecture and then we will have some