 Third lectures about the non-invertible symmetries by Sakura Great, thank you Okay, is this on? Yes, all right great, so let's start with a recap of last lecture and The main gist of it was we wanted to go back to the problem of studying the symmetries in two dimensions and we started with something very trivial in the sense that we just have a Zero forms of global symmetry in two dimensions Nothing fancy. It's completely invertible. It has essentially topological lines labeled by elements of a group Let's take g to be finite. So here g is finite But doesn't have to be a billion who shall we? And it fuses according to the group law the one g the one h is the one gh and Then we said let's gauge this zero form symmetry and one way of doing that was to actually Take the following perspective and this is I'm only repeating this perspective because this is the one that will use also nine higher dimensions So we take the two-dimensional theory and we take a product with a one-dimensional topological theory and this one dimension topological theory has the same g zero The same group g appearing here Symmetry what that meant is a one-dimensional topological theory has a bunch of a cure and Now there's a group acting on these back So the group acting on the back here will basically furnish some representation of this group It'll decompose the vacuums to some erupts and we're just taking the product of these two and then gauge this diagonal g and The point was once we gauge this this product t's is to be a product It becomes actually now sort of a topological defect This is this one d t q of t becomes now a defect in this theory and this is now as before labeled by representations and So of the group gene And so now if you're asking what actually is diffusion on these guys Well, this just descends from how these one d t q of t is used these were just representations And so they will fuse according to one of these defects with representation R1 our representation R2 will be a sum over some integer coefficients here times d1 R3 and Just to be very very clear these coefficients here are nothing other than the things that you do in your group theory lectures Right you decompose the reps R1 R2 This is not here the tensor product of representations and you have a clebsch Gordon decomposition into Ereps R3 say with some coefficients and multiplicities and you know very well that this is usually not just one term But it's a bunch of irreps and so in this sense the Symmetry after we've gauged for at least a non-abelian symmetry group G becomes Non-invertible, right? I did this example when G was a billion and essentially everything is characters and this actually turns out to be a group again but if G is non-abelian this Object which are called web G is just a representations of G Is a non-invertible Symmetry in two dimensions and it's this sort of thing that will not generalize to higher dimensions Where these kind of non-invertible structures are not that? Well understood yet, and it gives us a nice systematic way of constructing such non-invertible Are there any questions about this so far? This is just sort of the recap from last week not last week, last lecture It's been a long week Some time dilation in my head. Anyway, okay, so let's we generalize this to higher dimensions and If the idea will always be you start with a theory you take a product with a TQFT So this theory has some group G zero acting Still hear me something So this is a TQFT with G Symmetry and I'll explain what this is in concrete detail and in three dimensions and Then what we're doing is we gauging G and then this becomes a measure keep this So this is a TQFT. In fact, I want it to be a dimension One and this becomes a DD minus one defect Yep So once we are in higher dimensions While the TQFT that we take the product with the should have dimension D minus one It can also be of different dimension Then you will construct different types of defects here I will start with a zero form symmetry and I'm saying okay I would like to actually construct So to speak the defects that are actually now the analogs of these lines Like the dual symmetry so to speak and so this is basically then a D minus one topological defect I want to know what are actually these types of defects but indeed in the In the paper and I now try to actually spell my collaborators last name correctly This is the correct spelling And so in this paper you'll see lots of different things you can gauge even You can even have GP P form symmetries or higher group symmetries And then you have that defect of different dimensionality that you can construct in this way Just to make sure I understand so even keeping the zero form symmetry. Can we reduce the dimension of the TQFT? So you could in principle also for example in a tool. I will do actually the case of a 3d theory 3d T And I could also take up the set of topological lines and try to construct the rep and I'll get some lines And in fact indeed I do have these rep G lines in the gauge theory They're also there. I will get to them in a moment So in fact, this is what the abstract thing in these things this is the notation these will call theta defects For the reasons I explained yesterday That's like a theta angle Yes Are you assuming that G is a non-anomalous in the DQFT? Yes, so absolutely Can I take like an anomalous? Simmer in the DQFT and in this case I get like a defect attached to Yes, I guess you could yes something some sort of a non-genuine type thing. Yes, sure So I think this is literally just the simplest way to Implement this sort of construction, but as I said also yesterday you can now go wild with this You can do all kinds of things even non topological defects. You could construct in this way Okay, well, let's get now to the three-dimensional so T is a 3d theory with G0 So G0 is generated Basics I did some topological surfaces, but anyway, this is not important So now actually what I would like to do is in this three-dimensional theory This is theory. It has this G acting on it. I now take a product with a 2d TQFT that has That has G0 symmetry and So and then proceed in the same way So the first question you need to answer is actually what such 2d TQFTs with G7 tree and Then what is their fusion and then this will tell us the theta defects and the fusion in the theory T mod G By that sort of the logical progression and so that's what will this does now and actually um Will give us the first example of a nice Such structure so these TQFTs are Also known as G SPT phases SPT for symmetry Protected topological Phases so this comes out of sort of a cotton and smatter literature really and And they're actually classified at least in okay, some of these are classified The more general dimension this can be more complicated But the ones that will be interested they're classified by group cohomology our case. This will be Omega and H2 of G0 And I explained very quickly to give you an intuition why this is Sort of an interest why this could be correct and actually if you are Someone who's worked with string or defaults. Omega really is something like discreet torsion in Or before constructions and strings here if this is helpful So one way of thinking about this is we've seen they have sort of we're starting with a G symmetric theory So we have some lines say that are actually GNH and GH type lines and we're trying to actually ask what kind of Objects could be attached here some phases Omega GH Sort of right before we're just saying there's some kind of composition of these group Elements and it's just G times H, but now when I say actually I can attach a phase here There's some Omega in you one so what are they allowed Such phases and they of course need to be consistent with things that we also looked at Like these sort of diagrams If I do G at HK GHK GHK Why it needs to consistent with these numbers of decorating these these omegas and Now you can see they'll be a non-trivial consistency condition on these omegas because they'll basically say Omega GH first Times Omega and now here. I'm coming in with GH Right, so this is like Omega GH comma K That needs to be the same as first HK and then Omega and now here we have HK and so this is GHK So this sort of identity actually says that is this is sort of saying in some sense This is the Delta Omega and this sort of Choreological sense is zero So it's in this sort of Choremology it's a U1 valued map from G to G right so Omega is a map from G to G D times G into you one point and it satisfies these conditions and that's precisely the sort of thing Okay, I There's more to this story But that's what we'll need for the for the moment. So these type of sort of trivial TQ of T's right it they really are just characterized by the fact that they have this symmetry They will be the things that we can now attach Here if you're preserving the full G symmetry So I'll now tell you what actually is sort of the set of TQ of T's that we can attach to the 2D TQ of T's Basically that two sort of sets of things we can do we can say either they preserve and then there's one vacuum and There's one vacuum basically for each choice of these For each H2 But we all know if you have a symmetry and you have such a two-dimensional Theory you can also now look at lack of an extra break symmetry So you can have a G is broken spontaneously to H the subgroup spontaneous symmetry breaking and Then also When you do that you break to a subgroup and there's a multiplicity So now they're that G mod H Many vacuar But again once we have an H we can also turn on an H SPT phase So in fact what we have is G mod H vacuar with SPT phase for H I in element in H2 and let's call it Omega H and that actually gives you all the 2D TQ of T's and so Lo and behold what we find is these 2D TQ These 2D TQ of T's are basically labeled by Some H and some alpha H and H is basically any subgroup Including the group itself any subgroup So these are the 2D TQ of T's these are the 2D G TQ of T's Then we can use to play this game of these data defense and Now the question is when we have these what is actually now their fusion and Yes, yes So when you say that there are G mod H vacuar Do you mean that the number of vacuar is the order of that quotient right or because in the previous case in one dimensions We had generic irreducible representations of G So the number of vacuers not necessarily as many as the order of the group Yes, so here the multiplicity is really given in terms of the this coset So we'll do an example. It's that that determines the multiplicity of the so you essentially Have symmetry breaking and in each vacuum you have an H symmetry, but you have G mod H many such Suppose the G is completely broken. Why can't I have a generic representation of G? Ah, but we I haven't said anything about representation yet. I haven't told you anything about Representation this is literally when the number of vacuers fixed and I can they cannot form a more generic representation Oh, you mean on top of this? Yes, because in the one-dimensional case where we had G and we said the number of vacuers is the dimension of our generic Representation of G. So you're saying each one of these things could form something more complicated. Yeah, that's my question Because here it was essentially just Everything was just determined in terms of the decomposition into representations, right? And here I think it's really a statement about in each one of these So for example, you could have asked is there also maybe a choice of different SPT phase But that actually can because actually this group G mod H so acts on these so You will find in the actual Line also in this what we're constructing here at these surfaces. There is actually a representation decomposition Secretly in this which is in the lines. So this is actually happening again on the level of the line So maybe this is or it's hidden. Okay, let me let me think about it Okay, so We're saying our TQT is a basic these on H some omega H and I would like to know what is the Refusion of To such T. So actually what I will do is I will restrict Okay, I don't know. Okay. Yes, I will restrict in this for the moment in fact completely to G abelian In fact, you can do this also more generally and I think then when yours what you're saying is completely correct You could do something more complicated But even what we'll see is in the G abelian case There is now also some non-invaluable structure So G zero abelian Okay, so G abelian so for example think about it as just a cyclic group and we'll do Z2 or Z4 these type of examples So essentially what the what the in this case? tensor now Tk Omega K right so H and K are both subgroups of G and there are these Cosycles, but potentially attached so this actually gives just some multiplicity That's T H intersected K comma Omega H intersected K So let me unpack this and explain to you what is actually this multiplicity and If this multiplicity isn't one then that also is already a non-invaluable Right because this is not a fusion that you could then easily invert or invert at all Okay, so each of these TQT's I think I can think of essentially as Right we said there's a bunch of vacuums vacuums may one the G mod H and you want to take the tensor product with K and the vacuums here are say W1 WG mod K order and So now what you'll get is in the decomposition This will essentially mean the vacuums over here are just going to be something again like the IWJ and What is actually the symmetry in these vacuums? The symmetry That's unbroken is going to be Exactly H intersected K Right, so you have in one set of vacuums H and the other K and now you'd like to know in this combined set actually what's preserved is this subgroup H mod K H intersected K So in fact, you get H intersected K And then there's a story with the co-cycle That actually goes along for the right But now the question is what is actually the small duplicity, right? So here So we get these toggle vacuums and now what we have is a symmetry that's unbroken. That's H intersected K so we start with G and In these vacuums we have an H Intersected K subgroup so now we have of course a Decomposition and H intersected K. So in fact this number here will be simply So Reduction of how many of these you had how many these you had modulo the action of this group So this number here might be decomposed now all these like you are into Orbit of this new subgroup and that's this quotient G mod H Times G mod K That's just the total number of vacuums But then we have to mod out by the group that we're now using the orbiting into and that's H intersected. Does that make sense? So this is now a non-trivial multiplicity potentially and if this is unequal to one, this is definitely a non-invertible Inclusion So our story was These are now the TQ of T's they fuse in this way Now we actually put them into our 3d theory we gauge and we should be getting something Non-invertible as a topological defect in these theories. So now we basically do the map H Omega H It becomes now defect in this case as well as a two-dimensional defect of D2 and I actually it's a little bit more convenient to label it by the coset comma omega H All right, and these are now topological surface defects this theory mod D0 right and this theory here the 3d theory This is what we get over here So I want to make an example Because you might think whoa, but now you're talking about a billion groups. Are you sure this is actually going to give something on invertible and it does So are there any questions about the general sort of story? So it is important that the co-cycle is non-trivial like I think when the co-cycle is non-trivial it's projected representations So in fact if I were to do the Z2 times Z2 example, which is the first that has actually a non-trivial co-cycle This whole thing becomes a very very complicated category and you can look it up in the appendix of those paper and It's a very rich structure already And I'll restrict myself to Z2, but you already see in Z2. This is quite interesting Okay, thanks, but you're right with the co-cycle. There are more choices you get more defects And you need to also then track that this I didn't tell you what this is But there's a little bit of a story how you actually construct this co-cycle From the ones you have over there Okay, but let's you know First look at the simplest example at any other questions Prepare this example. So what are the subgroups of Z2? Okay, and this will sound very trivial because it's not particularly So there's the trivial subgroup. Let's call it one so this trivial subgroup and then It's a simple group So there is Z2 so these are the two subgroups and In the in this case H2 of Z2 U1 Is actually 0 and This is common. I just made a Christian is basically if you want a Non-trivial co-cycle if you want omega non-zero Look at the Z2 Because I'm too short for this Okay So this is a reason of the starting point when we have a relatively simple setup Now what we have is essentially so this is the case what is symmetry? So my age is sort of fully preserved G is fully preserved and this is why it's completely broken So let's look at the case H is equal to 1 So in this case actually Okay, so this is like one vacuum and G is unbroken and I want to call this vacuum zero and Let's call this defect Accordingly D to it. It is right. So this is So this is basically G mod H is like one So H is equal to 1 So the tool is completely broken But it is symmetry breaking completely to nothing. So in this case what we have is and the two vacuums And then so the gene mod H is two and I want to call them plus and minus And now what we have to do is Okay in the defect Associated to this. I want to call D to Right, so H. This is just that G mod H is now Z to now We would like to calculate the fusion of these two defects. So D to it With itself That's one vacuum on one vacuum. They're unbroken subgroup as the intersection of Z to with itself at Z to D to it With Right, so now we have here. This is one vacuum. We have here the two plus minus vacuums So We have here H is Z to and here H is trivial So the intersection of these is trivial. So this is gonna be D to Z to right because in this case H intersected K is just again trivial The question is what's the multiplicity? So we get two vacuums, right zero tensor with zero tensor one a plus plus zero minus but the Group act is one orbit. So we get just one And now we also have So far these look completely boring These fusions, right? There's a whole list super super nice and very Invertible looking and now the question is what is D to Z to with D to Z to remember This is the one right where H is equal to trivial K is equal to one. So I actually age intersected K is gonna be one as well So we get D to Z to But now we have got here say plus minus and also plus minus And now what we get here is if you decompose into these orbits we get plus plus minus minus Plus plus minus plus minus plus in terms of the vacua and now of course the Z to the two orbits of These I can also just stick it in this formula when this is to get two orbits and They are telling us there's a two. So this is our first truly devised non-invertible fusion and non-two-dimensional theory So some of you will notice and will say well, I've seen this before And actually this is not a coincidence, but let me just pause for question at this point about this example so very Quick question So the fusion rules you wrote are all abelian in a sense Does this depend on the group being abelian or it's Does this depend on the group being abelian or it's just a structure that is from, you know Yeah, so here these are all abelian things but For could I mention one defect that can't be non abelian this type of you know What I'll call triality defects condensation but here these are all appealing because everything is just determined in terms of the vacua So abelian in the sense of the the order doesn't matter. I mean here the order doesn't matter anyway, but okay Do you have a question? Yeah Forget for the technical question, but is this the higher gauging on the one? Exactly, so Francesco was just saying is this the higher gauging indeed these are exactly the condensation defects so These are the two two are also Constructible by Gaging a higher form symmetry a one-form symmetry in this case a one-form symmetry on a subspace In a two-dimensional subspace This is working. So this is what's called condensation defects And that's sort of what's been discussed in this paper here and introduce them and I want to explain a little bit how that all fits together So we started with a theory that had a zero-form symmetry We gauged it and then we get these topological defects now What actually we have is a little bit more than that so we started with a 3d theory t with G0 is equal to Z2 and Then someone should have at that point said to me wow hold on you told us in the first lecture It's a B at the end. There's a dual symmetry. So in T mod G0 we have The dual symmetry at the dual to a zero-form symmetry in three dimensions is a one-form symmetry So we have a dual as well, right? So this is Z2 one form and Z2 is Z2 one form is generated by topological lines D1G Okay, well D1 minus. Let's call it with D1 minus with itself Is the identity? I just align topological lines that generate a one-form symmetry at Z2 So if you now I should take stock. So what what is actually? What is the full symmetry of this theory T mod Z2 I want to do this over here. This is now a collection of different things, right? We reconstructed these D2 Z2 this these surfaces. We also have these topological lines and so this whole thing comprises the full collection of objects that We should call a symmetry. So this T mod Z2 zero has It has surfaces It has surfaces. It also has lines. This is completely invertible, right? This is this is just group like but these here have these non-invertible fusion And now the question is and then also secretly you have points You could think of like points like things on these lines, but actually not that interesting in this case so what you actually have is This collection of things and this is not new because in two dimensions We really had only lines and junctions points. Now we have surfaces lines and then this point. This is actually the full structure. So mathematically Okay, so let me answer two things. The first question is how does this fit with this sort of statement and the statement is these surfaces so in fact these D2 Z2 We can think of as we take the trivial surface and we gauge The Z2 one form, right? So these are the lines with the one-form symmetry These are the zero-form on On it. And this is also known as the condensation surface on some M2 for the V2 one So this is the connection between condensation surfaces and these data defects So these data defects give you the condensation surfaces for the dual one-form symmetry The data defects are the condensation for the dual one-form symmetry That's the first comment the other comment is Now if you are a mathematician and you know in all of these we have fusions, right? In addition, you have all of these as a fusion structure So some composition of these topological defects and what this structure actually is is Is something that people would call a two Fusion Category and so the two refers to the fact that we don't just have one layer of lines and then some points the points are like You know Things that sit between two lines. We actually have surfaces between two surfaces There can be lines and between two lines that can be points. So it's this sort of three-layered structure So we have surfaces and these are what they would call objects in your category These are morphisms or one morphisms and these are two morphisms But really this is just in some way that you could for the moment and think of it as terminology because and All it describes is what we just derived in terms of a physics picture and So I write one picture that's made useful is so you can think of you have these D2 for example D2 Z2 And then you can have topological lines Actually, let's not specify what so you have some kind of objects Let's leave it by a you can have two of these and then there's a lot a topological line here D1 AB D1 a a B prime and then here it is a D0 That is a junction between these two topological lines. I think maybe this is the moment that the color is coming in handy these here You have a line here and Okay, so that in particular This specific thing we construct that is in particular When you start with a three-dimensional theory T with G0 Remember in two dimensions. I just said is let's call this your back G It's vector spaces graded by G. These are just topological lines labeled by G with G group fusion so in three dimensions in analogy These are actually these D2 G surfaces right that we started with that were completely invertible and Their lines and there are actually the lines one G and points They form also now a two-category and this is the two-category two back G might start G back G and one dimension into two dimensions V dimension to a two-back G Now when we gauge this and not surprisingly we get the team on G theory and Here we have these D2 Z2 D2 it and V1 it V1 minus and then some points This object here is called to Rep Indeed you see that in fact this the lines here form exactly the representation the one rep And so if this one on a bill and you would also get non a billion lines here And that's exactly how you would see that there's more to this So this is just the label for what you just constructed Okay Okay, this practice this data defects coincide with the condensation defector is the most general situation I mean in in generic dimension Is it not true? so I will Do an example probably this afternoon where there's stocking with TQFTs We'll give you another condensation Okay, I see that you can get the old condensation defects Plus stacking of the couple to give these Because so here we are in two diamond the 2dT QFTs here you get this is what you get And this is you can always think of it as compensation But when we actually don't go to four dimensions, which is obviously interesting for various reasons You can stack with 3dT QFTs now when you have 3dT QFTs you can have non trivial topological order So it's not just labeled by not they are well basically a bunch of accurate and some representation theory on these Compositions, but you can have genuinely not interesting 3dT QFTs with g symmetry that you can attach and gauge And in that way, and I'll do it example. So for example If you have a mixed anomaly, you cannot even just stack something that is sort of trivial like these So 3dT QFTs, which is labeled by Cosycles you actually need to stack something to cancel anomaly So that's actually you want to Simon's theory for example in three dimensions and then the associate defects are not Actually in the different frame the duality defects It's still true that in the TQFT that you put the GCM that you must be non anomalous in the full Well, it must be non anomalous in the sense if the theory you start with is non anomalous Then yes, but in the case right when you have a anomalous Transformation the theory before you gauge I will do this example You will see that you actually need to cancel the anomaly first and then you use it the 3dT QFT to Stack it on top canceling anomaly, so then you can gauge the bike is working like an inflow for that Okay, but that's going ahead When we gauge our discrete symmetries usually we get also the magnetic symmetry on top of the electric one Which is in the lines that you constructed You mean this there's the dual symmetry There is the dual symmetry and also the magnetic one usually like if I gauge a Z2 symmetry I get two two Z2s after the gauging one is Generated by the Wilson lines And the other by the yeah, but then they are they are not genuine, right? Yes, I mean they have stuff attached So I really here I'm talking about symmetry generators. They're topological defects that are genuine For example if I gauge a Z2 symmetry in a trivial theory I get a Z2 gauge theory, which has two two symmetries one Z2 electric and Z2 magnetic Then okay, if I add matter maybe some some symmetry can be broken but generically I think Yes Okay, but do you agree that in this case when it start with a zero-form symmetry, right? Maybe the way I can answer this question is to write down the Right there's in this theory. There's a about 40 Tqft the same tft that has About 40 theory that has basically be to delta C1 And all I'm doing is I'm imposing different boundary conditions on that So I'm there is no room for an extra Z2, right? I mean, I started with something that was generated by flat But by C1 so C1 was the map background for the zero-form and then I actually Changed the boundary condition to be to being the background. That's the one Maybe we can discuss Okay, so this was a very abstract Analysis, I want to tell you an example of actual 3d quantum field theories where these gadgets are appearing in real life They're not complicated theories at all In fact, they're surprising So what are actually examples? 3d Qfts with such tool that tool or tool that G Generally symmetry For Z2 which equals to Z2 You can start with the zero-form symmetry and in fact The if I have gauge groups, so I just take pure G gauge Is equal to SO3? and I'll go to G gauge is equal to SU2 Yeah, mills It's not that this is the most interesting three-dimensional theory to look at but it gives you an idea of what actually The title series are where you would see these things and then four dimensions. They'll be interesting actually Series So the pure gauge theories this one has a Z2 zero-form symmetry So this is that has a two-back Symmetry and when I gauge I get a V2 one and all these conversation defect also, and this is the two-rep V2 Right in one way of seeing this is this SU2 has a one-form symmetry It's just a center. This is just a center symmetry and From the center symmetry. I know these I can construct either by conversation Or I can think of them as coming from the SO3 Where this one form actually becomes zero-form in three dimensions? But it's just the the analog of the magnetic one-form symmetry 40 where there was a one-form symmetry here It's a zero-form symmetry the dual and I can take that gauge it and then these are the theta defects I think and think of these as the condensation defects or theta defects and of course you can extend this to other things so in for example and more complicated examples which have also Symmetries that are a little bit more interesting. So for n equals to if you do a SU n You have a Vn One-form symmetry and so actually what the full category is is actually two-rep This is the symmetry and so now you have these these Defects here we construct the Z2, but you can repeat this for Zn And then you get again non-invertible defects for that It's also an example if you have actually n equals to 3 so SU3 It's a bit more interesting because in this case there's more structure because you also have Not just this but you also have a Z2 outer automorphism Zero-form symmetry you can combine this and then you can get actually something like that So that's a more common that has topological lines that actually are forming representation of S3 So these are examples of QFTs that actually Have these sort of condensation or theta defects as symmetries but That's just one class of these type of things and actually it's it's interesting to ask are they actually also and Non-invertible symmetries and in QFTs that are not of this time not of this compensation time When do I actually have to finish? 20 million So do you have questions about these 3d examples? Because I will now discuss in a bit more detail an example that isn't quite of this type where you get Non-invertible symmetries more from this sort of perspective of The gauging of the auto-automotive I'm thinking Okay, I will discuss this example that may actually fit in better and then I'll do this other thing later So in fact, let's continue along this this line of reasoning. So we had these theta defects And so now actually it goes to D goes to 4 Right, so you do care about what I mentioned quantum field theories and now What we can do here is Why the mantra was take this and now stack it with a 3d GT QFT or What is hearing? Engaged and that we can perfectly well do the issue is just so 3d QFTs More complicated So there's still they exist though These are analogs of SPT phases For so G SPT and again sort of in omega now in H Me G You want But they're also that genuinely non-trivial 3d QFT and so these are going to name a 3d topological order a Lot of coins matter theory is about understanding these better Measuring them in the lab. And so then they characterized by categories themselves and these are the so-called modular pencil category So in principle when we do this data defect construction, we are not limited to taking just these type of SPT phase QFTs that we could also be stacking something more non-trivial and in fact sometimes with with TQFTs That even non topological Boundary so if this is obscure I'll explain them on what this is on top of logical boundary conditions so for example Transiment theory for some group G with some level K and the boundary if I put them on some space M3 Where the boundary of M3 is sigma 2? We all know that this gives actually a carl CFT. That's a CFT 2 WDW at level K for example So sometimes you actually even need to do this and so there's obviously something non-trivial living even on these Boundaries of such theories and that's very different from these type of SPTs. They are relatively trivial And I'll give you an example. That's an interesting physics example that also illustrates That there's a much richer structure in terms of what we can use in terms of constructing theta defects What we call twisted theta defects because these are they would call these things This will give rise to theta defects organization and All of this here if you stack with that Call twisted theta defects and it's the same principle We take the TQFT we gauge the G and then the TQFT becomes a defect and the gauge theory And so actually most constructions of non invertibles actually That arise from some sort of gauging Fit into this framework. That's why it's nice to organize it in this way While then sort of discuss different constructions successively using different tools So the example I want to discuss is Is one that was first discussed by Heidi one of the two first papers We can also this was this other paper the duality defect paper But this construction here ties in exactly to these types of stockings with TQFT and it's and I apply it to 40 And it goes to one super young males say with SU2 gauge group or gauge algebra That's a theory we should all care about Because it has right so it's known that the SU2 group theory that actually has a into this confinement in the IR and very interesting vacuum structure So What actually characterizes this theory? I'm actually let's start with the the group as you to This theory is it's super symmetric That's only relevant for the following fact. So 40 and it goes to one super symmetry implies that you have a U1R symmetry But this in this case gets broken I'm too bi-normally to the Z to N Zero-form symmetry. So this is a zero-form symmetry in this theory And it also has a center. So actually doing it for N So let's refresh your end It has also a one-form symmetry That's actually quite crucial for the confinement Which is the N these two symmetries of global symmetries Those are the gauge symmetry but we don't worry about that too much and these have also mixed anomaly. So when we have a One-form symmetry that the background field B2 a zero-form symmetry There's a background field a one and this makes anomaly. We'll have the following one five E2 squared All right, so this lives in some five-dimensional space or the four-dimensionally. I'm not very on this boundary So this is the anomaly and What we want to do is the goal is gauge the one-form symmetry and go to the PSU N theory and The question is what is the symmetry of that PSU N theory? Just as a quick recap for the for the SUN this one-form symmetry Will persist With this mixed anomaly to the IR and I will confine and we'll have N vacuar right so the Z to N will break to basically Z2 and then there are N vacuar and They're all confining and the question is what actually happens when you go to PSU N so The first observation is if you gauge this It shouldn't be a problem to gauge the one-form symmetry because it's not that it's a one-form symmetry only anomaly It's a mixed anomaly But there is a problem with this particular setup because if you look at the zero-form symmetry generator This is some topological quarter mention one defect Let's say one the generator for this one on some three manifold If we actually do background gauge transformation in this background gauge fields And you shift this a one actually this picks up so background gauge Transformations then what happens is actually because a one goes to Delta lambda this however is not trivial. This actually picks up a particular term. So this goes to D3 With an extra phase it's just minus 2 pi i over N B2 square And now M. This is over an M4 and this is sort of M3 With an M4 attached and that's not good Well, you can't just gauge this symmetry So there's from the perspective of the zero-form symmetry generator. There is some anomaly. There's some anomalous background gauge transformation so When you actually now gauge the one-form symmetry, this will not be a healthy defect anymore. So what we need to do is before we gauge We actually need to cure this transformation there's an almost transformation of this defect and Well, we just learned we can before we gauge stack TQFT's and so now the question is can we can cure this? stacking a 3D TQFT such that cancels the same property it has a one-form symmetry and An anomaly which is basically this 2 pi over N These two squared anomaly might have told you there are in three dimensions for the one-form symmetry that is B2 square An anomaly so I can stack something that has precisely these properties a 3D TQFT on top and then gauge because then this Variation just will cancel out and In fact in a very nice paper sin lam Cyberg Essentially in 2018 classified. What are the minimal TQFT's in three dimensions that have a certain one-form symmetry anomaly and a certain at one-form symmetry and a certain anomaly so they classified the minimal 3D TQFT's with one-form symmetry and And a given anomaly one-form symmetry ZN say an anomaly P So P times B2 squared and they call them APN and all purposes We just need a N comma once So they're called a N comma P So for N ZN one-form symmetry and P is Anomaly that you have and so what we need is we need For this defect that generates this one for a zero-form symmetry. We need the an comma one And in fact they also showed Diffusion of these might be also always need the fusion of these TQT's these fusions are actually very simple When N is fine Please don't ask me what it is for N not right Let's stick to any goes to two All purposes is a prime number and now we can ask for SU2 What actually do we get now? So we take the defect D3 one and We just replace it With a defect. Let's call it curly D3 one on the same three manifold, which is just D3 one Times this a 2 comma one For SUN you would stack it with a SUN one and This object now Has precisely and I should have said Here there is a minor so it should have precisely the opposite anomaly This actually cancels precisely this anomaly. So this has No anomalous transformation and now we're ready To gauge Now we stack this with the TQT and now we know from our general sort of understanding Well, this should now become a topological defect in the gauge theory. So when you gauge the ZN1 Z21 form symmetry What we get is now a new defect. This is now one defect in the new theory and So the D3 one Three is now a zero form Symmetry defect and the question is what is its fusion and Because we have diffusion of these guys Right now we have two of these we get exactly that here. We can actually rewrite this diffusion is actually So I can actually write it on for the N case if you want If your N is prime, maybe you would like to form it D3 1 and D3 2 and there's another fusion. So now here you see this year is a TQFT That appears as a coefficient in this fusion and then there's the fusion of the defect with its so when we stack with sort of A dagger Dagger and this actually turns out to be the condensation defect on M3 of the dual one Condensation Okay, so we get in the PSUN or the SO3 for the N equals to case these zero form symmetry generators and have these non-invertable fusions so the Generally the PSUN pure And it goes to one Superannual theory and 40 has Sort of non condensation. So these are right. These are twisted data. They are really these TQFTs, right? These are oh, okay No one asked me. What the hell is the theory? This is actually very simple in this case. It's just you want to lend char time of theory So that's why you can then do all these calculations and determine it So here we actually have a theory that precisely fits into this framework We have a char time in theory and it's boundary is not a gap theory. It's actually a two-dimensional Carol boson I'm finishing in two minutes It has non invertible Symmetry Which is basically either coming from this perspective of this anomaly and these are basically what you could call these twisted Data right the non invertibility again comes from the fact that We had to attach or we were attaching these 3d TQFTs to this zero form symmetry generator and then gauged So this is sort of the theme that goes through out all these constructions that there is some sort of a non invertibility that comes Because of them The structure of these TQFTs that fuse in this peculiar Okay, I don't any questions one minute for questions. Yep This is not the example that I was referring to But if I'm understanding correctly this construction of a twisted data defect is not really as the data defect I mean you just put a TQFT and you gauge here you are taking a TQFT, which is anomalous So if you gauge you have to attack an SPT But you are using another defect of the theory which has an SPT attached and you are fusing together Sometimes you can't gauge without actually attaching a TQFT that has some anomalous Variation right because here. This was really the key thing Yeah, okay It's the same thing because you got you're attaching the other defect the other zero form symmetry of the theory which also have a An inflow of touch and they cancel together. That's right. So here It's not an option. I mean it's not an option either Here basically this forces you before you gauge to actually attach This TQFT so that you then get a consistent Zero form symmetry generator for that so But it still follows sort of in it's there are also examples, of course you can write down things like, you know, just So simplified versions of three rap Z2 for example, and that there you would just attach SPT's But the point is when you go to four dimensions There are more TQFT's and in fact you can even if it's not anomalous You can also actually attach a 3d say an MTC with a Z2 symmetry for example, and then gauge So this category three rep G is much much richer than what you would get from just looking at these things and Indeed sometimes in physical situations you can have sort of additional peculiar things that you have to attach TQFT's Which don't have these gap boundary conditions Sorry events are also another question even in the case in which the TQFT is non anomalous You can also have a situation in which Let's do an example for example you one level k times n square if k and n are not prime You can gauge the subgroup Zn In this case, do you get a condensation defect or not? You won't have a k n square Okay, this is your theory No, this is the TQFT that I want to attach In a in a 4d theory, which has a Zn1 from symmetry because in this case I mean if you if you take this theory This is not a pure gauge theory But and there is a non-trivial extension of the n times k n Zk n, so I'm not sure whether you get if you if you if you do the data construction for the end if you get you have a 4d theory where you have a Z you were saying Z2 at Zn1 from symmetry And you want to use the subgroup Zn It's not anomalous, okay, and you want to group use the subgroup Zn of Zk n square of the TQFT through the data construction But this here is Not a just an SPT phase right? No, it's not an SPT. So it's more. It's a definitely a twisted data construction. Okay Because this will have an anomaly of the type a k n square comma 1. Yeah, but the subgroup I'm gauging is not anomalous. That's true. Yes Okay, you're saying you're sort of doing Twisted slash So a hybrid thing. Yeah Okay. Yeah, that's interesting. I have There are many different variations of this and I don't think that has been explored yet, but it's an interesting case I have a question Which is somehow I'm follow up on this. Um, so when we discussed two and three dimensions The logic in this theta defect was so we gauge the symmetry and now we classify all the TQFTs that are the given symmetry But now in this example you're saying Now in three dimensions there are non trivial TQFTs So if I try to classify TQFTs with a given symmetry there's an infinite number So it looks like I can construct an infinite number of theta defects in a given way So there's still always this construction And then there's a question and I do actually have So you need to throw a TQFT that has a g symmetry, right? It's not just any I don't want to just attach anything it should still It can of course always attach a theory that has no g symmetry and then it's just a product it stays a product but I would like to have say um Some MTC with some g symmetry so it has some z2 symmetry thing So the question is then how many are there and when I touched and what do I actually get? Well, yes, because even with this condition there is an infinite number of TQFTs with g symmetry So it looks like I can generate in a given theory an infinite number of theta defects But are they really there? There is an infinite number of symmetry Yes, no the first question is indeed what actually so there's no question that you can't do this construction So then the question is when you have such a theory and you do attach this and you gauge And you have now all these different topological effects What actually do they do in the gauge theory? Like how do they act on the act of physical degrees of freedom and how is it actually distinguished? These different sort of you know if I attach one MTC or another one How are these these symmetries so but it indeed in 4d this this category becomes extremely huge um I think here there's also a maybe a comment in the similar sort of vein I actually write this The the main point of this paper was to classify the minimal such 3d TQFTs Of course, what all this is saying is These NP comma one They're basically minimum in the sense that if you have a TQFT with this one form symmetry and that anomaly Then you can always factor it into this Times something else some other a prime That doesn't have the anomaly so here even you could say you're attaching An arbitrary sort of TQFT now. This doesn't really talk to anything in this theory So it's completely decoupled right But in principle that also you could say well, why did you use the minimal one? Actually Pavel has a comment I was just going to look for Pavel because he was in fact saying it's sometimes easier to actually Also related to this restriction here to attach All of you know, actually you probably want to have a comment about Yes, he has a question and he's he's got this construction of these q mod z Symmetries where you actually take all a billion TQFTs and then you can compute all these fusions So we've been a closed right Yeah, first kind of a comment is not quite clear how this fact ran out is canonical of this Non-normal spot is second. Yeah in principle like yeah my comment is that if when you start fusing fusing them You will go out of the of the of the class or just minimal TQFTs Yes, so this is indeed that's why here I was saying Let's restrict the prime Then you really just you can write down the action for this you want level entrant simons and just integrate out some diagonal field And then you see this fusion In fact, if you just want to look at these an comma p's and do this More generically than actually you you see that there's some issue and Pavel has written this very nice paper on this q mod z But I still don't know what actually is now the the resolution should be now Attach all a billion TQFTs So you need to speak on the microphone. Otherwise, I can't hear you. I know I else can hear so the thing is that The the symmetry g no As to act faithfully, I mean it cannot If you which symmetry the g zero from symmetry, no If you put any tqft will not act faithfully on the full tqft, you know, we will leave some operators in variant No, the the the idea of amp is the same is that you mean here? Okay Because here this this is not zero from tonight. This is a one from some Yeah, exactly then some lines are invariant and Yeah, so there's a fine this factor So first point is that Pavel is saying this decomposition is not actually canonical And if you do this you go out of this class For generic But you can work in the equivalence class that that's a well-defined object So it's a It's called the with with class So you can you can work there and then it's well defined And I think the point I mean this is how we understand it is that If you kind of factor out or you choose a representative in which the symmetry acts faithfully like in in this case It's separating amp. I don't know what is in general the way in which you choose that nice representative Then you can do everything and Added data about the the couple thing is undetectable from the bulk. So it's not physical. I mean, I don't know to Measure it, no That's true, but Yeah, what okay, so Actually, this is maybe the right moment to make the following comment So I think this came this came up yesterday And when I think chris was asking me so when did this all happen? So this all of these developments basically um, but the solar star with this paper mentioned yesterday by this small plant people then this and in the in 2021 In the fall this paper and also the gordova omore choy lamb sin paper came out And so this is a very very young field and so some of these questions are not yet as There are loads of examples And one goal is to actually understand them in a slightly more systematic way, right? And one of these questions is so can you actually put them into sort of one story that we have a description that sort of gives you Mathematically, but also sort of on a more conceptual way a description of all these non-invertible symmetries And it's clear. We don't have that yet in higher dimensions. And one reason is that these These what we get here are three categories three fusion categories. They're very very complicated objects We don't really know the full structure of them but Exactly these sort of questions Right. Well at some point, hopefully tell us what are the things that we really should be Sort of, you know, taking into account what is physically relevant, right? Like what you're saying this with class We shouldn't be looking at each defect, but maybe some equivalence class of these things But that's all really part of everyday research at this moment But I think this was not clear to everybody. That's why I wanted to at some point say it now. It seemed a good moment I think now we're really over time. No And now we are already in the discussion session. Okay Okay, but then I think people should also ask