 Welcome back. So now we have the fourth lecture right by Sakura on non-invertible symmetries. Thank you. Okay so at the end of last lecture I think we've come to the point where we had discussed a construction of non-invertible symmetries using these stacking of TQFTs and engaging so these are these data defects and we did this in particular in 3D where we got these two web G0 defects and they were essentially you can think of them as condensation defects and in general data mentions you can imagine that there are sort of a notion where you have a d minus 1 higher representation of this group and here I put a little zero in because this is sort of the subtlety that once you hit for example and three fusion categories and you're actually stacking 3D TQFTs then this goes actually beyond the SPT phases right so this was sort of characterized these the defects here at SPTs and actually when you have higher raps you could have sort of non-trivial things and in fact there's sort of a G graded higher fusion categories that's sort of the general structure so this is for SPTs that that zero is indicating that and then I also discussed one example of twisted theta defects and that's really this construction of Caidio-Morin-Zing where you had an anomaly and a toft anomaly between zero and one form symmetry and the non-invertible came by actually stacking now a TQFT that was a U1 level entry on Simon's theory and that's sort of a slightly different class of non-invertibles in the sense that these ones here have an interpretation in terms of condensation so higher gauging whereas these ones here do not in general. There's also something which I will not have really time to get to which are duality defects I think it's okay not to discuss them because I think that's maybe one of the most intuitively constructable things and you can read the papers by by Choi, Cordoba, Sien Lam and Chao and this construction actually gives sort of the same defects in our slightly different way so and that's sort of inspired by this Kramas-Lani duality I discussed at the beginning in lecture one. So here you have a theory just to give you the brief idea you have a theory that has a self duality and then you can essentially gauge on a half space and then this Corda mentioned one defect is sort of wall that separates the theory from its gauged version becomes essentially this sort of non-invertible defect. There are also these disconnected gauge groups so I illustrated for you the O2 and then there are also loads of other examples for example the pin plus gauge theories in any dimension really and that's something that you can also sort of now I think you have some idea of how to access the papers or you just ask for this early up. Anyway so there are lots of constructions and so when I asked at the end of last lecture what you want me to perhaps focus on in the remaining lecture was how do you realize these type of generalized symmetries also non-invertible symmetries in homography and the other thing was how do you realize generalized charges. You can hear me? Good so let's actually and so we have talked about quantum field theories and many quantum field theories that we like have holographic doors including the one that had these peculiar twisted theta defects these non-invertibles for large enough SUN. So one question you could ask yourself is in particular a lot of you do holography and how do you realize these types of things in a holographic setting. So if you actually so just this is sort of an intercept the question you could ask yourself is given a QFT with some global symmetry maybe non-invertible or invertible generalized symmetry and let's say this theory with a holographic dual description how do you actually incorporate these symmetry structures in holography. How is let's say the symmetry call it S encoded the bulk one is an actual question and in many instances this is well I thought well known but then it turns out many people don't know it. They might be actually something useful to describe. So let's consider a vanilla ADS-5 times S5 dual to CFT-4 set up but this is not necessarily limited to CFTs. I mean so the example I'll talk about also for the non-invertibles will be 40 n equals to 1 so that's not conformal but there's still a holographic dual to some extent. So we have 40 n equals to 4 super young males well let's say with gauge algebra SUN. Now how in this dual do you encode the fact that you actually want to discuss SUN or maybe a PSUN super young males and you know maybe that's a question you have not asked yourself before but it's certainly an important question how you would encode that and this is something that already Witten tells us in 98 and this topological field theory and ADS CFT paper and it's a very beautiful story which actually underlies a lot of these symmetry structures already. So the key is look at the bulk couplings that are topological couplings. So what we're doing is we're actually reducing from type to be on the S5 and now we're just looking at this ADS5 and we're actually integrating of this here. Now we know in type to be there is a sort of topological coupling which is F5 B2 DC2 a ten dimensional thing and now we're integrating this over ADS5 times S5 and we all know that then in this solution to type to be super gravity there is 5 form flux on the S5 sphere. So in fact this coupling in 10D we can integrate over the S5 and note that the integral of the S5, oops, over the S5 and some nice normalizations effectively N and that's the N of the SUN. So what we're getting is some sort of coupling in 5D is sort of an ADS5N B2 DC2 coupling. So here this is the F5 is 5 form flux so that's all our flux this is the NSNS2 form and this is the R2 form. So we're getting this sort of topological coupling and one way to argue this is that this is sort of an interesting coupling but it's really the leading term it's like if you actually do a sort of expansion derivatives this is a linear derivatives term so even all the kinetic terms they all come later in that expansion. So this is the term that we want to look at and actually this is a term that I already discussed in one of the earlier lectures where I was saying well if you have a one-form symmetry it has a two-form background if I gauge I can think of it as an operation in a one-dimensional higher space where these are now the choices of backgrounds either electrical or magnetic and that's exactly what happens here. So what we should think of is we have sort of this bulk ADS5 and here this is the boundary here is the CFT4 living and now we are actually asking ourselves if we have in the bulk this theory what actually can we say about the types of operators topological operators we can write down now in terms of symmetries and what sort of boundary conditions we have to impose on this boundary. So first of all in this this is the BF theory for basically a VN for VN one background field I shouldn't write VN for VN field because here in the bulk there is a little dynamical field maybe I should put even little VN little thing anyway. So I can write down here two types of topological operators or two types of surface operators on some manifolds and two which are let me call this your B is e to the integral b2 and c2 and what you can actually show by looking at sort of the theory here that these two operators now in the bulk so these are surface operators in a five-dimensional thing they will actually do exactly what we discussed in the first lecture they'll do some non-trivial linking so in fact if you do something like insert them in some bulk correlation function you would like B to C and two then there's a phase which is the linking of M to N to over N and times the reversal so these are non-commuting surface operators that we have in this ADS space and now the question is what can what sort of things can we do with them as we push them to the boundary so B to couples if the N is an aspect field to form this couples to strings and this will couple to D1 strings so N is fundamental and this will couple to F1 and this will couple to D1 okay so what are now the consistent sort of things we can do so we can impose boundary conditions on this four-dimensional boundary the CFT lives at the CFT boundary and the first question we can ask is if you make both of these say Neumann so they just become surfaces in the four-dimensional theory then we hit this problem that those surfaces that if you insert them into collage they have they're actually mutually non-local this isn't a good thing so in fact we need to pick again boundary conditions so that what we get at least in the in the in the boundary theory is sort of sensible so one choice of boundary conditions is you put directly boundary conditions for one of these say E2B and Neumann boundary conditions for E2C so what this means is that we actually have this picture again and now we have these surfaces that these are orange so they actually will end and they'll also stretch into the bulk but this is sort of the boundary for the boundary of M2 is inside the M4 space time whereas imposing Neumann on the other guys so we actually saying they're actually becoming surfaces inside here so they're completely submerged and M2 is contained in M4 so this is now a fundamental string that has a boundary it's like an open string that has sort of ends on the boundary and has a world sheet that goes into the bulk in fact this is something we would identify in halography but as a Wilson loop and this is now actually still a topological operator it's a surface operator in four dimensions and because of this relation this actually will non-trivial link with this line so in fact there's a non-trivial linking between N2 and boundary and that's precisely calculating now the charge so this calculates the linking compute the charge of a Wilson line under one form symmetry generator which is now given in terms of D2C so if you actually have a holographic dictionary and you really want to be precise then you have to actually not to say ah I have an n equals to 4 so for your military with as you engage groups say on it is then that then to specify that you actually specify also boundary conditions for this topological operator now of course you can change this you can switch these around and then basically these guys become the top loops and these become the generators of the dual one form symmetry but it's all encoded in this BF theory so the mantra here is if you have this is not any kind of theorem it's by evidence if you have a bulk you have some holographic let's say it's you know you have some I think you can even start with just a D plus one-dimensional it can be as simple as a D plus one-dimensional gravity dual to QFT V in D dimensions I don't need a full thing and then you truncate look at the topological coupling for the form fields and then they should have some interpretation in terms of this theory that governs the symmetry defects because from this coupling where you have BF type couplings in supergrad you will call in transiment couplings very confusingly anyway so topological couplings such as BF couplings so you have some integral over D plus one dimensions of some B P plus one like DB so these will be the backgrounds for a P form symmetry and it's dual GP GD minus P minus two form symmetry this is the hat group so these are billion but it can also contain in general not just these type of topological couplings it can have also polynomials of these background fields some functions of these BTI so some combination BPI and some polynomial forms and then the the statement is you should be able to this topological action but this should be you should have and this should have this action should have the interpretation B plus one dimensional TQFT which admits boundary conditions in fact topological boundary conditions such that it's such a basic symmetry structure can I ask a question yeah so you told us that there is something that goes wrong if we fix Neumann for both of them in the example you mentioned before is there something wrong even also if we fix Dirichlet for both of them I think I think you can have sort of both of the issue then you would have some like you have two where both the Wilson and Toff line right and maybe you could have so then I think you can have some other combination that then would right so you maybe you're imposing something like a diagonal so you're looking at some subset of both of them ending it could be like a dionic line and then you have some mixed thing okay actually it's a good question I don't think there's anything completely wrong with that so what you can for example always do it also gauge some some sub part right but you're saying you have Dirichlet for both for both I think you basically then have Wilson and Toff lines and I think they would then still also have this kind of yeah right so they would still have a non-trivial linking in 40 right and then you would insert them into your colliders and then they you switch them around and I guess you would pick up this phase again and then that's inconsistent what you could do is have a Wilson plus Toft like you know for SU2 you could go into SO3 minus where you have a dionic line and that that you could do but I don't think you can just impose both of them then you will have line operators that don't have non-trivial because somehow they would link in the bulk but not in the bound there is something like that they will actually link in the bound in the in the 40 theory it's sort of the statement why can't you have Wilson lines and Toff lines in the gauge theory it's because both of them are it's like this non-mutual locality if you have the full set of lines it's in my first lecture this argument if you look at SUN you have Wilson lines and Toff lines and now if you had both of these sets then you could exchange them and pick up exactly a factor like that in correlation function say so that's a problem so you have to pick a polarization and what essentially this choice of different boundary conditions is is a holographic dual picture of what is the choice of polarization with different like this again a vanilla boundary condition the different boundary conditions where you switch them around if n is no prime then you can even gauge subgroups I'd have partially some of them be a Neumann Dirichlet so you can have mixed things so there's a lot of choice and that's exactly mirror in terms of what kind of polarizations you have in this lattice of line operators okay thanks this choice of boundary conditions for the B and C I didn't code it in how do you write the the BF so I'm just saying so this is the BF because you could have written I can write for example in this particular case I would say that um delta B2 is zero on the boundary right a B2 is zero on the boundary say just typically you can track these to a precise boundary term so maybe you have written C2 D B2 and four term where I just say that I have a delta function of B minus B so then okay not my notation reverse but I just fix it to a particular value the B and then for the C I will have to have uh I'll just leave it fluctuate and just remain it just restricts to the I guess my question was if you had written the BF term as C2 D B2 the difference is just the boundary term and if this boundary term is just making this exchange of and in leadership and boundary if you were referring to yes you could also encode it in there okay thank you okay so and in modern parlance we would call this the symmetry t topological field theory use tft all the time so symmetry tft or simply sim tft and this is really in the bill in the case of a billion symmetries with some type of anomalies that's precisely what it is it's somewhat coincidental but this is what what comes out of the examples that we have so I'll give you one example um which is the one that we had at the end of last lecture which is just to give you a flavor of how this would work and how you then can encode also non-invertible things and the many examples of these duality defects that's here the CISA group has worked on from Jet School and students and other people and on and actually constructing these holographic couplings and I'll just give you one example so an example is and this is something worked out with Fabio Brutzi leave my students from Beast and Dowell Gold who is in the second last row and and myself and then in a follow-up with Aputi, Ibu Ba, Bonetti, Federico Bonetti, a lot of Italians in this, Bonetti and myself and this is simply the dual and this is not a conformable so for n equals four this is all you have and then maybe there are couplings to the duality backgrounds right and that's also what was discussed with these duality defect papers but um so it's really important to have all the couplings for all the symmetries and that's sort of tricky to write down the symmetry TFT for your holographic dual so it's a dual to um 40 n equals to one so the angles with SUM and this is of course known this is Klevano-Schlußler and the interesting thing is if you look at Klevano-Schlußler and you restrict to the topological terms yeah you'll see that you get something right so there should be background fields for the one form symmetry so some B2s and C2s so these are the one form symmetry backgrounds all right because it's pure super angles and there should also be backgrounds A1 and it's dual C3 for the it comes from the U1R symmetry and then gets broken to Z2m and then there should be this anomaly and then the anomaly was the key to getting that non-invertible symmetry going and so now what you would like to see is take this Klevano-Schlußler solution in fact we have to do is look at the full consistent truncation which was a massive effort by Cassani and Feidl and so what we did is extract out the topological couplings and here the sim TFT is precisely it's a little paraphrase of if you're interested I'll give you more details so it has these couplings plus N2N A1 DC3 plus A1 D2 squared coupling which is precisely the BF couplings this is slightly yeah it's a little bit more complicated it's actually it's not just the F terms it's actually BF and Stuckelberg but more or less the king this is what you get so these are sort of the BF terms and this is the anomaly to top them up the imprint of the mixed anomaly and so now what you have to do is play the same game as we did here write down the topological defect and actually then this will have these sort of exchange relations in the bulk and then imposing boundary conditions will pick either the SUN or the PSUN but we already know this is essentially the same as what we would have gotten by looking at you know there's a zero form symmetry generator but it has a mixed anomaly with the one form symmetry so that guy can't be consistent we need to stack a TQ of T on top and so we get actually these non-invertables if you want to have this guy here to be Neumann so this is basically this in codes both sort of the invertible for SUN which is B2 is Dirichlet C2 is Neumann and A1 is Dirichlet and C3 is Neumann and also the non-invertable case where we have now B2 is Neumann and C2 is Dirichlet and this is just to tell you that these type of couplings when you're looking at your supergravity theory you'd like to study these types of effects you need to look at the topological couplings and from these you can then derive the symmetry structure okay so what is my time limit again 445 okay so I would like to ask whether there is a question regarding these sort of this is just to I thought it was useful given that many people are thinking about holography to at least connect a little bit to that but I want to actually return to field theory and discuss now something about these higher charges 45 minutes and return to our more field theoretic perspective and discuss what we'll call q charges and that's sort of just generalizing what are the properties of things that transform under generalized symmetries and that's in this paper that actually appeared recently at Laxabard Laj and then to other papers that are upcoming so we just discussed the invert invertible symmetries and that's already an interesting thing and then we'll also discuss non-invertibles and there's also paper by Bollinger Barge and Vigoledo I think Vigoledo was a student here of Paul's okay so what is the idea the question is given a we've talked a lot about these topological defects and you know some d minus p minus one g generating a gp p-form symmetry and when we had these invertible things in fact here we will focus on invertible symmetries for the moment the statement is always these p-dimensional that the objects that are charged charged objects so this is this is the charged objects are representations gp this is sort of the standard law now so you have a zero-form symmetry point operators are in representations of the zero-form symmetry you have a z2 one-form symmetry they'll be you know in z2 representations of that one-form symmetry the line operators and what I actually would like to make the case of this is sort of just so in this paper that Laxabard showed this is just a part of the story the representation of the charges under gp higher-form symmetries and namely so what I want to define is basically if you have a higher-dimensional operator so a generalized charge I something that's charged on a generalized symmetry of a p-dimensional operator is a q-charge right just in the same way that if you have a standard zero-form symmetry say ah this point part is actually this local operator is charged it carries a charge these are now q charges they're like high-dimensional things that are charged under certain symmetries another question is given a symmetry what are actually all its higher charges we'll just basically focus on zero-form symmetries and there are these interesting story coming out of this the question is what are the q-charges so for example 4g0 so essentially um one way you can look at maybe again the example that's been carrying through this whole series of lectures is let's look at the one gauge theory in 40 and essentially we know there is a z2 charge conjugation symmetry so this is a zero-form symmetry and we can ask what actually are the charges under this um so one thing that we can say is well it it certainly acts a goes to minus a and so it will act on local operators also and f goes to minus f we're actually interested in these higher charges so actually how does it act on line operators if q is equal to one what are actually now the how does it act so we know there were some lines remember these are just either then to go a so w alpha is e to the this will just go to w minus alpha except when alpha is equal to zero or pi then it's invariant and likewise we know this is a theory with one form symmetry so it has these topological defects let's just focus on the electric guys so these are a comma electric these are e to the integral and of course they will just go to d2 minus a and so they are going to be mapped into each other minus minus okay so there's an action on also the magnetic thing so the top lines will also go to minus so from this example we know a zero form symmetry of course also acts on everything else right so usually we would call then a particle that transforms in the representation of the z two is some kind of operator it can either have trivial charge or not trivial charge right so now of course also zero charges if you like so you can write down f that was the minus f and now you can construct operators but the the question is sort of how do we actually characterize this sort of structure so clearly g0 acts on extended operators who can act on extended operators there exist these types of q charges and now i want to actually ask what is sort of the structure that we should be looking at so what is actually what is the structure underlying these i this is like saying i have a group and i can write down all its representations in this kind of way this is sort of the level that we've discussed and i want to know what is the representation theory cool so let's look at one charges it's a one-dimensional defect of g0 that's the first thing so what actually is the picture so we have a this is the d d minus one g so this generates our g0 and a one charge is a line operator and when it acts on this essentially what it means i hit this i can't go here through okay so maybe go behind and then it comes out on this side and here i started with one operator one and here it can act with g times 01 all right that's what happened here w alpha went to w minus alpha as i had this topological defect right it's just acting on this line operator maps into minus itself okay so that's sort of one thing and now what i actually am interested in is first well there are also potentially group elements that just stabilize this so it just maps this into itself so let's say actually let's call h sub o1 the stabilizer of o1 so basically these are all the h and g such that h o1 is o1 and so essentially one thing that if i have such a situation and this line operator goes in and it comes out with o1 o1 then really there's a point where these two intersect and there's locally now some sort of group that that sort of localized here so this is some stabilizer some h some at at the intersection at o1 intersect at the d minus one g and we have essentially induced symmetry which is generated by point operators h h with h and h o1 so all this is saying is because this leaves it fixed i can also think of it as it does now a local symmetry that's induced on this that actually is h o1 work okay so basically let me repeat this so that d0 h with h is in h o1 a zero form symmetry intersection so if i now actually have for example two of these i can ask here's our line operator that goes through here and so o1 o1 then i can ask whether these guys um so here we have some d0 h and some d0 h1 some d0 h2 these are the right so here these are zero form symmetries in the bulk that we started with but in with the h i inside the stabilizer that's why the line operator doesn't change but now locally i have these induced zero form these point like operators that generate a local zero form symmetry i can ask when i now collapse this picture is there any kind of choice i have um this is just sort of like i can just multiply this and in fact i can sort of say if i collapse this picture it becomes a one but now there's a choice so here we have again d0 h1 h2 so this is just group like but there's a choice h1 h2 and this is exactly similar to these co cycles that we talked about this morning that i have a choice of a phase here and it needs to be consistent with you know associativity and so on and so in fact the sigma h1 h2 is a choice of co cycle sigma and h2 of h1 u1 okay so now i want you to pause for a moment and say well i've seen this before and you've seen this basically this morning what i'm what basically reconstructed here is something that we discussed this morning in a slightly different context so we have some subgroup that's the stabilizer of this line and that's the subgroup of g and then i have a choice of these two co cycles sigma here you call them omega because i call them omega this morning and so i have a pair of subgroup and omega on this subgroup so basically these type of one charges are characterized by a subgroup h and omega and h2 of h comma u1 where h is a subgroup of g and this we have seen this morning this is exactly what i called this morning a two representation of g right so recall i know if you've not seen this before this is probably a lot of two representations for a day but it's the same thing so this is basically a recall this is exactly a two representation of g this is how i said this category of you know defects when you gauge the zero form symmetry these are precisely forming two wrap and that's precisely what they were characterized by these were the tqfts this was basically the spontaneous symmetry breaking part and this was basically the part that was like the spt phase i see puzzled looks so let's recap we asked ourselves if i have a question yes i have a question the cycle is it something that we choose or is something that we compute in the theory is fixed in the theory you have to compute it i think it's given a no no so the setup is you have a zero form symmetry and then i'm saying what are the possible one dimensional objects one charges that are charged under this zero form symmetry and then the answer is everything that has this property so you would have if you have co-cycle and you would say i'm not going to do an example with a co-cycle but for z2 times z2 you would have non-trivial one charges associated to this co-cycle so the statement is you will then have line operators that will be labeled precisely by say a z2 times z2 subgroup and a non-trivial z2 co-cycle z2 times z2 co-cycle so it's it's like saying i have a group and then i have representations of that group and you're asking me do i compute the representations of this group in in the theory or are they given to us so it's like i'm here i'm answering the question given the group what are its representations given this zero form symmetry what are its one charges i mean its representations on line operators for instance if i take uh young meals and they look at the lines of young males and they have charge conjugation or maybe some larger group and vc2 is not enough i will do this exactly okay so i can compute for the lines what is the co-cycle given the lines of young the wilson line of young males i can definitely determine which which higher which to one charge it should be it will correspond i will write this down for you in a moment so this would be and i did all the purpose i only did it for z2 because as i said for z2 times z2 you would be still here writing down this the full category from this morning i don't think that would be in anyone's interest but we can we can look at it in the paper and we can actually see i actually haven't looked at the z2 times z2 but that might be very interesting so let's look at the example when g is z2 right and that's recap from this morning all right so h can either be one or z2 and this omega is basically zero in this case this is not answering your question but now we're looking at this example so we have a zero form symmetry and so let's say so what are now these these these higher charges so maybe i want to introduce one more piece of notation i'll label these this this by row and because this is our two representation with two so one charges our two representations okay so now for z2 we have which row twos all that's the one with the trivial h and z2 and and sigma omega is zero and the z2 one which is h is z2 and again omega is zero so this was the case where we had um case and we had an example there now let me identify which one is what so it's maybe the best way to do it and in fact it's right over there that was an example where we had a z2 charge conjugation and we will just like to now know what are now in terms of these these two two representations and how do i identify this so essentially we had the line operators w zero and pi w alpha and it goes to zero and pi and then we had also these these a comma e surface defects right so these are really for wilson nine somehow i always like to use w or these are these correspond to lines and these correspond to o twos i didn't say they had to be topological not topological they're just surface defects in the theory these are surface defects in the theory okay so what have we got so this one here is right so the zero and pi but these are fixed the stabilizer group is h is z2 so these are basically of this type so these are rosey two and these ones here right i had to basically the only so basically the only thing that sort of i can now write down these were basically uh the stabilizer group is trivial and it acts the group actually acts between these and these are the rosey two where it goes to trivial group two these ones here are again they're not invariant one has to actually look at the it's a similar sort of argument now for the surfaces d2a plus d2 minus a and these would be now three representations associated to so the trivial stabilizer group this is now sort of the surfaces right we get very similar to what we did there three representations so the general statement is if we have a zero form symmetry i would like to know its representation on the higher dimensional defects so the higher charges and more specifically one charges line operators are two representations and two charges are three representations or q charges or q plus one representations of g0 and we can of course also now replace g0 with gp and that will just follow through even for higher groups so this is the sort of general statement of if you have a symmetry an invertible symmetry and how do you actually construct the extended operate and how are they transforming under the symmetry and then this is the answer to that and i agree with you it's interesting to look at the case with with the core cycle you get a even more richer structure more defects yeah so is this choice of core cycle my choice the core cycle related to the g anomaly the anomaly of g0 because okay if you have an anomaly for the g0 it means in some sense that g0 acts projectively in some sense then i would expect that you cannot find the trivial and whatever it is q representation q plus one representations so yes like in this case for example you get a projective representation no in on this on this local operation exactly so if you can if i think of this local operators is more vector space it can't be one dimensional you can you can be actually higher so i will discuss maybe you're referring to the case when in fact there's a little bit more happening in the sense that that's not the full story and indeed this is true in higher dimensions for line operators this is all there is but for surface operators and higher there can be actually more interesting but so my question was is it possible to have an untrivial core cycle but a trivial anomaly for g for the zero for symmetry yes because i think i can i mean it's literally all i'm saying is if you have a z two times for example z two times z two and you're trying to construct all possible line operators then part of some of these will have an untrivial two core cycle okay that's the that's the you you're thinking theory i'm thinking symmetry and its representation right when you are when you're reading um uh falton harris okay oh is this not actually the representation that is realized in the theory you're not doing that right you're saying ah this is the representation of a lee algebra and that's the general structure and here i'm saying this is the general structure for q charges meaning q dimensional operators under even vanilla invertible symmetries so i should ask something like does is there an operator which has the trivial core cycle instead which is like saying uh do you have there is right because there are also even for the for z two times z two you can pick a trivial core cycle right because this group is z two or something and you can pick a trivial non trivial this is a choice of what is the set of possible represent two representations or one charges of a theory with a zero form symmetry i just like saying i have an s un gauge theory um what are the representations that matter can transform it it's the answer to that kind of a question it's a you have a zero form symmetry what are the line operators that i can insert into this it's that kind of a question and that's okay but you also now hinted at maybe you did that so for for a higher for a q better than one charges there is a bit more this is also what's called symmetry factionalization and i'll explain what that is and then for the last thing i'll manage today i'll find it here so for q basically bigger than one you can have so so basically what this means is in addition to symmetries that can basically be localized on this line right they come from the bulk and then there's a localized now symmetry on this this is induced on this line for lines right if this is a line it can't be now already something that's living on the line that's non trivial but on a surface for example q equals to i can have a non trivial line already on it this is say our o2 so we have we're looking at two charges the surface operators for such things i can have a non trivial line on the surface already um so they can exist already topological lines o2 right and these are sort of what i would call localized so this could be a d1 and i'll call it localized because it's letting on that surface operator and we have a theory that so this is maybe a three-dimensional theory and it stands out in here um but this actually is localized on this so this can't happen for q equals to one the q equals to one this is not possible because i can't have if i have a simple operator here i can't have some non trivial local operator sitting there then it would change this line there's no there's no such d zero that doesn't actually that's sort of you know if this is really a simple line that's actually non trivial and that's not coming as an induced symmetry so these are induced symmetries actually let me also define what an induced symmetry is just for completeness so in contrast we'll call symmetries induced they arise about so we have some symmetries we're looking at our d right i want to draw the picture like this sorry five seconds so this is our o2 this is the thing we're trying to characterize and now there's an induced symmetry which is basically i intersected with oh that is absolutely terrible okay wrong angle try again this is a challenge always like this and you try to do this in tixit okay so i have this is our bulk d minus one g just like these ones here these are zero form symmetry generators these are g zero and then when i intersect them with the surface there'll be some line here so this is a d one and this is what i call induced and that's different from something that already lives as a surface operator can have a already non trivial topological line on it okay so now what can happen is that i have a bulk symmetry g zero and i have a two charge o2 which has a localized symmetry but these guys which are purple and i have these guys which are red so these are here localized and now when i look at the full symmetry i will just do purple another purple so these are coming from the bulk and now i ask when i fuse these two lines and so this is d one g one d one g two and now the question is what is the fusion of these induced lines d one g two symmetry d one localized so in fact when i fuse them i go to this picture so i bring these two together in the bulk they just choose to g one times g two but in fact here they could just use to something that is sort of this line here say if this were z two in general this can be a bit more complicated let's say the z two it could choose into an induced line a localized line so this is localized okay so let me be a bit more clear about this select g zero is basically what is coming from the bulk you have g localized and what i'm saying is that these two may not just form sort of a product but actually there could be a non-trivial extension namely that one embeds into the full fractionalized and that project valves back so there's some exact sequence like this they have a fractionalized enhanced symmetry that happens on this effect on o two that isn't just the product of these two with fractionalized not the product right naively you would have said i have a symmetry that's localized i have a symmetry that's induced if i bring them together it's just a product of these things but that's not true there can be non-trivial extensions and these are sort of characterized by group extension classes g zero comma g and i'll give you an example how this can happen in a gauge theory let's look again at the z two zero form symmetry and then for example we have in the bulk the z two and we actually construct this is actually a theory which is su four not z two and essentially what's happening here is that the two z two so the z two zero which is the bulk this is what i called g induced and the z two localized actually extend towards e four so the fusion of these lines actually is you can think of business essentially is equal to i and this is equal to minus whereas in the bulk this was just z two and you would have thought these two fuses to get just the identity that's right yes so here basically this is coming from the fact that you can think of this is coming from gauging so one way of understanding this is you can think of this is coming from gauging and psu four which has a z two z four zero form symmetry and you gauge the z two and you go to ps to su four not z two with a tough denominator and this actually is an example that we already knew from this paper with um lea boutini a pool of tibari well actually this i should mention this because so this is really the first time i've worked as a condensed matter for this and it's sort of part of the the appeal of this whole field that it's very interdisciplinary okay so now one other thing that is very very cool is now we start with completely invertible symmetries um but actually what can happen is i will finish don't worry and is that in fact you can also fractionalize to a non-invertible symmetry so even though everything was completely invertible so this is sort of the last cool thing that you can sort of observe is that even if your your g zero is the tool that can fractionalize to a non-invertible symmetry in this case that is the tool can actually fractionalize to the easing fusion category why so this was the category started right so it's nice to come also circle so here you had sort of to line the d minus which square to one that is e2 but then there's also the d2 s squared which becomes d1 it plus d1 minus so this is the easing fusion so basically you start with a with a bulk z2 uh symmetry that's invertible but on a two charge it can symmetry on symmetry on a two charge it can fractionalize to the easing okay so thanks very much for your attention maybe we could take one question before the break i have like half an hour of questions yeah one hour of questions yeah so okay let's go to a break and we come back at five for the informal discussion thanks