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


Adventure Ahead Adventure Ahead presents Toby Tyler, or Ten Weeks With A Circus. The circus classic by James Odin, a yarn which will bring nostalgia to most of us older folks, and we hope pleasure to those younger friends who are meeting Toby for the first time. It's circus day in the American city of Springfield, and a small and perspiring boy carries a heavy pail of water in each hand. As he struggles manfully under the double load, an old man comes out of a nearby tender pail. And pretty hot work, eh, boy? Gee, I'll say, mister, they're mellifant sure are thirsty. I don't know how many pails of water they've drunk. And you better put those pails down rest, or else the sun wouldn't want you to get sunstroke. Yeah, I guess I will for a while. You with the circus, mister? Yeah, been with it a long time, sonny. Gee, must be pretty exciting all right. I sure like to be a circus man. Oh, you would, would you? What's your name, son? Tommy Davis. Tommy Davis, huh? Well, you know Tommy, you remind me of a boy I used to be acquainted with once.  You know what he did? No, what? Yeah, he kind of felt the way you did. So he ran away with the circus. Gee, he did? Yeah, name was Toby Tyler. Oh, it's quite a story. Do you like to hear about it, Tommy? I sure would, mister. Well, all right, son. I guess we can let those elephants wait a while while we sit down on this plank and kind of take it easy. Yeah. Well, now about this young fellow Toby Tyler. He came from a little country town called Guilford, much smaller than Springfield it was. The circus was a big thing back then. There wasn't a boy around who didn't hanker to be a circus man. Anyway, the circus came to Toby Tyler's town, Guilford. And sure enough, down went Toby to the circus grounds. And the first place he headed for was a refreshment stand where one of the circus men was selling peanuts, candy, and lemonade. Yes, peanuts, candy, and lemonade right here. Get it where in the mouth. Ice cold lemonade. Fresh roasted peanuts right here. What's it be, young fellow? How much are your peanuts, mister? A nickel a bag. Well, he got a nickel. Only a cent. How many peanuts in a cent for it? I said I'm six for a cent. Oh, couldn't you give me more in that? Nope. Six for a cent. It was and six it is. Well, all right then. What's your name, boy? Toby Tyler.  Where's your father and mother? I don't know. Uncle Daniel says I'm an orphan. Uncle Daniel? Yep. He ain't my real uncle, though. I just live with him. Oh, so you're an orphan, huh? Yep. The same mister. Here's a bad peanut. Gonna give me another? Oh, yeah. I guess I'd better. Say, the way you're eating those peanuts seems to me you haven't had any vitals for a week. Oh, yes, sir. I just had dinner when I am an orphan leader. I just can't seem to help it. Somehow, I'm hungry all the time. Say, mister, I don't suppose you'd like to sell me some candy and lemonade, and let me tell you when I get older, would you? No, I don't do business that way. Well, I didn't suppose you would. Say, young fella, about your uncle Daniel, do you work for him? Yep, and I work hard, too. When Uncle Daniel says I eat four times as much as I earn. Well, then maybe you'd like to work somewhere else, huh? Well, I might. I see. Would you like to work for Joe Blore, boy? Who's Joe Blore? Well, that's me. You mean, travel with the circus? Yeah, that's the idea. This is your chance to be a circus man, young fella. You can help me sell peanuts and candy and lemonade. Gosh, me? A circus man? Well, boy, how about it? Is it a bargain? Oh, you bet, Mr. Lord. I should say so. All right, then. You'll be here at ten o'clock tonight. And mind you, not a word to your uncle Daniel. You see, he might not understand and stop you from coming. I mean, you can imagine how excited Toby was. This was his big chance, all right. He ran for home almost as fast as his legs could carry him. But as night came, he began to feel pretty scared, too. You see, son, Toby hadn't lived anywhere else but Guilford all his life. And this was a pretty important decision for him to make. His uncle Daniel noticed at the dinner table that something was wrong. Well, Toby, what is it? You haven't eaten a thing? Oh, there ain't nothing the matter, Uncle Daniel. Nonsense, boy. Nonsense. Are you sick? No, sir. I ain't sick. The first time I ever see you leave anything on your plate. Anyhow, you'd better go upstairs to bed. Nothing like a good night's sleep to set you straight again. I'm going into the parlor and read the script. Well, son, it got pretty close to 10 o'clock, and over at the circus grounds, Joe, Lord waited, and wondered if Toby had managed to get away from his uncle Daniel's farm, or whether the boy's nerve had failed him at the last moment. As Lord watched the wagon leading to Guilford, an old circus wagon driver who everybody called old Ben joined him. I hear you hired a new boy, Job. That's right. Been waiting for him now.  What do you mean? Well, I don't seem to keep them very long. They all run away sooner, lady. And good riddance. They were spoiled brats every dang one of them. Well, maybe they run away because you don't treat them right, Job. Maybe you'd try and mind in your own business, Ben. Well, it seems to me, if he's up on this new boy, why, he might be able to keep him alive. Now, look, Ben, I don't need any advice from an old broken-down teamster like you. I'll handle this boy the way I see fit. Well, in that case, I hope the boy don't show up. Well, go ahead and hope, Ben. Won't do you much good because here he comes now. On those days, Tommy, the circuses weren't very big, and they didn't travel on trains and trailer trucks like they do now. Everything was loaded on big, high wagons. And in this way, the circus moved from town to town. Well, as I said, old Ben was one of the drivers. He arranged to have Toby ride with him through the night. The two struck up a friendship right away, and the veteran teamster gave the young fellow some good words of advice. Yeah, Betsy. Come on, man, lift them hooves. Well, Toby, I reckon you're pretty excited about being with the circus, aren't you? Yes, sir, I sure am. Well, I ain't never met the boy who wouldn't be him. But now, Toby, listen to me and listen sharp.  But now that you're here, well, you got to make the best of it.  You shouldn't have no trouble. It was a pretty long ride before they reached the next town, and Toby managed to fall asleep in spite of the rough motion of the wagon. When he awoke, it was broad daylight. He saw that the wagons had stopped in the middle of a broad, grassy field. The whole place was buzzing and humming like a high-powered beehive. And he saw men putting up tents and pounding stakes and leading out the animals. Well, when Toby saw this, he scrambled from the wagon and walked over to a brook at the edge of the field. He was just leaning over and washing his face when he... Good morning, Mr. Lord. Good morning, your little brat. Where have you been all this time? Well, sir, I just woke up and walked over here. I thought I'd wash and tidy up a full breakfast. Wash up?  Do you think I hired you to take up my time washing? But, Mr. Lord, it ain't right to eat breakfast without washing. It ain't proper. Oh, so I picked up a preacher. Have I now listened to me, my banter? I do all the preaching around here as well as the practicing. And I've got a nice, quick way of making you understand. Come here, boy. Mr. Lord, what are you going to do? Well, when nothing told me nothing at all, I just lost my temper for a minute.  I wouldn't hurt you. Come closer. That's it. Ah, this will lawn your respect, I want it. Ow! Oh, please, Mr. Lord. Don't hit me with that cane again. I'll lamb base some manners into you. My back! Oh, my back! Hey, old Joe, Lord. What's that? What's going on here? Here, give me that cane. Why, you give me that cane, I say. Why? Mr. Boy, is that your job? I'll bust up this cane here so you'll never use it again. Now, look here, then. I told you to keep your nose out of my business. This boy works for me, and I'll do as I please.   Toby, boy, I'll see you later. We've got some human folks around here. I'd like to have you meet. Toby, meet Bosco the Clown. Bosco, this is Joe Lord's new boy. Hello, boy. Gosh, hello, Mr. Bosco. You don't look well, boy. Undernourished, if you ask me. Well, I ain't never had enough to eat, but I feel all right, Mr. Bosco. It's them who feel all right to go sudden. Here today, gone tomorrow. Take care of yourself, boy. You'll never come back. Gosh, Ben, is Bosco really a clown? One of the best in the big show. Well, why is he so sad? Oh, he's repressed, Bosco is. You see, Toby, he always wanted to be an undertaker. Oh. Oh, and Toby, meet Sam Treat, the living skeleton. Hello, son. Gosh, Mr. Treat, you sure are skinny. I'm the skinniest man in the world. Yes, sir, I'm the skinniest bank of bones there is. Say yes, you're Joe Lord's boy, aren't you? Yes, sir. I've heard what the old tyrant's trying to do to you this morning. Lucky you happened along, Ben. Yeah, like I told the boy, if Job tries it again, he'll answer to me. I guess Lily's looking for you, Sam. Here I am, my dove. Is all that your wife, Mr. Treat? She sure is. She weighs over 400 pounds and still gaining.   We make a pretty good thing of it with the big show. Samuel Treat, where's your head? Look at your death and the mnemonious cannon out in the cold like this.  I didn't see you. Lily, darling, this is Toby Tyler, Joe Lord's new boy.  Isn't he small, Samuel? Yes, and small as he is, Joe Lord laid a cane across his back this morning. Oh, that Lord man is a precious one he is. I'll give him a good shake in one of these days. Morse me. Well, Tommy, old Ben introduced Toby to many other circus performers, and they all became Toby's fast friends. But the best friend of all, the closest in Toby's confidence was an old monkey. The old fellow, as soon as he saw Toby there with his nose pressed against the bars, jumped down from the ring on which he was swinging and reached out a hairy hand through the cage. Wow, hello there. So you want to shake hands and be friends? Well, I do too, and here's my hand on it. You know what?  Yes, sir? You know why? Well, because you look so much like a fellow named Stubbs I knew back in Guilford. Yep, Mr. Stubbs, you're the limit image of him. Although he don't have quite as many whiskers as you do. You know, Mr. Stubbs, it's mighty funny.  But you seem to know what I'm saying, and I know what you're thinking. Hey, you! Boys! Well, Mr. Lazy Bones, what entire nation are you doing out here? Get away from that monkey cage and get into that main tent quick. Be sharp about it now. There's a good crowd in there, and you'd better let them know you're selling lemonade.  You know what, Mr. Stubbs? I'm going to run away from this circus the first chance I get. Now remember, Mr. Stubbs, this is a secret. Nobody knows but you. Some of the folks give me extra money when I sell them lemonade. And I'm going to save and save. And when I save enough, I'm going to run away and take you with me. At that very moment, other plans were being made for him. The circus riding master, and face fellow by the name of Castle, dropped in to see Toby's employer. Hey, Joe, but I've got a proposition for you. About that boy of yours, Toby Tyler. Well, what about him, Castle? I'd like to train him to be a bareback rider.  Maybe, but he's doing pretty well where he is. He always looks hungry and sad, and the customers buy. I'll be a fool, Joe. You'll make ten times more out of him if you ride in the center ring. He'll draw big salaries a performer. Well, that sounds fine, but what's in it for me? You own him? I'll train him. We can split his earnings 50-50, what do you say? All right, Castle, it's a deal. You can start to train him tomorrow. I don't know whether you know it or not, Tommy, but riding bareback is one of the hardest things there is. For days, Toby dangled in the leather belt trying to keep his balance and his courage as his feet slipped about on the horse's back. And Mr. Castle, by the way, was almost as hard a taskmaster as Joe Bluord was an employer. Now, boys, stand up on the saddle pad and try to keep your balance.  Yes, sir, Mr. Castle. I'll try. Keep your wits about you now. Stand up on the horse. Yes, sir. All right. Now, here we go. Stand up, boys. Stand up. Bet you don't look at the ground. Look ahead. Faster, Sultan. Faster. Stay on your feet, boys. That's not it. Whoa, Sultan. Whoa.  Can't you learn to stand on that horse? Oh, it's no use. You'll never learn. Get to bed now. But, Mr. Castle, I ain't had any supper.  Well, my bad, and you won't get any either. Maybe you'll learn how to ride tomorrow on an empty stomach. Toby didn't know that a little golden-haired girl had watched the whole performance. Old Bennett introduced them when Toby had first come to the circus. She was a performer, a bareback rider, who used the professional name of Mademoiselle Jeanette, but whose real name was Ella. When Toby left the riding circle after Mr. Castle had dressed him down, she bothered him. Don't cry, Toby. I saw your ride just now, and I think you'll be a fine performer someday. No, I won't. I'm scared to ride. Well, you're not scared of the horse. You're scared of Mr. Castle. Tell you what, Toby. What? I'm going to ask Mr. Castle if he'll let us ride together. I can help you a lot. And he won't be very alone before we can do an act all by ourselves. And then we'll want the people to ride together. Oh, gosh. Do you really think we can ride together, Ella? Why, of course I do, Toby. Whatever this fellow Castle was, Tommy, he was a showman, and he was quick to see the money-making possibilities of this new team. He put Toby and Ella through their paces hour after hour, day after day. Stay close, you two. I straight ahead. Feet flat on the salad pad. Fast as softens. You're ready now.  Gosh, Ella, I'm scared. You needn't be, Toby. I'll be with you every minute. Now, you just hold my hand tight. Hold till we ride out to the ring. All right, Ella. I guess I ain't scared anymore. I was there, and I can tell you Toby and Ella were a sensation. In fact, the owner of the circus was so pleased that he gave Toby Mr. Stubbs the Monkey as a reward. Toby was now a successful performer, yet never for one minute kept him from planning to run away and take Mr. Stubbs with him. Finally, he told his secret, the secret that only Mr. Stubbs and he knew to Ella. I'm going to run away from the circus, and Mr. Lloyd and Mr. Cassellella run away. Yep, I'm going home. Don't go down. What's the matter, Ella? If you run away, I'll never see you again. Oh, yes, you will, Ella. Honest. You know what? I'm just a runaway boy now, but Sunday I'll be grown up into a man.  Who are you, Toby? Honest. Cross your heart and hope to die. Cross my heart and hope to die. Toby, Toby, I'll miss you. I'll miss you just also. It's dark, and I don't think there's anyone following us. Mr. Stubbs, we've got to be careful not to make any noise and wake anyone up. Once we get away from the circus grounds and down the road, you can talk all you want. Yes, sure, Mr. Stubbs. We're going home to Gilbert and the farm. Won't you have a good time when we get there? You can run all over the barn and up the trees and... Toby, stop!  Toby! He's gaining on us, Mr. Stubbs. If it's Mr. Lord or Mr. Castle, we'll get whipped. Toby, you don't have to be afraid. Go ahead, Ben. Oh, Ben. Oh, gosh, he scared me. I thought maybe it was Mr. Lord. I got the note you left on my bed, Toby. But I wanted to see it before you left. Oh, gosh, Ben. I left the notes because I wouldn't wake you. I know, I know. Now listen to me, Toby. Go into the woods for a day or two before you take the train back to Gilbert. You're too big a prize for Joe Lord or Castle to lose. And they're going to do their level best to catch you. Do you do that, son? Yes, sir, Ben. I will. Well, goodbye, my boy. And if ever the time comes and you want to remember that, well, that you had a friend. You think of old Ben, whose heart this beats as warm for you as if he was your father. Goodbye, my boy. Goodbye, and may the good Lord bless you. Goodbye, Ben. Goodbye.  Dear apples, Mr. Stubbs, Mr. Lord and Mr. Castle, they've got dogs and they're following us. Come on, Mr. Stubbs. We better run deeper in these woods. We've got to get away. They're closing in on Mr. Stubbs. I'm so tired. I just can't run anymore. Maybe, maybe we better climb up these trees and hide. Maybe they'll go past us. Looks like we've finally treed the little farmer. Yeah, there he is, Lord. Up there. Holding a monk. Come down out of that tree, boy. Come down and be quick about it. There goes the monk. Good shot, Joe. You got him proper.    Come down out of that tree. Sit, my boy. Come on down. You let us a pretty face, but we're willing to forget me and your soul, baby. Yes, I did. What do you mean running out on us after all? We've done for you. Seems to me the boy needs a little lesson, Joe. Yes, Castle. That's just what I'm thinking. Yeah. I'll hold him. You let him have it with your keen. Go ahead and square him, boy. It ain't going to do you no good. You need a thrashing and you're going to get it. Right out on us, will you? No, sir. Castle, somebody's coming.  What do you think it is? I don't know. I'm going to give you some of the fashion you'll never forget for whipping that boy. Oh, now you're down, Joe.  Look, it's old Ben.  You killed him. No, no. I didn't mean to do it, Castle. I didn't mean to do it. The gun went off.  We haven't got time for that now. We got to get out of here quick. Come on. Well, Tommy, they found Toby in the morning crying his heart out over Mr. Stubbs and the old Ben. Then they took Toby back to the circus grounds and the circus folk got together a collection, put Toby on the train for Gilford. And so it was after ten weeks with the circus that Toby Tyler entered Uncle Dannell's familiar power again. The old man was sitting by the window. His face was sad. Uncle Dannell. Uncle Dannell. Wait, wait. It's Toby. Toby, you've come home. Oh, bless my heart, boy. You've come home. Please, Uncle Dannell, forgive me for being wicked and running away with the circus. Please let me stay here again, Uncle Dannell. Oh, my dear boy. My little lad. You've made me a happy old man. Why, of course. Of course you can stay here.  And I know now why you did run off. Uncle Dannell. Oh, now, Toby lad. Toby, I want you to stay here. And Toby, I'm glad you came home. Well, Tommy, I guess that's about all, that they finally tracked down Job, Lord, and Castle. Those two gentlemen got what they deserved. Well, that's the story of Toby Tyler, just as it happened. As I said, you reminded me a lot of him. That's why I told you the story, I guess. He'll feel like being a circus man. Gosh, I ain't so sure now. Maybe I'd better wait a little while. Yes, I guess maybe you had. Well, Tommy, we'd better get back to hauling water. Those elephants are beginning to look thirsty again. Oh, uh, there's a couple of pastures for the big tents for your trouble. Gee, two pastures. Reserve seats, too. Can you show you can spare them, mister? Oh, I guess I can. You see, son, I own the circus. Toby, Toby Tyler. All right, Ella dear. I'm coming. Toby Tyler, or Ten Weeks with a Circus, the children's classic by James Otis was adapted for Adventure Ahead by Max Erlich. Special music was written and conducted by Henri Nosko, and the production was under the direction of Joseph Mansfield. NBC and its independent affiliate stations present Adventure Ahead as a public service.


It's going on everyone Sam here with late sports cards. We are ripping 2023 Panini gold standard football hobby six-box half case break number 18 change the name Pick your team stop read off list of our names in our teams We've got Alan B with the commanders Barry wide dolphins void as Panthers Brent C Rams cardboard legacy with the Niners Charles G Broncos Eagles Corey S. Colts Daniel D. Bills Drew B. Falcons Texans Eric R with the Seahawks Greg M Titans Jacob F Packers Joseph C Vikings Justin C chiefs Keith B with the Lions Marcus Pete Cardinals Michael J Saints Nate S bucks that is Nick H with the Ravens Pascal V Raiders Phillip L with the Jets Russell P Chargers Ryan P Giants Ryan C with the bangles Scott F bears Sean O cowboys Timothy W Brown Steelers Patriots and Vaughn with the Jags So good luck here everybody. Let's do it. I'm gonna say personals are probably close to full right now Think we'll get to everybody in queue tonight If we don't Kyle's doing them tomorrow night, so there'll be plenty of room Plenty of time for them to be ripped tomorrow Good luck here everybody. Yeah, Jordan Addison to 75 for the Vikings It Justin Jefferson to 75 for the Vikings nice one there Wanda Robinson to 99 to colored patch going to the Giants. Yeah eight to go in the first gilded PYT break tomorrow night That's gonna be so much fun, dude Zach Charbonnet to 399 go into the Seahawks At your car pick up the Mabs and no our photo. Thank you, buddy Cd lamb to 49 on the jersey for the Cowboys Nice one there. Got a rookie auto for the Patriots to 199 Marta Mappu Grats on that one there Patriots and going to the Giants Jalen Hyatt rookie auto to 199 congrats Giants Like I said case hit is still alive in this Should be seeing it soon DJ or a Derek Hall to 75 and 99 DJ Moore George Kittle nice one there to 199 heart of gold for the Niners Got to tank Bigsby going to the Jaguars to 399 Dak Prescott to 299 going to the Cowboys. This card's so cool.  How about an AJ green to 24 10 karat gold auto for the Bengals Again, I in my opinion will go down as one of the most underrated receivers in NFL history The my least favorite receiver I'd ever have I've ever had to play in my entire life of being a Ravens fan Was AJ green?   You can tell it really drives me crazy. It drives me. You're genuinely insane That all right, we've got a George Kittle to 24 Nice one there and Paris Johnson to 49 No, Derek Mason was great Don't get me wrong like we've had our share of great receivers Derek Mason and then Anquan Tori Smith was really good in his tenure. We like Steve Smith was great for those three years So we've had good receivers. We just never had like, you know a guy that like you have to completely game plan for Nice one here JK Dobbins to 49 three colored patch. There you go. Kwame Mike Wallace really rich Mike Wallace Yeah, I would say Anquan Anquan and And and what's his face? Yes, Sammy Watkins. Yeah, how the Sean Jackson Jeremy Macklin Kamar Aiken Marlon Brown the list goes on and on of just bots dude good like good players But just like never again never the top top the receiver in the NFL Devin DuVernay. Yeah Jalen Hurts nice one there for the Eagles to 299 But yeah, I agree with that Anquan and and what's his face in Derek Mason were the closest Anquan never had a thousand yards as a Raven though That's the only thing that sucks. I don't think he did at least Look that up. Let me fact check Baltimore Yeah, no never had a thousand yards had Last two years in Arizona a thousand a thousand then never over a thousand and then in San Fran had a thousand and thousand We've got a Clayton tune rookie Jersey auto going to the Cardinals what rich Wallace was good. He had a thousand yards of the Ravens one here was our leading receiver one year But he just he was never quite the guy Mike Wallace. We had Mike Wallace like three years after he was really dominant to Quadrate is mill. Oh my god, dude And now we're just sit now. We're just saying names Colesha can see the 199 for the Bucks Rats on that their Tampa and the nice one here Hinden Hooker at a 199 for the Lions. I have a quandary ishmael rookie Floating around my PC somewhere. I do one of those die-cut ones Okay, this is the case hit You can already tell this is the case hit heck We've got a that's nasty Calvin Ridley to 99 That's nasty going to the Chargers L Dorado to 24 Justin Herbert congrats LA nice one there at chages Nice one there Charges, I'm not gonna take away Mike Wallace. Mike Wallace was good. Mike Wallace was good with the Ravens Mike Wallace was good with the Ravens, but he still was not a top three receiver in the NFL. You know what I mean? That is a TJ Hawkins into 299 We have it right during the Super Bowl years The most recent one it was Anquan Tori Smith like we had Pitta tight end Who the hell else was on that roster it was that like those were the two main guys I think mom brown wasn't there yet. Oh God who else was there? Oh This is gonna bother me. Let me look this up. He was gone at that point. He was good, but We've always had good tight ends. That's that's what I'm happy about Ravens my receiver bold in and Tori Smith were the main two Tandon Doss was there. He was never that good. Jacobi Jones. He was more of a Go be Jones is more of a return man. No, David Reed Tori the Ante Thompson the Kwan Williams Appreciate that John Thanks, man. Hell, yeah, that's Andy ish. Absolutely We've got Tyree Wilson on the rookie jersey the 399 appreciate that John. Thank you, man. Thank you. Thank you Nice one there. Now. I think Clayton was gone at that point. Yeah, Clayton Clayton wasn't there He was gone in like 2010 No, I mean we only we've had the best kicker for a while We had Stover in the early years and then we had then we then we stunk for a bit with Haushka and a few other guys And then we had what's his name again? She's good To attack of my low heart of gold to 24 Of course Tucker I'm gonna open I'm slower now because he said that Fire and young to five nice one there for the Rams Anthony Richardson 13 of 25 Golden debut going to the Colts. I swung there congrats on that Colts I think that's a Corey very nice very nice. Get that into a mag as well China sharp fair the case. It was the El Dorado the Justin Herbert El Dorado. That was the case it I'm trying. I'm trying they I'm trying Jordan battle to 30 and CMC to 75 Let's go. That's right illegal. Go Phillies, baby T white hill in as a cowboy. That is so weird to 49 congrats Dallas. That's cool, but that's weird, dude That's so weird Roshan Johnson to 399 for the Bears Jalen hurts heart of gold to 49 for the Eagles The hit of the break free. I mean hey it is man. It's weird, but it is Theoretically it is yes Dante cold pepper 24 karat gold to 24 for the Vikings. That's cool.   Yes, I do remember Travis Taylor. Yes, I do. Yes, I do and Sam LaPorta 18 of 24 going to the Lions nice when they're Lions three color two color two color. I was so ampt when we signed Jeremy Macklin. I was like this is this is what's gonna do it right here But this is this is it man was Jeremy Macklin and that was the complete opposite complete opposite Yep That is Breton strange to 49 and Jalen Ramsey to 99 Goldgear 299 Traylon Berks for the Titans Nice one here is Erica Zucanma. Nice patches on that to 24 Little logo up there patch patch patch and more patch. No, I was a great eagle dude. He's a pretty decent chief to There you go, Barry Barry again with the 299 Jalen waddle. Let's go mate We got a cool one coming up Nate really cool one really sick one Really sick one for the bucks Vinny test of Ernie to 25 on the 24 karat gold Yes, sir. He asked for it. We got it good stuff Nate cool one there man Yes, we'll have Jones now an eagle which is strange as well true and Jake Heiner rookie auto to 199 for the Saints got that exact card last night Brandon. That's dope though. These are sweet man I love the 24 karat autos quick recap here everybody start with our jerseys We've got Jalen waddle Jalen Hertz T. Y. Hilton Roshan Johnson to a tag of I low Tyree Wilson T. J. Hawkinson El Dorado case it of Justin Herbert Jalen Hertz J. K. Dobbins that Prescott Tank Figsby George Kittle Zach Charbonnet CD lamb Wanda Robinson Few more coming as well start with some of the autos here. We had the Vinny test of Ernie Side now. I'm sorry Drew Jake Heiner Dante Cole Pepper Sam LaPorta Byron young and in Hooker collage. You can see Clayton Tune Chase Brown AJ Green That is a Erica to Conma Traylon Burks Marta Mappu Jalen Hyatt and Anthony Richardson to 25 that'll do it for the great guys. Thanks again for the filp up next is NT