 Thank you. It's nice to be here. Sorry to disappoint those who wanted to hear Simeon. Hopefully you will get your chance later. Well, I guess I should start by congratulating Samsung and I have known him for many many years. And actually we are very lucky people because sort of professionally we are surrounded by lots of smart people. From whom we learn lots of lots of things. If you think from other side about your interactions with people whom you knew for many years, you realize that beyond kind of narrow technical aspect of it, there aren't very many memorable things they have said. And this is definitely not the case about Samsung. I'm sure everyone can kind of remember funny, idiosyncratic, surprising things coming out of him. Moreover, it's a completely different category. He also somehow provides quotable things. So I'll share one samsonism not because it's the best or most characteristic because it's personal and because also of some sort of guilt feeling. I have been plagiarizing it for more than 15 years. Pretty much during any collaboration I had for those years and most likely I will continue doing so for as long as I do find collaborators. So years ago we were writing a paper with Pierre and Samsung and it was coming to the end. Somehow I volunteered or I was volunteered to have a first go at the conclusion which I was doing and either there was something I had to run away or I just simply run out of steam. So comes the last sentence, wasn't sure what to write. So I wrote something which sounded along those lines and finally some of the authors are more skeptical than others about blah blah blah. So I sent the file out two days later. I get this was before Dropbox. I get a file from Samsung. I go through the pages seemingly nothing has changed and then I reach the very very end and there it comes. A new paragraph which was just one line written in boldface possibly in all capitals. And it said the authors shall agree. So and this is the one I've been using ever since and my birthday wish for Samsung is that he will continue finding people who will sort of continue agreeing with him while preserving all of his idiosyncrasies, originality and other things which make Samsung-Samsung. All right. And now I'll talk about anomalies and inflow mechanisms to which and this end notably to which Samsung had as much contribution as any other person. So it's a work with a very nice set of people. Ibu Ba, Federico Bonetti and Emily Nardoni. And I am sorry there are no credits on the transparencies otherwise, but I will be happy to provide the references. And this is a talk which is kind of very much if not by the goal but by kind of a regional point connected to Cumbulon's talk, actually. So anomalies. I realize the audience is mixed so I will start from the very very beginning. And the very beginning is the 11-dimensional supergravity which was mentioned already. So this is the theory in highest dimension with highest amount of super symmetries. Once you know the the field content, actually the probably most remarkable thing about this theory is that it exists or that this action has been constructed. But otherwise it looks fairly innocuous, you know. Hilbert Einstein term, kinetic term, the usual fermionic terms. So actually the next most interesting thing about it is this triple coupling, let's call it Chern-Simons. And this is pretty much anything I am going to tell is one way or another traces to this particular coupling. And of course in the supersymmetric in just supergravity, I mean this field, the G field strength is just simply taken to be the field strength of the three-form potential. So we have local symmetries. There is a bare potential appearing in here. But nevertheless there is a symmetry under C goes to C plus lambda. So there is some kind of gauge transformation, okay? Now one other old thing which we knew for many many years about this this theory is that it admits ADS 7 vacuum. And so there is a four-form. And if there is a flux of this four-form field strength through S4, then somehow there is a stable maximally supersymmetric solution which displaces of 5 symmetry, which is of course directly related to the isometries of the sphere, okay? Now, I'm not sure how far I will manage to get in my talk. So the sort of ostensibly what I was planning to talk about was about kind of a program of let's say systematic geometric derivation of anomalies in four and possibly two-dimensional theories starting from from M theory five-pray, okay? So there has been a lot of progress and there is a continuous progress. Lots of anomalies provide a nice robust information. So this is I mean there are lots of theories in four dimensions without for example Lagrangian description and things like that. The anomalies, okay, they have been some of the results have been derived by field theoretic methods, but it would we felt it would be nice to have some kind of top-down approach and the classes of geometries which would appear here which you could kind of I mean again we know that ADS, the boundary of ADS, there is some theory living there and if we talk about ADS 5 then we will be talking about four-dimensional super conformal theories and the two large classes of solutions that that one can consider this way are the following. So either we kind of take this ADS 7 roughly and embed a Riemann surface there and then this way we will get to ADS 5 and this is a more brainy picture for string theorists and this way we can get theories with 8 or 16 supercharges in the bulk. So respectively n equal 1 or n equal 2 in the in four dimensions or there is another way of doing it is if you consider a solutions which involve non-compact seven manifolds compact and Minkowski 4 in such a way that there are some sort of that the solution is conical and it involves fluxes and such solutions do exist then we can find kind of an ADS limit to those and this is a rather large class of solutions and in many of these in this class we don't even have brain realizations. So we cannot really talk about five-brain wrapping this or that or we don't know it in in detail. Okay, but I'll again hopefully I'll have some time at least to comment on those. So for the moment what I my first thing to do is as I said everything is going to be about the CGG but I am going to add another counterpart to it and the counterpart is of this form C wage x8 where x8 is some curvature polynomial and 8 form it's an 8 derivative term the exact the concrete form of the polynomial is is written here. It's just some combination of p1 square of the Pythagorean class is p1 square and and p2. This guy appears okay, not directly in 11 dimensions. It is computable in string theory in 10 dimensions in type 2a string. It's just a result of some five-point function calculation in in 10 dimensions with one b field and four gravitons and it can be lifted to 11 dimensions. This x8 is kind of a remarkable polynomial it seems to have a feature of sort of showing up in lots of not necessarily not immediately same contexts okay. The way we see it in in in string theory is actually sort of a product the way it comes out is a product of four gamma two matrices and one finds a following expression which is actually relating this x8 to a lower number on any manifold which on any eight manifold which admits nowhere vanishing spinners. So this is something which has been used very very much in in in string theory okay. But one way or another we we add this guy to 11-dimensional supergravity and this is the kind of the beginning of m-theory okay this is how m-theory starts becoming or 11-dimensional supergravity starts becoming an m-theory. Once more notice that because x8 is closed we do keep our gauge invariance c can be shifted by by some closed form. All right and now we will talk about five brains so there are objects in this theory kind of solitonic. The first goal is you can try to find them as as classical solutions but okay why first the nomenclature it's called a five brain because I mean what what it does it provides a source to these four form flux so if we write some kind of five form delta function which means it's an object that when you integrate anything wedge this in 11 dimensions it will be putting you on this transverse six-dimensional space time and that's the nomenclature you know membrane is three dimensions so five brain will be extended to one time and five transverse dimensions okay. Now if you do a zero mode expansion of this guy you just try to determine in spectrum what lives on this theory then you find that what this theory supports is actually a supersymmetric six-dimensional multiplet with 16 supercharges and the multiplet is in question is a tensor multiplet so the field content is given by a b-menu a tensor field however its field strength is constrained and it is of dual okay and then there are fermions and there are five scalars which parameterize the five transverse directions this multiplet has an s o five r symmetry again there are just five scalars which are in the fundamental of s o five and the fermions live in the spinner representation of s o five and what we know is in principle we don't know much about the non-Abelian theory but in principle it does admit a de classification so there exist kind of non-Abelian brains okay and this is still like kind of a direct description of this is still rather mysterious okay now let's start moving gently towards kind of inflow okay who is flowing into what so first we will have inflow without anomalies so I kept saying that this theory okay is symmetric in absence of the brain under c goes to c plus d lambda and the the spacetime diffeomorphisms so let's look at this gauge invariance okay you you compute this and now remember I mean okay c goes to c plus d lambda you do some integration by part when d hits g now you will have a delta and then you end up with something which lives on w five on the world volume so what five brain does it breaks the gauge invariance of the bulk of the eleven-dimensional theory but it's not a big deal because we can we can fix it by having a coupling of the tensor field this little h three is the field strength of this this beta written here so we can fix this by writing some couplings between the three fields which live on the brain with pullback of the fields which live on the in the bulk okay but the moral is something that you have to retain is that the bulk and the brain are not separately gauge invariant anymore so there are some symmetries the overall symmetry is preserved but it works in a combined way you have to have the full system and the things are not separately symmetric okay now we can look at diffeomorphisms okay so this was simple there is no anomaly all i have been doing was just simply classical variation okay just checking the classical invariance so again similarly i can take this coupling c wedge x eight and also look at its variation so i'll integrate by parts and rather write it as g wedge x seven where x seven is just some local object whose derivative is x eight so here we start getting into descent and then this x seven is no longer gauge invariant or diffeomorphism invariant and so if you do this i mean you will end up again sooner or later hitting this g with the delta with the derivative that will give you this delta phi form so you will end up with some object which is pulled back to the world volume and formally it looks like gravitational anomaly once more i've been i looked at some coupling and i just did classical variation of it okay i'm just studying the variation of it under diffeomorphism but this is this is good why is it good because because the theory on the brain as i showed it's a chiral theory and you can actually check that it is anomalous so you just check what lives on this brain world volume there is a chiral two form and we know it's an anomaly which is given by herzerberg polynomial and then there are the the world volume fermions and if we compute this total anomaly in the case of trivial normal bundle i mean we get exactly the same combination of p one square and p two which i was showing so again very nice if the normal bundle is trivial then this is the classic the canonical anomaly inflow mechanism is that classical variation of the bulk fails on the brain so you had a symmetry you introduce a brain the brain breaks this symmetry but the breaking is confined to the brain itself but luckily the brain is anomalous by itself and so the two two currents two anomalous currents can actually just cancel out okay there is just one little thing is that okay we generally should not assume that m theory allows only trivial normal bundles right so the triviality of the normal bundle comes noticed that here when i am computing the anomaly on the brain i am writing characteristic classes on the tangent bundle to to w which is the the manifold the sub manifold in 11 dimensions while the original coupling here is written in the spacetime okay so if if you do it with non-trivial normal bundle when you are restricting the classes from the spacetime to the world volume you will be picking pieces which depend on the normal bundle it's also not a big deal because you can go and actually be a little more careful about how you compute the anomaly on the world volume there isn't much change about this this chiral two form but there is a big change about the fermions because the fermions do live in the non-trivial representation of SO5 so they are sections of the normal bundle so you have to recalculate a little bit and once you do you really you you see that kind of magically pieces start cancelling out again except for one very last piece so you end up with a rather unpleasant surprise that the anomaly which was the coupling which was anyway undisputed because it's computed in string theory but which was also very well working with the with the inflow in general case has an extra piece which has to do with the non-trivial normal bundle so you have this p2 of n divided by 24 okay um so where do we look for the answers and the answers are while I already kind of told you that the only interesting thing is this charm simons coupling okay in the 11-dimensional supergravity so we should look at it a little closer and you see this charm simons coupling was having cgg so you would think look c wedge x8 was okay there is one delta function hidden inside cgg is too many delta functions this guy is somehow too singular let's try to to make it a little bit more regular okay so what we do is we look for another representation of of tom class which we can find by by writing this poincare dual to the brain in the following form okay we introduce a bump function something which so you cut out a small disc around the brain and you you look you define the radial direction away from it and then you can have some bump function some profile which interpolates let's say between minus one and zero its derivative will will be approximating a delta function as as this bump function gets kind of steeper okay but the the big point is to to to make the formalism so five invariant okay so we introduce something which is called global angular form and um and okay there is there is a bit of dimensional dependence whether you you do this procedure with the transverse dimensions which are odd or even which has to do essentially with Euler class Euler number of a sphere being either two or zero and in the case that we are interested in I mean our global angular form is e4 and it has to be closed okay and then there is a remarkable result by Katanao and Bot which kind of showed that homologically this global angular form squared is a pullback of some okay it works for any n but for case of n equal to which means e4 square is a pullback of p2 over again p2 is important here because that's that was our uncancelled anomaly and moreover there is another formula is that when you take e4 cubed and you integrate along the fibers you will get p2 again we have p2 and notice we have a cube cher Simon's coupling was cubic okay so these these are the ingredients okay I show you the explicit expressions I mean they normally should follow from general kind of formalism but in case you want to see the explicit expressions they are not really super hard to to derive in some ways I mean these things very much are along the the steps in Sharon's proof of of Gauss-Bonethiore okay I think he was constructing similar kind of objects but roughly speaking take a volume form so why why had think of it as stereographic coordinate and the volume form will be will have epsilon a1 to a5 with four d y's and one y okay and now take the derivatives and covariantize them but that will make this guy non-closed so you you start adding terms in order to make it closed okay simple for low I mean you have general formula for any dimensions but doing it for n equal to just explicitly is not very hard just to show it we can also do a descent on it so we can write it as d of some local expression and we can calculate the variation so everything we do for kind of skeptical physicists we can actually everything can be verified all right I'll skip you some of the steps the important point is that eventually we can rewrite this equation with just simply saying you know so we started with g equals dc and now we shift it around by something so that dg is no longer zero okay but somehow we can still preserve the chair simons form of it so we do the shift in in all three of them and then if we do it and we can compute the variation then you see essentially what contributes here is this e cube and e cube essentially eventually gives us the p2 of n the answer reaction did not change just correctly yes yes the same action it just correctly expanded it right there is a brain so the end result is this is the formula to retain that the chair simons coupling in the presence of the brain when the non-trivial normal when there is a non-trivial normal bundle is no longer diffeomorphismy variant okay and this contribution should sum up and here is the importance of having zeros zeros are good indeed we want to sum things up in order to have a zero so we have the m5 anomaly what I call bulk is just this contribution from c wedge x8 and the contribution the variation of chair simons they all set add up to zero okay great fantastic so we saved m theory it's no longer suffers from anomalies but I was telling you about ads in the beginning of the talk and then you could object that ads would require large n and what I am doing is notably I talked about anomaly of m5 I explicitly computed the polynomial so I am actually talking about single brain so you would ask what does it mean the anomaly for multiple brains a theory which has an ad classification a theory about which we know strictly nothing okay other than the fact that it has ad classification and here is a kind of a rare case in life when you are kind of lucky that you get almost out of nothing something which looks rather nice you stare a little bit at this cancellation mechanism or long enough and then you say look if we replace one brain by many brains there are two parts of this formula which change very little I mean eventually everything was about dg equals something and now we we stick a charge there that's all that changes and there doesn't seem to be anything else happening from the point of view of inflow so why don't we assume that the mechanism still holds if you assume it then it's a very trivial calculation to show that an anomaly of q coincident five brains and this is the case of suq or uq in enhancement will become q times anomaly of the single five brain plus this p2 of n multiplied by qq minus q over 24 okay and then you try to test this formula in all possible cases but essentially you just really got a knowledge of anomalies of two zero non-abillion theories with doing very very little just simply requiring that m theory with the brains is not supposed to suffer from anomalies now in order to to talk about a series you have to take away the center of mass which you can do and the change is very little okay then a calculation has been done by replacing r5 by r5 mod z2 or by sphere by rp4 okay it requires a bit of care but you do get a formula okay for for d series for e series we do not have a brain realization not at least that i know but by now there is also a proof using five-dimensional gate chair assignments theories and it does confirm actually that there is a general pattern that the coefficients always look the same way okay and since then i mean this is about two zero theories since then the things have been extended to to also different one zero theories the first step for example is to do the so-called e strings and there for the string theories in the audience i would like to remind you that there is another place where c wedge x8 and cgg play together very very well to to give us something and that's the horjavan witten mechanism so if you look at the green schwarz polynomial near a boundary so i here is one or two one eight or the other eight it also has this remarkable form that it's a four-form polynomial wedge x8 the very same x8 i was talking about plus the cube of that polynomial that's actually the original this is horjavan witten two so just deriving green schwarz when witten tries to cancel f5 very normally could not explain in f5 he went to 2a this is heter this is heterodic no so this is just simply saying that the way to do to start getting into one zero theories is combining horjavan witten with the with the cancellation mechanism that i was describing above okay and this way you start seeing one zeros and then of course there are many more others than e strings it gets a bit more complicated okay now i put a transparency to make it sort of contact with cumbrun's talk which unfortunately are the parts where which he didn't really cover okay so he didn't talk about six dimensional one zero theories but i will so if you put the five the m theory on a columbial threefold you will get five dimensional theory with eight supercharges if the threefold in question is elliptically fine but that theory leads to six dimensional one zero theory okay now if you pick one vector field in the in this five dimensional supergravity then you see this looks very much similar to two eleven dimensional supergravity where you replace the three form by a one form so generically there are couplings of this form with again just like was cumbrun was saying a priori unfixed coefficients okay so there is a kind of a five dimensional assignments term and then there is a which p1 so it's a little lighter kind of instead of an eight form polynomial you have just just a p1 and just really doing variation of this and assuming that there is some kind of charge source yeah a is is a billion then then you can you can immediately see the relation of these coefficients to to the brain anomalies so there is a there is a chiral string living in this theory the anomaly of this chiral string is given by the following form it's very clear that only one sector knows the n here is so three bundle or su2 only one sector knows about it so only sirite knows about p1 of n or at all about normal bundle while the the tangent bundle has the difference it's just encodes the difference of central charges so it immediately tells you that in order to have consistent supergravity you better have quantized coefficients it also kind of bounds you see you have two extreme cases where alpha or bt is equal to zero and then you can you can see that there are there are bounds between these things this is realized in m theory if you look at five brain which is wrapping a very ample divisor and in that case you can express this alpha and beta in terms of intersection numbers of this divisor and the signature okay these strings do not lift into strings in six dimensions so the things that kumbru didn't talk about were not really exactly those things they are slightly different in that they would have q square there okay but morally you get the flavors and again if you had seen the formula with normal bundles and stuff you would know where it's coming from okay all right now i have few minutes to talk about gauge theories but before i want to kind of introduce a formal object so i want to say that from now on what we will be doing we will be dealing with some kind of formal 12 form which will be integrating down and sort of okay you probably saw the flavor of what's going to happen if i have this 12 form polynomial and it is defined in such a way that if i take the f m theory action and i vary it i get some 10 form and that 10 form comes from this i12 via descent okay so it is a formal construct its motivation is to give via descent something which does agree with the variation of 11 dimensional action and if we integrate this guy on a sphere let's say if we are interested in six dimensional CFTs or another higher dimensional surface if we are interested in lower dimensional theories so what we want to know is or what we want to claim is that you you should always get the zero that the the inflow should agree with the CFT anomaly up to up to the coupling modes okay so one example of this where things become non-trivial and also rather technical is but in a way the very simplest one is take this s4 and and and fiber it over an over a remand surface and so in other words what you have is you have a five-brain wrapping the remand surface and this way you will be getting four dimensional theories so preservation of supersymmetry tells you that s05 can be broken either into s02 times s03 or s02 times s02 this is n equal 2 or n equal 4 case but the basic idea always stays so what before I was showing you the formula with subscript 8 and now it's the formula with subscript 6 that the direct integration gives us an overall result for the anomaly of the CFT plus the decoupling mode you look at the explicit examples you want to compare with cases you know and normally you find the perfect matching with kind of every single calculation which we have done it does require special treatment I mean what you are interested generally is in remand surfaces with punctures and things like that so the bottom line is it seems to work okay now all right so here is a anomaly expression for a generic CFT and here is somehow the identification that you get from the inflow and once more uh if you check it on examples it does seem to check every time you can do the calculation but there is a different different things you can do here that the anomaly is also equal again let's go back and think about the p2 of n the p2 of the normal bundle what we would be saying is that this six-dimensional 2-0 theory is also the singleton which leaves on the boundary of ADS so exactly the same calculation uh or computing this anomalies is the same as computing chair simons terms in this case in seven-dimensional supergravity so you would be asking seven-dimensional supergravity is a theory with 11-dimensional is with two derivatives you compactify somehow and you find a theory which has higher derivatives at just not our procedure gives you chair simons terms you it's not hard to count derivatives af cube so it starts with three derivative terms okay where do these guys come from well I mean you you just look at it and you say okay what our vacuum configuration was was this that we took the the flux and we said we equated it up to a charge q to the volume of the s4 okay but now we should we should excite we should expand around the vacuum but we should kind of stick to some rules we should we want to stay invariant under s05 we want it to be closed we want it to be quantized so again you stare at least of properties and you say oh we know such an object this was our friend the global angular form so if we if we take it and and use it for kind of as an answer for excitation around the vacuum actually it does give us the the the wanted chair simons term with the wanted coefficients this was the calculation which is let's say the topological sector of the field and if you really want a full supergravity you start moving into something which is a science of its own which is called kind of consistent truncations you have to incorporate the scalars okay but in some ways you also have a way of thinking about it because you are asking about the space of deformations of this global angular form and then you can conclude that this is given by essentially a coset or by symmetric matrices so it's an sl5 mod s05 coset and that's exactly the coset of the five-dimensional super of the seven-dimensional supergravity and you can check this thing in the lower dimensional case so if you have an e5 and that's ads7 type 2b on ad on s5 then you will be getting sl5 mod s05 coset but then the theory had also an sl2 to start with and two together they built the exceptional group e6 and then if you do the the the seven sphere reductions you see an sl8 which is a maximal subgroup of of e7 the the missing ingredients here are associated with the s7 being parallelizable okay so in the rest of the talk what i will want to do is actually since i am just for now interested in anomalies to sort of say look cut it short and just go and compute the char simons couplings in the theories especially that given that i i know how to produce ads5 in many cases where i do not have kind of a direct brain interpretation so say look take all the geometries you know and you can control and try to compute the char simons coefficients and that's essentially going to give you uh the anomalies you want so again we have a list of properties we know that g should be generally be replaced by some global angular form sort of the our sphere has to be fibered over some space in order to provide a bigger space on which we are compactifying the 11 dimensional theory so we are always in a situation like this again the list of properties of this e hasn't changed i just made it capital and the reason i did is because the general these general manifolds are not spheres anymore so they have more stuff there what do they have they have isometries so there is some a priori non-abelian group of isometries and then there are two forms which give rise also to vector fields so i'm interested in four-dimensional vector fields where i get vector fields notice that this is an ansatz taken by hand if i had taken kind of other forms presumably i should be getting towards the higher symmetries which i suppose sohar will be talking about okay i'm not doing any of it here so if you do it you can you can argue that the most general form of e4 is the following where i have kind of joined together the isometry objects and the objects which are associated with harmonic two forms into a single object x the superscript g means things are gauged which means you take the differential forms that you would be writing otherwise and every time that you would be writing dx dy or whatever you replace them by appropriate covariant derivatives there is an extra kind of subtlety that in d equal four one of these modes associated with the with the harmonic two forms becomes massive and so there is an extra constraint now that consists of the closure and invariance of this e4 in this form sums up to a bunch of unpleasant looking equations which however you just realize that okay if you introduce an equivariant object so a four form which is homogeneous degree four but in such a way that you are adding differential forms plus two times the degree of the polynomial form of the objects which are valued in some Lie algebra that all this invariance and closure end up just being a condition that the object defines an equivariant class this e4 okay this is also a convenient object language to analyze the deformations so we do not want the things which are equivariantly exact we just want the closed ones so we want a a cohomology which weeds out lots of redundancies of this e4 not completely if you want to fix the the ambiguity is completely what we found a little bit by trial and error is that okay imposing the following condition that this object is trivial in cohomology does the job and agrees with again all the known examples its moral motivation comes from the 11-dimensional supergravity equations of motion which says d star g4 equals g4 with g4 plus x8 once more you are replacing g's by e's but you notice that this object is trivial in cohomology so if you integrate it on any eight surface this better be zero okay so now okay i probably don't have that much time to talk about a particular example which i was planning to do which which is made of so there is a large class of geometries which provide this suitable six manifold on which by compactifying the 11-dimensional theory we get ads5 and that is by having an s2 bundle over a product of s2 and the Riemann surface with g holes somehow things better work they don't quite work for torres for the reason that there are accidental symmetries so it's safer to take this sphere which is the best studied example so the product of two spheres with a sphere bundle on top of just or just a higher genus Riemann surface okay so you go through some steps you look at the quantization you figure out your redundancy and again we have three-dimensional three harmonic forms but only two independent fields and i chose an f2 and f3 to write my final result in which ends up this okay i don't imagine or it wasn't for you to to particularly appreciate it it's just somehow to tell you that okay this simple compact formula in 11 dimensions once you put them through the machinery are able to produce stuff like this okay in order to figure out the central charges you have to go through through a maximization you end up with okay equally cumbersome looking central charge which this expression however at least we can check so it has not been computed in generality but for for the sphere and one particular flux being zero we can we can check that this does agree with the known results okay so in principle there is a potential inside that you can sort of handle cases which you couldn't do otherwise or haven't been done otherwise okay so i am close to the end then so what we hope is that well once more the chair simons couplings in the in the in the ads via holography give us kind of a direct access to anomalies in super conformal theories and the program which we had been kind of trying to develop is just kind of to systematize this computation and essentially we are seeing that everything is determined by by the topology of the space and by some class e4 and there is a class of theories which we cannot get in m theory but they require type 2b and there we kind of there we are developing a similar formalism notice that i mentioned that there is some kind of z2 story here the even global forms and odd global forms behave very differently so here for the five sphere you will need e5 and e5 is not closed okay and then there is extra complications having to do with self duality but in principle we can define an equivalent or analog of this class i12 which will be a class i11 for type 2b theory which again seems to give seems to allow us to do calculations one i mean there are many questions one could ask thinks about discrete symmetries higher symmetries and also kind of doing a more systematic calculations which take us beyond these topological limits and actually construction of real physical theories in terms of super gravities and in some ways maybe one could actually try to think again going back to cumbron's work not a question which has been asked but possibly that for example the good match of discrete symmetries and higher symmetries might start imposing constraints on ADS or ADS theories which we have not seen before so that if there is a any kind of swamp land also in the ADS context but okay these are dreams for the moment only and i'll stop here thank you this mysterious 12th class that you showed in the talk you say some more words on it right you you said it's a formal construction but what what do you think it is right this 12th well again the inspiration comes from the fact that when you have i mean from like guesumino models you have cgg of course you can think of 11 dimensional theory as a boundary of some 12 dimensional world and then you would be just dealing with ggg actually i was dealing with e4 cube throughout so that's the that's the basic underlying idea but once more it was just simply constructed as something whose descent gives us the variation of the 11 dimensional action for the general like p-form fields do you know the boundary conditions in ADS no so it's not fixed by this construction we we haven't tried yet i mean we start thinking about it so maybe ask me in a while and hopefully i'll tell you something about it why do you think it's not fixed by this construction well for instance in ADS 5 times s5 there are many options yeah you can have the you can have ps you engage theory s you engage theory you engage theory and they correspond to different choices of the boundary conditions of the two form gauge fields so i think it would be an interesting thing to kind of put that through this machine theory and see if there are any constraints thank you