 Okay, so in the morning I tried to explain why if we are trying to uncover some larger unbroken symmetry phase of string theory, ADS-3 is very special and so in the rest of these lectures I will concentrate on a particular vacuum of ADS-3, namely the ADS-3 times S3 times B4 vacuum of string theory or so type 2 string theory. So this, so I will say a few words about this but just so that you get an idea of where I'm going I'll just give you a sort of a preview of what I will try to cover in the rest of these remaining couple of lectures so that you don't get lost so at least you have the broad picture even if you get lost in the technicalities. So firstly this will be a vacuum with very large exponentially bigger exponentially larger than the Vecilia unbroken gauge symmetry. So that is the first thing I will try to explain or equivalently the dual safety will have a very large number of exponentially larger number of conserve currents than the bilinear ones that I showed you that's of course if you know the CFT that's not a great surprise that's it's a free CFT and there are lots of these things there the really interesting thing about this this unbroken gauge symmetry is the way you can organize it so it can be organized in terms of representations of a higher spin symmetry or a W infinity symmetry. I will use these somewhat interchangeably sometimes one needs to be a little careful to distinguish between the two as I said in the morning the asymptotic symmetry is usually the same brown Heno like announcement of the higher spin symmetry but there's a very definite relation between the two if you wish this is the so-called wedge algebra for those who know what that means that this is the wedge algebra of that any case you can organize it in terms of a Vecilia type higher spin symmetry in terms of representations of this which I will sort of schematically write like this there's a sort of a infinity of them which can be labeled by an integer n so if you wish n equal to 1 n equal to 2 so on so that each of these is a representation so of W infinity in particular the n equal to 2 corresponds to the W infinity algebra or the highest pin algebra itself and each of these columns is in this way of drawing representing it is a is a representation of this so it's a there's a highest weight vector and there's a sort of a tower so that's what a column is supposed to denote so there is a whole set of whole set of higher spin generators whole set of currents are equivalently gauge fields but of which the Vecilia one the corresponding to the bilinears like I talked about in the morning it's just one and there's a there's a there's a whole tower of them and and we'll call this the vertical W infinity or highest pin algebra has been this the vertical highest print algebra according to which we are classifying things but the somewhat remarkable thing is that there is a novel additional highest spin symmetry which so this is the full set of highest they exponentially larger set of gauge symmetries have been organized this way but there's actually a novel additional high spin symmetry which also organizes these gauge symmetries are this of these conserved currents and this one is a higher spin symmetry which organizes these generators which here are organized in columns it organizes them in an orthogonal way in terms of rows and in which again labeled by an integer and in which this that say the top row will correspond to this other W infinity or higher spin symmetry which we'll call a horizontal W infinity symmetry so there's an additional one this is the horizontal one it's different from the from this vertical one and in terms of which you can organize things in terms of again representations of these so again there's an integer let's call it M so M equal to 1 will correspond to this horizontal symmetry then there is M equal to 2 etc so the whole thing sort of forms what what we call a higher spin square because the algebra between any two elements in this in this is determined by both these two in terms the commutator between two elements in this in I should actually use maybe two different colors these are highest rate unitary representations of the W infinity algebra I don't see any color but anyway so so there's so the commutator between any two elements here can be reduced to commutators of both these horizontal and these vertical higher spin symmetries and this forms I think a quite a novel structure which we call the higher spin square so I should say this is work with Mathias Gabbardial so most of it is in a paper that came out in February that builds on something that came out last June so so so this higher spin square is a somewhat at least to us it was somewhat unusual and novel but we realized that there's a sort of a toy model of this which is probably familiar to all of you I'm sure which we'll call the Clifford algebra square and which will sort of illustrate this and so which is a sort of a baby version of this of this structure just to give you some intuition so that's roughly the plan of what I will try to tell you in the next this lecture in the next so let me start with this vacuum it is three times s3 times t4 and it's dual CFT so so this is the this is one of the canonical examples of ADS CFT actually in fact in the original paper of Maldesena itself this was one of the examples and it comes from it's obtained in the same way as many of the other examples it was it's argued for in much the same way you consider type 2 string theory d1 and d5 brains let's say r4 comma 1 times t4 times a circle which so this will be a sort of a small t4 small meaning stringy size so you'll mostly not have to worry about this but this will take to be sort of a large circle anyway these details won't be so important so we'll consider d1 d5 system with so wrapped on the d4 times s1 namely the d1 brain is wrapped on the s1 and the d5 brain is wrapped on all these so so you can write down the supergravity solution corresponding to this again I won't write all this but these are standard things that there's a supergravity solution corresponding to this brain configuration and its near-horizon geometry is ADS 3 times s3 times t4 so this t4 is essentially sort of that t4 but the the remaining six directions assemble into a ADS 3 times s3 and so this is the sort of the gravity picture or the closed string picture of the string background but as in the usual ADS CFD derivations the duality comes from the existence of a dual gauge theory picture of these brains in this case this brain system so the near horizon geometry is some kind of a low energy limit of the string theory so at low energies the d1 d5 system is one plus one dimensional gauge theory so basically because that's why you need the t4 to be small so that those directions you will those will be stringy energy states and and this one plus one dimension this is sort of the s1 direction and this is the time direction and you have a one plus one dimensional gauge theory which even further in the IR flows to two-dimensional CFD and though there isn't a complete derivation of this it is believed that there's good reason to believe that this CFD is a deformation of a symmetric product CFD symmetric product or before CFD so what is that so this so this is the symmetric product where you take n copies of the t4 and you mod out by the symmetric group acting on these n copies and so this is a target space for this two-dimensional CFD so you consider where n is q1 q5 q1 and q5 are the number of these d brains and so there's good reason to believe this though it's never been completely sort of proven but among the reasons to believe it of course are various things like the matching of the moduli space and more importantly this is the context in which the original Strominger Waffa derivation of the microscopics of the black hole entropy of the five-dimensional black hole that you get when you put some momentum along this s1 this is the context in which the microscopic entropy of that black hole from this geometric picture was matched from the counting of states in particular of the elliptic genus of this two-dimensional CFD so there's good reason to believe that it's somewhere on the moduli space of this CFD so this is a free CFD because we basically have in this t4 so this is actually of I forgot to say it's a supersymmetric theory it's in fact a 4 comma 4 super conformal field theory for left and right super symmetries and this so this is essentially n copies of the simplest 4 comma 4 CFD you can think of namely with four bosons and their partner fermions you take n copies of that and and mod out by the symmetric group so it's an orbit fold of a free theory which is again can be is described by the is essentially a free theory so this is this is the basic CFD and at the generic point in the moduli space of this of this of this background corresponds to a deformation of the CFD so the this CFD is if you wish the analog of the free angles so this is the analog of of free angles that we described earlier so sort of lambda equal to zero point and then there's some deformations corresponding now it's not just a single there are 20 marginal parameters of which four are sort of interesting marginal directions and so there are so those are deformations you can go away from this free theory you can turn on marginal operators and that that space of deformations it leads to leads you to sort of the general background of this kind of it in the gravity limit will be one extreme just like we had in the morning for free and mill theory so this is the analog of the free angles it's a free theory so so this is the point you would naturally associate with the sort of tension less limit of of this background and indeed as we will see that like in the higher dimensions there will be a higher spin symmetry at this point there will be a higher spin generators but not just them and there'll be all these other infinite ones but certainly there will be the higher spin generators the Vasiliya type gate symmetries so so it's good reason to associate this with the so this corresponds to the sort of the tension less limit in the in the sense we talked about in the morning okay so so that's the background we are going to talk about and so it's a this is a very explicit two-dimensional CFD about which you can say many things and so exactly and in particular we will be interested in the unbroken the conserved currents or the from the bulk point of view of the unbroken gauge in variances so we can easily address that in this theory because it's a free theory and as I said it's an orbit fold of some free bosons and free fermions and so so that's what we want to first address so so we want to look at the chiral sector or what you would call the chiral algebra of this two-dimensional CFD at this orbit fold point okay so in particular we I mean actually in our in the first paper that we studied we looked at the full partition function and this paper we actually looked at the full partition function of this CFD which is actually known and I'll maybe just explain that briefly but you can specialize then to the set of states which are chiral namely which have h bar equal to zero or equivalently the antiholomorphic part so but it's not very difficult to study the full portion function as well at the orbit fold point since it's a free theory and in fact one can organize all the states in the CFD in terms of this vacillia unbroken symmetry and not just the chiral states but I will focus in this lectures only on the chiral states but I wanted to just tell you that you can actually organize all the states of the CFD in terms of representations of the higher spin this vertical algebra and but but again we will mostly focus on this but let me first quote for you the expression for the full partition function because it's simple enough so there's a simple formula for the full partition function now when I say partition function I'm not talking of anything like a supersymmetric partition function I'm not talking of elliptic genus or with an index or anything like that I'm talking about the full genuine partition function trace q to the L0 q bar to the L0 and if there are some chemical potentials something like this no minus one to the F's or anything like that so we are looking at the full partition function so of course this quantity will not be protected as you go away from the orbit fold point but our interest at the moment is not to go away from the orbit fold point but to understand the structure at the orbit fold point so so there's a simple formula for this which was derived many years ago by digraph more valende and valende and let me give a reference for seeing there's some nice pedagogical lectures by digraph which if you want to learn more about this you can look at this reference so this partition function of it's a general formula you can write for the partition function so if we ha if we know z of x where x is some CFD and if we know this if you can write this as some c of h h bar then let's say r r bar and q to the h q to the h bar so if we know this partition function z of some this sum is overall the h h bars supposing we know this full partition function we know all these coefficients then the partition function of x to the n copies of x mod out by SN is is given by the gen following generating function so this generating function is I'll write it as z of p let me write the generating function the generating function is of all these for any n so it's more convenient to write in terms of this generating function with p being the with p being the well if you let's start with zero so that there's a one so p is just a formal parameter for this generating series and this is given in a very explicit way in terms of product an infinite product so there's an infinite product this is p to the n and q to the h q bar to the h bar why to all these chemical potential these are some r charges so it's given by this and in terms of these coefficients that appear over here in this in the original so you can read it there's actually a condition as well h minus h bar should be congruent to zero mod n and this n labels various twisted sectors and greater than one labels various twisted sectors so this is an explicit formula proved by these people I mean they their main interest there was for the elliptic genus but the same argument works for the full partition function and so there's an explicit formula so if I know these coefficients and in the free theory x in our case is just t4 we know these we will write down explicitly the answer for this you can write down the generating function for the nth symmetric product of of these for of this CFT so I mean this may look a little bit mysterious but just to demystify it a little bit let me just point out a simpler case which may be more familiar to all of you yes there was a question can yeah so these are chemical potentials wise and white bars for the our charge so so this this symmetric product taking the symmetric product is like multi-particling a single particle wave function a single particle partition function for instance in statistical mechanics you may have often encountered the problem where you have n free particles you know their individual single particle a partition function so supposing we know q is e to the minus beta h let's say so supposing we know the single particle wave function the partition function then a natural question is what is the multi-particle partition function and and if these are for instance a bosons let's say if we consider the case of bosons let me write it here let's say this are bosons then then the multi-particle partition function is something you can again write a generating function for it sort of a grand canonical ensemble for it where p is the fugacity so far for those you can write the grand canonical for a partition function with fixed number n and this is in fact just given by it's just given by these coefficients but they appear in the exponent over here and this is because of the boson the usual rule for boson multi-particling when you expand this you it's not difficult to convince yourself if you have not seen this before that putting these up in the numerator essentially counts so these are the degeneracies of the of the single particle states and putting this up in the exponent basically accounts for you the number of ways you can distribute bosons amongst so many degenerate states so that's what this is for fermions you would just put this thing in the numerator so that's the fermionic case so this thing that appearing in the in the numerator or denominator is is is a reflection of this fermionic statistics or bosonic statistics and and this is a sort of a simpler version of how things work in the general case except that here you have also twisted sectors and so on but in the untwisted sector which is where the which was where the chiral part will lie we don't have it's basically this this logic so be so essentially the chiral sector of the t4 symmetric product is in fact just the multi-particling the chiral sector of of a single t4 theory I should say is is not I shouldn't say is the but is in okay I'll just clarify as of the single t4 theory by that what I mean is that if you want to know all the chiral generators of the symmetric product theory but you want to know if you want to know that so so this is the full partition function the full partition function contains both single particle and multi-particle states in a general field theory and at large n if you are interested in so this duality over here we are going to look at it in the n goes to infinity limit to remember the diagram that I drew in the morning this is the classical limit so so the n in this case is this so we will take this to infinity and so in this large n limit the the chiral sector is just the multi-particling of the single t4 theory meaning that you you you just you can associate the single particle generators so let me maybe write that the single particle generators of the chiral algebra of the symmetric product of the chiral sector of the chiral algebra of of the symmetric product is in one-to-one correspondence with the chiral the full chiral algebra of a single t4 so so what do I mean by this what I what I mean to say is that you can consider a single t4 CFT and write down all the currents that means the holomorphic and anti- holomorphic quantities in this theory that's what will give you the chiral algebra of this t4 theory and I am claiming in the large n limit these so as n goes to infinity in the large n limit the generators the single particle generators of the chiral algebra of this are in one-to-one correspondence with this and the logic is essentially that I'm going I'm not going to go into the details but the logic is essentially that of this multi-particling that the fact that the symmetric product is some kind of multi-particling you can so the lambda going to zero limit is I mean I'm assuming that we are at the free point and I'm taking that as the definition of the lambda goes to zero because yeah so there's some point in the modular space of the CFT where it's a free theory and I'm calling that the lambda equal to zero point and there I have an additional parameter n which I will need to take to be large it depends on yeah what you treat as the deformation away from this free point so in some ways it is indeed natural to take the limit that you said take you five equal to one for instance and Q1 to be large and in some ways that's what we effectively do and not in this way though I'm not hundred percent sure that's strictly necessary I could take the point of view that I just stated namely I'm at this free theory point and this n is just a parameter in which case I'm agnostic about how to take Q1 Q5 but I suspect that and this is probably related to the confusions about where exactly in the modular space the symmetric product lies in the D1 D5 modular space and perhaps to make that connection you might need to take the limit in the way you said but at the moment yeah okay so what do I mean by this one is to one correspondence note that I'm not going to quite say I'm not it would be wrong of me to say that the chiral algebra of this theory is the same of the single particle generators are the same as the chiral algebra of the t4 theory but I just want to say that there's a one is to one correspondence so let me just illustrate that because in fact and I will mostly work in the particular case not of the t4 so I can make the similar statement with say t4 go replace by a single boson for instance and the logic is much the same it is just for the symmetric product so let me just illustrate it for this example so that you don't have all the complications of supersymmetry and extra bosons and so on so if I just consider a single boson the so when I say I can make the same statement here with this t4 replaced by R this would be just R R to the n-mod SN so what I'm saying is that corresponding to some so for instance this phi is a single boson this is an element of the chiral algebra of a single boson I just take del phi del phi is holomorphic quantity this I can raise it to some power that's a holomorphic current of the single boson of the single boson that's an element of this chiral algebra but I will associate to that the following generator of the symmetric product chiral algebra so this is now the symmetric product chiral algebra has n bosons and you're mod out by the symmetric group so this is an element of that chiral algebra which is of course by construction invariant under the symmetric group so it belongs to the untwisted sector of this orbital and is built out of the n boson so I can always make a correspondence you give me an element of the chiral algebra over here I just symmetrize it over all the elements and I give you an element of the chiral algebra over here and so I can do this for any current any spin as current I just do the same thing I take the n copies of this and I have an element of this chiral algebra why I want to stress that this is this is why I wanted to stress that this is only a correspondence is that in this in this single boson case or the t4 case not all the elements of the chiral algebra are independent generators so in a single boson theory del phi square is of course the stress energy tensor and del phi to the fourth is in some sense the square of the stress energy tensor in some in some appropriately defined way so these are not independent generators of the chiral algebra in the in the single boson theory but here del phi but this too will this corresponds to the stress tensor of the symmetric product theory and this corresponds to a similar generator but this is not equal to t of z the whole square in the symmetric product so they are independent generators in the symmetric product theory they seem to be there the single boson theory of course these are these are all just the same but here they these are all independent generators and so so basically the single particle generators of the chiral algebra over here you can write them down in one is to one correspondence with this but I wanted to just stress that I'm not saying that it is the same as the chiral algebra of in terms of the algebra structure okay so and in fact this is where the large n limit will be important at finite n there will be relations amongst all these generators but at infinite n these will all be strictly independent generators so because there will be no relations amongst these so at so that's important to appreciate okay so so what I have told you is a way to enumerate all the generators of this unbroken algebra and I've already in fact exhibited to you generators which are not just bilinear so for instance this del phi to the fourth this sort of this is an element of the chiral algebra single particle generator of the chiral algebra the chiral sector of the symmetric product theory and it is not so it's definitely not bilinear and I can write out many many such things so obviously the single particle sector states you can see are much bigger than just that of the bilinears which are a small subset so so this enumeration tells us that gives us the generating function for for the for the single particle generators so the the generating function is nothing but the partition function in the case of the t4 theory going back to the t4 theory there's just the partition function of a t4 of the chiral sector of a t4 theory which is as I said for so this is just equal to chiral sector for bosons plus for fermions and that's nothing but the sort of the usual free fermion partition function for the free bosons they carry opposite our charge two of them carry one charge into the other so that's why this is the this is the the fermionic part so the chiral sector of the fermionic part and then you also have for free for free bosons which is this this because this is the partition function of a single boson and then one so the four corresponds to the four for bosons and these are the four fermions with equal and opposite our charge so so this is a generating function which counts it enumerates all the chiral generators for you because this is just the chiral sector for free bosons and chiral part of for free fermions there's only q's no q bars so that's why these are just the this is just the chiral sector so this is an enumeration of the number of generators and you can see that this is exponentially large a number of generators at any given spin because spin is now so for a given delta which is equal to s there is a cardee growth of the number of states so the degeneracy this is an enumeration and it tells you that at any given so for q to the s that's going to tell you the number of generators at spin s there will be a cardee growth so it will be essentially e to the square root s times some coefficient because it's effectively like some c equal to a sixth theory or something it will have some coefficient and times e to the root s so it's a very large it's unlike the vacillia of growth where in the vacillia of theory remember I said that I emphasize that there's only a fixed number of states fixed number of currents or gate equivalently gauge fields for a given s and whereas here the number number is growing exponentially so that's why I was saying that this is exponentially larger than the vacillia of growth so you have a much larger set of single particle generators or cons or conserve currents or the chiral algebra compared to the vacillia of case is this clear okay so so we've so we've understood sort of there's a large number so in the by the gauge gravity duality as I said in the morning each of these single particle generators remember it's always the single part why did I focus on the single particle generators because those are the ones which you will associate it to the fields in the bulk the multi particle generators are multi just multi particle excitations of those fields in the bulk so that's why it was important to focus on the single particle ones which as I argued are in correspondence with those of a single t4 and this is indeed the enumeration of all the single of all the chiral currents of a single t4 theory okay so so now we want to understand how to organize these so so this is maybe yeah I mean if you have thought a little about the symmetric products here at any point it would be obvious to you that there's a large chiral sector and this this is just that but the interesting thing is to organize this sector this is a large algebra and the large this thing it's not very useful to just say it in this way if you want to use this unbroken symmetry and to relate it to the vacillia of symmetry which is present generic I mean which as we saw earlier is the generic unbroken symmetry if you want to see this as some enlargement of the vacillia of symmetry you should organize these in terms of the vacillia of symmetry and indeed the vacillia type higher spin symmetry is a sub-sector of this chiral algebra and and and and one can show that but actually I will to remove various complications are just additional baggage coming from just having four bosons and four fermions and so on I will restrict myself to a simpler case just to illustrate but things will you will see go exactly the same way just like I did over here I'll make the same restriction of restricting to a single boson is also good to because it show you that nothing depends on super symmetry in all this it's just something that would exist for single boson it's nothing to do with super symmetry so so we'll so to organize this in terms of the highest spin symmetry we will restrict to all the statements that I make here are just as good for a single boson and so we will stick to that so for a single boson symmetric product as the chiral algebra single particle generators single particle symmetries one to one into the chiral sector of a single boson as I describe the enumeration of that is just the single boson partition function namely one minus q to the end so it's just I have gotten rid of most of the baggage all these fermions I've gotten rid of and instead of the four I have just a one so it's just so if we were to do the same thing for a single boson this is the enumeration of all the single particle chiral generators and we can now understand this shown of all complications so so this single boson chiral algebra is so the generators are things like this I mean for instance this was an example of such of the chiral algebra of a single boson in particular the stress tensor was this quadratic one but the there is a W infinity or as I said W infinity or higher spin sub algebra from the quadratic or bilinear bilinears so they are basically so you can write W at spin s is schematically some so basically it's del to the k phi del to the s minus k phi so they're quadratic in phi but with s derivatives sprinkled with some combinatorial coefficients and some combination of this is a primary and which is a spin s primary and this is a subset of this or sub algebra of these of this big chiral algebra because these are clearly holomorphic they just involve the del del is del del z by the way so so these are this is in fact what is called W infinity in terms of the general family of W infinity of lambda this is the particular case when lambda is equal to one and it's the Brown Hino extension of the HS lambda equal to one higher spin algebra which is it's so-called wedge algebra and so and so this is the subset of these and so I had made the statement that we could organize all the states in terms of representations of this W infinity so in this case of the single boson we'll see that all the states well I'll at least motivate why all the states can be written in terms of representations of this algebra so in fact following monomials to the k del to the m1 phi to the m2 phi I can consider these monomials where these mi are greater than or equal to one these are elements of the chiral algebra because they are all holomorphic quantities del phi is holomorphic a further derivatives will make so these are monomials in the chiral algebra of a single boson and as I said you can make it in correspondence by the symmetrization to the to the symmetric product so so I'll write write down just the one in the single boson theory and you can always translate it to the other case so I claim that these monomials are in so so these monomials for a fixed n so this is something there are n phi's and with arbitrary numbers of derivatives on them these for fixed n transform in an irreducible representation there's W infinity of one of those W infinity algebra under these so that's not difficult because it's a free theory you can convince yourself that these are in a irreducible representation in fact the general representations so we can even say which one the this is the representation so a general representation of of W infinity of lambda can be labeled by sort of two labels which are which are if you wish dinkin labels or young diagrams and and and and and this particular so this is a general so a general representation is labeled by these labels and these monomials transform in the representation where lambda plus is basically anti-symmetric representation with sort of n boxes of a young tableau in a in a vertical young tableau and lambda minus is zero is the trivial representation so it's a very canonical representation and so I claim that that's the representation and just to sort of flesh it out a little bit more let me so and so of course this is for a fixed n the same n that appears over here so if I fix n so the things which are sort of with n phi's and arbitrary m's obeying this they form one representation now if I let n vary from one to infinity that covers all the generators of the chiral algebra because the arbitrary generator you can write in terms of it will lie in one of these this thing so so I so that means I'm claiming that this so this is the n that I was mentioning here these columns and to to flesh it out a little if this is indeed the case then my enumeration of my partition function the free boson just the chiral part of the free boson partition function should become a character decomposition of these representations so indeed that is true and the character decomposition is the following the character decomposition of the character of this representation I claim is is just this there this is the character of this representation so this is what this is what is called the edge character of this representation so this is equal to this character I claim and this identity this is actually an identity it's I will in fact prove it for you with this identity in a moment it's not a very difficult identity it goes it's it's a special case of what is sometimes called the q binomial theorem and you will find it in combinatorics books it's an identity due to Gauss but so this identity is is this decomposition is is is basically this identity and how do you see this you see this by by looking at these monomials so these monomials are in correspondence so by the state operator correspondence you can if alphas are the sort of chiral modes of the free boson these are in correspondence with with states like this in in the fox space and as I vary my m's and my n's I generate the full fox space but you see that I can arrange these are boson there's a single boson I can arrange this fox space in the following way remember these m is are greater than equal to 1 what I these are bosons so I can arrange them in some ascending order of their mode numbers so these keys these k i's are greater than equal to 0 and therefore this is in sort of non decreasing order of their mode numbers and all the way up to summation k i up to n so the mode numbers are just increasing and so and what is the so so this these are clearly just different ways of relabeling the same thing and and now you can count very easily what how many of these there are it's just the total mode number is so so the total mode number is n because there are there's a minus 1 I've put everywhere because there has to be at least these m i's are greater than equal to 1 so there's a minus 1 in each of them so the total mode number is n plus n plus kn into 1 plus kn minus 1 into 2 plus kn minus 2 into 3 all the way up to plus k 1 into n right because k 1 appears in all of them k 2 appears in all but one of them and so on so this is the total mode number how many ways do you have to distribute a given mode number into this that's precisely this there's a q to the n and then this just counts the number of ways of distributing this so that's the reason that so this identity is just this is the proof of this identity it's just a matter of organizing this and if you look and this are indeed I mean these the characters of these representations are known and you can go and check and see that these are indeed this so this decomposition is is really a character decomposition of all these other of these representations and as I said the n equal to 2 corresponds to so n equal to 2 over here corresponds to the bilinears which are the w infinity generators themselves and yeah and and and so so you can think of the whole thing as as I said the whole chiral algebra is organized in terms of these columns each of these has this w infinity of 1 or this vertical w infinity of which this was the this was the original w infinity and these are all representations which transform in a particular way this is if you wish the adjoint representation by the way the n equal to 1 is rather trivial it is just del phi a single the single del phi in fact these are just l minus 1 descendant so that's a little trivial and my n equal to 2 is the first non-trivial one and then these are all other representation so this is the first thing that I was saying that you can organize this chiral algebra in terms of representations of this vertical w infinity algebra and so so this tells you in fact so therefore if I wanted to take the commutator of some object here with something over here the commutator is determined by the fact that this is transforming in this particular representation of this vertical w infinity and so so it's just given by w infinity representation theory so so that is of course not the most general thing we don't want just commutators of this with elements here we want commutators of this with this and that I mean you can of course use the chiral algebra to compute that but that's not we want to do it in a sort of in a more in a nicer way to sort of see the least the minimal structure that is there and that's where this other horizontal w infinity will come into play but that I will describe tomorrow so thanks