 Okay, so let's pick up the thread again from where we left it yesterday. So let me very briefly recapitulate what we saw yesterday, which is that the central points were the following that, so we were looking at this example of the string theory on ADS three times S three times D four as a candidate example where we might be able to see an enlarged gauge invariance over and beyond the vacillia of gauge invariance and the symmetric product CFT, which is believed to be dual to this vacuum is a symmetric product of a free theory, namely of T four or four bosons and fermions. And this theory being a free theory has a very large chiral algebra. So the first thing we showed was that the single particle generators of the chiral algebra of this symmetric product was in one-to-one correspondence with the chiral sector of single T four. So the single particle, so this is of course not an isomorphism of chiral algebras, it's just correspondence between the elements of the algebra and if you remember it was essentially something like this. So this was one observation, not perhaps a very surprising one, but the first observation to make and this was important as I stressed that these generators on the symmetric product these are all the single particle generators are independent generators. At finite end there would be relations between them but we will be working at large end so we'll consider, so these will all be independent single particle generators. Now the next question was the algebra of these single particle generators and of course you could say it's just the chiral algebra of the T four theory which is one answer but that's not an answer, one would I think be very helpful if we wanted to try to use the Wesselier theory and things we know about the Wesselier theory to learn about the unbroken gauge, unbroken symmetry phase of string theory. So we tried to organize this so to understand this algebra we organized it in terms of representations of the highest spin algebra or the W infinity algebra. In fact it turned out that it was I showed you that there is a W infinity algebra of free bosons. I mean to do this as I said I would restrict to the simpler case where instead of a T four we just have a single boson, just R. So R four or T four we just strip off all the complications due to supersymmetry and talk about the simpler case of a single boson because all these structures are present even in the symmetric product of a single boson. So we organized the chiral sector in terms of representations of a higher spin algebra which is this W infinity algebra and the structure we found was the following that the W infinity algebra itself this was the W infinity of one algebra so when you organized it in terms of representations the representations were labeled by one integer the infinite many of them basically they were labeled by a completely anti-symmetric representation with n boxes and the n equal to two representation was the one which corresponded to the W infinity generators themselves so that was this column so all the W infinity generators form if you wish some kind of adjoint representation but then there were all an infinitely many other representations as well and these were roughly speaking so the top component of these was del phi to the n so these were the bilinears so the top component here is the stress tensor del phi square but you have each of these representations comprised of n linear quantities of these bosons and I showed you that the chiral partition function of a single boson corresponding to this decomposition was the sort of cubinomial identity this was the character decomposition of the full chiral algebra so this enumerates the chiral sector and so this was the full chiral sector of a single boson and that can be decomposed in terms of these and each of these corresponds to the character of this nth representation so this identity which I showed you how you can derive from this sort of decomposition of the fox space this cubinomial identity is essentially this character decomposition so we had this sort of vertical so we called this the vertical w infinity because we have arranged things in these columns and the w infinity action is sort of vertical there is some highest weight representation and then you have the action of the w infinity generators okay so any questions? so this of course this way of organizing it is useful because it tells you how the so it firstly tells you that the vessel of higher spin symmetry is subset of the full algebra and it tells you moreover how the other generators transform under this w infinity symmetry how they are charged under this vessel of gauge symmetry and therefore if you want to take the commutator some arbitrary element here with something over here that's just given by w infinity commutators of the representation theory of w infinity but as I said that's what we are interested in is the full algebra the commutators between arbitrary generators something here and something here how would you do that? and that's where we will see the presence of a new higher spin algebra will enter so the basic observation behind that is quite simple it is what maybe many of you have seen it is the property of two dimensional field theories which is known as bosonization which is the statement that in our context it's the statement that the CFT of a single real boson is equal to the neutral sector of two fermions sort of carrying opposite charge so in fact this is captured by the identity so at least at the level of the partition function you can write this as the following statement that this partition function is the same as a fermionic partition function restricted to the neutral sector that is the y is chemical potential so you have two fermions carrying opposite charge that's why y and y inverse and if you restrict to the neutral sector that is the sector which is independent of y the charge zero sector then this is the sort of partition identity which is a special case of a more general identity relating bosonic fermionic partition functions so if you haven't seen this sort of thing before you can just look at Ginz-Parek's review on 2D CFT and you'll see the proof of this statement and others like it so that's the thing that we will exploit that basically the boson partition function can be written in terms of well we'll exploit the fact that the full partition function but the full theory itself you can map it to this from you can map the bosons to the fermions and what will be important for us is the following sort of dictionary that del phi to the s plus some lower order terms with some coefficients but those will not be so important this is equivalent to a bilinear of fermions with s minus one derivative sprinkled over this so the simplest case is when s equal to two the stress tensor del phi square of the boson is equivalent to psi bar del slash psi which is the stress tensor of the fermion theory but more generally there is a mapping which goes as follows and this will be crucial for us but before I really use that let me tell you how this identity this bosonization identity will be very helpful for us in reassembling this chiral algebra in a different way so for that we will take one of these building blocks these fermions and I'll write down another decomposition of the fermions which is very similar so let me look at one of these building blocks and so I claim that this is given by a very similar decomposition to this one so this is in fact yet another example of the Q binomial theorem but you don't even need all those fancy this thing just like I showed you a combinatorial proof for this we can very simply show a combinatorial proof for this that goes as follows again I'll sketch it out and if you have any problems ask me in the discussion session and I can flesh it out some more so basically what's the fox space of a single fermion we can assemble the fox space of a single fermion in terms of modes of these fermions acting on the vacuum where each of these MIs are greater than equal to zero but because they are fermionic oscillators we can actually regroup these or reorder these and write it in the following way we can order it in terms of the modes I can easily equally well label this sort of general state in terms of these it's very similar to what I did for the bosons except that here because they're fermions no two oscillators can have the same mode number so I've taken that into account by this one starts from minus half this is minus three half minus five half and so on and each of the keys are then greater than or equal to zero and these will all be distinct oscillators and a general oscillator state can be written in this form as you can with a little thought convince yourself and then if you see now each of these each of these has a r charge equal to one so it will have a factor of y to the n when you're counting its chemical potential and what is its dimension the minimal dimension is half plus three half plus five half all the way up to two n minus one by two that gives you n square by two and then all the different oscillators just give you this factor, combinatorial factor so basically the fact that you can organize your fox space of the fermions in this way tells you that this fermionic partition function written this way should be equal to this but there's a mathematical identity there's a physical proof of that mathematical identity but again if you want if the details are not completely obvious I'll discuss that later so in particular the bilinears so the... well okay let me just say that these okay so what we will use is so this was just one of the fermionic building blocks whereas a boson is built out of two fermions so there is this and this but they are very similar just y replaced by y inverse so if you are looking so this if I use this identity then it's very easy to see that this combined with this implies the following identity so basically it's very simple if you are looking at the y it's the neutral part because here this is a series with increasing powers of y and the corresponding one with y inverse will have y to the minus n so the only terms which will have y to the zero are when you multiply the y to the n term with the y to the minus n term and that's what I have done here so you just get the square of this so q to the n square by 2 becomes q to the n square and the denominator is also square so this is a simple consequence of these two identities and so this is another decomposition this is just a mathematical decomposition at this stage of the chiral sector of a single boson but I claim now that just like this was the character of a W infinity algebra with this n boxes and the trivial representation this was the character of this representation this is the character of representation of a W infinity algebra but a fermionic W infinity algebra because we are now looking in terms of the fermions so this is a decomposition of the fermionic W infinity algebra but with sort of something like this so what I mean by this something well it has dinkin labels n00n so there's sort of n fundamentals and n anti-fundamentals so it's a particular representation of a fermionic W infinity this is the character of such a representation and we can understand where this comes from because in fact the n equal to 1 term in this decomposition maybe I should have I mean it's just notation but just so that I don't I should call the dummy label but I'll just call it m because I'm using n for say the column so that may not sorry so for now if we consider the m equal to 1 term in the series that corresponds to the 1001 that's just something where you have 1 fermion so remember so here also we would have had m and so on so so this so remember that m was the number of fermions over here so the m equal to 1 case is the one where you have 1 fermion of psi and from here you get 1 fermion of the anti-fundamental kind psi bar so the m equal to 1 term corresponds to bilinear of the fermions so something like this psi bar del to the s minus 1 psi so in fact you can see the m equal to 1 term over here gives you has 1 quantity for every spin and these are precisely these fermionic bilinears which as which as I mentioned over here correspond to the top row of this of this table so so the bilinears of the fermions so this is a different way of organizing so this fermionization of the bosonic theory gives a different way of organizing the chiral algebra by what these mathematical identities really mean is that I can organize things now in terms of the number of fermions here this index m labels the number of fermions whereas in this way of organizing it here n label the number of bosons m equal to 2 were the bilinears etc but here in this way of organizing the number of we are doing it in terms of the number of fermion oscillators in the occupancy of the fermion oscillators and m equal to 1 corresponds to one fermion of this kind and one of its opposite kind and that is so that the fermion boson dictionary tells you corresponds to del phi s so it cuts across all these columns and is in fact just the top row the one with lowest dimension so so that's the that's the sort of picture that this decomposition you have so this is these generate the fermionic infinity or rather it's what is called w1 plus infinity because actually there's a spin 1 when s equal to 1 psi bar psi generates a spin 1 field as well so there is so this is what is called w1 plus infinity which in the in the general taxonomy of w infinity algebras is the one which corresponds to this parameter lambda equal to 0 so these generate these bilinears generate this and the other terms which are over here are representations of this fermionic w infinity and they transform into in this representation labeled by the stinking labels of this lambda equal to 0 w infinity algebras so what do we have over here this tells us that we have a different decomposition now of our chiral algebras so it's the same chiral algebras but we have decomposed it now in terms of so there is this which is the which we will call the w infinity the horizontal w infinity algebras and then there are other rows so this was m equal to 1 m equal to 2 and so on general m so this is another way of slicing the whole chiral sector in terms of the fermions so so we see this interesting sort of two alternative ways of slicing the chiral algebras both of which are organized both of which organize the elements in terms of representations either of the vertical or the horizontal w infinity algebras so this is the one we'll call the horizontal highest spin algebras or the vertical highest spin algebras the horizontal highest spin algebras horizontal w infinity algebras and each of these when you go along these rows those are w infinity the horizontal w infinity descendants when you move along each of these rows as I said yesterday moving along the columns is by the vertical w infinity algebra so these two algebras together sort of generate what we call this highest spin square because now knowing both these algebras and their commutators effectively allows you to compute commutators between arbitrary elements like I said let's say between here and here how is that because so these two w infinity algebras effectively generate the full unbroken stringy algebras string asymmetry of this background so we call this the highest spin square because these two are sort of orthogonal w infinity algebras and they generate this whole square but at the same time you should remember this notation when I say it's square it doesn't mean that the algebra is some tensor product of these two it's a slightly more unusual unusual structure which I don't know it's proper mathematical terminology or it's home in fact I've tried to ask a few mathematicians and I've not gotten anything very useful but if any of you know what kind of structure this is and in fact I'll make it a little bit more concrete in a baby example very soon an example which all of you would have seen so if there's a name for it please tell me but we will at the moment we go with our coined term of a higher spin square and so why is the fact that you have these two w infinity algebras how does that help you in getting the arbitrary commutators oh sorry yeah most sorry this is of course the number of fields here are much it's a stringy spectrum and the chiral algebra is also stringy in the sense that it has a cardy like growth so the number of so yeah so I wouldn't say that we are reformulating the Vasilyev theory it is we are trying to I would say it's a way of organizing the string theory in terms of Vasilyev like higher spin symmetries using and with the idea of eventually using the Vasilyev structure to maybe write down the because Vasilyev has solved the problem of how to construct interacting theories of an infinite tower of these and to some extent he has coupled some matter fields to that here we have a problem where we have the Vasilyev fields of course but we have infinitely many others transforming in some representation but in fact these are all gauge fields so it's much bigger than Vasilyev and string theory combines them into giving you a consistent description of all these so it's somehow larger than Vasilyev but the fact that there are these two so there's not just the original Vasilyev symmetry but there's another newer emergent Vasilyev symmetry when you organize in terms of the horizontal W infinity combining both these two I think so you should be able to I think the fact that there are these two different Vasilyev like symmetries combining should be the key to understanding the symmetry of the full structure because as I said Vasilyev tells you how to organize each of them so presumably you should be able to take that to understanding this too but that's something of course is for future work but at the moment I'm just uncovering a certain structure that is there in these in this symmetry algebra so as I said so this higher spin square structure earlier by as I said by organizing in terms of these representations it was easy to take the commutator of some arbitrary element with some element of this column and that was given by W infinity commutators because these were specific representations of this algebra but now we can now that we have this other W infinity the vertical the horizontal W infinity and each of these are also representations under this so this first row is the is the horizontal W infinity since we know both these if you want to take commutator between two arbitrary elements you can do one of two equivalent things you can so this is some descendant of some element here by the horizontal by the vertical W infinity so there's corresponding to this element this element comes lies in some particular column there is the highest weight state of that representation in that column that's related to this by action of some W infinity descendants so knowing if I know the commutator of this with this then I know the commutator of this with this because this is just given by some further descendants of the W infinity so it's equivalent to computing the commutator of this with this but that we now know because this will lie in some row and you can use the fact that this generates the horizontal W infinity so the commutator of this with this is given by a horizontal W infinity commutator or you could have done the other way around you could have taken this to this column this is related by some horizontal W infinity action to some element over here and then you compute the commutator of this with this in fact the closure of these two the operations also I think gives you some very strong constraints on the commutator the fact that both these operations should give you equivalent commutators I think gives a strong constraint on the nature of the full algebra so what I mean to say is that this higher spin square is a very tight structure it's a very tight structure which is governed completely by the horizontal W infinity and the vertical W infinity knowing just commutators of representations of these these two W infinities is enough to specify all the commutators of the algebra so okay so this structure at least was sort of unfamiliar to us so we thought a little bit and realized that there was something very similar to this in some ways very analogous to this which is which all of you can appreciate and this is what we call the Clifford algebra square so let me describe that since it's very neat so so as the name suggests we consider the Clifford algebra or gamma matrix algebra in 2k Euclidean dimensions so basically there are the gamma i's which go from i equal to 1 to 2k and they obey the usual the Euclidean gamma matrix algebra so so now you must have all studied when you studied Dirac equation and so on that you have the antisymmetric bilinears of these gamma matrices generate the algebra SO2n in fact that's why this is the spinner representation of SO2n so these are the generators SO2n right so so these are antisymmetrize so but of course these are not the only combinations you can construct is that a question oh sorry SO2k yeah at some point I switched so if I make this mistake again just to remind me yeah okay so so so these are the antisymmetric bilinears of course the symmetric one is just delta ij so that's trivial that's the identity matrix so but we can organize we can in fact consider the following combinations just gamma i then gamma i gamma j so we can consider all the antisymmetric combinations of these and we know that each of these I'll draw as a column because each of these forms an irrep of so the general gamma i1 this transforms in the antisymmetric representation of SO2k right these are they each of the so as you let the gamma i's range over their values the each of these forms a representation and the SO2k is the vertical algebra that acts on these so there's a highest rate state of this representation and then there's a whole tower and each of these has their appropriate dimensions so so you have so you can consider all these and if you wish you can even add the identity so each of these is decomposition of I mean each of these is an irrep of SO2k with this vertical action so this is what we will call the vertical algebra but because these are so again we have the similar thing where the adjoint of the n equal to 2 case is the SO2k itself generates the SO2k itself and all these are under some representation so I can take commutator of this with this and I know that it transforms in the appropriate representation but because these are gamma matrices they are formed from an underlying Clifford algebra I can take commutators across any two rows any two columns and in fact if I just denote the vector space of a particular column by Hm then schematically you can see that the commutator of the mth column with the nth column is sum and with some structure constants something like this there you get back things between so these there's a sort of a horizontal algebra between the different columns the mth column and the nth column give you some linear combination of columns which go from some r min to some r max which you can write in terms of m and n but that's not important and there are some numbers here but so this is in fact this algebra is sort of very similar to a W infinity fermionic algebra which in which the generators have very similar sort of the W infinity of spin s and spin s prime have have a similar sort of commutation relations but in any case so this is what we would call a horizontal algebra or a horizontal action and so you can so this whole space of these gamma matrices you can organize in terms of these you can slice it in terms of these vertical ones or equivalently with sort of in terms of SO2K acting vertically or in terms of sort of a horizontal algebra like this but the interesting point which I think is very makes it very close to this particular case is that you can ask what is the full algebra here of all these gamma matrices it's not SO2K of course but you know what the answer is because there's something again you would have learned when you studied the Dirac equation that the gamma matrices can be realized by the matrices of size 2 to the K because it's in 2K dimensions so 2 to the dimension by 2 so it's of size 2 to the K and you can by the sort of arguments that you learn and you do gamma matrix algebra they are all traceless you can make them Hermitian and so on so they in fact generate SU2 to the K so the full algebra here the full stringy symmetry algebra if you wish of the square is SU2 to the K or if you include the identity you can you have a plus identity but it's and you see this has the same feature that we saw in our stringy algebra namely the vertical algebra this had rank which was like K whereas this has exponentially larger rank right it's like 2 to the K so it's not some kind of tensor product or anything like that it's exponentially larger in size and this is what we saw in the higher spin square as well that each of these vertical or horizontal algebras had essentially one one generator at each spin but the full thing had a hagedon sorry a cardy kind of growth of states exponential in the number of in the spin and so this is this enhancement here is very similar and so in many ways this sort of I think it's it's a reasonable analog of this and at least give some sort of intuition for this structure but but this is not the case here I must say it's not a clifford algebra so it's not it's not quite the same but not some infinite dimensional version of this so what exactly the relation of the two is I don't know but it seems to have a similar mathematical nature to this and as I said I don't really know what the terminology for this should be so that's so now so this highest yeah so that big group we don't know what to call it I mean it's as of now it's some highest spin square you can if you wish call it W2 to the infinity or something I don't know if that's a name it's not because that would have had generators which had just the product of the two whereas this is exponentially larger so it's generated in a in a more I mean you see it's again that's why I'm not sure what is the exact structure because these are representations of the original one and but the representations themselves form elements of an algebra so each of there are things transforming in these yeah so I would very much I think it would very much help to understand this mathematical structure or at least place it in some familiar context to be able to to to get more leave a the properties of this so okay so that's sort of more or less where we are at the moment so I'll just conclude by with some remarks and so this this higher spin structure so we so I remember I started off with the question of is there some kind of larger gate symmetry and and so we have some we see that these ADS3 vacuar do have some kind of exponentially larger unbroken gate symmetry than the Veselyev ones than in higher dimensional ADS where only the Veselyev gate symmetry and so so clearly this should be powerful if this is the unbroken gate symmetry it should we should be able so you can ask what can we do with this higher spin square so there are many possibilities so this is all just things that we are in the midst of doing and understanding so no I won't give you any complete statements but the first thing is move away from lambda equal to zero in a controlled way so remember this was the motivation with which I started off these lectures that we want to sort of move away from say lambda equal to zero and that was the motivation was this symmetry that will enable us to sort of do that in a controlled way so what can we say about that question in the light of everything that I've said right now well there you can make a few statements firstly so in this symmetric product case where we can ask how are you going to move away from lambda there are many marginal deformations in this CFT in fact there's a 20 parameter family of them but many of them 16 of them are very boring they are just the shape of the T4 etc there are four more interesting ones but in fact one of them is really the one which you would think of as really going away from which would I think be justifiably called the lambda and this is the so called Z2 twisted sector marginal operator which is a singlet under the r symmetry so this is the thing that blows up there's in the Z2 sector within any within the symmetric group of course there's always a Z2 in the symmetric group of any n elements so there's a you can consider Z2 which sort of permutes two elements and there's a twisted sector mode in the twisted sector of that in the Z2 twisted sector there's a lowest dimension operator which is a marginal operator of the CFT which corresponds to what you would call as the blow up mode of that Z2 the desingularization of that Z2 and this is the thing that is most interest in as the deformation parameter and while I didn't describe this in these lectures I mentioned at the beginning that it was not just the chiral sector of the symmetric or before but the full spectrum of the full partition function the full spectrum of the symmetric product that could be organized in terms of representations of the higher spin algebra so this Z2 twisted sector itself is transforms in a very definite way in a definite representation of the this vertical W infinity that we described this one so though that of course is a it's not in the chiral sector so it doesn't appear in this square it's a matter field but which transforms also in a specific representation with some lambda plus and lambda minus but now both of them are non-zero unlike these examples so it transforms in a definite representation of this so that means that means that when you turn on this marginal operator you're giving a wave to a field which is charged so from the point of view of the bulk theory the operator the field that couples to this marginal operator of the CFT is a field which is also because this one is in non-trivial representation of W infinity the corresponding field is also charged under the Vesselier vertical W infinity higher spin and therefore also the higher spin square under this bigger algebra and so you're giving a wave to a field which is charged under this symmetry that you're breaking that is a classic thing that you do in a Higgs mechanism you're giving you have a large unbroken gauge symmetry now you turn on a wave for a particular operator or a field which is charged under a specific representation of that so this is what you call Higgsing so you're Higgsing this Vesselier symmetry so you're Higgsing the W infinity or the higher spin square so it's not some uncontrolled breaking of the symmetry it is a very definite pattern of the symmetry breaking in terms of a particular representation and so you might think that you might be able to move away from it in a very controlled way in that the symmetry should still govern the matrix elements in the broken phase so just like in the standard model when you Higgs the SU2 times U1 you can still use the underlying water identities of the gauge symmetries the SU2 cross U1 gauge symmetry water identities still hold even though in the broken phase and that strongly constraints the structure of the theory so so you should be so of course these are fond hopes should be able to to calculate or to control the matrix elements the perturbed theory if you wish it's like the Wigner Eckart philosophy that you break rotational symmetry you turn on something in a particular representation nevertheless the rotation symmetry governs the matrix elements because it relates matrix elements of different quantities all in the same representation of this bigger symmetry so you should be able to use the fact that you are doing this Higgs and use the sort of Wigner Eckart philosophy to to control moving away from this so that's one thing one can hope to do with this and that would be a realization of the original idea of the question that motivated this talk and another thing I would like to understand is understand the representations of the higher spin square itself the full higher spin square we understand how to group the spectrum in terms of representations of the W infinity but now that the chiral algebra is much larger that the higher spin symmetry is the sort of governing symmetry we should be able to understand the representations of this and the full spectrum should be organized in terms of representations of this rather than just the higher spin representations so many higher spin representations will combine to form one representation of the bigger symmetry algebra because this is after all a small piece of this big square so it's like over here we had under the SU2 to the K there will be many things which if you have things now which are representations of SO2K which secretly are also representations of this then of course many many representations of SO2K will have to combine to form representations of SU2 to the K so so this is in fact related to this the power of moving away from this will be so we understand it's in a definite representation of W infinity but that will mean that it's also in a definite representation of this bigger one so that will give even more sort of group theoretic sort of constraints on the different states so we would be able to group different W infinity representations into one bigger representation but for that you have to understand a bit more about the representation theory of this higher spin square we understand representations of W infinity but as of now we don't have any clue about the representations of this bigger bigger object but but one sort of preliminary result which we are trying to completely form up is that but so preliminary is that the entire untwisted sector so of the symmetric product seems to be one representation and it's multi-particles of the higher spin square of this bigger thing it seems to be one in fact in some ways the simplest non-trivial representation we can find the analog of a fundamental representation so the entire untwisted sector seems to be one quantity so so and then it's multi-particles no binding on T4 you mean no no no so the T4 we are taking no charges under the T4 so when I say untwisted sector of the just of the I mean the T4 theory yeah so in a way this is very similar to the Veselyev description in the cosets coset models where all the non-trivial primaries of the coset could be at least the so called perturbative sector of the primaries of the coset which was an infinite set of coset representations was one scalar field and it's multi-particle states of the of the of the Veselyev theory this is a bit analogous to that except now instead of the Veselyev theory we have the full higher spin square so I think this is if this is indeed right that's a sort of illustration of how powerful this can be how nicely it can organize the things of course the interesting thing is to ask about the twisted sector can we organize the twisted sector in to say a specific set of representations very definite representations then I think you can use it to make a statement even more powerful than we have over here because it would be some definite representation not just of the W infinity but of the full high spin square but that we still haven't made much progress so I will leave over there, thanks