 So, a few years ago when I lectured in the spring school, I started off with drawing a certain diagram of the parameter space of ADSCFT dualities, if you wish, a very schematic parameter sketch in which on the vertical axis you have something which I'll call 1 over n, being the number of degrees of freedom, the rank of the gauge group and the gauge string dualities, and on the horizontal axis something which you can call lambda, the Toft coupling. So, I'm putting these in quotation marks because in different theories the definitions and the exact parameters will be slightly different, but in the canonical example which probably most of you are familiar with of n equal to 4 super angles, this is the same 1 over n that we are talking about. So, of course there's a dictionary which relates this to, these are gauge theory quantities, the coupling and the rank of the gauge group, but there's a dictionary which relates it to something like the geometric parameters on the ADS side, so r refers to the radius of the ADS in string units, so r square by alpha prime is dimensionless, alpha prime is the usual string length square, so this lambda is in fact in the canonical ADS example, it is actually related to r to the 4, but roughly speaking there's a relation something like this, and this 1 over n is roughly speaking like the string coupling, that's the dictionary on the vertical axis. So, in this dictionary we mostly talk, I mean in this parameter space we mostly talk about the region where n is close to infinity, so that's what I'm drawing over here, this is sort of the, if you wish in this language, this is the classical regime, because you're looking at the regime sort of in very large, but in fact most work focuses on a corner somewhere over here for lambda very large, so this is what you would call the gravity regime for a very strong coupling from the gauge theory point of view, and the regime where the radius is very large in string units so you can apply to derivative Einstein gravity, so most work in the ADS CFD is roughly in this region, and so classical gravity, classical because G string or G Newton is essentially zero, so that's the familiar region, and of course you can do a systematic expansion, you can include higher derivative corrections, so that's why I've sort of drawn a blob over here, so if you wish that lambda goes to infinity, but you can do some corrections which are in inverse powers of lambda, but there's another regime here, which is also under very good control from the field theory point of view, and this is the perturbative regime, this QFT or CFD regime where your coupling is very weak, and so you use conventional textbook quantum field theory, of course the perturbative regime extends all over here, but we won't be looking at that so much, so these are two regimes which you know how to do computations in, but most of this parameter space is still a mystery, and in fact in this central region is sort of the most mysterious part, if you wish it as where you need quantum string theory, this part is at least the good thing about this part is that it's at least classical string theory, you don't need to know too much about quantum string effects or details of higher genus string perturbation theory, so given that you know these two regions, I mean when people try to test ADS CFD or to make some headway in understanding it better, what do you do given that you know only sort of these small little corners, well first people do sort of two things, where people started off by looking first at quantities which are not renormalized, so non-renormalized quantities of course are very nice because the same at weak and strong coupling, and so you don't have to, you can compute it at weak coupling and non-renormalization theorems tell you that it's the same at strong coupling and therefore you test it, test by a gravity computation and see whether that is indeed true and there are many examples of such quantities including certain chiral spectrum, some of the three-point functions etc. So these are the simplest set of things in which you can sort of go from here to here without much and test the ADS CFD. So many of the initial tests were like this. Of course once you've done all that you've exhausted the class of non-renormalized theorem, non-renormalized quantities, of course you still want to do things. So what do you do? Well there's a more clever exploitation of supersymmetry. So of course I didn't emphasize but all of you are aware that this is something that you see in supersymmetric theories. So this class of properties are very special to supersymmetric theories but in supersymmetric theories you can do sort of a more clever thing and which is you can, many quantities which are in some sense protected but they are renormalized. There are things which you can compute using localization techniques which allow you to reduce the path integral of certain quantities to sort of finite dimensional integrals. And so the examples are the Wilson loop, circular Wilson loop, a half BPS Wilson loop, the partition function of ABJM theories etc. So there are localization computations which give you exact answers for quantities which you can compute for any value of the coupling and in fact they have quite non-trivial dependence on the coupling if you have seen any of the expressions for the Wilson loop or these ABJM functions and maybe I don't know if Marcos Marino might say something about the latter part and so if you've seen some of these expressions that quite non-trivial functions of the coupling and it is quite miraculous that you can reproduce some of these quantities, these calculations from the gravity side by extrapolating to large values of the coupling. So that's a second set of computations you can do but once again it exploits very much the properties of supersymmetry which is what allows you to localize the infinite dimensional path integral to finite one. There's a somewhat different development which people have also used to go from moving this parameter space which is integrability. So integrability of the world sheet theory of the dual string theory are equivalently of the better spin chain of the gauge theory or the spin chain description of the gauge theory. So there's a somewhat unexpected integrability in fact its origin is still not completely well understood at least from the field theory point of view from the world sheet point of view you can sort of understand it in the sense that the sigma models on anti-disseter spaces have a classical and a quantum integrability but the two notions of integrability have not been completely I think matched with each other. In any case there appears to be a certain hidden integrability at least in the planar sector which is what we are mostly interested in and the n goes to infinity limit and this gives you in principle you can exploit the integrability and many of the powerful techniques of so this is 2D integrability of both the underlying world sheet theory or the spin chain picture. You can exploit the 2D integrability to again compare so you can compare weak and strong coupling for instance of the spectrum and this has been very successful in the case of the N equal to 4 Young-Mills theory and some other similarly highly supersymmetric theories so you can again have a non-trivial dependence so now we are talking not of just quantities which are not renormalized arbitrary operators in the theory whose anomalous dimension may be a very non-trivial function of the coupling which is not necessarily given by localization computations but by more sophisticated ones and that allows you to compare again successfully weak and strong coupling so this is the way in which these are sort of roughly speaking three categories in which people have compared weak and strong coupling how you have tried to move in the parameter space between these corners that one knows how to understand but one unfortunate aspect if you wish of these is that these seem to be restricted to special gauge theories special cases of course these involve supersymmetry and even here we seem to understand it best for the antideciter case and it does seem to some extent that it's tied to supersymmetry so these seem to be quite restricted in the applicability so if you believe gauge string duality is very general and applies to strongly coupled gauge theories very generally then these three approaches while very powerful and able to give you a lot of insight seem restrictive if you for general applicability so you can ask the question could there be a more general symmetry which governs the sort of theory as one moves away from weak or strong coupling which governs the movement from weak to strong coupling so this is the question I would like to ask whether there's a sort of a more general symmetry which controls motion in lambda so I'm not going to be able to give you a definitive answer to this question but it's I think a good question to ask and let me tell you why this may not be a stupid question or why there's a chance I mean this of course is an interesting question if the answer is yes I mean of course if the answer is no it's not an interesting question so why is there a chance that the answer may be yes well so answer may be yes because well I'll try to tell you a few reasons why answer may be yes because in some ways one very general reason I'll start with is string theory is unique the uniqueness of string theory string perturbation expansion is very highly constrained there is there are few free parameters few things you can adjust there's the string coupling which of course is also the web of appeal and these are some kind of these are parameters of your of the string vacuum and the string expansion which is what this is this expansion is so I should have said this is the classical string theory I probably said it yeah well I said this is classical gravity but this whole regime is of course classical string alpha prime expansion so it's a world sheet expansion and the structure of the string interactions on the world sheet is very is quite unique and so yeah and whenever there's a unique structure in physics we are always we have seen by experience that it's often a symmetry that that that determines that uniqueness so so in fact it has been a speculation that string theory that this well maybe this is really a part of this is an elaboration of this which is that there's a notion that there's a large unbroken there's a large symmetry in string theory which is mostly broken so there's a large sort of a when I say symmetry I should really say gauge symmetry in string theory which is mostly broken or Higgs and in terms of which you see the usual phases of string theory but there might be phases where this symmetry is partially restored partially or perhaps fully but let me be conservative and just say partially so so what do I mean by this so of course when you quantize string theory in flat space as you study in your textbook you see that the film of ism in space time and even other gauge symmetries in space time like young mill symmetries are manifest so those gauge invariance is come about come arise in a manifest way from the usual the usual quantization of string but there isn't there isn't much more that you see so what is the reason why people believe that there might be a large gauge symmetry well there are a couple of reasons to believe this there were there were some reasons from the early days of string theory people believed that this might be the case and the reasons were roughly as follows one is that well I just I won't elaborate on these reasons just sort of bookmark them for you to go and read upon them if you're interested which is that open string field theory which is sort of a more non-perturbative formulation at least of open string theory open string field theory is on sort of a more sound footing than close string field theory so open string field theory at least exhibits in its usual in its sort of most natural formulation in terms of sort of a non-commutative star product it exhibits a large gauge invariance in fact open string field theory in its simplest form in its bosonic form is some kind of a non-commutative version of a non-commutative stringy version of a Chen-Simons theory in which there's a sort of a gauge invariance in terms of a star product so there's a sort of an open string field if you wish who's who's with a gauge invariance which is like your normal non-Abelian gauge invariance except that the role of the D operator is replaced by a BRST operator and commutator is with respect to the non-commutative star product of open string field theory so there's some natural formulation even of bosonic string field theory has some very large gauge invariance so this is a string field so it has all the stringy modes and this is another sort of a string field it's a string field like valued sort of object so it has all the oscillators of string theory so there's a very large gauge invariance a sort of a string gauge invariance and while closed string field theory which is what we are mainly interested in our closed strings there's no such a neat formulation but it is believed I mean you can sort of see similar sort of gauge invariance is again with the BRST operator playing the role of D so but at least in these cases you see and this I want to emphasize is something you see even in the bosonic case in fact it's simplest to see in the bosonic case so this more general symmetry which we are hopefully after is something we would like to see in both bosonic supersymmetric cases etc so this is one hint in that direction another hint was again hints of an enlarged symmetry in high energy scattering in flat space so if you look at the scattering amplitudes in flat space of some highly excited string modes in the very when the central mass energy is very large it is possible to do some kind of saddle point analysis and estimate the scattering amplitudes and what one finds is that and this was pointed out first by gross and later by various and elaborated on by various others like Witten and Sanyoti and so on that there seems to be in these high energy limit the scattering amplitudes there are sort of simple relations between them so some kind of word identities if you wish what might be interpreted as some kind of word identities which relate the string scattering amplitudes at different oscillator levels so there are some simple relations between different amplitudes and this was taken as sort of a hint that there is that these relations are some kind of word identities for a larger symmetry so these were all some hints that were there in the 1980s itself but these are sort of schematic and not I think very precise not so precise but at least it goes in the direction of sort of suggesting that there might be some large structure that leads to the uniqueness of string theory but coming back to our original ADS CFT since the discovery of ADS CFT there's been more concrete hints that there might be a more general symmetry there was an important thing I should have stated over here which is that this high energy scattering is effectively like some tension less limit because what do I mean by high energy scattering energy there's the only dimensionless scale that you can construct in a flat space scattering amplitude is alpha prime times energy square and so if you take its inverse that goes to zero that's a way of saying that so that's the limit that you are really interested in the limit where this parameter is going to zero so that's what you would call tension less because effectively with respect to the energy scales you're doing the scattering the string tension is irrelevant because it's very small so you can think of this as some kind of tension less limit so gate string dualities so the second reason why this larger symmetry seems to appear is in gate string dualities in the tension less limit so the nice thing about string theories in anti-disseter space as opposed to string theories in flat space is that you can take this tension less limit in a more uniform way here this is not a limit of the theory it is a limit of certain amplitudes in the theory you're focusing on a class of amplitudes and then looking at them that's not the same as taking a smooth uniform limit of the theory but in ADS there is a natural sense in which you can take the string theories in ADS you can take the tension less limit and so that's why you have this parameter which is the radius of ADS in alpha prime units which I talked about over there and so you can take this to zero this is a genuine dimension less parameter of the theory itself because when you specify the string vacuum of ADS you specify its radius and this is the dimension less parameter that characterizes the theory which in fact controls the alpha prime expansion so this parameter you can take it to zero that's a uniform way of taking tension less limit as opposed to the flat space case so then you can try to see whether this hints that were there over here are realized in this limit and you find in fact that they are because by this dictionary we saw that this parameter r square over alpha prime is roughly like the coupling so these correspond to free gauge theories and so this is the limit from the point of view of the gauge theory leads to just a free theory a free gauge theory is not as trivial as it sounds because it still has the Gauss law constrained it's a non-trivial theory in fact it's a dual to a string theory but so this is quite an interesting point in the parameter space of your ADS CFD dualities and you indeed see that free theories it's been known for a long time that free theories these have a large number of conserved currents the typical example in fact they have a large number of conserved currents of spin higher and higher spin of spin greater than two so of course every local field theory relativistic local field theory has a spin two stress energy tensor which is conserved and they may have some spin one flavor currents as well but typically you don't have conserved currents of spin greater than two and in fact there's a no-go theorem for interacting quantum field theories and d greater than two such existence of such currents but the free theory being free you can write down large numbers of conserved currents maybe you have seen some of these before so I'll anyway be schematic so you have for instance just for concreteness I will take a gauge theory which has some adjoin scalars like n equal to four young mills adjoin scalars phi but this is not to be confused with the string field phi maybe I'll call it psi and then various derivatives acting on the left and the right so you can take a single trace operator which is a bilinear in the basic fields it could be the scalars or fermions or even the gauge fields you can construct bilinears with various numbers of derivatives if these are scalars then s derivatives gives in a symmetric so this is again short form for something in which I would have some del mu one del mu k acting on phi on psi and then del mu k plus one del mu s acting on psi and sprinkled in various orders so with some coefficients so it's some linear combination of all these linear combination of such things that's what I'm just abbreviating and these are symmetrized so symmetrized overall indices so these are symmetric and you can choose them to be traceless tensors of rank s so they generalize the stress tensor the stress tensor you would write in terms of some similar bilinear in the free theory similar bilinear but with only two derivatives when acting on the scalars but you can write down more general ones like this with s indices and these are exactly conserved using the equations of motion which are in this case just free the equations of motion for one of these scalars is just del square psi equal to zero and the indices are such that when you contract it you will always get some factor of del square psi so it will be trivially conserved so these are conserved but they use crucially the fact that these are free theories and the equation of motion is just simple so these free theories have a large number of conserved currents and therefore so what does this mean from the point of view of the bulk theory from the point of view of the bulk theory by the gauge bulk boundary dictionary of ADS CFT conserved currents are operators of the gauge theory they couple to sources bulk sources which have to have a gauge invariance for this current to be conserved I mean it's the usual argument that a field that couples to a conserved current for the equations of motion whatever to be consistent the field to which it couples or the source to which it couples has to have a gauge invariance so if you wish there has to be a source there has to be a gauge field the source so these are the conserved currents which are boundary operators they couple to bulk sources and these sources have to have a gauge invariance so the sources will also have to be spin S objects and they'll have at least at the linearized level gauge invariance like this and this is what when you integrate by parts this there lacks on the J and ensures that the current conservation is satisfied so this is so what you conclude is that there are bulk sources or bulk fields so these are bulk fields which have this kind of gauge invariance and in fact they have to be gauge invariances for fields of spin S with S greater than 2 because here you can have anything S equal to 2, 3, whatever so in fact so what this means is that the bulk string theory must have a sector of massless fields massless gauge fields of spin S greater than or equal to 2 of course S equal to 2 we always know that string theory has as a graviton and that corresponds to the stress tensor but all the other gauge fields so massless is I put in quotation marks but you can say the definition of massless is that they have gauge fields so in ADS of course it's a little ambiguous what you mean by mass but there's a value of the mass parameter at which they have a gauge invariance so what you conclude is that it must have at least a sector of course these are only typically these are only a sector of the operators, single trace operators of your gauge theory and it's only the fields dual to that sector which are massless there are many other currents and many other operators in the theory which will not be conserved and so they will have they will be dual to massive fields but there must be at least a sector corresponding to this and in fact this is a closed sub sector you can show in the free theory that this sector closes its OPE closes amongst itself so this is a closed sub sector and indeed quite miraculously this sub sector there's a very natural description of this sub sector from the point of view of the bulk which was independent of string theory which is the which is the description by Wesselier so Wesselier over a heroic period of 15 years constructed the dynamics of such a sub sector well the classical dynamics but then again we are only looking at classical theories of such a sub sector of interacting gauge fields of spin S greater than equal to 2 in ADS D plus 1 so in general for that matter he showed that you can construct he constructed the full non-linear equations of motion which have gauge invariance so which have a combined gauge invariance for all the fields which is a non-linear generalization of this linearized gauge invariance and it was quite a remarkable thing that such such system exists and he constructed them constructed these equations independent of any string theory or gauge gravity dualities or anything like that just purely from the consistency and that's actually the lesson there because it shows how the gauge invariance can be sometimes powerful enough to constrain the form of a theory so Wesselier just used the gauge invariance and found essentially unique there are a few parameters but essentially unique equations of motion so the gauge invariance completely fixed the structure of the theory this is what you would like to see in the full string theory itself this is the sort of philosophy over here whether there's a large gauge symmetry which uniquely fixes the structure of string theory Wesselier's example is a sort of a baby version of that it tells you that at least for an infinite tower of fields namely all these massless gauge fields in ADS for spin greater than equal to 2 there is a essentially unique classical theory which is fixed by the gauge invariance so this is sort of a positive fruition encouraging sign that maybe looking for the stringy symmetry might help in in fixing the stringy symmetry might be might be the thing that's behind the uniqueness of string theory in any case coming back to our original tension less limit what we see is that this tension less limit indeed there is a so since the logic of gauge string duality so let me review the logic again the logic was the tension less limit corresponds to the free theory the free theory has all these conserved currents the conserved currents gauge fields which are massless and have a gauge invariance that forms a sub-sector so this sub-sector therefore must have a there must be a close dynamics for this sub-sector in the bulk and indeed Vasilev has constructed such a sub-sector so that was the logic which tells you that there is a enlarge so what you learn from this logic is that it's a large unbroken so at least a larger unbroken gauge invariance for tension less ADS theories so larger than at least flat space in flat space there were hints but as I said was sort of more schematic but here the gauge gravity duality at least gives you much stronger logic that there's a larger unbroken gauge invariance for for these string theories then this is the Vasilev gauge invariance so there is at least this much which is unbroken in a particular point of the parameter space namely this corner this small lambda equal to zero or at least precisely at lambda equal to zero and there's this larger unbroken gauge invariance and so that's very good it tells you in somewhat counterintuitively that you see ADS manifests more of the unbroken gauge symmetry than string theories in flat space you might have thought flat space is where somehow that's the sort of most nicest if you wish vacuum of string theory but some ways it's the other way around ADS seems to be a nicer vacuum of string theory in that at least from the point of view of its gauge invariance it seems to manifest more than flat space does so that's good already that we are seeing a bigger gauge invariance so at least maybe there's a partial restoration at this point of the string theory but I must emphasize something that this logic is indirect and that this has not been obtained through the quantization of string theory in ADS spaces which we still have yet to directly do but it is through the logic of gauge string duality and the bulk construction of Wesselier which is independent of string theory that we are arriving at this conclusion but I think we can be reasonably confident of it since we are since we believe gauge string duality I mean and and there is this explicit construction so okay but but let's then see this larger unbroken gauge invariance get a better sense for it so the first thing is that the number of massless gauge fields is only is fixed at any given spin so in the simplest case where you have some one scalar or something there's essentially one but if you have say an equal to four Yang-Mills there'll be some 10 or 16 or whatever number of currents fixed at a given spin but that is independent of the spin it doesn't grow or decrease just remains the same for all spin so that's one point note and in fact that's a very that's in some sense very good in a sense because it tells you that this can be associated with the leading rege trajectory of the string theory so the leading rege trajectory remember if you have seen in some of the historical books on string theory or the historical chapters in the beginning of all string theory books there were things called the rege trajectories from which of mezzons from which people guessed string theory but basically it is the lowest mass states for a given spin so the leading rege trajectory is those set of oscillator states or stringy excitations which are the lowest mass for a given spin that's the operational definition of a leading rege trajectory and so you see that these for a given spin there's one or some finite number of them which are part of the leading trajectory multiple it so it's one trajectory is becoming massless but there are string theory if you've seen these diagrams there are all these different rege trajectories and so this spin is on the on the vertical axis and mass square so there are so this is the leading rege trajectory and then there are all these others and so that's there here as well it's only that this leading rege trajectory is becoming massless but there are many other infinitely many more rege trajectories which in this case correspond to all the other operators in the same equal to four young mills theory or whatever the gauge theory you're looking at which are not conserved currents and therefore don't have protected dimensions and whose dimensions are greater than minimum bound in fact you can ask whether whether I wrote down for you these set of bilinears and it's easy to write down the specific form and check using the equations that they're conserved but you can ask are there could there are there more I mean I just showed you these but could there be more in fact there are unitarity bounds in a CFT which tell you that the dimension of the conserved current must be basically the spin plus d minus 2 in a CFT in d dimensions the the dimension of the so the currents by being conserved have null states in their representation because del of j is 0 because del j is 0 that means in that amongst the descendants there are states which are null so that null states you can use to show that the dimensions must saturate a certain bound and that bound goes roughly like this delta is s plus p which is d dependent and this in d equal to in d equal let me write it in d equal to 2, d equal to 4 is the statement that delta is equal to s plus 2 and that's so in d equal to 4 delta is s plus 2 so this quantity delta minus s which is the twist is equal to 2 so these are in fact these operators are examples of such twist operators in d equal to 4 a scalar field has dimension 1 and there are derivatives but there are s derivatives so the delta is s plus the dimension of these two scalar fields which is which is 2 so delta is s plus 2 that is why so these are an example of the so these so called twist operators which saturate that bound and more generally you see that this delta is s plus 2 times d minus 2 by 2 and so it's this bound is saturated by so these s come from the derivatives and and this from two fields the fundamental fields so this is a free theory and so you know the dimensions of all the in the free theory there are all the canonical dimensions and so you can see that you can get this bound only by only by so for the scalars you can get it only from two bilinears from a bilinear of the scalars for fermions it's just slightly different there will be s minus 1 derivatives and the bilinears of the fermions so on so this unitarity bound in the CFT tells you that conserved currents these are the only conserved currents they can only be formed from bilinears of the fields so this is the complete set of conserved currents that you can have in a theory at least for d greater than 2 you can only have so the conclusion is that for d greater than 2 only bilinears of the basic fields of the fundamental fields of the theory so two gauge fields or two fermions or two scalars so that's all that you can have in the there are no other conserved currents but this is special to d this is this conclusion you can make when d is greater than 2 when this is not equal to 0 and this d minus 2 by 2 is not equal to 0 that's why we could I could write it like this in fact in 2d this is evaded so in 2d many more conserved currents but before I go to 2d let me just emphasize again so this was a conclusion from the field theory this was a conclusion from the field theory that the only conserved currents are the ones that I showed you which are bilinears and they correspond in the bulk to these vasiliy of gauge fields of which there are a fixed number at any given spin and plausibly are the leading trajectory so the fact that these are the only ones tells you that free theories in adsd plus 1 sorry free theories in d dimension which are dual to tension less string theories in adsd plus 1 at least for d greater than 2 they only the maximally restored gauge symmetry is the vasiliy of symmetry okay so you can't have anything more than the vasiliy of symmetry for d greater than so it is great that more than flat space we do see an enlarged gauge symmetry so we do see larger unbroken gauge invariance but this is the extent of the unbroken gauge invariance that you can have you can't have any more and the free theory in d greater than 2 is the place where you can have the maximal number of conserved currents because of the no-go theorems and theories in d greater than 2 and in the free theory unitarity bounds tell you that these are the only ones so we have a watertight case that in at least in adsd plus 1 for d greater than 2 the maximally restored gauge invariance is the vasiliy of gauge invariance but the situation is different in two dimensions in two dimensions you can evade this unitarity bound and not just have bilinears because this factor here becomes 0 in d equal to 2 and indeed it has been known that you can have many more conserved currents than from bilinears in a free theory and we'll construct examples of this for example so we can have many more conserved currents in fact in 2d there is more that is true but which we won't be using here which is that in fact can have conserved currents of s greater than equal to of s greater than 2 in interacting theories as well so the no-go theorems the sort of Coleman Mandela like theorems are evaded in two dimensions so both two-dimensional massive theories as well as conformal field theories there are interacting examples where there are conserved currents of spin s greater than 2 so but we won't be using this we won't be sticking to the free theories but I just wanted to alert you to the fact that in two dimensions even that is different and in fact in some ways this is sort of related to that of local degrees of freedom in the bulk are you saying in the bulk in the bulk yes so the corresponding spin s fields don't have local degrees of freedom yes that is so which you're asking about this this statement yes in ADS 3 the fields dual to these currents don't have local degrees of freedom yes so it is related yes yeah so so that's if you wish reflection of of in the bulk of this because the degrees of freedom of the are only boundary degrees of freedom which correspond to these conserved currents and so so in D equal to 2 so why can you have many more conserved currents in a free theory and because the unitarity bound is just delta equal to s and in two-dimensional CFTs which is what we will consider in two-dimensional CFTs this is just the statement that h equal to s or h bar equal to 0 h plus h bar is delta and h minus h bar is s so if you wish I should really put this as equal to mod s or s let me define s is equal to mod of h minus h bar so this means h equal to s and h bar equal to 0 or h bar equal to s and h equal to 0 so these are two-dimensional CFTs and as you know any holomorphic current is conserved del bar of that is 0 so you can have many holomorphic quantities in 2d CFT which are not just bilinear of your fields you can have a large number of such holomorphic quantities and we will write examples of them so we will see things which are not just bilinear so the hope will be that therefore ADS-3 string theories so I will just stop in a minute I guess I started maybe 5 minutes later so ADS-3 string theories may have an even larger unbroken gauge symmetry ADS-3 string vacua if you wish may have an even larger unbroken gauge symmetry and and and that's something we'll in fact see is indeed the case so ADS vacua in general as I said are nice nicer than flat space in that they have larger unbroken gauge invariance but ADS-3 vacua seem to be even potentially nicer from this point of view because they have potentially many more conserved currents and we'll see that they have exponentially larger number of conserved currents in specific examples that we understand we'll see they have exponentially larger number of conserved currents and therefore massless gauge fields in the bulk compared to the higher dimensional cases so so that's something very nice about ADS-3 which I would like to emphasize which I think has not been fully appreciated so amongst the ADS vacua ADS-3 is nicer ADS-3 is also nice is also special ADS-3 vacua are also special because of another somewhat at least seemingly unrelated reason because of the so called brown Heno enhancement enhancement of global asymptotic global symmetries so they have both this a large unbroken and may have I haven't showed them to you but I will show it to you probably in the next lecture but they also have another feature which perhaps many of you have seen which is this brown Heno enhancement which is a priori at least I don't know of a clear relation between the two but maybe they are not unrelated in any case so this is the enhancement that any that is firstly true for one second but the theory of gravity in ADS-3 will at least have the ADS-3 isometries so SL-2R 2 copies of SL-2R and what brown Heno showed was that this is actually enhanced to verisoral in that the even though there are you might have expected the states to be in representations only of SL-2R the global conformal group they are actually in representations of verisoral and this has been generalized to theories with higher spin which typically can be based on SL-nR or what is called HS-lambda more generally where lambda equal to n corresponds to SL-nR but lambda can be a more general parameter so in these cases this is enhanced to a W-n or HS-lambda more generally to what is called the W-infinity of lambda so there is an enhancement of the asymptotic symmetry algebra to make it even bigger so that there is even more powerful states are in representations of even bigger symmetry groups these are actually infinite dimensional SL-2R is of course finite dimensional SL-nR is finite dimensional HS-lambda is actually infinite dimensional the algebra but these things on the right are anyway infinite dimensional even for SL-2R and SL-nR but they are bigger infinities if you wish so there is this other feature of ADS-3 which is also very nice so we will see that these two features so what we will see in the rest of the lectures is how these features interplay and how we will see a concrete example in two dimensions of an ADS-3 vacuum and a dual 2D CFT in which you have many more conserved currents which will give rise to a much bigger symmetry but which we will be able to organize in a useful way in terms of the vacillia of symmetry and in fact two different vacillia of symmetries will in a sense generate two orthogonal vacillia of symmetries will in a sense generate this whole stringy unbroken stringy symmetry which will make it exponentially larger and potentially therefore more powerful and perhaps will go towards the goal of constraining the string theory in these space times so at least it tells you that there is a much larger symmetry that can be restored at least in a class of string vacuums but that we will wait the afternoon to start on that so thanks and sorry for going over time