 Please. OK, well, can you hear me? OK, so thanks very much for the invitation. It's a great pleasure to be back in Trieste. And what I want to do is I want to give you some review of what's been happening in our understanding of string theory in ADS-3. And since not all of you may, so I'll try. I mean, I hope you'll stop me if I say things that you don't understand. It's a bit difficult for me to judge your background. So please slow me down and ask me more specifics if you need more details. I'll try to be pedagogical and don't assume that you know all the ins and outs of 2D CFT. And let's see how that goes. Anyway, I want to start by explaining you the motivation and then motivation. And then I'll finish the plan of the lectures once I've explained to you what this is all about. So the basic question is, so we all know about the ADS-CFT duality. So let's review on a sort of a 30,000 foot perspective how the ADS-CFT duality works. And let's concentrate on the most familiar case, which is a super strings on ADS-5 process 5 being due to n equals to 4 super angles. This is not ADS-3, but you'll get to ADS-3 in a second. So n equals to 4 super angles, what you have in mind is that you take the SUI engage group and then you have to understand what's the relation between the parameters that characterize string theory on this background and the parameters that appear on the young mill side. So it's strings on ADS-3. So it's strings on ADS-5. So the basic idea is that you have a string coupling constant here and the string coupling constant is going to be identified with 1 over n under this dictionary, where n is the rank of the SUI engage group that appears in the gauge theory. So that's one part of the dictionary between string theory and super angles. And then the other part of the dictionary is that, I mean, we will mainly be interested in the larger limit. In the larger limit of these gauge theories, as many of you may know, there is an effective coupling constant that controls the perturbation theory. And the effective coupling constant is this two parameter, which is g squared young mills times n, which is the combination from the coupling constant. So this is a rank and a coupling constant. And that's the effective coupling constant at large n. And you have to ask, what is this to be compared to from the point of view of ADS-5 plus S5? And it's roughly something like the radius of, say, the five sphere divided by the string length to some power. And in ADS-5, it will be the fourth power. So what does this tell you? Well, what this tells you is, if you want to be in the world of supergravity here, what does the world of supergravity look like? The world of supergravity means the string is very, very small, so you can approximate it by a point particle by the graviton. So this will mean that this parameter, the string is very, very small, which means the space in which it propagates, it's much, much bigger than the typical size of the string. Unless you should think of as being the typical size of a string and R as being the radius of S5 or the parameter that captures the cosmological constant from ADS-5. So in supergravity, this will be large because the space is much larger than the size of the string. And what this tells you is that supergravity corresponds to strongly coupled gauge theory. So this is good and interesting. This is what has motivated much of the developments of this field because the ADS-CFT correspondence gives you access into a strongly coupled gauge theory from an alternative perspective, namely by doing supergravity calculations in ADS-5 process 5, that's something you can actually calculate and thereby learn something about strongly coupled gauge theory, an area which you have very little access to otherwise. Now, this is great, this is good, this is fine, but if you are a skeptic like me, then that somebody tells you that this is true, this may be not enough for you. You would like to understand it more conceptually. You would like to, in some sense, derive this duality. I mean, part of the motivation for trying to derive it is that there are many versions of the ADS-CFT correspondence, not just ADS-5 process 5 to n equals to 4-3-0. People have tried to apply it to condensed matter systems, to gotten what not. And you would like to understand which features are essential for this duality to work and which of them are accidental. So you really want to understand a little bit more how this duality works in detail. So if you want to understand this in detail, or maybe if you want to prove it, I mean, proving is always a big word, but let's use it anyway. So if you want to prove it, then how can you go about trying to derive or prove this duality? Well, I mean, n equals to 4-3-0 or the analog of the gauge, there we will only have under control if this is weakly coupled. So we need to be weakly coupled here. i lambda has to be small. I mean, we also want n to be large. So this we want the n large so that G string is small, that makes life surely easier. And then, but in order to have control over the gauge theory, we need this parameter to be weakly coupled to be small. And what this dictates for you is that if you are in that corner, then also this parameter has to be small. I mean, that's part of the dictionary between the two sides. If you are weakly coupled in the gauge theory, then the consequence of it is that R over LS, that the radius of the five-sphere or whatever is of the same size as the string length. Put differently, the string is very, very large. I mean, think about the space as being given. You can think of it as a measure for the size of the string. So it's the opposite to the supergravity limit. It's not the limit where the string is tiny. It's the limit where the string is as large as it can be while still fitting inside the space. I mean, it can't really be larger than the space. Otherwise it'd have to curl up. And you should, I mean, I've deliberately written till this here, you shouldn't take this too literally. This comes out of some, you know this to be true in the regime where supergravity is applicable, but there'll probably be corrections when you go to the regime where lambda is being small. Okay, so what you have to do is you have to look at the regime where the string is very, very large. And that means you have to be in the regime that's sometimes called the tensionless regime of string theory. And what you mean by that is that the string tension is very, very small. I, the string tends to be very, very large. It doesn't cost much energy for the string to be large. And therefore it's going to fit sort of explore the whole space. Now, these are nice words. So you can say, okay, I can prove ADS CFT. I just have to study tensionless string theory on ADS spaces. The problem is that you don't know how to study tensionless string theory on ADS spaces because you can't use supergravity methods. Now you have to use exact world sheet techniques, but strings on ADS spaces are notoriously hard. Nobody is really able to write on the world sheet description of strings on ADS five. So, I mean, on a certain level, these are just nice words. So you can ask, is there a chance that we can fill these nice words with something that's more than words? Can we work out, understand some example in detail? And the spirit or the idea or the inspiration of what we are trying to do is that in some sense, this seems to be like a bizarre problem. You see weakly coupled gauge theory as it's simple as it goes. It has lots of symmetries. So therefore it should be particularly simple. And on the other hand, you seem to think that string theory in ADS in the tensionless limit will be highly complicated because it's a regime you don't really know how to approximate. But if the dual theory is really simple, doesn't that suggest that also the string theory will be very simple if looked at from the right perspective? I mean, it's not simple if you look at it from the perspective of somebody who is used to supergravity and trying to extend it to more and more stringy backgrounds, but there must be a sense in which also the string theory is very simple because it after all has to reproduce a very, very simple theory. Free Super Young Mills basically. So you would expect that while this is inaccessible from the point of view of directly constructing it maybe, there should be something simple about this theory. This should be a simple theory. It should be a highly symmetrical theory because the dual CFT, the free Super Young Mills is highly symmetrical, has lots of conserved currents. So there must be something happening on this side and the spirit of what we are trying to do is somehow find this theory using all the bits and pieces of information we can get being inspired by the fact that after all it should have a simple World Sheet description and trying to in some sense to write down this World Sheet description that will reproduce what free Super Young Mills will describe. So this is the basic strategy. Try to use all the constraints that you can get your hands on to get a handle on what on the face of it looks like a horrendously complicated theory but fundamentally must be a simple theory because it's due to a very simple theory. Now, these are still again words. This is the basic strategy. And so far while we have made some attempts to solve this for N equals to four Super Young Mills the example where we've managed to put more into this than just words is the case of ADS-3. So for ADS-3, we believe we've really managed we've really managed to find a solution to this problem. I have managed to identify the World Sheet description of this theory, which is due to the analog of free Super Young Mills for the case of ADS-5. So let me review for you what the situation for ADS-3 is. So what's the folklore about string the ADS-CFT duality for the three dimensional ADS case? Well, the folklore is that if you look at super strings on ADS-3 cross S3 cross T4 that is still so this is a super string or say let's strings. This is a belief to be dual. And now you have to watch my words carefully. This is due to a 2D CFT. It's going to be dual to a 2D CFT because the boundary of ADS-3 is two dimensional. So that's what you would expect. So what, which 2D CFT is it? Well, it's a 2D CFT that lies on the same modular space. And I'll explain to this in a second of CFTs that also contains the symmetric orbit fold the symmetric orbit fold of T4. So what's the picture here? The picture is here. We have some modular space of 2D CFTs. So every point in this diagram is a two dimensional conform a feed theory. There's one point here, which is the special point that is the symmetric orbit fold of T4. And I'll later review for you what this is. This is a very concrete and specific 2D CFT that we can solve in great detail. And what you should take away from this is that the symmetric orbit fold of T4 is the analog of free super angles. I mean, it'll become apparent when I'll explain it to you in more detail, but it's basically a free theory subject to some global constraint. So it's the analog of free super angles and the N is the analog of the N appearing here. So also in this context, the N of the symmetric orbit fold is related to the string coupling constant of the string theory by this description. But so what did I mean by this here? Well, here you have one specific conform a feed theory in the symmetric orbit fold of T4, but this theory has many exactly marginal operators. An exactly marginal operator is like a parameter that you can choose. And for whichever value you pick, you get a CFT. So there isn't just one CFT. There's a whole modular space of CFTs. And from a point of view of two-dimensional conform a feed series, you describe them by starting from the one simple CFT you have under control and perturbing it for all the exactly marginal operators. But as a whole, yes. Would there be an analogous statement where the T4 is replaced by K3? Absolutely, yes. K3 works the same way. I'm just concentrating on the simplest case. It'll already be complicated enough. So let's just try to understand the simplest example and that is the example for T4. But you're absolutely right. You can replace this by K3. In fact, there are probably other versions, there are in fact other versions where we can also do it, but I'll concentrate on the simple. Thanks a lot. So this is a sort of one CFT in the modular space of CFTs. And the way you should think about it is that this is the analog of free super angles. And you should ask which worldsheet description corresponds to this point. Now, what do we know about worldsheets descriptions of strings on ADS-3 cross S3 cross T4? Now, the advantage of going to three dimensions is that in three dimensions, so as you know, you can produce the ADS-CFT duality by starting with a bunch of brains and looking at the decoupling limit. And the appropriate configuration of brains for the case of ADS-3 cross S3 is the so-called D1 D5 system. So for ADS-3 cross S3, the way you understand the string theory is that you start with a D1 D5 system, the D5 wraps the T4, and then you look at a decoupling limit and you get the 2D CFT. Now what's special about ADS-3 is that in, there's not just a Ramon Ramon, there is a five brain, there's also the nervous sports five brain, and correspondingly, there's obviously the fundamental string. So there's some sort of S-dual version of that that involves the fundamental string and the NS5 brain. And as a consequence, if you have this sort of problem in mind, that has a much simpler perturbative worldsheet description because you don't have to fight with the Ramon Ramon flux that's always the reason why this ADS background is difficult to deal with. So what we're concentrated on are going to be ADS-3 backgrounds with pure nervous sports, nervous sports flux. The sort of configurations you would get by starting with a fundamental string NS5 brain configuration rather than a D1 D5 brain. Obviously you can also start with some mixed configuration. So there's a whole zoo of ADS-3 cross S3 cross T4 dualities, but we'll go to exclusively concentrate on this background. And the reason for that is that in that case we actually have a solvable worldsheet theory. And the solvable worldsheet theory is, so you can think of this as being some sort of slice inside this modular space. This is the slice corresponding to pure nervous sports, nervous sports flux. And along this slice, you have an exactly solvable worldsheet description. And this is the Maldesino or Gorori, and I'll review this. In fact, that's what I tried to spend basically the rest of today's lecture on explaining to the Maldesino or Gorori the theory, describing the exactly solvable string theory that's the pure nervous sports, nervous sports background for ADS-3 cross S3 cross T4. But here you start from D1 D5 because in this way you get these correspondences. No, I mean, you get these correspondences in either way, right? No, I mean, but if you started from another thing like D1 NS5, you get another kind of correspondence. I mean, D1 and NS5, this I don't know. So what I meant was that you have a combination of D1 D5 or F1 NS5. And you can look at some combination of the two. But it's not the only possible choice. Probably not, but so what I'm going to do is I'm going to exclusively concentrate on that case for the simple fact that that's the situation where I have a worldsheet description I have under control. Now, obviously nobody tells you that the symmetric orbit fold of T4 will have anything to do with this specific choice. And in fact, the conventional wisdom was that this is not true, that these two things are totally orthogonal and you shouldn't even think about this background having anything to do with the symmetric orbit fold. But if you're an optimist, you find it's a... Physics would have to be pretty unfair. You see, there's one special point here that you have under control and there's one special line here that you have under control. So why on earth should that special line have nothing to do with that special point? I mean, if there's something simple on one side and something simple on the other side, wouldn't you expect that somehow the simple things ultimately match with one another? At least that was sort of my working hypothesis. I mean, it would be cruel if you could solve this theory on some priests that had nothing to do with this theory despite the fact that both of them are simple. Anyway, so what I'm arguing is for whatever reason, let's just concentrate on the background just pure in the visual, so visual flux. Now, in the spirit of what we explained here, we'll have to look at the regime where the radius is small. That's the regime where we would expect it to be due to a free super angles or free super angles should replace in the context of ADS-3 with the symmetric orbifold of T4 itself rather than some deformation that has broken the orbifold symmetry. So this specific 2D CFT should correspond to the tensionless limit of this world sheet descriptions in terms of these vasomino written models. Now, these vasomino written models as I'll review for you, these are vasomino written models based on the Lie algebra SL2R and vasomino written models always have a parameter which is called the level which I'll also explain to you. And this level you should think of as basically being the radius squared in string units. So K is to be identified with the radius over LS squared. So if you believe that this somehow fits into this general picture and if you believe that a symmetric orbifold is the analog of free super angles, then what this should mean is that the symmetric orbifold should be due to one of these vasomino written type models when you take the level to be small because that's the regime where the dual CFT becomes weakly coupled and surely this is weakly coupled. So the proposal was we should look at this CFT for the smallest possible value of K. That's somehow what it means to become tensionless. Yes. Sorry, just about the kind of curious of the historical perspective. So was there actually a argument or some strong intuition for why people thought this locus does not contain this measure over the whole point? Yes, yes. It has to do with the fact that so backwards if you have a short flux have this long string solutions near the boundary of ADS, they're stabilized by the fact so that a long string would like to contract. So you have to turn on the R flux. It's stabilized. But then what it means is that you have a fluctuations of this and if you go to the boundary they get redshifted and as a consequence, you get a continuum of excitations. Right. And in fact, when I will explain them all to see an algorithm written model, we will see this continue. But on the other hand, the symmetric overfield of T4 it's not a rational CFT but it's almost a rational CFT, right? It has a discrete spectrum. There's no sign of any continuity. So therefore the conventional wisdom was that this discrete or quasi-rational CFT has clearly nothing to do with the back ones with pure Nervish-Waltz, Nervish-Waltz flux. And generically that is true. But what you would expect is that somehow if you make the level very small maybe something special happens. And so what our proposal is and what I spent these lectures trying to explain to you is that if you take this level and now I use a small piece of a colored chalk. So if you take this level to be equal to one which is in some reasonable sense I'll also try to explain to you the smallest possible value you can pick. Then this Wesselminen written model has a very different flavor than if K is bigger than one. Some of this continuum disappears for representation theoretic reasons. It becomes a much more quantum rigid system. And what we've managed to show is that this worldsheet theory if you calculate the space time spectrum of it it reproduces exactly the spectrum of the symmetric orbit fold of T4 in the large end limit where it qualifies the single particle spectrum because we're doing perturbative strength theory only seeing the single particle states. But this comes out on the nose and it doesn't just come out for the BPS spectrum it comes out for everybody. So this theory at level K equals to one reproduces exactly the spectrum of the symmetric orbit fold of T4 in the large end limit. So this is already a very good indication that this is an example of where you really have this worldsheet theory being exactly due to the analog of free super angles. Now what I also want to explain to you is that it's not that we can just show that the spectrum matches we've also managed to show that the structure of the correlation functions of this theory are correctly reproduced by this worldsheet theory. And this is actually quite intricate. And I think it also suggests in some sense how this picture may fit into something that is ADS-5, process five and N equals to four. So I think what I'll try to convince you is I'll try to explain to you this correspondence in detail. And at the same time I want to convince you that this is not just some low dimensional accident it has many of the features you would also expect to be present for ADS-5, process five and it should be a good blueprint towards identifying the worldsheet theory that's due to free super angles. And in fact, that's what we've been working on for the last year. We've made a proposal along these lines probably the last word hasn't yet been said about it as much more to be done. But I think this is, we are very confident that we are on the right track towards identifying the worldsheet theory that's exactly due to free super angles. And the inspiration comes from the three dimension example that I can explain to you in great detail. And that's what I want to explain to you during this lecture. So what's my plan for these lectures? My plan for these lectures is that I'll first want to explain to you the NSR formalism a description of ADS-3. So this is really reviewing the work of Maldesino Oguri. The Maldesino and Oguri, they developed the description of the Vesemino written model describing strings on ADS-3, versus 3 cross T4. They developed this for general K. And then what I want to explain to you is why 4K equals to one this model the spacetime spectrum that you can calculate really reproduces the symmetric orbital spectrum. But I can see that you're standing up. So there's probably a question. Yes. Maybe I'm going too far, but is the case for K larger than one corresponds to any known CFT, for example, in the same modular space. So there is a proposal for what that you will see if he is, but it is quite complicated. And if you ask me, if I look at it, sort of unbiased, I would have thought it doesn't lie in the same modular space. But I think the conventional wisdom would say it does lie in the same modular space. I mean, who knows, right? It looks pretty different. It has a continuum, but who knows that once you've walked a long way in this modular space and crossed many rivers and bridges and stuff, maybe there is a CFT that you can reach that looks like what you get if you take K bigger than one. So K bigger than one, roughly speaking is the symmetric orbit fold of sort of Leoville, N equals to four Leoville. It's not exactly like that. So there's a recent paper of Laurence Eberhut in which she worked it out. So it's the linear Diller-Tron times T4, the symmetric orbit fold, and then you perturb it by operator that sits in the two cycle twisted sector, but also does something to the linear Diller. It's quite a complicated construction. It's given good evidence that that's the correct description. The spectrum surely looks like the symmetric orbit fold of N equals to four Leoville. That's the paper that Laurence and I wrote some time ago. And on the face of it, that looks very different than the spectrum of the symmetric orbit fold of T4. So if you're naive, you would say it is unlikely that they lie in the same modular space, but who knows, right? I mean, could be that there's many things happen as you walk around. Thank you. But here I'll concentrate exclusively on K equals to one. Well, I'll explain to you the Wessler-Mino-Witten model for general K and then I'll explain to you what happens at K equals to one. K equals to one in this description is actually a little bit subtle and I'll explain to you why. And because it's a little bit subtle, the description inside it, so in Nervysch-Bordermond is what all of us laugh because that feels like flat space. I mean, that's like, that doesn't seem that different from flat space, but there are some subtleties about the K equals to one theory and therefore there's an alternative formalism which is the so-called hybrid formalism, which is in some sense you should think of, this is Nervysch-Bordermond and a hybrid formalism is like green Schwartz. So you go to a description in which spacetime supersymmetry is manifest. So what this means is you'll have a P as you one comma one slash two Wessler-Mino-Witten model together with some topologically twisted T4 and whatnot. So I want to explain to you a little bit about this for the extent that you need to know. And then this description becomes totally clean and we can prove that we get exactly the right spectrum. So this is the clean version, that's the cleaned up version of this, that's the easy to understand version, easy to understand version, that's harder to understand but cleaner. And then the last lecture, I want to try to explain to you how correlation functions emerge. And that's actually a very beautiful story because correlation functions in the symmetric orbital theory are characterized in terms of holomorphic covering maps. And what we see is that this duality actually doesn't just work, it does work in the larger limit but it reproduces correctly all one over N corrections where the one over N corrections come from the higher genus contributions from the world. That's what you would expect, right? The genus of the world sheet, if you look at a torus amplitude, its contribution is suppressed by powers of G string. So the one over N effects in the dual CFT should come from higher genus world sheets and discovering maps have a natural covering surface involved and what will turn out, and we can make this very precise is that this covering surface is the world sheet and thereby it reproduces correctly the one over N corrections of the symmetric orbital from the world sheet perspective. So this is really in some things going beyond the planar limit of N equals to four because we also systematically understand all the one over N corrections rather than just the leading term as N goes to infinity. But I think there's a question here. Yeah, this is perhaps a very elementary question. So in the way zoom in a written models for compact groups, the level key is quantized but SL2R is a non-compact group and it has no issue to subgroup. So I suspect the level key won't be quantized anymore, right? If you just look at SL2R in isolation, that is correct. But as I'll explain to you, you see, we are looking at this background. Okay. And in this background, the SL2R, vessel amino written model comes together with an SU2 vessel amino written model describing strings on A3. And then the requirement that the string is critical requires that the level for the vessel amino written model here is the same as the level of the vessel amino written model here. And while this isn't quantized, this is quantized and therefore in the context of the super string, K is indeed quantized and therefore one is the smallest number. You may have asked why not a half or one over 10 or whatever, but it is quantized on account of the fact that you're really having an S3 factor and S3 is only consistent if the level is an integer. I see two. But I'll come to that in due course. Thanks a lot. Yes, there's another question. But so is it possible that K correspond to R over L since R is a continuous parameter? Well, as I said, you have to treat these statements with a grain of salt, right? I mean, this is true for large K. For large K, you approximate some, I mean, the radius only makes sense in some sort of supergravity description. I mean, classical geometry makes sense if the space is much larger than the string length. If you're going to the regime where the string is as big as the space, the notions of classical geometry are likely to break down one way or another. And in fact, in one way, we will see this is as you may know, the SU2 vasomino written model is quantized, right? I mean, it has nothing to do with ADS3. If you just ask what does string theory in ADS3, or S3 look like, you would say this is the SU2 vasomino written model and then K is quantized and you would learn that somehow this is only consistent if the radius takes a specific ratio in terms of the string length. You can't choose the radius arbitrarily otherwise somehow the theory is ill-defined, not unitary or whatever. So this is some sort of string effect. For large K, you don't care, right? I mean, for a very, very large level, the level spacing is infinitesimal, but when you come to very, very small radii, then this becomes more pronounced. And also, as you know, or as you may know, SU2 at level one, which is describes the three sphere at the smallest possible radius, SU2 level one is actually the same theory as a single free boson, right? That's something people may know. It's this free field construction for the SU2 level one theory. So what this tells you is that when you think the string is propagating on a three sphere, if the three sphere is small enough, you can't distinguish it from propagating on a circle. That's how you should, that's the level to which your geometric notion will break down when you go deep into the stringy regime. Deep inside the stringy regime, the string doesn't just see classical geometry. It's the same with t duality, right? I mean, a small radius is the same as the big radius and the critical radius is the same as the SU2 level one theory. So you would thought the string propagates in three space time direction, space direction, but it actually only propagates in one. It's quantum equivalent to it. So it's not something you easily see on the level of geometry. So we are in the regime where all of these effects play a role. So your notions of classical geometry are good in the regime when this parameter is large. And we are in the opposite regime and there are string theory rules and string theory, world sheet string theory rules. Okay, are there any further questions? Now let me start raising the blackboard. So now I want to explain to you a little bit how, I mean, now I've made all these big claims and words. So now I have to deliver, right? I have to explain to you how you really do this. And yeah, so there's a certain danger I'll get to technical. So please slow me down. If I say things that you don't understand and I'll try to explain at least the key features in a way that even people without some good background in 2D CFT will be able to understand. Okay, so what we want to start with is trying to understand strings on ADS-3. So let's start with, let's first of all just look at the ADS-3 factor. Okay, so ADS-3, one way of writing ADS-3 is as a hyperbolic in four dimensional space. So this is the space where, so you can write ADS-3 as the manifold that's characterized. So you look at, you think of this as a subspace of R4 rather R2,2 and you impose the condition X minus one squared plus X zero squared minus X one squared minus X two squared equal to zero. That defines for your hypersurface, sorry, equal to one. Yes, thank you. That defines for your hypersurface inside this space. And this is the hypersurface that defines ADS-3. I mean, it's the hyperboloid inside R2,2. And obviously I need to take this one. I could also take it L squared, but for simplicity, that doesn't really matter. Now this actually is in one to one correspondence with group elements in SL2R. So if I look at the group element, if I prioritize my group element of SL2R as X minus one plus X one, X zero minus X two minus X zero minus X two and X minus one minus X one. And you see the determinant of G is just, well, I have to take this. So this will be X minus one squared minus X one squared minus, and then you see it's plus X zero squared minus X two squared. So the determinant equal to one is exactly this condition. So describing this hypersurface in R2,2 is in one to one correspondence with looking at two by two real matrices that have determinant equal to one. So this wants to be an element of SL2R, which is another way of just saying that it's a real two by two matrix with determinant equal to one. And the determinant condition is exactly the hypersurface condition for ADS-3. So that's one of the reasons why ADS-3 is so simple. ADS-3 is a group manifold. Namely, it's just the group manifold of SL2R. But it's not exactly like that, but it's almost like that. You have to go to the covering group, but I'll come to that because that will actually play an important role. Okay, so this is simple. Now, in order to understand this a little bit more geometrically, if you want to describe in terms of global coordinates, we're going to parameterize this group element inside of SL2R as e to the i times u times sigma two. So unfortunately, I have to introduce some notation now. So please bear with me. So we have e to the row times sigma three. And then we have e to the i times v times sigma two. The sigmas are the Pauli matrices. So sigma one is equal to zero one one zero. Sigma two is equal to zero minus pi i zero. And sigma three is equal to one zero zero minus one. And u and v. So here, so this g is a function of row u and v. And in terms of global coordinates, row will be the radial coordinate of ADS-3. And u and v are related to the time and the angular coordinate as u is equal to a half times t plus pi. And v is equal to a half times t minus pi. So then if I write this out, if I write out this group element, then it takes the form g is equal to cos row cosine t plus cos sin row cosine pi. And then likewise here, so it's cos row sine t minus sin row sine pi. And then over here, it's minus sin, minus cos row sine t. Minus sin row sine pi. I mean, this just comes from plucking in this formula, right? I mean, these are two by two matrices. You exponentiate them, you multiply them, you rewrite it in terms of row t and pi. So this is not rocket science. It's just, I'm just telling you what the answer is. And then, and this row pi and t coordinates are designed in such a way that the metric, I mean, the metric is induced from the r two comma two metric here, if you go to the subspace and you parameterize it in terms of this coordinate. So you notice that this has determined one as you can easily confirm. So this solves this constraint. I mean, it's a parameterization of all of these group elements just like you can parameterize SU two in terms of explicit parameters. I can parameterize SL2R in terms of row t and pi or sometimes I write row u and v. U and v and t and pi are interchangeably used. And then the metric is so that you have a sense of what this is, is cos squared row dt squared plus the row squared plus sin squared row d phi squared. And what this describes is a cylinder where row is the radial coordinate of the cylinder. So row is the radial coordinate. So row equal to zero is in the center of ADS-3. Row going to infinity is at the boundary. And at the boundary, our degrees of freedom is time and phi. So it's the, so along the cylinder, it's the time. And then the rotational axis is the phi direction. And sometimes it's convenient to use u and v. So u and v are the light cone coordinates on the boundary cylinder whereas t and phi are like the time and the spatial coordinate on the boundary cylinder. And I'll switch between them from time to time. Now, the reason I'm writing this down in some detail is that you see this description, we want phi to be two-by-periodic. So phi and phi plus two pi should describe the same point of ADS-3 space, but t and t plus two pi shouldn't, right? I mean, unless you also identify this and turn this into a torus. But on the other hand, if I look at this explicit description of this group element, you see this group element is periodic in t goes to t plus two pi. If I replace any t here by t plus two pi, I obviously get the same matrix. So it's not exactly true that ADS-3 is the group manifold of SL2R. What I have to do is I have to unfold the time direction so that I don't artificially identify t with t plus two pi, which I don't want to do. My time should just run forward and there shouldn't be somehow some identification after time has elapsed for periods of two pi. So this is a point I'll have to come back to. If I just treat it as an SL2R, I mean, a written model I have to keep in the back of my mind that I'm not exactly doing the right thing. At some stage, I will have to include solutions that will undo this periodicity in t, which is an undue requirement. Okay, so this is the geometry of ADS-3 and now I want to write down a conformal field theory. So the sigma model on ADS-3 is not conformal. On SL2R is not conformal, but you know how to make it conformal. You just add a vessel minoterm and then it becomes conformal. So the well sheet theory that describes that sort of the conformal version of strings propagating on this group manifold is the resummoner written model action. And what does it look like? Well, it looks like, and I'm not deriving this for you, but this is what you find. So you have some, this is the well sheet integral. So it's a sigma model. So it's maps from the two dimensional well sheet into the target space. And the action is trace g to the minus one dg, that's the sigma model, right? So you have, so g is a function of sigma and tau. Sigma and tau are my well sheet variables. So this is well sheet. So I'm looking at maps from the well sheet into SL2R. And then this is the metric that is induced from SL2R, g to the minus one dg is just the tangent vector. So the trace of the tension vector is just the metric of the space time pulled back. So that would be the sigma model action. And then you have to add this so-called Decimino term, which I'm not going to explain to you in detail. That's a bit subtle to calculate. And that's where your question comes in, that the well-definedness being an integer, but that's maybe the topic for another lecture. I'm just telling you that if you want this to be conformal, you have to include this term, otherwise it won't be conformal. So let's include this term and let's think of it as being sort of the conformalized version of the sigma model on SL2R. Now, what does this buy you having added this vessel minoterm? Now, again, I'm not really deriving this for you because so this vessel minoterm requires going to a three-dimensional surface with boundaries, the well sheets, then extending it in the interior, writing down a three form and so on. But the upshot of it is once you vary it, it leads to the equations of motion that only involves the two-dimensional well sheet theory. And the advantage of adding this term or what it buys you is that you get the following conserved currents. So you get two conserved currents, one purely holomorphic and one purely anti-holomorphic. What about holomorphic now? I mean, so on the well sheet, we have the variable sigma and tau. And what we're going to do is we're going to introduce the well sheet, light cone coordinates, which I have to do this right, tau plus or minus sigma. So there are two light cones happening here. So this is on the well sheet. So tau and sigma on the well sheet and X plus and minus is on the well sheet. And on the target space, we have T and phi and UNV. So T and phi are the coordinates on the boundary. There are two two-dimensional CFTs living here. One is living on the boundary of ADS-3. Its coordinates are T and phi with light cone variables UNV. And there's a two-dimensional CFT living on the well sheet and its variables are tau and sigma with light cone variables X plus and X minus. And we have to keep them clearly separated because they're totally different animals. They're going to get related to one another eventually, but at the moment, they are just very different types of coordinates. Now, adding this mesomino term, what this implies is that you get the holomorphic current. So you get the holomorphic currents, which are the right moving current, which is only a function of X plus. And this is defined to be K times the trace of TA times D plus G to the minus one, where the D plus is the derivative with respect to X plus. And what the equations of motion tell you, once you've added this mesomino term, is that this is only a function of X plus, namely that the X minus derivative of this expression is zero. That comes out of the equations of motions, once you've added this term. And you have one left moving current and you have one right moving current or one right moving current and one left moving current. There's another current who is purely a function of X minus. And it is defined by taking a similar trace, but not exactly the same trace. Actually, for reasons of convenience, we take the complex conjugate of the TA matrix, but that's just a relabeling of the indices. But what's important is that you consider G to the minus one, D minus G. I mean, there's a subtle point here. You see, these look deceptively similar, but because this is a non-Abelian group, it matters whether you consider DG, G to the minus one or whether you conduct G to the minus one, DG. And once you've added the mesomino term, this is a purely a function of X plus and this is purely a function of X minus. That's what this adding this mesomino term gives you. It's maybe not obvious at this stage why that means it's conformal, but and in fact, I probably won't really explain this, but what it gives you is that you have a, that these currents satisfy an interesting OPE. So when we, so these currents are functions of X plus and the world sheet is periodic. So therefore we can expand, say JAR of X plus in the Fourier series. You see X plus because it's a function of tau plus sigma and sigma is two pi periodic. We are in the world of closed strings. So everything is two pi periodic and sigma. This must be two pi periodic in X plus. Yes. But I was just to be sure. So when you speak of the second model here on SL2R, are you already taking an universal covering? Well, at this moment, I'm sort of agnostic about it, but you'll see the universal cover appearing. Because the CFT will be different. I mean, based on the format. Yes, yes, I'll, I'll, I'm not sure. Wait, wait, I have another 11 minutes. So there's a fighting chance I'll get to it, but there's a fighting chance I'll explain it tomorrow, but I'll explain it in detail. So at the moment I'm not trying to address this question but it'll come back to me and then I'll address it. At the moment, I'm just trying to read off, what does this by me? What does this by me that I have this currents that are purely a function of X plus and a current that's purely a function of X minus? Well, what I can do is I can Fourier expand them, right? Because there must be periodic in X plus. So it must be of the form e to the in X plus because it's two pi periodic. So I can make a Fourier decomposition. And then I can calculate the Poisson brackets that follow from that action. And then I can quantize it and that turns the Poisson brackets into commutators. And lo and behold, if you do that, what you find is that these currents, this moment, these modes of these currents satisfy the Katsmoody algebra. And in my convention, the Katsmoody algebra will be the following. It'll be plus or minus m plus n. And then we have a J plus with J minus, minus two times J three plus K times m times delta m minus n. And then J three with itself is equal to minus K over two m times delta m minus n. So what you find is that the momenta, the moments of this right moving currents form what's called the SL2R affine Katsmoody algebra at level K, where this parameter K is the parameter K that appears here. You see, because it appears here, it's basically proportional to the radius squared in string units. So this is a parameter that labels how classical this space is, how big this space is. And from the point of view of the Poisson brackets, it enters by virtue of the fact that it multiplies what's called the central term. So what does this look like? So if you are a little bit sleepy, then ignore these funny delta terms and it basically looks like a le algebra of SL2R. So three with plus and minus gives you plus minus, plus with minus gives you J three. Don't mind the signs, the signs are assigned so that it corresponds to the real form of SL2R. And then you have these funny labels and these labels basically just add up. So it looks like, in fact, it's the loop algebra. It's the le algebra of the loop group into SL2R, but who cares? And then you have these pieces that are just numbers. So if you want to impress your friends, then you can say this is a central extension and you can think of K as being an operator and K is an operator that simply commutes with everybody and a physicist can never tell the difference between an operator that commutes with everybody and the number. So for physicists, K takes the value for mathematicians. It's an operator that's central that commutes with everybody else, but in an irreducible representation, an operator that commutes with everybody else will just take one value because, I mean, every state will have exactly the same eigenvalue under K. And this K parameter is sort of the central terms here. They go like m and delta m minus n and you can check that that le algebra satisfies the Jacobi identities. That's actually, in fact, a good exercise to convince yourself that that's a consistent le algebra. And that's what the moments of these currents that come out of this vestromino written model do. And you do this, you can obviously do this for the right movers and you can do obviously the same thing for the left movers. So you get a right moving SL2R level K. I find Katsmoody algebra and a left moving one and they don't talk to each other. The Poisson brackets of the life movers and the right movers is 0. So there are two commuting copies of an SL2R vestromino of an SL2R affine Katsmoody algebra. And the reason why this is conformal is whenever you have an affine Katsmoody algebra, you can construct it by the Shugavara construction, everywhere, so algebra out of it. But I'm not going to explain that in detail because we won't really be using it. So that will not be that important. But that's the reason why this guarantees that this is in fact a conformal feed theory because you get a bero-saurus symmetry, which is the hallmark of two-dimensional conformal feed theories. Okay, so what I want to comment on though is one important feature. You see, every affine Katsmoody algebra contains a copy of the originally algebra inside it. Namely, if you look at the generator with mode number equal to 0, they just form a conventionally algebra because you see 0 plus 0 is 0. So you always produce 0 on the right-hand side. And if M and N are 0, then these central terms disappear because they get multiplied by M. If M is equal to 0, it's obviously 0. So this algebra always contains a copy of SL2R. And the copy of SL2R is generated by JA0. And what does this mean? Well, the zero modes are the guys that don't depend on X plus. So these are the rotations where you sort of globally rotate the SL2R space. So if you think about it from the point of view of the boundary, that's also globally rotating your boundary. And therefore what you, the way you should think about is that this SL2R is to be identified with the Möbius group of the two-dimensional conformal feed theory. So this will be to be identified with the Möbius group or the Möbius generator, I mean the Lie Algebra, so it's the Möbius generators. So more specifically, L0 of the space-time theory will correspond to J3 0, L minus 1 will correspond to J plus 0 and L plus 1 will correspond to J minus 0. And L0 and L plus minus 1 are the generators of the Möbius transformation. So this is the scaling sum. So this is the space-time. This is the generators that act on the dual 2D CFT because these are the global rotations of the whole space. And because they globally rotate the space, they also rotate the boundary. So therefore they act like, and they generate a Lie Algebra of SL2R. And if you look at the generators of the Versailles Algebra and restrict the mode numbers to zero plus and minus one, then it also closes. And it is exactly the Möbius group which are the holomorphic maps from the sphere to globally defined holomorphic maps from the sphere to the sphere. So this is the scaling transformation. This is translation. And that's the generator of the special conformal transformation that maps the sphere into itself. And you can check. So for example, why does this work? You see, if you look at L plus 1, L minus 1, then from the Versailles, you know this is M minus N. So this should give you two L0 space-time. But if you believe this dictionary, this should be J minus zero, J plus zero. And if we look here, J minus zero, J plus zero. But if it was J plus zero, J minus zero, it'd be minus two times J is three zero. But because I reversed the order, this will be plus two times J three zero. And therefore that just agrees. So you can check that these generators just make up a copy of the Möbius group. And that's the first indication that you're really getting. It's the global conformal transformation of the boundary. You're not seeing the full Versailles algebra, but you're seeing the global transformations of the boundary surface. And that will be important because you see what this allows us to do is to read off the space-time conformal dimension from a world-sheet perspective. The L0, so this will be the absolutely central identity. You see what this tells you is the conformal dimension of a state from the perspective of the 2D CFT is equal to its J three zero eigenvalue as calculated on the world-sheet. So when we enumerate all the states, all the physical states on the world-sheet, we are not just getting a whole bunch of stuff. We are getting them filtered by their eigenvalue with respect to an operator that we know what it means. It's the conformal dimension in the dual CFT. So that gives us a chance to really see what sort of spectrum we are generating. And as I said, you have this on the left and you have this on the right. So you have a corresponding statement for the BART modes which correspond to the Mobius group acting on the anti-holomorphic coordinate in the space-time CFT. Okay, so this is, and I'm coming back to the problem that Yvonne asked the... So normally when you, once you have a resumino written, once you have an affine-clad smoothie algebra, then you say, ah, okay. So now I'm going to look at this problem differently. I could try to understand all the classical solutions and so on, but actually there's a smarter way of describing what this CFT looks like because you see, I have this enormous symmetry and therefore I know that my space of state will fall into a direct sum of representations for the left-moving SL2R level K resumino written model, tensor has some representations with respect to the right-movers. I mean, this is sort of typical way in which 2D CFT technology works. I mean, you could go ahead and try to calculate the Feynman rules of a two-dimensional conformity theory. You would be mad to do so because while you can do it, you know that there's a smarter way of getting the answer because you know the answer after you've summed up all these complicated calculations will have to fall into representations of your symmetry algebra. So you should use your symmetry to describe the answer without trying to painfully reconstruct it from doing Feynman-Graf calculations. So you know that your space of states, however you get it at the end of the day has to lie in representations of these commuting Affin-Katzmoudi algebras and they are not that many representations. So typically what you would say is that I should just sum over some set of highest rate representations and the set of highest rate representation after sum over will be effectively described to you by the Pieterweil theorem describing through what sort of representations will appear and that will give me the answer to this conformity theory, sort of the cheap way without trying hard to classify solutions and all the rest of it. I'll just use the fact that it has these enormous symmetries or it has to fall into this pattern that respects the symmetry and therefore it basically has to be of that form. There's a question. But the states that you construct, they still have to satisfy the Wiedersruhe constraint, right? Well, so at the moment, I'm just talking about a world sheet model as a conformity theory haven't yet imposed the physical state conditions, right? So this is, so what I'm doing here is I'm describing the world sheet theory like you would construct the world sheet theory of free strings propagating in space-time but I haven't yet imposed diverse or conditions. I'll impose the very zero conditions later on. At the moment I'm describing the theory before I've imposed the very zero conditions. Now before I've imposed the very zero conditions normally would have here just some set of highest rate representations. So what does highest rate representations mean? Well, basically it means it's a, I'm running out of, let me just say that and then I'll finish. So a highest rate representation is basically a Fox space of the following form. You take any of these generators all with some negative notes and then you apply it to some set of ground state and these ground states are characterized by the property that any positive mode kills it. And then this set of JMs will form a representation of the SL2R0 model. So any resumédo written model known to mankind is basically of that form, any highest rate representation will basically take the highest rate states, i.e. the guys that are killed by the positive modes because they're killed by the positive modes they'll typically sit in the representation of the zero modes because the commutator of zero mode with positive mode gives you positive modes. So the zero maps will map these states into one another and then the full Fox space is produced from applying all the negative modes. So that's what you can do. And then the only thing you have to specify is which representations, which highest rate, which representation of the zero mode algebra appear and there you use geometry or Peter Waal theorem to determine it. And for the case of SL2R, that will be the continuous and the discrete representation that I explained tomorrow. And then I'll come back to your question, namely, that would be if I was looking at SL2R without going to universal cover. Then I have to think about what's the new effect going to the universal cover. And the new effect going to the universal cover is that that is not enough. You have to include additional representation that I explained to you where they come from. They come from the fact that you demand that you also have solutions that are not periodicity. And this will be from the point of view of the representations, there it is spectrally flowed representations. And we'll see very explicitly how this spectrally flowed representations appear. That's what I'll explain to you tomorrow. And then once we've got this model under control, then we just go ahead, we open bridge what's written and we follow the rules of determining the physical spectrum, we impose the virus or condition. And then we simply enumerate all the states that satisfy the virus or condition. And what I'll try to explain to you tomorrow is that that reproduces on the nose, the spectrum of the symmetric or default if you said K equals tomorrow. But that's what I'll do tomorrow. And I think my time is up, so I'll better stop here.