 The courage to ask questions. Just shout and then wait that I will bring you the microphone. And for the people on zoom. You can just unmute yourself and ask your question. So we'll now start with the recording. Recording in progress. So our first speaker of the afternoon will be Matias Gabertijl from ETH Zurich. And he will tell us about the strings in YES3. Pistim. OK, bomo da se pravno. OK, tako če vidimo, da je to vse. To je več del, da sem v Tresti. A vse, da so, da sem da vse, da vse, da sem vse, da vse, da vse, da vse, da vse. Vse, da sem vse, da vse, da vse, da vse, da vse. In zato njič sem... Zato sem sem vse, da sem vse. Vse, da sem vse, da sem vse. Zato vse, da sem vse. To je zelo več zelo, da sem vse. 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... ...propatiation, and then I'll finish the plan of the lectures once I've explained to you what this is all about. Zelo taj zelo je... Zelo so vsi vse ne znam, da je adc eft. doalati. Zelo vse rečim na tudi 30.000-daj glasbo, kako je adc eft. doalati. Pozirajte, da je vse najfamiliarne vse, ki je, če je super strine, na adc 5, proz. 5, je doličil po n njega 4 super in milz. On je inovalo srednji na subrednjih. To je všeč, da je ne adiS3, ne zelo srednji v adiS3 in drugi. Nenekaj ležite vstavljenje in vse, da se predaj tudi, nekaj ne možete, zelo se vzalo, zelo se z vstavnjih. Srednjih je tudi, nekaj je tudi, nekaj je vzalo, nekaj je vzalo, nekaj je vzalo, nekaj je vzalo, nekaj je vzalo. Nekaj je vzalo, nekaj je vzalo, nekaj je vzalo, nekaj je vzalo, vzlušaj, da povoljšel mesej, ki je, da aqui je vse strin kurplijančju n-korljuče vseh, ko je n-korlče vseh, kot na del na zađernega delarjega delarjega. Čekaj, zato kšeljnje z konč players ko je povoljskim ternimi, kaj je ternimi dželji, je joinedi dželji, ko je začelje n-korlče, ber n-korlče in ljubi ljubi ljubi ljubi ljubi ljubi ljubi ljubi. 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 ADS5 cross S5, and it's roughly something like the radius od, kurim, 5-zvirkega v lzinku kuzej. Na 5vrkega ABS sem začnega po techniques. Kaj, da prejste to postave? Zato je, da boš kar, v spajku braču, začnega supajku trafi, tako, da je, kaj začnega supajku, kaj bah pravič généralne pristike, zato pravitega začnega. to before a before means that this parameter and the string is very, very small which means the space in which it propagates is much, much bigger than the typical size of the string. LS 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 super-gravity this will be large because the space is much larger than the size of the string 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 versus 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 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 versus 5 to n equals to 4, 3, 5 mils, people have tried to apply it to condensed matter systems, to gotten whatnot, 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 super mils or the analog of the gauge theory 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 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. Or, 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 would have to curl up. 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 will probably be corrections when you go to the regime where lambda is being small. OK, 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 sometimes called the tensionless regime of string theory. And what you mean by that is that the string tension is very, very small. 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, OK, I can prove ADSEFT. 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 down the world-sheet description of strings on ADS 5, 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 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 and 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 for all has to reproduce a very, very simple theory. Free superangmils, 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 superangmils, is highly symmetrical, has lots of conserved currents, 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 worldsheet description and trying in some sense to write down this worldsheet description that will reproduce what free superangmils will describe. So this is the basic strategy. We 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 4 superangmils, the example where we've managed to more into this than just words is the case of ADS-3. So for ADS-3, we believe we've really managed to find a solution to this problem. We've managed to identify the worldsheet description of this theory, which is due to the analog of free superangmils for the case of ADS-5. So let me review for you what the situation for ADS-3 is. Is there a folklore about the ADS-CFT duality for the three-dimensional ADS case? Well, the folklore is that if you look at superstrings on ADS-3, cross S3, cross T4, that is dual. So this is a superstring, or, say, lead strings. This is a belief to be dual and now you have to watch my words carefully. This is dual to a 2D-CFT and 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 which 2D-CFT is it? Well, it's a 2D-CFT that lies on the same modularized space and I'll explain to this in a second of CFTs that also contains the symmetric orbifold, the symmetric orbifold of T4. What's the picture here? The picture is here, we have some modularized space of 2D-CFTs. So every point in this diagram is a two-dimensional conforma feed theory. There's one point here, which is the special point. That is the symmetric orbifold 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 the symmetric orbifold of T4 is the analog of free supering mills. 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 supering mills and the N is the analog of the N appearing here. So also in this context the N of the symmetric orbifold is related to the string coupling constant of the string theory by this description. So what did I mean by this here? Well, here you have one specific conforma feed theory in the symmetric orbifold 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 modularized space of CFTs from a point of view of two-dimensional conforma feed theories. You describe them by starting from the one simple CFT you have under control and perturbing it by all the exactly marginal operators. So there's a whole, yes. Would there be an analogous statement where the T4 is replaced by K3? Absolutely, yes. Okay, if he 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, but we can also do it, but I'll concentrate on the simple case. Thanks a lot. So this is sort of one CFT in the modularized 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 worldsheet 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 the 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's literally a 5-brain, there's also the Nervyshvods 5-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 these ADS-backgrounds are difficult to deal with. So what we're concentrated on are going to be ADS-3 backgrounds with pure Nervyshvods 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're going 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 Nervyshvods flux. And along this slice you have an exactly solvable worldsheet description and this is the Maldesino or Grori and I'll review this. In fact, that's what I tried to spend basically today's lecture on explaining to the Maldesino or Grori the theory, describing the exactly solvable string theory that's the pure Nervyshvods background for ADS-3 cross S3 cross T4. Here you start from D1-D5 because in this way you get this correspondence. No, I mean, you get this correspondence 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-NS5, this I don't know. So what I meant was that you have a combination of D1-D5 or F1-NS5. You can look at some combination of the two. But it's not the only possible choice. Probably not. 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 that the symmetric orbifold 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 orbifold. 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 is 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 is 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 priest that had nothing to do with this theory that both of them are simple. Anyway, so what I'm arguing is for whatever reason let's just concentrate on the background with pure in the visual and 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 dual to a free super angles or free super angles that 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 these worldsheet descriptions in terms of these vasominoviton models. Now, these vasominoviton models, as I'll review for you, these are vasomiton models based on the Lie algebra SL2R and vasominoviton 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 dual to one of these vasominoviton 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 somehow what it means to become tensionless. Sorry, just about kind of curious of the historical perspective. So was there actually a argument or some strong intuition for why people thought this loka does not contain this metric orbifold point? Yes, yes. It has to do with the fact that so backwards in the period of a short flux have this long string solutions near the boundary of ADS. They are stabilized by the fact so the long string would like to contract. But then what it means is that you have fluctuations of this and if you go to the boundary they get redshifted and as a consequence you get a continuum of excitations. And in fact as when I will explain the Maldesino-Auguri Resumino-Witten model we will see this continuum. But on the other hand the symmetric orbifold of T4 it's not a rational CFT but it's almost a rational CFT. 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 Nervišvod-Nervišvod 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 now I use a small piece of 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 Resumino-Witten model has a very different flavor than if k is bigger than one. Somehow 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 spacetime spectrum it uses exactly the spectrum of the symmetric orbit fold of T4 in the large end limit where to qualify is the single particle spectrum because we're doing perturbative strength here you're 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 1 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 that is very intricate and I think it also suggests in some sense how this picture may fit into something that is ADS5 cross S5 and n equals to 4 super nil. So I think what I'll try to condens you is I'll try to explain to you this correspondence in detail and at the same time I want to condens you that this is not just some low dimensional accident it has many of the features it would also expect to be present for ADS5 cross S5 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 there's 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 dimensional 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 description of ADS3 so this is really reviewing the work of Maldesina Oguri Maldesina and Oguri they developed the description of the Resomino written model for things on ADS3 cross S3 cross T4 they developed this for general k and then what I want to explain to you is why 4k equals to 1 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 1 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 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 1 so k bigger than 1 roughly speaking is the symmetric orbital of sort of Liouville n equals to 4 Liouville it's not exactly like that so there is a recent paper of Lawrence Iberhutt in which he worked it out so it's the linear dilaton times T4 the symmetric orbital it's in the two-cycle twisted sector but also does something to the linear dilaton it's quite a complicated construction it's given good evidence that's the correct description the spectrum surely looks like the symmetric orbital of n equals to 4 Liouville that's the paper Lawrence and I wrote some time ago and on the face of it that looks very different than the spectrum of the symmetric orbital of T4 so if you're naive you would say it is unlikely that they lie in the same modular space I mean, could be that there is many things happen as you walk around thank you but here I'll concentrate exclusively on k equals to 1 I'll explain to you the Wesseham in a written model for general k and then I'll explain to you what happens at k equals to 1 k equals to 1 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 so in Newish word Ramon and because that's 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 1 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 Newish word Ramon then 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 have a p as u1,1 slash 2 Wesseham in a written model together with some topologically twisted T4 and whatnot so I want to explain to you a little bit about this to 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 1 over N corrections where the 1 over N corrections come from the higher genus contributions from the worldsheet that's what you would expect the genus of the worldsheet if you look at a torus amplitude its contribution is suppressed by powers of G string so the 1 over N effects in the duality of T should come from higher genus worldsheets and this covering 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 worldsheet and it reproduces correctly the 1 over N corrections of the symmetric orbifold from the worldsheet perspective so this is really in some things going beyond the planar limit of N equals to 4 because we also systematically understand all the 1 over N corrections rather than just the leading term as N goes to infinity but I think there is a question here this is perhaps a very elementary question so in zuminovaten models for compact groups but SL2R is a non-compact group and it has no SU2 subgroup so I suspect the level K won't be contentized 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 and in this background the SL2R vesominovaten model comes together with an SU2 vesominovaten model describing strings on A3 and then the fact the requirement that the string is critical requires that the level for the vesominovaten model here is the same as the level of the vesominovaten 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 1 is the smallest number you may have asked why not a half or 1 over 10 or whatever but it is quantized on account of the fact that you are really having an S3 factor and S3 is only consistent if the level is an integer but I'll come to that in due course thanks a lot there's another question so is it possible that K correspond to R over L since R is a continuous parameter as I said you have to treat these statements with a grain of salt this is true for large K for large K you approximate something the radius only makes sense in some sort of supergravity description classical geometry makes sense if the space is much larger than the string length if you are going to the regime where the string is as big as the space your notions of classical geometry are likely to break down one way or another and in fact in one way we will see this as you may know the SU2 vesominovaten model is quantized it has nothing to do with ADS3 if you just ask what does the string here in ADS3 look like you would say this is the SU2 vesominovaten model 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 for a very large level the level spacing is infinitesimal but when you come to very small radii then this becomes more pronounced and also as you may know SU2 at level 1 which describes the 3 sphere at the smallest possible radius SU2 level 1 is actually the same theory as a single free boson that's something people may know there's this free field construction for the SU2 level 1 theory so what this tells you is that when you think the string is propagating on a 3 sphere if the 3 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 a small radius is the same as the big radius and the critical radius is the same as the SU2 level 1 theory so you would thought the string propagates in 3 space direction but it actually only propagates in 1 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 worldsheet string theory rules okay are there any further questions let me start in 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 I have to explain to you how you really do this and there's a certain danger I'll get too technical so please slow me down if I say things that you don't understand and I'll try to explain 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 hyperboiloid in 4 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 or rather r2,2 and you impose the condition x-1 squared plus x0 squared minus x1 squared minus x2 squared equal to 0 that defines for you a hypersurface sorry equal to 1 yes, thank you that defines for you a hypersurface for you a hypersurface inside this space and this is the hypersurface that defines ADS-3 I mean it's the hyperboiloid inside r2,2 and obviously I needn't take this one I could also take it L squared but for simplicity that doesn't really matter now this actually is in 1 to 1 correspondence with group elements in SL2R so if I look at the group element if I prioritize my group element of SL2R as x-1 plus x1 x0 minus x2 minus x0 minus x2 and x1, x-1 minus x1 and you see the determinant of g is just well I have to take this so this will be x-1 squared minus x1 squared minus x0 squared minus x2 squared so the determinant equal to 1 is exactly this condition so describing this hypersurface in r2,2 is in 1 to 1 correspondence with looking at 2 by 2 real matrices that have determinant equal to 1 so this is so this wants to be an element of SL2R which is another way of just saying that it's a real 2 by 2 matrix with determinant equal to 1 and the determinant condition is exactly the hypersurface condition for ADS3 so that's one of the reasons why ADS3 is so simple ADS3 is a group manifold namely it's just a 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 so this is simple and I understand it's 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 sigma2 so unfortunately I have to introduce some notation now so please bear with me so we have e to the row times sigma3 and then we have e to the i times v times sigma2 the sigmas are the Pauli matrices so sigma1 is equal to 0 1 1 0 sigma2 is equal to 0 minus i i 0 and sigma3 is equal to 1 0 0 minus 1 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 ADS3 and u and v are related to the time and the angular coordinate as u is equal to a half times t plus phi and v is equal to a half times t minus phi so then if I write this out if I write out this group element then it takes the form g is equal to cosine t plus sin rho cosine phi and likewise here so it's cos rho sin t minus sin rho sin phi and then over here it's minus sin minus cos rho sin t minus sin rho sin phi this just comes from plugging in this formula these are 2 by 2 matrices you exponentiate them, you multiply them you rewrite it in terms of rho t and phi so this is not rocket science I'm just telling you what the answer is and then and this rho, phi and t coordinates are designed in such a way that the metric I mean the metric is induced from the r2,2 metric here if you go to the subspace and you parameterize it in terms of this coordinate so you notice that this has determinant 1 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 su2 in terms of explicit parameters I can parameterize sl2r in terms of rho t and phi and sometimes I write rho u and v u and v and t and phi are interchangeably used and then the metric is so that you have a sense of what this is cos squared rho dt squared plus d rho squared plus sin squared rho d phi squared and what this describes is a cylinder where rho is the radial coordinate of the cylinder so rho is the radial coordinate so rho equal to 0 is in the center of ads3, rho going to infinity is at the boundary and at the boundary our degrees of freedom is time and phi 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 and the reason I'm writing this down in some detail is that this description we want phi to be 2 by periodic so phi plus 2 pi should describe the same point of ads3 space but t and t plus 2 pi shouldn't 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 2 pi if I replace any t here by t plus 2 pi I obviously get the same matrix so it's not exactly true that ads3 is the group manifold of sl2r and the time direction so that I don't artificially identify t with t plus 2 pi which I don't want to do my time should just run forward and there shouldn't be somehow some identification after time as elapsed for periods of 2 pi so this is a point I'll have to come back to if I just treat it as an sl2r so in 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 do this periodicity in t which is an undue requirement ok so this is the geometry of ads3 and now I want to write down a conformal field theory so the sigma model on ads3 is not conformal on sl2r is not conformal but you know how to make it conformal you just add a vesomino term and then it becomes conformal so the worldsheet theory that describes that sort of the conformal version of strings propagating on this group manifold is the vesomino written model action and what does it look like well it looks like I'm not deriving this for you but this is what you find so you have some this is the worldsheet integral so it's a sigma model so it's maps from the two dimensional worldsheet into the target space and the action is trace g to the minus 1 dg g to the minus 1 dg that's the sigma model so g is a function of sigma and tau sigma and tau are my worldsheet variables so this is worldsheet so I'm looking at maps from the worldsheet into sl2r and then this is the metric that is induced from sl2r g to the minus 1 dg is just the tension vector so the trace of the tension vector so that would be the sigma model action and then you have to add this so-called vesomino 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 next 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 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 vesomino term now again I'm not really deriving this for you because so this vesomino term requires going to a three-dimensional surface whose boundary is the worldsheet then extending it in the interior writing down a three form but the upshot of it is once you vary it it leads to equations of motion that only involve the two-dimensional worldsheet 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 un-to-holomorphic whereby holomorphic now I mean so on the worldsheet we have the variable sigma and tau and what we are going to do is we are going to introduce the worldsheet 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 worldsheet so tau and sigma on the worldsheet and x plus and minus is on the worldsheet and on the target space we have t and phi and u and v 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 kone variables u and v and there is a two-dimensional CFT living on the worldsheet 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 are totally different animals they are going to get related to one another eventually but at the moment they are just very different types of coordinates now adding this mesomeno term what this implies is that you get a 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 1 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 mesomeno 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 of the ta matrix but that's just the relabeling of the indices but what's important is that you consider g to the minus 1 d minus g I mean there's a subtle point here you see this look deceptively similar but because this is a non-abelian group it matters whether you consider dg g to the minus 1 or whether you consider g to the minus 1 dg and once you've added the mesomeno term this is a purely a function of x plus and this is purely a function of x minus that's what adding this mesomeno term gives you it's maybe not obvious at this stage by that means it's conformal and in fact I probably won't really explain this but what it gives you is that you have that these currents satisfy an interesting OPE so these currents are functions of x plus and the world sheet is periodic so therefore we can expand say j a r of x plus in a Fourier series you see x plus because it's a function of tau plus sigma and sigma is 2 pi periodic we are in the world of closed strings so everything is 2 pi periodic in sigma this must be 2 pi periodic in x plus just to be sure so when you speak of the second model the SL2R are you already taking the 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'm not sure maybe 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 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 buy me what does this buy me that I have these 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 because there must be periodic in x plus so it must be of the form e to the i and x plus because it's 2 pi periodic so I can make a Fourier decomposition and calculate the prasant brackets that follow from that action and then I can quantize it and that turns the prasant 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 j plus with j minus minus 2 times j3 plus k times m times delta m minus n and then j3 with itself is equal to minus k over 2 m times delta m minus n so what you find is that the momenta 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 unit 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 prasant brackets it enters by virtue of the fact that it implies 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 then it basically looks like the lealgebra of SL2R so 3 plus minus plus minus gives you j3 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 in fact it's the loop algebra it's the lealgebra 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 so for physicists k takes the value for mathematicians it's an operator that's central that commutes with everybody else but in irreducible representation an operator that commutes with everybody else will just take one value because 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 the operator satisfies the jacobi identities that's in fact a good exercise to convince yourself that that's a consistent lealgebra and that's what the moments of these currents that come out of this vesselmino 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 katsmudi algebra and a left moving one and they don't talk to each other so there are some brackets of the life movers and the right movers is zero so there are two commuting copies of an SL2R vesselmino of an SL2R affine katsmudi algebra and the reason why this is conformal is whenever you have an affine katsmudi algebra you can construct by the shukavara construction a verosora 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 that's the reason why this guarantees that this is in fact a conformal field theory because you get a verosora symmetry which is the hallmark of two-dimensional conformal field theories ok, so what I want to comment on though is one important feature you see every affine katsmudi algebra contains a copy of the originally algebra inside it namely if you look at a generator as with mode number equal to zero they just form a conventionally algebra because you see zero plus zero is zero so you always produce zero on the right hand side and if m and n are zero then these central terms disappear because they get multiplied by m if m is equal to zero it's obviously zero 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 mobius group of the two-dimensional conformal field theory so this will be to be identified with the mobius group or the mobius generator obviously le algebra so it's the mobius generator so more specifically L0 of the space time theory will correspond to J3 zero L minus one will correspond to J plus zero and L plus one will correspond to J minus zero and L0 and L plus minus one are the generators of the mobius transformations 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 le algebra of SL2R and if you look at the generators of the Verozova Algebra and restrict the mode numbers to zero plus and minus one then it also closes and it is exactly the mobius group which are the holomorphic maps from the sphere the 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 one L minus one minus n so this should give you two L zero 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 well if it was j plus zero j minus zero it would 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 mobius group and that's the first indication that you're really getting well it's the global conformal transformation of the boundary but you're seeing the global transformations of the boundary surface and that would be important because you see what this allows us to do is to read off the space time conformal dimension from a worldsheet perspective the L zero 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 worldsheet so when we enumerate all the states, all the physical states on the worldsheet we are not just getting a whole bunch of stuff we are getting them filtrated by their eigenvalue with respect to an operator that we know what it means it's the conformal dimension of 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 barth modes which correspond to the mobius group acting on the antiholomofic coordinate in the space time CFT so this is and I'm coming back to the problem that Ivan asked the so the so normally when you, once you have a resomino, once you have an affine cut 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 resomino written model, tenser there's 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 dimension 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 this 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 graph 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 this commuting affine-Katzmudi algebras and there 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 representations 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 just 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 Wiedersruhr constraint, right? so in a moment I'm just talking about a worldsheet model as a conformity theory I haven't yet imposed the physical state conditions so what I'm doing here is I'm describing the worldsheet theory like you would construct a worldsheet theory of free strings propagating in spacetime but I haven't yet imposed the Wiedersruhr conditions I'll impose the Wiedersruhr conditions later on at the moment I'm describing the theory before I've imposed the Wiedersruhr conditions Thanks Now before I've imposed the Wiedersruhr conditions normally you would have here just some set of highest rate representations so what does highest rate representations mean? well basically it means it's a let me just say that and then I'll finish so highest rate representation is basically a fox space of the following form you take any of these generators all with some negative nodes and then you apply it to some set of ground states 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 mod algebra so any resumino written model known to mankind is basically of that form any highest rate representation you basically take the highest rate states i.e. the guys that are killed by the positive modes because they are 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 mode so the zero modes 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 representation of the zero mode algebra appear and there you use geometry or petaval theorem to determine it and for the case of SL2R that will be the continuous and the discrete representation and that I'll explain 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 and I'll explain to you where they come from they come from the fact that you demand that you also have solutions that are not periodic in T and this will be from the point of view of the representation theory the spectrally flowed representations and we'll see very explicitly how the spectrally flowed representations appear that's what I'll explain to you tomorrow so we'll do this model under control then we just go ahead we open green Schwartz-Bitten and we follow the rules of determining the physical spectrum we impose the berozovo condition and then we simply enumerate all the states that satisfy the berozovo condition and what I'll try to explain to you tomorrow is that that reproduces on the nose the spectrum of the symmetric orbit fold if you set k equals to 1 but that's what I'll do tomorrow and I think my time is up so I'll better stop here