 So remember we have on the one side string theory and ads three cross s three cross. And that we described in terms of an SL to our resume in a written model at level one. And equals one version say, I mean I'm a bit. I'm writing this again in the NSR formulation. I had the impression that the hybrid formalism didn't reach enormous level of enthusiasm so that's just stick with this and you can understand this as well and that's description as in the hybrid description. So this is a string theory on this. So this is the world sheet. And this is still to the symmetric orbit fold of T for which is the CFT. And what we want to do is we want to calculate a correlation functions in that theory. And the boundary of ADS three you should think of as being a cylinder and if you compactify it, it will be a sphere so these correlators will always live on the sphere. So this will be things like you, you take your space time sphere, and you insert various points where certain fields from the symmetric orbit would be reset. And throughout today, the coordinates in this space will always be denoted by X. So X will always be a coordinate of the correlation functions of the symmetric orbit fold of T for. So this will be distinguished from the parameter on the world sheet. Now the world sheet will not always be a sphere, although for most of today I'll concentrate on the case where the world sheet is a sphere. So in general here I will have a world sheet of whatever genius is appropriate. But for most part I look at G equal to zero. And here I will denote by Z. So, so just to be clear, whenever you see a Z, it will be a coordinate on the world sheet. Whenever you see an X, it will be coordinate in the symmetric orbit fold theory. And I'm looking at sphere correlators of twisted sector states in the symmetric orbit fold theory so this will be something like, if you if you wish. So the simplest case would be a correlator of this sort of type, the three point function x one x two x three are three points on the sphere. And this is say the twisted sector ground state in the w one w two and w three cycle twisted sector. So that's the sort of thing we are going to calculate in the X space in the symmetric orbit fold space. And I first of all want to explain to you how you calculate this correlate correlator in the symmetric orbit fold, irrespective of whether it comes from ideas or not just I mean how would you calculate this. And then I want to relate it to a calculation in the world sheet theory. And I'll explain to you how this correlation function of X will be reproduced from the string theory perspective. The string theory perspective life is a bit complicated, because as you know in general, we'll have to integrate over the Z variables. The insertion points of the vertex operators on the world sheet are modular you have to integrate over, say for a four point function you would have to integrate over the cross ratio of the Zs. And what we should manage to achieve is that after we've done this integral, we should reproduce the correlator as we calculated from the dual CFT. That's the roadmap we have in mind. So how do you calculate this correlate. I mean there are different ways of doing it, but there's one smart way of doing it. That will fit very nicely with what we have in mind. So how do you calculate that. Now remember what a sigma w field looks like. Remember, this has the property that if I start. So let's take w to be the cycle from one to w. And then if I start remember in this simple case of a single boson, if we take the Xi around this field, we ended up with the Xi plus one. So that was the effect of this twist field right it was to make sure the ice one, the first one goes to the second the second one goes to the third, the third, the w one goes to the first one. So it's a w. It's a it's a multi story car park, where if you go to the top floor and you go up one level you reach at the bottom. That's the structure of of this twist detector fields and now how what's the smart way of calculating it. So the smart way of calculating it is to change coordinates, and to think to go to the covering space. So we want to find the covering space locally. And what do I mean by the covering space. Well the covering space will be a map gamma, and I'll write it in a map of Z. So this said at this stage, it's not obvious that we will have anything to do with this. At this stage, it's just the parameter I call that to distinguish it from X, but I call it that because ultimately it will be connected to that set but that's far from clear stage. So gamma of that will be a map from some neighborhood of gamma to the minus one of X, who a neighborhood of X. So let's call this X zero, which is the point where the twist fields it. And this map should be such that that gets mapped for X zero plus a times X as that minus and let's call this point is at zero zero to the top. So what you're looking for is a map that sort of w fold covers the point X. So why is this a smart thing to do. Well think about in the coverings. So here's the covering space. This is the covering space in the covering space we have the point that zero. This gets mapped down in our real space. This gets mapped to X zero. And this map has the property that if I go once around up here. If I go, if I multiply that minus that zero by each of the two pie is that minus that zero, then I go w many times downstairs right because of the power of w. I circle once around here. I circle w many times around here. And you see after going around w many times, it's become single value. So this is the map that locally maps the individual coordinates the Xi single value. Right because as you go once around upstairs in the covering map, you go w times around downstairs. You go from X one to X two to X three to X four up to X w back to X one X two, each of them gets mapped to itself so from the point of view of the covering space. This map will be single value in terms of the individual the Xi's. So in some sense you've removed the effect of this twist by going to this covering space. Now locally, it's clear how you do it. I mean you just write down something like that. What you do there is, if you calculate the correlator of that kind. What you should be doing is you should patch together the local covering maps of that type into a global covering map, where you have some remand surface on top. That has the property that near it needs near each of these insertion points. It's doing exactly that and it's therefore undoing the twisting that happened downstairs. So first you may think is this possible. Okay, so let's give an example. The simplest example is you take three points with w equals to three. So let's take w equals to three at X one w equals to three at X two, and w equals to three at X three. So let's say you set X one equal to one X two zero zero one and infinity. And let's choose also the sets, the pre images to sit at zero one and infinity. That is without loss of generality, because you see by the movie symmetry, I can always move three points to three points. So the corresponding map, the corresponding covering map in this case is of the form z to the four minus two times z to the three divided by one minus two z. Okay, I learned this by heart, I wouldn't be able to spot it as I'm not that good a calculator but what you can do you can, you can either be smart and seed in your head or you can type it into mathematical. So let's go back with this how this behaves how does this behaves near zero. Well near zero you see the first time goes like minus two times that to the cube. So this is exactly of that form that map zero to zero plus some coefficient which turns out to be minus two in that case, times the third power of that minus zero, and then what happens near one. So if I calculate gamma of the Z minus one, but let's write a gamma of Z is of the form if I expanded around one it goes like one minus two times that to the minus one to the power three for that near one. And then at infinity what I have to look at is gamma to the minus one of one over you. So, caught in a chart around infinity and you map it back to run zero, and then you can read off that it goes like minus to you cubed, which describes it for you equal to zero, which is the same as that goes to infinity. So this is an example of such a map. I mean it's holomorphic. It's in fact a map from the sphere to the sphere. It's fourfold. I mean every generic point has four pre images, because it's a fourth order polynomial divided by a fourth order polynomial, the fourth order polynomial in the denoted denominator has been degenerate but it's fourth order over fourth order. So it has, it has four pre images, and it maps zero to zero one to one infinity to infinity. And at each of the points, it has a threefold covering. So it will undo this correlator. And the idea is that instead of calculating this correlator, I applied this covering map, which is a holomorphic map correlation functions transform nicely under coordinate transformation and under holomorphic maps. So I lifted up to a map of the covering space, and then on the covering space, this map is this correlator is effectively trivial because you see once you've moved it up to the covering space, then everything is single valued, any trace of this funny sort of insertions where things got sort of a twisted have disappeared. And because I'm looking at this twisted sector ground state. The idea is that once I've lifted it up to the covering space. That's the vacuum correlator, every, every single trace of this twist field has disappeared. The only purpose of the twist field in life was to impose this twisting, but by this local coordinate transformation I've undone the twist things or nothing is left. So therefore this correlation function is just a conformal factor associated to this conformal transformation. So I just have to calculate the conformal anomaly that comes from this coordinate transformation and then I've calculated this correlation function. That's my idea. That's the smart way of defining of calculating the correlation functions of symmetric orbital theories. No question. Yes. Shouldn't we expect some remnant of the parameter capital N. Oh, sorry. What I'm saying here is is what, well, no. Sorry, yes and no. Yeah, I'll come to that in a second. Let, let, yeah, you're right, but let me explain one thing and then I'll explain. Okay, so, so the point is, so in this example, the covering map, I've written down is, is a sphere. So in this example, this example is a map from the sphere to the sphere, but in general, the covering surface need to be a sphere. So the covering surface. So the covering map. If you allow yourself a possible covering maps, and you allow yourself your covering space to have arbitrary topology, then you get different answer you get different covering maps there's not a unique covering map. There's a finite number of covering maps, and in general the covering space that covers say this configuration needn't be a sphere. In fact, there is the Riemann Hurwitz formula. The Riemann Hurwitz formula tells you exactly what's the, what the genus of the covering surface is in terms of the parameters w and the number of covering so the Riemann Hurwitz formula is I is equal to one to n. So if an endpoint function w i minus one over two plus one minus G, where G is the genus of the covering surface is equal to M and M is the number of pre images number of pre images at a generic point. For example, for this example, you see w is equal to three, so three minus one over two is equal to one. There are three points so this gives you three plus one is four is equal to four so the genus is equal to zero. So this is an example of a map that's from the sphere to the sphere a covering map. It's a fourfold cover, each point in the downstairs sphere has four pre images in the upstairs. But in fact, there's also a torus covering. There is in fact also a torus covering where you choose the number of pre images to be three, and then you find out that the genus has to be equal to one. There are a number of different coverings, in particular for this configuration is at least a sphere covering in the torus. Now, coming back to Francesca's questions what is the role of the different. What is the role of the genus of of this covering map, and what you discover is that a correlation functions in their large and expansion. The one over and dependence of the correlators is controlled by the genus of the covering space. So the way you should think about it is that if you calculate these correlators, what you have to do you have to sum over all possible covering maps. You get some conformal factor associated to the covering map gamma. And if you expand this in a one over and expansion. What you realize, and this was an observation of rastelli and rather much is that a correlation function, if you ask. So the diagram of for given gamma, it contributes as n to the one minus G minus and over two. And then is the end of the symmetric overflowed. So this and so I'm thinking of taking n large. I'm calculating this correlation function in a larger and expansion. And what I'm saying is that the prescription this is some of all possible covering maps, from the point of view of conformity theory that's summing over the different conformal blocks. Given covering map contributes, if you think about it in the one over and expansion in a way that depends on the genus of the covering. And so that's a fact that you can just read off from the structure of the symmetric or before current. I mean, there's a, there's a theorem that you can calculate them in this fashion and then if you look at each term there you can say each term goes as n to the one minus G minus and over two where G is the genus of the covering map of the covering surface that I'm considering. Now, remember that n is to be identified with a one over n is to be identified with the with the genus. Now, from the point of view of the dual string theory that the genus should go like the string coupling constant should go like one of a square root of n. So if I write this in terms of the genus this will go like the genus of the string, sorry that string coupling constant to the minus two G. Now, it'll, it'll suggest that the genus, but that is exactly how the genus expansion of a string theory should appear so what it suggests is that somehow the contribution that comes from a specific genus here, if I interpret it, why are the ADS CFT reality. That should be the contribution that from a world sheet perspective comes from a world sheet of the corresponding genes. The contribution that's natural here that comes from just expanding out the one over X and expansion of the symmetric orbital correlators should let should match with the genus expansion as you write it down and conform a feed theory because the one over and expansion go behaves exactly like the G string expansion under the usual correspondence between the string coupling constant and one of a thing there's a question there. Okay, when we do an orbit for the usually today we, we get some symmetry, and that we usually call quantum seemingly corresponding to this orbit folder. And usually when we have a symmetry, the correlators will will fit in some, we will follow some selection rules or something related to the symmetry. In the year we can understand something of this, what we get from from this or. Well, first of all, I think the quantum symmetry is usually you can undo the orbital by taking the orbital with respect to the quantum symmetry that only works in the abelian case. You can't undo non abelian orbitals. But obviously you're still have you may so so the obvious quantum symmetry just work for a billion orbitals, because the twisted sector just you can characterize the twisted sector by some face. And then the faces give you selection rule and if you're all before by that to go back to the original theory from the non abelian orbital you can't do it. So it's this complication that the twisted sectors are now labeled by conjugacy classes rather than group elements so there's no easy way to reconstruct the full original theory from taking the orbital of the orbital. At least that's as far as I know that's not known. There is no the story about non invertible seamless in this case. So maybe there's something like that but certainly none of the conventional or befalls will do the job. Now maybe maybe there is some more general construction but I'm not aware of a general construction. And I actually I talked to you fun about this and I think I think that would be interesting to see what one could use non invertible symmetries here but that's certainly not not now obviously there are selection rules so for example, this can only be non zero. If you can find three, three cycle permutations that multiply to give you the identity. So, but that you can do. Right, I mean you could for example take three times the same one, but that would give you the tourist covering here you have to take one that involves for four different numbers. There are different ways of multiplying three cycle permutations together to give you the identity but beyond that I don't think there's any global symmetry for a non a billion or before like this metric or before. Thank you. There's another question over there. So the cover coordinate you called Z. Yes, and then the same Z as the word. So this is anticipating what I just explained. I mean, at a covering space at a moment I shouldn't have called that I should have called baby Z term because it does at this stage nothing to do with the Z. But over there I explained because of the one of our independence you will expect this Z to be the Z and the upshot of my talk today is that this that is this. So I'm I'm I'm I'm sort of suggesting to you already where I'm going, but I'm using a notation that will go there but at the moment this has nothing to do with the world. This is just some abstract covering space. So I want to be sure I understood this sum over gammas. So triple and function. It seems that there are possibly a unique. So are you referring to higher point function where you want to take. But three point function. I mean the question is how many channels do you have right. Yeah, so you have mine the four or higher point function I guess. I mean they know P where you have an infinite number of triple and function then an infinite number of gambas something like that. I mean for a fixed three point function there. There are different channels so I'm probably going to screw this up but so so there's one obvious channel which is so you pick the permutation 123 123 and 123 right. That would be one solution right. But remember in the symmetric or default, you have to pick representatives. So you. So when I say a w cycle to detect that what I mean you I take any w cycle. But I haven't specified that it's necessarily 123 right because I'm labeling the states by elements in the conjugacy class. So this will be one configuration that will contribute to the three point function, but there's also another. And if I can find my notes, I can tell you what it is. And maybe I can't tell you it is there is one. There's another one. 341324 and 123 that will also do the job. Right. So one goes to two, two goes to four, four goes to one, two goes to three, three goes to two, three goes to one, one goes to three, four goes to three, three goes to four. And that's also another solution. So there are two conformal blocks for the three point function of a three cycle twisted sector operator, because that is a different channel than that. Right. Because you see that involves three, three elements in the covering space that develops that involves a fourfold cover. So that's the genius one contribution and that's the genius zero contribution. And when you calculate this correlation function you have to sum over the conformal block coming from here and from here. And what I'm saying is in the large and limit this conformal block dominates over this conformal block, because this conformal block will be down by a power of one over N, because it has genius one relative to the zero. So if you look at more complicated correlators you get finite number of contributions from different genius, and they're one of our independence is controlled by the genius of the corresponding covering surface. Thanks. So another mild question is, there are other correlators right this is twist fields. Why are you focusing on the specific set of correlators matching and not. So first of all, I'm looking at a single cycle. I'm looking at a single particle states from the string theory perspective so I'm only going to look at states that live in a single cycle twisted sector. So they'll always live in a cycle in a cycle of one length, because otherwise they would be multi particle states and I wouldn't see in perturbative string theory. So I don't necessarily have to look at the ground state. I couldn't look at some excited state, and the punchline is, if you look at some excited state, then you don't get a trivial correlator on the covering surface but you get a correlation function of the seed theory on the covering surface. So you can relate it to something relatively simple and the interesting bits that I'm trying to explore to explain to you really hinges on which cycle it comes from rather than exactly which state you look at the covering map is always the same Okay, any further questions. Okay, good. So now, now I've written here some. Now, people know that when I write sums I don't necessarily mean sums maybe sometimes I mean integrals because I've been sloppy when I wrote down the spectrum of the SL2 hours we know written model. But the point is here I actually mean a sum covering surf covering maps are rare. And they don't exist covering maps are something that's, they're really a finite number of covering that says not a continuum of covering. And in order to explain that and that will be crucial for what we are going to see. Let's look at the case where you look at the genius zero covering. So let's look at the genius zero case. So let's lose use the remind who of its formula for the example of G equal to zero. So then this time isn't there. So, if you have on the sphere, what does the most general holomorphic map look like. Well my function gamma of that will be of the form minus PM M of Z divided B by P plus M of Z, where P minus plus of M of Z is a polynomial of degree and M. So this is the most general M to one map from a sphere to the sphere, right, I mean it's, it's just the ratio of polynomials. Now what is the condition for this to be a covering map. Well the condition has to be that gamma of Z should be equal to Xi plus something of order that minus that I to the w i. So I have in mind. I have an endpoint function. So I'm having endpoint function, and the point sit at coordinate Xi, and they have winding w i, and I want the, the covering the pre images to sit at position that I. So I'm asking. So what this means is that near that equal to set I it has to map to Xi, and then it has to be order that minus that I to the w i. That's the covering that condition. So you see, we can just plug this answer into this formula. So what does it say, but now maybe people can see anymore so maybe I'll continue over them. So what this means is that minus PMM of that divided by P plus M of that is equal to Xi plus order that minus that I to the w i. So we multiply by P plus M of that this is some standard generic polynomial, and then bring it is to the other side and what do we learn we learn we get the equation PM minus of that. Plus Xi P plus M of that must be of order that minus that I to the w i. So that means I have to satisfy in order for this to be a covering, right. I mean if I multiply this rule, this is a generic polynomial it will generically not have any zero or anything at z equal to z i. So therefore, you can just multiply through so Xi multiplies P plus, you bring P minus to the other side, and then has to be of order that minus that I to the w i. So let's count how many parameters we have in our ansatz and how many constraints we have to satisfy. Well how many parameters do we have well this is not polynomial of order M. So it has M plus one parameters. This is a polynomial of order M so it has another M plus one parameters so I have to M plus two free parameters, but one parameter drops out because I'm looking at a ratio. The overall scale drops out. So the number of parameters in my ansatz is equal to two m plus one. On the other hand, how many constraints do I have. So the number of constraints. I have to satisfy is equal to well at each point Xi. So at some for each I so some I is equal to one to N. I get how many constraint, I get w i constraints, because I have to make the order zero term vanish the order one term vanish up to the order W minus one So I get w i many constraints. But now let's look at the Riemann Hurwitz formula. You see the Riemann Hurwitz formula tells you that the sum over N of w i is equal to two M minus two. So this is equal to two M minus two plus and that's the number of constraints that you have to solve. If you want if you so what we are specifying here is we're specifying the coordinates Xi, we specify the W is and we've specified the Z is. So with all that information we have two M minus two plus and constraints, but we have only two M plus one, three parameters. So the number of parameters minus constraints is equal to two M minus plus one minus two M minus two plus N. So that this is equal to three minus N. So what does this tells you is that you see generically if N is bigger than three, you have more constraints than parameters, which is another way of saying that generically the covering map doesn't exist. And that will only exist if the Z is satisfied and minus three constraints. I mean that the Z is must must lie on a hypersurface of co dimension and minus three in order for you, at least generically to have a chance to find a covering map because the number of parameters is simply too small to satisfy all the constraints that you have to satisfy. So that's a very suggestive number. Right, because, you know, if we are doing world sheet theory on the sphere, if we're calculating a string diagram, we have to calculate, we have to integrate over the Zs. Three of that integrals are for free because of the furniture symmetry. So we always have to do n minus three Z integrals, we have to cross integrate and minus three cross ratios. So there are exactly n minus three constraints that allow us for the existence of a covering map. So this is the structure. So if you. So this suggests the following, the following way in which this wants to work. So suppose I can, I can write these constraints in terms of the variables is that for up to ZN, then I would expect that from the world sheet point of view. So, so, so far I've done symmetric orbital analysis now I make a leap of faith. Now I say how could I possibly get it from the string theory side. And there's a very, very natural way in which you can get it from the string theory side and the natural way is the following the vertex operators on the string Well, they're going to depend on Xi and that X one and that one, and up to WN X and and that and, but they should be a sum over all the covering maps, labeled by Jay, and it's for each covering map. So you get an n minus three delta functions, and the delta functions will be of the form, gamma j to the minus one of X for Xi minus z i times some conform the factor. So I propose, that is a natural way in which the world sheet theory should look like, and why is this natural. Well this is natural because once you do string theory. Remember in string theory we have to do the integral over DZ for up to the ZN of this correlation function. Right, because if you calculate an endpoint function on the sphere there and minus three more July we have to integrate over here I've for simplicity I've chosen that one the two and that three to be fixed. So then I have to integrate over that for up to ZN. And then when I do this integral, you see this delta function will just kill the integral. So this is exactly the configuration of z for which the covering map exists, and then I will end up with the sum of our covering maps with the conform the factor of the covering map, which is exactly what the symmetric or before the answer is. And if you stare at the symmetric or before answer given the fact that the covering map only exists on co dimension and minus three, what it suggests is that a world sheet correlators are actually delta function localized on the loci, loci, where the covering map exists and then the integral over the modular of my world sheet will automatically produce the structure of the correlation function that's calculated by the symmetric or before theory. It's a bit of a, I mean, probably takes a moment to get used to it I mean it's very unusual to think about a correlation function to be delta function localized that's not if you open a yellow book on a random page. That's not what CFT correlators look like CFT correlators are rational functions so this looks very far fetched. But if you think about what would the world sheet theory have to be like in order to reproduce the symmetric or before answer. That's the only thing that can really work right it has to. I mean, I mean everything smells very strongly that that's how it wants to be. At least, if this was true, then it would manifestly reproduce the symmetric or before the answer because the integrals over the free parameters in my string theory just kill the delta functions that appear here and I'm left with a some over covering maps as calculated from the symmetric or before the perspective. So that's our claim our claim is that this world sheet theory has exactly this property. Now this as I said this is a, is a bold claim because this is not what normal conformity is looked like. But our conformity there isn't entirely normal. It has this funny spectrally float sectors it has all this funny stuff, and unfortunately I don't have that much time left but what I want to do in the rest of my lecture is to convince you that we have actually proven that this is true for our world sheet theory our world sheet here has exactly that structure. And thereby it manifestly reproduces this metric over for the answer, and this does not only work a genus zero it also work at higher gene so if you do the corresponding analysis here you get a minus G term from here, counting for the, for the genius, the genus of the genius, and they, they are always exactly getting fixed by these constraints, and it, it reproduces exactly the structure that I was outlining here, namely that the world sheet gives you exactly the contributions, the world sheet of genius G gives you the contributions that come from the covering surface of genius G. That's exactly how it works. Now, this is a little bit. So, so what I want to spend the rest of my. So, unless there are questions, what I want to spend the rest of my lecture is to explain in how you can prove that this identity is true in our specific world sheets here. And obviously I won't give the full proof because I don't have enough time but I want to sketch how this works. So if you ask, ask me anything if this is unclear what I'm trying to do. I'm not sure whether that means that I've lost all of you or whether it's abundantly clear. But if you don't ask me there's nothing I can do. Okay, so, so how do we, how do we try to prove this from the symmetric from the world sheet perspective. So there's one key key insight, which in retrospect is a pretty obvious key insight, but maybe it's something that people didn't quite appreciate in its full significance. So in the first three lectures what I explained to you was how the states of the symmetric or before theory can be reduced from a physical state on the world sheet. So I established a map from a vector space of states here to the vector space of states here. Now, as you're probably familiar in two dimensional conformity in fact in higher dimension conformity as well. There's what's called the operator state correspondence to every state, you can construct and operate. But here we have to be a little bit careful, exactly what we mean by. So suppose I pick a state in the symmetric or before theory. This is a state now. And in the symmetric or before theory, I do the usual operator state correspondence which means I identified states with the corresponding vertex operator at x equal to zero. That's the usual way in which you do it and conform a feed theory at x equal to zero the vertex operator creates the state, but this is at x equal to zero because this is the symmetric or before T for that lives in X space. Now, what I explained to you is that this state corresponds to a state that which I call Phi hat in the world sheet theory. And in the world sheet theory, I also have the operator state correspondence. So to that state, I can associate it a vertex operator which I'll denote by Phi head of Z. So that's the vertex operator that's associated to the state. And again, I make the association as is conventional to say that the vertex operator at zero, that light on the vacuum produces for me the state, just like you always do into the CFT. Now, let me remind you how you can characterize the Z dependence of such a vertex operator. The vertex operators actually totally obvious, because you see we know what the translation operator is on the world sheet the translation operator is e to the Z times L minus one. And therefore, this vertex operator is really necessarily of the form e to the Z L minus one, the vertex operator associated to Z equal to zero, e to the minus Z L minus one right the the Z dependence is always trivial it just comes from with the with the translation operator and the translation operator on the world sheet is L minus one where L minus one is the real server generator on the world sheet. But it so far this describes the vertex operator that corresponds to the state being inserted at X equal to zero. So this is written here. This is the vertex operator corresponding to X equal to zero, and that, and this is the vertex operator associated to X equal to zero, and that is that equal to zero. Because you see I've identified it with a state and the state is really associated to the vertex operator in the dual CFT being at X equal to zero. So you can ask what is the vertex operator associated to this symmetric or before state if I inserted at an arbitrary position X, and an arbitrary position X in the dual CFT, and an arbitrary position that on my world sheet. Now, given that I've just explained to you how it works for the world sheet CFT. It's now obvious what I have to do. Also in space time, there is a translation operator, there's a mobius group, and therefore the X dependence of the space time vertex operator comes from conjugation by the translation operator in space time. What's the translation operator in space time but it's L minus one space time, but L minus one space time from the point of view of my world sheet theory is simply J plus zero. So obviously this is equal to e to the X, a plus zero e to the Z, L minus one, the vertex operator associated to the state X equal to zero at coordinate zero, and then it's just conjugated by the corresponding. And what I'm using here is that J plus zero is the same as L minus one in space time. The vertex operators on the world sheet will depend on X and on Z. And there is no ambiguity in how to define this once I've identified the states. It's clear how it will depend on X and Z independently because I know what the translation operator is on the world sheet. And I know what the translation operator is in space time. And note, there is no ordering ambiguity here because J plus zero commutes with L minus one. So there is no minus one and J plus zero commute so there is no, there's no funny ordering I have to choose whether I first translate in space time, first on the world sheet and then in space time or first on the world sheet and then space and doesn't matter. They commute with one another. And that's, that's how the vertex operator will depend on X and Z. And this is what I meant with this symbol. This is the vertex operator that will depend on where I insert a state in the dual CFTI where this crosses is an X space. And it depends on that, that is where the corresponding vertex operator is inserted on my world sheet on my world sheet surface, and the Z is the stuff I have to integrate over when I do a string theory, so that at the end of the day. I'm only left with the function of the exits. What I have to do is I have to calculate the correlation functions of these vertex operators. Okay, so now I at least know which problem I have to solve. And now I can go about trying to solve that problem. Now the, the, the key trick. So, so you may think, oh, what's the big deal about this funny exponential, but this funny exponential has a very significant impact. It has a significant impact of that exponential. Well, so I was planning to, it's most easily explained in terms of, if I can find my notes. It's most easily explained in terms of introducing the free field realization of SL2R. But, or should I do it, can I have maybe a few more minutes. Yes, yes, sure. Okay, so. So I have to, so at this stage I have to do introduce a fee for this is for the purpose of making this a little bit more pedagogical otherwise it becomes a little bit complicated. What I claim is that this has a free field realization in terms of what I like to call symplectic bosons. And string theory people tend to call this a beta gamma system. So these are bosonic coordinates, and they're characterized by the following commutation relations psi alpha beta beta s is equal to epsilon alpha beta delta r minus s. So beta take values in plus minus epsilon plus minus is equal to plus one and epsilon minus plus is equal to minus one and the other epsilon zero. So these are, this is like a beta gamma system. And then I claim this beta gamma system realizes the SL2R f and cut smoothie algebra at level one. So I write the plus the j plus and j minus generators that are bilinear of the form. So j plus is eta plus psi plus j minus is eta minus psi minus understood as as fields, and then j three is equal to a minus a half times eta plus psi always normal order minus a half times eta minus psi plus. Okay, so I hope that this is not not scary for you I mean these are free fields this is like a beta gamma systems or this is not so different from stuff you've seen. And what I claim is if you look at these combinations, they all spin a half field so these guys has been one fields. And when you calculate the OPEs you realize SL2R at level one. That's what I claim. For example in string. Sorry, is it exactly the beta gamma system or for the boson ice to like in string theory we have this. Well, I mean it's it's bosonic field to have a simple pole OPE right that I thought is called beta gamma system right. I mean it's, I mean the corresponding field psi alpha eta beta of Z and W goes like epsilon alpha beta over Z minus w. That's the OPE right I thought that's what you call a beta gamma system. I mean that's what I mean, that's what it is. Right, and if I write it in terms of mode that's what it looks like. And then if I look at the normal order products that gives me SL2R at level one. Thanks. Okay, so that is that is. Now I've lost my notes again. Okay, so this is, so this is something you can check and it's not, it's not a rocket science to check it, and it's true. It requires maybe a little bit care but it's true. Okay, so let's take that. And then remember, in the world sheet theory don't have just the SL. So I'm looking at just the bosonic piece of it and just looking at the SL2R. I'm ignoring the fermions. So the SL2R bosonic algebra can be generated by these fields. And then the other thing we need to know is what spectral flow does. And I explained to you a spectral flow does to this fields, but actually this spectral flow comes from a natural spectral flow that you can formulate in terms of the, in terms of the three fields. And in terms of the three fields, the spectral flow is that sigma of sigma of psi plus minus r is equal to psi plus minus r minus a half minus plus a half, and sigma of eta plus minus is eta plus minus r minus a plus a half. So this is how spectral flow acts on the three fields and that just generates the spectral flow as we've seen in the generator. So you see the plus components get shifted downwards, the minus components get shifted upwards. So the plus component gets shifted downwards by two minus a half, so it gets shifted down by minus one. So the minus component gets shifted moved up by two plus one, so it gets shifted up by plus one, and then the zero mode stays the same and a normal ordering term gives you the shift in the zero mode. Anyway, I mean it's, if you don't trust me on that then I probably have no chance to explain you the rest. This is not, this is not rocket science. This is something to do. So why, why is this funny formula, why am I so excited about this formula about the correct definition of the vertex operators. So why is this such an important formula. You see what we want to do in order to understand these correlators, they want to understand the world identities with respect to these three fields. What we're going to do is we're going to look at the correlators where we insert one of the symplectic bosons and see what the poles are of the symplectic bosons with respect to everybody else. So what we're interested in is the pole, the OPE of psi plus of zeta with a vertex operator of this kind. What's important is you see j plus is proportional to eta plus and psi plus, and eta plus and psi plus commutes both with psi plus, because the only commutator that's not trivial is between psi plus and eta minus, but there's no eta minus inside. So this psi plus doesn't care about the fact that there's an e to the xj plus zero standing here, because it just goes through. So it will behave exactly as it originally behaved. And because it gets shifted downwards, what you find is that this OPE you can calculate. So this goes like the sum of our of the mode numbers and it'll be so let's specify the spectrally flowed sector by this upper index. What you find is that this is going to be psi plus r minus w over two in this w fold spectrally flowed sector you shift this down by w times minus a half acting on fine hat, and then x and z. And then it goes like a zeta minus z to the minus r minus a half. That's just the usual way the OPE works and this is independent of the x variable because it commutes with the e to the xj plus zero term. And then if you look at it what's the first time that survives. If I had this the highest rate state, then this will survive is R is a is a less than w over two so the leading term will go like zeta minus z to the minus w plus one over two. And then it'll go like this term. But that's the structure of this OPE. So the XI plus OPE with this is this form. Okay, so that's how XI plus works. But now what about XI minus. So when you try to do the same calculation with XI minus you see previously we didn't have this factors of of x here after we didn't have this factor of e to the xj plus. So it's like the OPE we would have calculated previously, except we have to ask what happens when I move XI minus past e to the xj plus. And that you can easily calculate what is XI minus over zeta with e to the xj plus zero. But if you think about it, you can expand this out j plus zero has a term that matters. Because there is a XI minus is going to eat the eta plus and spit out XI plus. And what you're finding is that this is equal to e to the xj plus zero times XI minus of zeta that's the term where nothing has happened, and then you get a correction term, the correction term is goes like minus X times XI plus of zeta. And that just comes from the fact that you are moving to see as you move this through, you pick up a commutator term of the XI minus with the j plus zero, and that produces you have XI plus term. And that's really algebraic right I mean there's no, there's no guesswork involved here this just follows from the commutation relations of these modes with one another. So, so white XI plus is invisible to the e to the xj plus the XI minus isn't. And therefore, when I calculate this OPE. If you pass the e to the xj plus zero, I have to calculate the OPE of minus x times the OPE I've just calculated, and then I have to calculate it so this will go like minus X times this OPE, except I sort of moved out the, but actually does there it doesn't matter because that commutes with X. It has a term that goes like plus the OPE I would get from XI minus, and the OPE I get from XI minus is actually much more regular than normal because under spectral flow XI minus modes get shifted the opposite way. So what they did what you find is, it goes like the correction term goes like plus zeta minus z to the power w minus one over two times the w of XI minus of zero on Phi X and Z. So, so the important fact is that XI plus and XI minus have a different OPE when you bring them near these vertex operators and more importantly, this OPE depends in an intricate way on the coordinate X, where you've inserted the vertex operator in the dual CFT, and algebraically it comes from this identity. This is the key identity. The key identity is that the translation operator in space time has an impact on XI minus geometrically what this means is you see you've picked the spectral flow direction. But under translation in space time the spectral flow direction isn't really invariant and you pick up correction terms, and the correction term you pick up is this correction term. Obviously when you said X equal to zero this goes away, but for X not equal to zero, you get this correction term. I mean it really comes all out of just doing things properly. I mean there was no ambiguity in how to define this operator. There's no ambiguity in calculating that. There's no ambiguity in calculating these OPEs. This is just dictated to you once I've defined the theory. There's no freedom here. Okay, so now, now there's one small additional fact I have to mention and then we'll, and you'll see what happens. Super string theory and super string theory correlators are awkward, because there are things like picture changing and stuff. So we have to incorporate picture changing, and the correct way of incorporating picture changing in this context is that you, is that you can use the real correlator as some additional picture changing operators inserted on and what this means is that the, the real correlator that you have to calculate is not just the product of all of these vertex operators, but the real real correlator that you have to calculate is the product that has alpha equals to n minus two of these sort of picture changing type operators. And then you have this product of I is equal to one to N, VWI of Phi hat I xi and so that's the physical correlate that's the physical correlate. So what we're discussing here the only thing you have to understand is the OPE of xi plus and minus with the W's and the OPE of xi plus and minus of zeta with W of u alpha goes like order of zeta minus u alpha to the power one. They're all, they're all regular. Okay, I've accumulated all the data that I need in order to explain to you the water identities. Yes. My question is about the correlators that do not necessarily preserve the winding number. So the spectral flow variable. So, they are totally singular. These are the vertex operators where you've moved the points here all into one position. These are non-zero, right, so you can compute them. But they're singular. They are not, they are not the right correlators to consider. These are the, so I mean if you ask about it from the point of view of the of the symmetric orbit fold, the winding number will only add if the two points sit on top of each other, then the winding number obviously adds. So then don't sit on top of each other it never adds. But what would be the string shear interpretation because from the string shear side we know that the winding numbers can be in principle violated in this case. So these are correlated so previously people have calculated these correlators but they didn't introduce the X variable. If you don't introduce the X variable what this means is that all your points are bunched up on one point. So then from the point of view of the dual CFT you're calculating not a correlation function, you're calculating the limit of all the points coming together then obviously winding number adds. So if you want, if you're interested in the honest correlator of the symmetric orbit fold then you have to evaluate them at different points, and then winding number will not add as we will see and it's all characterized in terms of covering. But that's the data that really controls the correlators of the symmetric orbit fold that controls the one over N dependence and so on. So, so the correlators that were looked at previously are sort of the funny singular limits. And it's not the correct I mean in some sense you must be able to retrieve them from there but they are very very singular that's the limit where the covering surface degenerates. So the regular correlators where they sit at different points winding number can never add and you get the correlate you get the, the, the structure of the fusion rules that are appropriate for the symmetric orbit fold. That's exactly how people calculate symmetric orbit fold correlators. Okay, so now our plan is that we're looking at these correlators, and we want to analyze the world identities and in order to analyze the world identities. What we have to do, we have to insert side plus and minus and now I've managed to mislead my crucial piece of paper. But maybe I find it here. Yes, I find it is the final one so the end is near. So, I propose to define the following functions. So, okay, let me let me just write it down it's a product of one to N of zeta minus Xi to the w i plus one over two divided by the product of alpha is equal to an n minus two of zeta minus you alpha. And then it's the correlation function where I insert zeta xi plus of zeta, and then I put the product of the w and the product of the piece. So it's basically this correlator. And what I do is I insert side plus and minus inside the correlator here, and then I multiply it with this funny prefactor. Okay, so. So now what is what are these functions. Well, let's think about it. So let's ask, where, where do these functions have a pole. Well, originally you would think these functions have a pole and Xi plus and minus go close to one of the vertex operators. But if Xi plus comes close to a vertex operator the Polish of order minus w plus one over two. So Xi minus comes to occur to the vertex operator the Polish of the same order because of this term there's also a pole of that order, but in those types, I've exactly removed that pole by multiplying it by zeta. So this is zeta minus z i with a w i plus one over two. So this factor is designed to kill this pole. Right. I mean, Xi plus near each of these vertex operator has a pole of order w plus one over two, and I multiply it by a pre factor by hand that is just a zero of that corresponding order. So there's no poles at zeta equal to the z is and at zeta equal to w it has a simple zero. And I've cut divided by the simple zero so it still doesn't have a pole. So this function doesn't have a pole in zeta, and a function that doesn't have a pole in zeta and is holomorphic is called the polynomial. So P plus and minus of zeta are polynomials, because they don't have any zeros and they don't have any poles. So this is the degree of the polynomial. Well the degree of the polynomial is what, well I have to calculate the degree of the pre factor so I get the sum from is equal to one to n w i plus one over two. Then I get minus. The pre factor coming from here that obviously adds to the degree and then this subtracts to the degree. So then I have minus the sum of alpha is equal to one over and minus two times one. These are the things I divide out through. And what is the degree of this. So the degree of this you can determine by asking how does it behave when zeta goes to infinity, but you know how it behaves when zeta goes to infinity because zeta is a spin a half field. So it'll go like one over zeta. So there'll be a minus one from the behavior. That's the zeta behave the large zeta behavior of this correlation function which simply follows from the fact that zeta does xi plus does out at infinity. So now if I find my, my help sheet, then I will realize that this expression here is exactly equal to. So now I can't find my sheet anymore but this expression is exactly equal to M. So this goes exactly like, I can rewrite this this goes like this goes like the sum over is equal to one to N of w i minus one over two, because I can subtract these two these things out, and then plus one, because I'm only checking out N minus two ones and I have to add in another one so this is the degree of this map. And this is exactly M as coming of the Riemann for its formula that I described before. So these are polynomials of degree M. Right. I mean they have no, they have no poles. That's the degree and the degree is exactly the degree as predicted by the Riemann for its formula. So what about the characteristic identity that characterizes the covering map. Remember, what we have to look at is P minus of zeta minus x plus xi of P plus of zeta. But look at this, you see if you look at this combination, then see minus of the C minus of zeta plus x time P plus this kills exactly this term. So what's left behind is it goes like something to the order. So this goes like order z minus zeta minus z i to the power w i minus one over two. And remember there's an additional pre-factor standing outside that goes like zeta minus z to the power w i plus one over two plus w i plus one over two. So you see that this combination goes exactly like zeta minus z i to the w i because w i minus two plus w i plus one over two is w i. And therefore, these functions are polynomials of degree m that satisfy the characteristic equation to be defined in the covering map so I conclude that the covering map gamma of zeta is actually equal to P minus divided by P plus. So this is the minus sign for button and somebody remind me where I dropped my piece of paper. So this is my so this is minus. This is minus P minus of zeta divided by P plus of zeta. What this tells you is that the worksheet theory allows you to to recover the covering map. The covering map arises naturally out of the worksheet theory by virtue of looking at these correlation functions and just dressing them up so that they become polynomials of the right degree and the construction guarantees that these correlation functions of the worksheet theory produce for you exactly the covering map. But remember, the covering map doesn't generally exist. But if the correlators produce you the covering map, then something with this construction must go wrong for those configurations where the covering map doesn't exist, because we know the covering map generically doesn't exist. Now if you trace through this argument carefully what you see is that the thing that has to go wrong is that P plus has to go to zero. So the question must break down, and then you can show that if you plus goes to zero then the correlator without the insertion of psi plus must also go to zero. So you prove from that that because you can reconstruct the covering map from the correlators by inserting an additional field, and you know that the covering map will sometimes not exist. You can prove that whenever the covering map doesn't exist. You can prove that the correlators, even without the insertion of the xi fields must be equal to zero. I the correlators on the world sheet can only be non zero at the location where the covering map exists. And if you do a little bit more work you can show that it's actually a delta function type localization property, so that when you do the integral over the world sheet modular you really recover the sum of covering maps. So that's the reason why this world sheet theory knows about covering maps, and what it really tells you is, you see this identity. This identity tells you that psi plus and psi minus want to be the twister variables. And this is like the incidence relation that recovers the space time coordinate from the twister variables. What I strongly suggest is that the symplectic bosons played a role of the space time twisters, and the localization, the incidence relation of the space time theory is encoded by the world identities of this symplectic boson theories. And this is the idea that has then subsequently motivated us to try to imitate this also for ideas five and, but obviously I don't have any time anymore to sketch that but here in the ideas three case we can really nail this down. And this is exactly with what you would expect from the symmetric or default answer. So my time is truly up so I'll stop here and thank you for your attention. Thank you very much for this beautiful set of lectures.