 Yeah. Oh, now you can see everything very good, very good. Okay. Okay. So we go to the next page. Can you still see? It's a little out at the top from now. Now it's good. Now it's good. Yeah. Okay. Now the integrable highest weight g-height modules with central charge is small c. So, so see this is small c was the level of the central charge which was fixed once and for all. So now let's look at integrable highest weight g-height modules. So g-height is the finally algebra with central charge c and central charge just means the action of this capital C, the central element. So that is in bijective correspondence with the set dc which was introduced on the last page. So dc was this set consisting of dominant integral weights such that lambda theta check is at most c. Okay. So, so this dc parameterized edge all the integrable highest weight g-height modules with central charge is small c. And let's denote given any lambda in dc, let's denote by each lambda the corresponding integrable highest weight module. Now, so this is the fundamental object in this conformal field theory. So this is the space of dual conformal blocks. And there is also a similar conformal block, but that's just the dual space of conformal blocks. Okay, so now we have already taken the S pointed curve sigma p, where this p vector p consists of as many points, all of them are smooth and distinct. Sorry, I should have mentioned that there is smooth and distinct points. Now, we take one more ingredient here. And that is this vector lambda, which consists of lambda one through lambda s exactly as many as smallest many. And this lambda i is attached to the point p i. And what is this lambda i so lambda i is an element in DC. So we are thinking of that each point p i is attached to a representation h of lambda i with same central charge so we are not changing the central charge c. So we define h lambda to be the tensor product of all these. So h lambda one tensor up to h lambda s. So that's now on this one. So this is the algebra. And what is this the algebra g tensor c sigma minus p. So what is that. So we are removing the points of P one through PS from Sigma. Okay. And now we are taking the affine coordinate ring there. And we tensor it with G. Now this is the algebra. Okay, so so I should define the bracket first. So the bracket here is maybe maybe again it might go off the screen. So the bracket here is extensor f y tensor g. That's the plane and simple bracket x y tensor f g where f and g are functions on sigma minus p. Now there is one technical thing I'm crossing over. And that is the choice of P should be such that every useful component of Sigma must get at least one point. Sigma minus p and a fine variety and that will that will give up for us a ample supply of regular functions on Sigma minus p. In any case, now the basic object is this dual conformable blocks. And that is, we take this H Sigma, sorry, it's Lambda, it's vector Lambda. So that's the tensor product of these representations. And we mod out by action of G tensor c Sigma minus p to each Lambda. Now I have not told you exactly how it acts on each Lambda, which I'm going to tell you a minute. So take this le algebra act it on each Lambda. So that will generate a sub space of each Lambda take the quotient. And that is the basic object in conformal field theory that is called a space of dual conformal blocks. Now the conformal block is nothing but a dual of this space. So dual dual is the conformal block. Now, now let me tell you what is the action of G tensor this C Sigma minus P on each Lambda and that action is given by is it in focus. Can you move it to the left and a little bit down also. Yes. Yes, I think we can see it now at least for a while. Okay. So X tensor F where X is in the le algebra and F is a regular function on Sigma minus P that acts on this tensor product V1 tensor vs where Vi is in H Lambda I and F is a regular function on Sigma minus P. So this action is, we take the sum from one through S V1 and on the ith factor we act by X tensor F. Mind that F is really a regular function on Sigma minus P. So what we do, we take it's, we fix a local parameter at the point Pi. Okay. Now Pi is a smooth point. So we can, we will have a local parameter. So let's fix a local parameter Ti on Sigma Pi. And now we take F at Ti is the Laura series expansion of F at Pi through the coordinate Ti. So we use this coordinate Ti and get a Laura series expansion of F at Ti. So that will give us so F Ti becomes an element in the Laura series in Ti. So that will act, see the H Lambda I was going to be acted upon by G tensor CT are the central extension of that. So that's the action. And we sum over from one through S. Okay. Now, there is this lemma, which says two things. First of all, this is space which I defined the space of dual conformable block. This is finite dimensional vector space. So that's fact number one, which is not too difficult to prove but that it does require a proof. It's, it's not totally straightforward, but it's not too difficult either. So this is a finite dimensional vector space over complex numbers. Moreover, it does not depend upon the choice of the local parameters. So I said that let's make a choice of local parameter Ti at the point Pi. It does not depend upon the choice of the local parameter up to a canonical isomorphism and that canonical isomorphism is canonical up to a scalar choice. Now, in this whole theory, there is the whole thing depends upon a choice, which is usually okay, but only up to a scalar multiple. Now, this is a finite dimensional vector space. Now, it depends upon the curve sigma, the points P, and the choice of these weights lambda one through lambda S in DC. And it's a finite dimensional vector space. So if you have a finite dimensional vector space, the most important question, the most basic question is what is the dimension of this space. As I said, this is based depends upon the choice of sigma, the curve, the points and the choice of Lagrange. Now, Eric Warland made a conjecture in 1988. And actually it's interesting. He made this conjecture. I think while he was at the Institute for Advanced Study, where I am right now. Okay, so, so now I am going to tell you the basic idea of the proof. The full proof really is fairly long. It has several other ingredients and many of those ingredients take several, several pages. Actually, if you write the full proof, it will be, it will occupy 100 pages. But I'm going to give you the basic idea of the proof as to what are the main ingredients and how does one really prove that. So, so I'm going to list some of the theorems and I will explain to you. Okay, so again, it might have gone out the focus is the last lines readable. It's a little bit to the right, can you move it to the thing to the left. No, I move it to the left or I move it to the right. You can see your mouse, instead of your paper. We can see your mouse instead of your paper. Okay, okay, okay, okay. I see, I see. Okay, okay. Okay, so, so I know what do we do it. Can you see the paper now or it has gone even worse. Move it where the mouse was. Ah, okay. How about this. You're still missing something at the right side of your paper. Oh, right side of the paper. You moved the camera to that. Yeah, now we can see it. Now you can see it. Okay. So the first theorem, it is called propagation of Vecua. So what it says, so this V Sigma is the space of dual conformal blocks. So V Sigma P Lambda is same as we attach one more point. Okay, we attach one more point. Let's call it q zero. This should be a distinct point and it should be a smooth point. That's the requirement that it should be a distinct point from peach and it should be a smooth point. And to that point to be we have already attached these weights lambda I, but to the point q zero q zero, I am going to attach the weight zero. The central charge is still small C. So the corresponding representation of a finally a zebra is not the, the trivial representation is the representation with G highest weight zero but central charge small C. Okay. Let's move to the next page. So that's one theorem, and it says that you can add more points without changing the dimension as long as you attach the weight zero to those points. Now, next comes the factorization theorem. Is it visible. No, it needs to go towards the mouse again to the left. Yes. Okay, very good. So the next theorem is called factorization theorem. And it is a very important theorem in this whole theory and you will see why, but let me first explain the theorem. So, so let's start with a nodal car. So it can have one node and it can have several nodes. So no basically means that it's a singularity of the type x times y equals zero locally. It just means that it has a singularity of x times y equals zero. So let's start with a nodal curve. But, but keep in mind that the, the, the points P vector, these are all the smooth points. And let small q be that nodal point, which we are just going to choose one of them. I mean, it might have several nodal points, but I'm going to choose one of them are fixed one of them to. So that's the node nodal point. And now I'm going to take the normalization of the curve only at q. So we are going to normalize the curve only at q. Sigma height be the normalized curve. And now this q will blow up into two points q prime and q double prime. Now this, even though Sigma was connected, there is no guarantee that Sigma height will be connected but but never mind. Okay. So now this factorization theorem, what it asserts that this V Sigma P lambda that was the basic space of dual conformal block that space is canonically isomorphic. And again this canonical is up to a scalar this whole theory is canonical is up to a scalar so projectively everything is canonical, but not otherwise. So now we are going to take the direct sum where mu runs over all the weights in DC, they're only finitely make many weights in DC, because lambda theta check has to be at most C. So they're only finitely many weights. So now we take the space of conformal block not for Sigma but for Sigma height, where we have now this point q has been resolved, and it has obtained two smooth points. Now I'm going to take the pointed curve Sigma height to be the original points p p p vector. So the smooth points in Sigma, they remain of course is smooth in Sigma height. In fact, we have not done anything to them. But now I add two points q prime and q double prime. Now they have become smooth points in Sigma height. So this is completely legitimate. The original one attached to the points p, but now to the point q prime I attach mu star, and to the point q double prime I attach the weight mu, where mu runs over DC, that's a finite set it's running over that. So now I take this finite sum of dual conformal block, and this factorization theorem says that the original space of conformal block is isomorphic with this direct sum of these conformal blocks. Now, let me tell you right away what is its importance, and we will tell, I mean we will discuss it in more detail in few minutes. If the arithmetic genus of original Sigma was G, okay, so arithmetic DNA genus just means that H upper one of Sigma with coefficient in the structure sheath. So arithmetic genus G is nothing but dimension of H one Sigma or Sigma. That's the definition. Okay, so now. So, when we start with Sigma with arithmetic genus G, the Sigma height has an arithmetic genus G minus one. So, right away what we see that when we are calculating this dimension of conformal block for a curve of genus G. We reduce the problem to a genus G minus one car and eventually we will inductively reduce the problem to a genus zero curve, which is nothing but P one. Anyway, I'm going to come to this in more detail in few minutes. Okay, let's keep up. Okay, I am two more minutes before now. So, so far we talked about conformal blocks are dual conformal block for a single car with some mark points P, and weights attached to that lambda. Now we are going to shift if I the situation and how we are going to shift if I we are going to take Sigma T over T a family of S pointed curves. So basically what it means that T is let's let's say T is a smooth scheme, although it does not have to be just for convenience let's assume that T is a smooth scheme. And Sigma T to T is a proper morphism and a flight morphism, proper and flight morphism. And what we are requiring that this pie this map from Sigma T to T had sections now P one through PS, and PI are sections of pie. They are mutually disjoint at every point. And if I take a small team capital T then by university is a S pointed curve for any geometric point T in capital T. So what we are doing we are just if you find the earlier situation. Now we are allowing a family of curves with. I mean family of marred family of curves with Mark points, but we are keeping the same lambda we are not changing the lambda. So we will still keep lambda one through lambda s, and we will attach at every point is multi in capital T, the same lambda one through lambda s. So this gives us chief of conformable blocks. And so we have Sigma T pre lambda. So this becomes a sheaf over T now. And the sheaf has the property that if I specialize the sheaf at a small tea in tea, then we get the original sheaf up, sorry, space up to a conformable block, but we can shefify the situation that's the sheen. And I see that I have just 25 minutes. So we can have a few minutes break at this stage, unless anybody has any question.