 We should get back again. Okay. Okay. So, so let's continue. And edge. Yeah. So I am going to, to use my pen to point to the place. So, right. So we had this sheaf of conformal blocks. Now one important theorem is, so let's assume that. Can you? Okay. Maybe not. Yeah. Okay. Does that work? A little bit to the left to watch the mouse. Yes. Okay. That was worse. That you were also moving the camera sometimes. All right, right. Okay. How about that? Is it worse? I can see T belongs to capital T and geometric with C. Right. Right. But okay. Okay. How about that? That's worse. Okay. Now. Now I can see to me to me where I could see geometric before. Okay. But theorem. Can you read the theorem? Yes. Okay. Okay. So let's assume that T is a smooth and pie is a smooth morphism. Yes. Yes. I can see the, the theorem. Okay. Very good. Very good. Okay. So that's where we are. So let's assume that this parameter family of S pointed curves has the property that T is smooth. I assume knowledge is smooth. And pie is a smooth morphism earlier. We only asked it to be a proper and flat family, so let's assume that actually it's a proper and a smooth morphism. Then what happens? So then, okay. So can you see it? It still needs to go. Yeah. Yeah. That's better. That's better. That's better. Actually, right. Right. Okay. So this is space. Now this is the, the, some shift over T. So this shift admits a projectively flat connection. So this shift, which is a shift over T and T is a smooth variety now. So this shift admits a projectively flat connection. In particular, V sigma T, this family is a vector bundle over T. So when you have a variety and you have some, some coherent sheaf over that, which admits a projectively flat connection or a flight connection, then that she becomes actually a locally free sheaf. So this sheaf of conformal block now becomes a vector bundle. And the consequence it has that the dimension of this one only depends upon the genus of sigma and the weight lambda. So it does not depend upon the choice of the actual curve within that genus. Let's take, let's say that sigma was a smooth. So, so it does not depend upon the choice of the curve sigma within that genus. And it does not depend upon the choice of the points, the mark points there because given any other mark point, you can move in a smooth proper family to go from one set of points to another set of mark points. And from one particular curve within that genus to another curve, because the model I of genus G curves is connected. So you can go from one genus G curve to another genus G curve. And also you can go from one set of mark points to another set of mark points. So this theorem has a very important consequence that this dimension does not depend upon the choice of actual curve within that genus. And, but it does depend upon the weights and the genus. So now I'm going to introduce this notation mg lambda vector, which is the dimension of v sigma p lambda. And I said it does not depend upon the choice of sigma within that genus and the choice of the mark points. But mark points have to be smooth. Now, let's observe a corollary of the factorization theorem. So what it says that mg lambda is same as mg minus one lambda, but you attach two more points, mu star and mu. I mean weights, mu star and mu. And you run over mu in DC. So let's write here. So we run over all DC. So mg lambda is now mg minus one lambda mu star mu. And you keep going inductively and you get to mg lambda, but you have introduced two g many points, two g times, two times g many points. And the weights mu and star mu and mu two star mu two, mu g star mu g, where each mu i is an element in DC. So this one sum which we are interested in has become a sum of all these quantities, but the advantage is that we need to make the calculation on a genus zero curve, which is P one. Now there is, right. Now, actually here I have stopped over one small point that when I'm going from mg to mg minus one, we have to introduce a node in the curve. And then we have to normalize it. And what I said that this diamonds only depends upon the genus. That was only for the smooth curves. But actually what happens that there is some extension of this result, which says that the theorem remains true even for non-smooth curves, as long as they have only nodal points. So those are called the stable curves. Okay, so that's what I'm remarking. Let's remark that the last theorem remains valid when T is the modular space of S pointed stable curves. So I stated the theorem only for a smooth family, but actually this result remains true for the modular space of S pointed stable curves. And then what happens that this coherent sheaf of conformal blocks is becomes a vector bundle, not only on the family of smooth curves, but also on the whole modular space of S pointed stable curves. Okay, now let's see. Is it visible the last line? The very last line is visible. The top above is out to the left, out to the right. Okay, yes. So let's see further by the factor. No, no, no, no. Now we can see. Yes, correct. Actually now I should. Yeah, yeah. Now I figured out that I can see some picture. Okay. So now, so the problem of calculating these diamonds and MG lambda reduced to a problem of computing M zero that is a P one, but million more points now because it will depend upon genus. So if genus is 1000, then we have to attach 2000 points. And of course the original lambda. And then we have to sum over all these weights. So that becomes a very large sum. So the question is, have we really gained anything? And the answer is yes. So the factorization theorem is also valid. So what we do, we take P one and we, we, we. We deform P one to make a node. I think you got out of you now. Oh, sorry, sorry, sorry, right. Sorry, sorry, sorry. Yes, sorry. One minute. No, let's. Yeah, yeah. No, it's good. Yeah, yeah, correct, correct. Okay. So what we have done, we take P one and we pinch it so that we introduce one node point. And now the factorization theorem is still available for this situation. And when we normalize, we don't get a connected P one. We get two copies of P one where this node point has become one point on one P one and another point on another P one. And let's analyze the situation. And we can apply the factorization theorem again. Let's see. Yeah, it's visible now. Yes. So we can apply the factorization theorem again. And now we take M zero as many points and we can break it into M zero, lambda one, lambda two and mu and M zero, mu star and rest lambda, lambda three, lambda s. This could be any number. So I have broken this number into product of two numbers. One number has only three points and another number has S minus two plus one point. So S minus one points, but it's still have to sum over DC. Now we can inductively do this process and this M zero with one million points becomes product of things with three points only. Of course, it will still be a big sum, but now we only have to understand in theory and actually in practice M zero with three points and three weights attached to that. So if we apply factorization theorem successively. So first we have reduced from MG to M zero. And now M zero with several points, I reduced it to M zero with three points. Okay. So now this tells us that we are reduced to calculating M zero, lambda one, lambda two, lambda three. And this is achieved by the fusion product on level C, level C representation of G. Now this thing is guided by what is called the fusion product at level C. So fusion product is. Okay. I mean, maybe all of you know about that, but just like tensor product of two representations, we can define fusion product of two representations at level C. And then this is not tensor product. This is fusion product. So what we get is only representations at level C. So we discard some representations which have level more than C, but also we have to discard some mirror images of that. So let's just say that I'm not being very precise there, but fusion product of two representations at level C is like tensor product, but it produces only components of level C and it discards all the representations of level more than C, but also it discards some representations which are mirror images under the affine while group of C. Okay. So, so this is the end of it. So we wanted to calculate mg of lambda bar or lambda a vector. And now this problem has reduced into calculating M0 that means on P1 with three points where it's guided by the fusion product. And in fact, it's not only a theoretical reduction. This becomes really a very calculable problem. And in fact, there's a very closed expression. I'm not writing down the closed expression for mg lambda, but it's a very, very precise closed expression very similar to the wild character formula. But probably it's not much interest to really write down that formula. But I'm saying that there's a very precise formula which comes out of that for any mg lambda. Now let me say very briefly the history here. So, as I said, this conjecture was made in 1988. He made a precise conjecture. And then Shuchia, you know, and Yamada in 1989, they made a major progress in this project, major, major progress. And then I'm just going to list some of the names. So Bobby Laszlo in 94, they did some work for SLN. Faltings did in 94 for General G. Myself, Narasimha Naramanathan, we did something in 94. And Pauli did it for the parabolic case in 96. Laszlo Surgar did it for the modularized type in 97. And then Bismuth and Laburi did it in 93, actually. Direct proof for holomorphic sections, which I have not really talked about, but they have restrictions on the weights. And then there was some contribution by Terimaan in 97. And then for SL2 actually predates all these works, Bertram, Zenas and Tidius did it for 91, 93 and 94, but for group SL2. Now, what I have not talked about for lack of time, that there is a completely parallel picture for the space of conformal blocks. Maybe I am again out of sync. Yes. Yes, good. Yeah. So there is a parallel picture for the space of conformal blocks in terms of holomorphic sections of line bundles on the modularized space or modularized type of semi-stable parabolic bundles on Sigma. So I will have no time to explain more about this thing, but this space of dual conformal block or conformal block is canonically isomorphic with the space of holomorphic sections of certain line bundles on the modularized space or modularized type of semi-stable parabolic bundles on Sigma. And that makes this whole theory even more interesting for classical algebraic geometers because this line bundles on the modularized space of G bundles was studied for a very long time and its dimension was a mystery, but with this work now you can calculate the dimension in a very precise manner and because this space becomes isomorphic with the space of conformal block. Okay. So any questions so far? Anybody? So actually I see that I have 10 minutes and now as I said I am going to so this is so far this was not really I mean there was some part of it was done by me with other collaborators and several other people, but now what I'm going to talk about is a work with Juju and this is the Galois picture. So I'm going to very briefly talk about that. Let's see now. Now we can see the first half at least. Exactly. Okay. So what is the Galois picture? So earlier we took a reduced projective curve Sigma, but now what we are going to do, we are going to take still a reduced projective curve, but with the action of a finite group Gamma. So we'll let the finite group Gamma act on Sigma and let Sigma bar with the quotient of Sigma by this action of Gamma. Now we assume that any Gamma, I mean it's a technical assumption and we can even gloss over that, but let's say that we assume that any element in Gamma other than one does not fix any irreversible component of Sigma point wise. I mean it's a technical assumption. You may just forget about that. So the requirement is that Sigma basically does not act trivially on any irreversible component of Sigma. Now, as before, we are going to take a simple Li-Agebra over complex numbers. Now, except that now Gamma is also acting on G as Li-Agebra automorphisms. So Gamma is acting first of all on the curve Sigma and Gamma is acting on the Li-Agebra G and we have one technical restriction for at least some of the results. So let me say, but many of the results are true without this restriction. So we assume that Gamma is fixing a Borel-Saba-Gebra in G. That's not automatic, but if Sigma, sorry, if Gamma were a cyclic group, then it's automatic that it will fix the Borel-Saba-Gebra. But otherwise it's not automatic. So we just put that assumption and I said this assumption is for some of the results which I'm going to talk about, not for all the results. Okay. So now we want to talk about this twisted conformal blocks. So a S-pointed Gamma curve is by definition, again just like last time Sigma with bunch of distinct points. These are smooth points. These are distinctly smooth points, but we also require that Gamma orbits do not intersect. So Gamma PI is not equal to Gamma PJ for the I different from J. Now to PI, earlier we attached the same simple L-Gebra, sorry, same affine L-Gebra. But now to each PI, we are going to associate not the same affine L-Gebra, but a twisted affine L-Gebra. Okay. So what is this twisted affine L-Gebra? So let's take the isotropy of Gamma at PI. I think we are out of view now. Oh, sorry. Yes, yes, yes. I see that. I see that. I see that. Yeah, I think it's okay. Thank you. So I take Gamma PI, isotropy of Gamma at PI, and I take G-Hat as before, but now I'm taking it's twisted in the lock. So G-Hat at PI is nothing but G-Hat. And now Gamma PI, the isotropic group acts here. And we are going to take the fixed point there. So this is what is called the twisted affine L-Gebra if you are familiar with the theory. So G-Hat was the usual affine L-Gebra, but now we are going to take now Gamma PI. Okay. I should mention that Gamma PI is a finite, I mean Gamma was a finite group and it's isotropy. So of course it's finite, but actually it turns out that Gamma PI is a cyclic group because PI very smooth points. So the isotropic group is a cyclic group. And this is what is called the twisted affine L-Gebra. So now to the point PI, we are going to associate this twisted affine L-Gebra, not the affine L-Gebra. Okay. So the next thing is, now we are going to take, so now I'm going to take integrable highest weight module, not for the affine L-Gebra, but for the twisted affine L-Gebra, which I'm denoting by G-Hat PI. And this has its own parameterization, which I'm just denoting by DC PI. Now Siege fixed is still the central charge. I am not changing the central charge. Central charge is fixed for all the PI, but this DC PI depends upon the choice of PI. And now I'm going to take Lambda vector, Lambda 1 through Lambda S, where Lambda I is now in DC PI. That means the corresponding H of Lambda I is not a representation of G-Hat, but it's a representation of G-Hat PI. So still it's the integrable highest weight module for the twisted affine L-Gebra at PI. With highest weight Lambda I, I'm central charge C. And now again, as earlier, I take their tensor product, H Lambda 1 tensor H Lambda S. But mind that H Lambda I is not a module for same G-Hat, but for G-Hat PI. Okay. Right. So I'm just remarking that H Lambda vector is a module for G-Hat P1, direct sum, direct sum G-Hat PS, where the ith factor acts only on the ith factor here. Okay. So now I need to define the twisted dual conformal blocks. So to this curve Sigma with the gamma action, the points, the marked points P and the marked weights Lambda, we have this V Sigma, which depends upon the choice of the action of gamma P Lambda. So this is called the space of twisted conformal block. And this is by definition, similar to earlier, but we take H Lambda, but we divide by G of Sigma minus, not only P, but the whole gamma orbit of P. We removed the whole gamma orbit of P, but now we take the gamma invariance here in this Lear-Gebra. And that acts on H Lambda vector, similar to the previous here. Now the action of, okay, it's again out of focus. So let's put it out of focus. So the action of G Sigma minus gamma P on H Lambda is as before, except that, okay, okay, no, no, no, it's exactly as before, that we take a local parameter and since I'm taking gamma invariant, the local parameter will push us to the twisted-affined Lear-Gebra at that point and we let that factor act on the H Lambda I. Okay, so then we come to the Lemma. So this is space of twisted conformal block, is again finite dimensional. So we're missing the right half of the Lemma. No, no, no, thanks, thanks, thanks. Yes, now we can see. Okay, so this is space is again finite dimensional and it does not depend upon the choice of the local parameters TI at PI. Now theorems, and these are the results which I proved with Juju. So the analog of propagation of Vaku, I'm not going to say exactly what it is in this setting, but analog of propagation of Vaku is true. Factorization theorem remains valid, again with proper formulation. Existence of projectively flat connection on a smooth family remains true. And in fact, extension of the connection to the what is called the Hurich stack, which I don't have time to explain, on Hurich stack, it's similar to the modular space or modular stack of curves, but now here it's modular stack of curves, together with the marked points, PI with the monodromy fixed at that points. So we are fixing now two things. So we are fixing, no, we are taking three things and putting them together, modular space of genus G curves, points there, which are smooth points, and also we are fixing the monodromy at those points. And if we combine all three, then what we get is what is called the Hurich stack. And now there is extension of this connection, not only on the smooth family, but on this Hurich stack, which is not smooth, but this connection has logarithmic singularity at the boundary. And then what we prove is that identification of twisted conformal block with the global section of some modular stack, which were conjectured by 5% RACOPORT. So again, similar to the dual conformal block in the classical picture is identified with the global sections, it's based on global sections on the modular stack of G bundles with respect to some line bundle. So there is a very similar picture here where we identify the twisted conformal block with the global sections on some modular stack which were conjectured by 5% RACOPORT. And then we prove the precise diamonds and formula very similar to the whirling diamonds and formula. And this was proved by me and Juju and also independently by Deshpande and Mukhopadhyaya. In fact, Deshpande and Mukhopadhyaya proved it before us, but our method is very different, very geometric method. So we can see the bottom part. Yes, sorry. Yes, I am out of focus. Yes, correct. Yes, now we can see the bottom part. So I just want to end by saying there were earlier work in the twisted conformal block before us by Frankl Zeni. I don't know how to pronounce this. Shershni. Yeah, okay, okay. And then Kurobi and Takobi and Kira Demiolini. So there were some earlier works in this direction. I will not say exactly what they did. But actually I am over time by few minutes, so I will stop here. Thanks very much for a very interesting talk. Let's thank Sravan.