 All right, so welcome back to the second half of the seminar with Changlong Zhang telling us about elliptic losses by the periodic module and the 3d meter symmetry. So please, please go ahead. Thank you. Yeah, so the next part is about this, the materialistic operator in elliptic case. So I use the same notation for, I mean, both of them are denoted by T, but since they are really in different context, I think there should be no confusion. So, yeah, we have just constructed this t-state group algebra and also this periodic module, which are vector bundles over A cross H check. Okay. Again, let me remind you that A is equivalent to a module point and the H check is equivalent to a module G of a point where the torus have to be has to be used replaced by this T check. Torus for the London's dual. Okay. So you really have to consider things over A cross H checks in the same time. So then we can define this the materialistic operators for elliptic case. But for that one first, we have to define the coordinate function just like when you write on this k-series the materialistic operators you have to use this e alpha notation. And so they are basically coordinate functions. So for the elliptic case, the coordinate function are defined as follows. So the ability a is t low star tensor e. We know this is basically e to the n spa and the coordinates are defined as follows. So I give you any alpha in t alpha star, any root or any character, you can define a map going from this ability to e by sending this tensor to this pairing between alpha and mu check. So mu check is a core character. So you take the pairing with alpha and alpha check, which gives you an integer and then you can multiply with t. T is an element in capital E. Okay, so this linear map extends. And so if you take an element C from A, you denote its image to be so the c alpha is denote to be chi alpha of C. Okay. So that's like the coordinate of C at the alpha direction. You can do the same thing for the other ability variety. So if you have Linda in mu cross t, mu tensor t, you can again. So after you fix alpha check, which is a core character, then you can consider the pairing of our check with mu, and then which gives you an integer and then multiply with t. So that's the coordinate of Linda at this alpha check direction. And that's chi alpha check of Linda. Okay, so that's the meaning of the notation C alpha and Linda check. So keep in mind that both of them belongs to the elliptic curve. Okay, so with that, then, first, let's recall this data function, Jacobi's data function, which is the product of so for you in C star invertible complex number you can define you are one half minus you to the negative one half, and then take the product, this infinite product with this one minus q to the s u multiply with one minus q to the s u to the negative one. And then there's an actual piece here. Okay, this has nothing this is does not depend on the variable. So here q is the parameter that you choose to define this elliptic curve. So that's basically the multiplicative version of the Jacobi's data function and we can, that's a homomorphic function over some double cover of this C star because of this one half exponent. And the first we really like to work with the additive version so we can replace this theta by the other theta. So this is the curly theta that's the normal theta. Okay, so in this case, this x belongs to C. Okay, so many other way, but they usually use this theta but we like to use the other one. That's more convenient for us. Okay, with that, then we can define this term as realistic operators. She alpha. So it's basically again, it contains two parts. So here I can. He's the casey reversion. So it has two fractions. So similarly here we also have two fractions. So all the fractions involve the theta function. Additive version of the theta function. And then the variables H, which belongs to E that we fixed at the beginning. See alpha belongs to E, then alpha check belongs to E. So this is difference of two elements in capital E. And so that's why we have that's why this theta makes sense. So we consider this fraction. Okay, and we have this fraction. And also notice that for the denominators, the denominator, denominators are the same which is similar like the case theory case. Denominators are the same. And also if you look at the zeros of the pose of C. You have a zero, you have a pole at zero, you have a pole at zero, now you have a pole at E alpha equal to one, a pole at E alpha equal to one. And the other thing is like, so for this part, you also have H minus C alpha, this corresponds to this one minus Q E alpha. Okay, also I made a mistake in the notation that is this Q, in the case theory case, corresponds to the C star action on the flag right. So this Q has nothing to do with this Q. It's just, I shouldn't have used Q here, I guess. This is more standard. Okay, the two Q have no relation. And also for the case theory case, you have this one, the one group action S alpha. For the elliptic case, you also have this S alpha. Okay, the actual piece is this S alpha D, which appears in both of the two fractions. Remember that S alpha, X on A, S alpha D, X on A check. Okay. So this is the formula obtained by Emmanuel Weiber in their work in the geometric construction of the elliptic classes. They use Poisson resolution and they studied the recursive properties of the elliptic classes, they obtained this formula. One can also obtain this formula if you studied the work of Feather, when he was talking about this, studying this dynamical quantum Young-Baxter equation. He also come up with some solution and the solution involves some fractions, which will be just the two. I believe the his work is for type A, I'm not familiar with other types, but for type A, if you look at his solution, he will have some fractions, the fractions can be put in here to obtain this T alpha, this operator. Okay. And one can also define the non-linear version. So we call it T alpha D. They are very similar with each other. Okay. There's a symmetric property here, which I may skip for now because of the time, but they look very similar with each other. So for instance, you see the numerator that's C minus lambda, this is lambda minus C, and that's H minus Z, that's H plus lambda. Okay. So there's some similarity between the two. So the elliptic version of the modularity calculator and now they satisfy T alpha square equal to one and T alpha D square also equals to one, and also they satisfy the graduation. So that's the elliptic version. So with all the construction of this, the modularity calculators and this periodic module, we can start to mention the main result. The first part is that we prove that T alpha and T alpha D, both of their original sections of S. Let me remind you, S is this drug sum of W inverse alpha star, V inverse D star L, inverse tensor with L. Okay, that's the twisted group algebra. So this T alpha and T alpha D, both of their original sections of this S. Okay. So to prove this, one has to study the zeros and posts of this T, and also by using the property that over elliptic curve, if you have a rational section of some limando, the zeros and posts of the function uniquely determine the limando. Okay. So the other property is that we can define the other result is that these limani webers. Elliptic classes can be obtained by applying this TW inverse applied on C. Now what is C? C is just some rational section of L that you fix at the beginning. Okay, after you fix this, you can let TW inverse act on this periodic module. You will get elliptic classes and then one can prove that this elliptic classes corresponds to the limani webers. Elliptic classes and also so that means they are also equal to agonistic or concoves. Elliptic stable envelope. Okay. I don't know that this C is always there because in their definition in agonistic or concoves definition, they have to fix a polarization. This polarization corresponds to the C. So in other words, if you choose suitable polarization, you will get the result that you can identify this with their work. Okay. And you can do the London's dual version. Okay, so that's the second main result. And now the last one is more related to the 3D middle symmetry that we mentioned before. That is, you can take the sum of all the demodulistic operators. You can also take the sum of the demodulistic operators for the London's dual version. And both of them can be realized as maps between periodic modules. The first one can be think of as maps from M to MD. The other one is a map from MD to M. Okay. Also note that in the definition of the T, there are a lot of the denominators. So they are not, so the T here are not global sections, they are only regional sections. Okay, because they contain holes. So the result we have is the following. That is the T and TD, they are dual, they are inverse to each other. Okay. So with that one can recover, so then recover 3D middle symmetry result of Riemannian Weiber. Okay. So that's the main result. So any questions? We're good. Okay, so I have 12 minutes. So first let me mention some remarks and then if there's still some time I can talk about the idea of the proof or maybe some ideas. So the first one is that in Riemannian Weiber's work, they also have another recursion. So if you have this Bohr-Samson recursion, they have another one, they call it our matrix recursion. We believe that that recursion can also be obtained in this way. That is, it's more like if you have TW, inverse, ectome, something. Now, if you apply a T on the left, you get some recursion. That's the Bohr-Samson recursion. If you apply T in the middle of the two objects, you get the other recursion. Okay. So this is not verified, but I believe this is true. So one just have to write down the calculation and then to verify with their calculation. Okay. And the other one is this Bohr-Samson, sorry, this free Makai transform. So one can define HD to be the elliptic HEC algebra with dynamical parameters. So this will give you some coherent shift over A cross A check. Okay. Some algebra object over this A cross A check. And then there's also the definition of the Skinsburg-Caplanov-Vasserot elliptic HEC algebra they defined in 1996. That's the reconstruction of HEC algebra. They define some elliptic version of the HEC algebra. There's no demodularistic operator because they do not satisfy the operator ratio, but the algebra is well defined. So you can look at the HEC modules and they are, this gives you a subcategory of the coherent shifts over A. Similarly, you can do the Laland-Stewell version, which gives you a coherent shift, a subcategory of the category of coherent shifts over A check. Now in the middle, you have this coherent shift over A cross A check. And then this competition is called the free Makai transform. That is, you take F, you transfer with F, transfer with the concrete lambda. Sorry, pull back. So if you have a shift over A, you pull back to the product, and then you transfer with the P, the concrete lambda, and then you push forward. Q, so Q low size, push forward. Sorry, that's another Q. Okay. So the competition of the two gives you an equivalence of categories between the two categories, sorry, the derived version, you have to derive them. Okay. It does not give you an equivalence of categories on the obedient categories. You have to use the derived category. That's the so-called free Makai transform. What you can prove is that one can start from the hack, the GKB hack module, and then you can pull back to get a module over A cross A check, and then transfer with L, and then you push forward to this category. Okay. And then this competition will prove that this is the equivalence of categories. And in this case, you don't need to take the derived category. Just the current category is good enough. Okay. That's some work of myself. And this should appear soon. So that's the second remark. And the last one is that about properties about like models. So we know that like models over E are uniquely determined by their sex, their sections. And the zeros of status zeros and the pose of data functions are uniquely determined by quasi periodicity of the data functions. So all the four parts uniquely determine each other. Okay, so that's why if we can determine the zeros and posts and some elliptic classes, you can recover the line bundles. Conversely, if we know the line bundles over E, we can recover the zeros and posts of the expression section. So they're almost, so I mean, one can use the information from one side to get information about the other side. Okay. So I have eight more minutes. So any questions? Very good. Good. Okay. So maybe I can say a bit more about the idea of the proof. So this is the construction in this tensor triangle. Sorry, this tensor category. So remember we define this twisted group algebra to be this one. So SWV is basically the, the piece at the grading WV. And the tensor structure is defined as follows. You just define to be so the tensor of this part, this WV piece with this XY piece to be this one. And by definition, this tensor is seismic to this, this WX, VY piece. Okay, now if you write down the rational section of this, I mean, for people in commentaries, they like to work with actual elements. So what you can do is you can look at the rational section, say a, sorry, a rational section here. But if you want to indicate the degree, WV, you can use constant Kuhlman notation. Similarly, if you want to take a section of SXY, you can take a rational section B, but if you want to indicate X and Y, you have these delta X and delta Y. And of course things are confusing here because this V are really the dynamical part. So that's why in this case we have to put a D here to distinguish between the two. Okay. And the product, if you write on the product, it will be equal to the constant Kuhlman twisted product. That is, it will be equal to A, WVD of D, delta WX, delta VY. And there's a D here. Of course, it's easy to see that the two Y group actions should commute with each other because they act on different pieces. So that's why it doesn't matter how you write WVD or VDW, it doesn't matter, it will be the same. So this product is really the constant Kuhlman twisted product. The only difference is that here you have two Y group actions. Okay, but they commute with each other. So that's why this is still well defined. That's the meaning of this tensor product. Okay. The other point I want to make is that, remember we defined the T and the modulostics for London dual as regular sections of S. But if you can, you can change the S a little bit. TWD will be still there and they can be think as functions from maps from this to here naturally because remember we defined SWV to be this tensor product. Now if you apply W up style on both sides, you're going to get this identity. W up style SWV equal to this line model tends with this line model, but that's the inverse so you can automatically write it as maps from the first line model to the second one. So this model is saying that you can think of it as a map from M to MD. And similarly, if you apply VD style on this identity, you are going to get this tensor which will be giving you this one. So that's why you can think of it as sort of like maps from MD to M. So that's the reason why this TW and TWD if you take the sum of them, they will give you maps from M to MD and the other one maps from MD to M. And then as I said, they are inverse to each other. Okay, so that's the construction. Okay, so, so that's why this T and this TD they are inverse to each other. I'm sorry. I mean, they are maps going opposite direction and then one has to prove that they are really inverse of each other and this proof does not mean it's not technical because one can just use some functionality, sorry, canonical properties and a little bit calculation to prove that they are inverse to each other. Okay, there's no technology involved. Okay, so I think three more minutes. There's also the temporary pairing. So I think I don't have time to go into detail here so I will be very brief here. That is, so we know that this Twisted Group Algebra act on this periodic module. And also, there's another version that's not the London dual just the same version. We are still in the same context of the new bundle of GMP. There's a variant form of this as we call it S prime. And you can also change the M into M prime. So then temporary pairing is really tearing between this M and this M prime. So it's not over the same M is M tensor with M prime. Okay, the tensor of the two will be asymptomatic to this line bundle. This is a line bundle over A cross A check. And this line bundle is defined as follows. This O tilde is defined to be the following. So we know A cross A check you can project to the first component A and then we know we have this coordinate map chi row. So then this O of 2H, this is a line bundle degree zero line bundle over E. The corresponding real divisor is 2H minus zero. 2H is a element in capital E. So this real divisor gives you this line bundle, degree zero line bundle, you can pull it back to get O tilde of 2H row. So the tensor of the two will be asymptomatic to this degree zero line bundle. So that's why if you fix a rational section of this degree zero line bundle, which is this fraction. One can use this to define a pairing going from M plus M check, M prime, sorry, to O of A cross A check. Of course, we know A cross A check is a billion varieties. So this is basically constants. Okay, so that's the meaning of this Pancré line bundle. So why is to use the Pancré line bundle to, sorry, Pancré pairing to transform between elliptic classes here. And there's another version we call EW prime. So then we have that. So if this is EV, this will be equal to delta WV. So this is the Pancré pairing in this elliptic case. Okay. So basically the construction and the proof of the main result uses the two things. One is this Pancré pairing. And the other one is this construction of this crystal group algebra. Okay. So my time is up. So I will stop here. Thank you. Thank you very much for a very interesting talk.