 謝謝 very much. I'd like to talk about the MOOC modular forms and representation theory of affinely super Jebus. There is a very simple definition by Zagie and other papers. A MOOC modular form is a homomorphic part of a real analytic modular form. In the case of affinely algebra, there is a very beautiful theory by Katz and Peterson. They found that the character of integrable modules, the linear span of characters of integrable modules are S L to the invariant. And this result was extended to a much wider class to admissible representations. And so in the case of affinely algebra, there is a very beautiful modular invariance. But in the case of affinely super algebra, the character formula, it is not established, it is still conjecture, but there is a factor in this denominator. So because of this existence of this factor, the property of characters is very unclear. But the very remarkable breakthrough was given by Tzubekar. He proved that the super character of L lambda 0 for S L to 1 hat module, affinely S L to 1 module is a mock modular form. And he gave an explicit formula for it to a real analytic modular form. And so extending his method, we recently obtained the following cases. One is V super algebra of rank 2 and any integrable representations. Rank 2 V super algebra, S L to 1 and S L to 2 divided by center and OSP32. And this result, it is possible to extend this result to admissible representations. And then we go to the quantum reduction and we get N equal to and N equal to 4 super conformal algebras. We obtain the representations whose characters are mock modular forms. And how it can be done? So I will explain it in the simplest case, in the case S L hat 1, affinely S L to 1. The general integral weight, which is vanish at alpha 1, alpha 1 is the odd simple root, is given by this formula, is written in this form a minus S lambda 0 plus S lambda 2. A method works for any S, for general this weight, but for simplicity in this talk, I would like to consider the case lambda is equal to M lambda 0. The supercharacter, generally supercharacter has a better property than the characters from the view of the modular transformation. So we would like to consider the supercharacter. Supercharacter of M lambda 0 for affinely S L to 1 is given by this formula. This is a numerator and divided by superdeluminator. Superdeluminator is a product of a root. It is given by such formula. And to make calculation, we introduce the coordinates. And then supercharacter is given by this formula. And divided by superdeluminator. This is a product of Yakovic data functions. So the denominator, this part is it is very easy to compute the modular properties. So the problem is how to compute the how to know the property of this function. So we put this function tm tau z1 z2 t. Namely, we want to study the modular property of this function, the numerator of the supercharacter. Modular property means the behavior under the action of S L to z. And t action is simply tau plus 1. And S action is just inversion minus 1 over tau, z1 over tau, z2 over tau. And in the case of this function, fm, the t action is very, very simple. Just it is fixed by t action. So we want to know the S action. What is fm applied by S? So namely, in other words, fm minus fms. What is this function? And first, we'd like to note that the difference, this difference is a harmonic function. The proof is very simple. First, consider case t1. Then it is very easy to know poles are g plus tau z. And also poles are tau plus tau z. And by simple calculation, we see that the residue is the same. So the difference is harmonic. So for further calculation, it is very convenient. We use much better variable, u and w. So we write this function in terms of u and w and put this function small fm and difference is gm. So we want to know this function fm. And gm, the property of gm, it is a harmonic function. It is very simple. Just as we have looked at it. And we consider w plus fm minus w. Then it is not so difficult. It is very simple calculation. It is written in this form. And also we compute w minus tau difference between w and w. It is also written in this form. There is a misprint. s is equal to 0. At first I wrote in the general setting so s remains, but s is equal to 0 in my talk. And the harmonic is a very, very good property. And gm is determined uniquely by the above three conditions because it is due to the rubric theorem. W-periodic harmonic function is constant. So we can, it is directly, the uniqueness follows directly. And one more thing. The point in this formula, u appears in this form theta minus theta, theta j minus theta minus j. And also theta j and theta minus j. So the behavior of the function gm with respect to u is very good. It is written by tater function, uter tater functions. But the behavior under w is a bit more complicated. So we'd like to solve these equations. And it is possible to solve these equations. So we want to find gm of the form. So at this stage, at this point, we don't know whether it is gm is written in this form or not. But we put it. And we'll show that there exists just a j type. There is a function, a harmonic function and satisfying this condition. W plus 1 minus w is just taking just the coefficient of the previous formula. So it was taken from just this form and also taken just this form. We get these three conditions. And by the condition property, homomorphic. So if there exists a j is determined uniquely by this condition. So to construct, there is a very, very interesting Tzubega's functions. So Tzubega introduced such kind of functions. So I made some modifications. But essentially this is Tzubega's functions. Rm plus j is a signature of this minus e is a function of this. And important is the derivative of e prime. E prime is just equal to this function. So we make this property. And after a grand city it looks very curious, but this function has very good properties. Rm plus j w minus w minus tau is just gives just and also w plus 1 minus w by simple calculation we get this. And also important point is differential w bar derivative is this is not homomorphic function because imaginary part w imaginary part. So this is a real analytic function not homomorphic. And if we differentiate by d bar, then it is a tether function with respect to anti-variable variable minus tau bar and w bar. So anti-homomorphic tether function. Then by using this function tether function Rm plus j we put Lj tau w is this function times some factors and s transform modular transformation of Rm plus k. Then these functions we can prove that this function is homomorphic with respect to w. So in its place in its proof it is proved by using the transformation formula of Jacobi tether function tether j and plus 1. And also this function satisfies all conditions expected for A, J, W. And then we put additional function by using this function Rm plus j and that u-part is just tether function and this is we call it additional term fm plus this function we put tether m. This is modification of fm and it is a good modification because this function is modular invariance this function is a modular function and fm is its homomorphic part and this function satisfies also elliptic properties, elliptic invariance u plus a w plus b is invariant u plus a tau w plus b tau just get such a factor so then the problem naturally arises what is the representation theoretical meaning of these additional terms and Tzvega's type functions so it is a problem now by using this result we consider admissible representation admissible weight is by definition it is integrable with respect to some sub-route systems so it is a maximal sub-route system so integrable with respect then the super character of this highest weight module is the character formula super character is tau is replaced by m tau m is just and g1, g2 is just g1 plus k1 tau g2 plus k tau tau so this is also our conjectural character formula for admissible modules and then we modification it is not a modular function so we consider the modification of super character of the numerator of super characters so this was honest numerator so we put replace it modular function and put some suitable factors and we consider epsilon prime so epsilon prime is 0 or 1 half 0 is super character work character and don't twist it then we consider 4 types of characters then they form an invariant family SL to the invariant family modular transformation properties then express transformation formula S transformation this is transformation matrix and T transformation produces just just this factor and then we consider the quantum Hamiltonian reduction so given the data finite dimension of the super algebra and the important element and the level of representation then we get W algebra so in the very beautiful case is F is a minimal important element root vector of the lowest then in the case if G is a cell to C then W algebra is biracero and OSP1 to then W algebra is here then so so if we so in this program so affine the super algebra give very very very many information to give super algebra for example characters are obtained computed from G transformation calculated with the transformation of characters of SL21 so in our case we are considering just this SL21 so we get equal to super conformal algebra any equal to super conformal algebra is spanned by biracero ln and Jn even element and order G is plus and G minus and central element in the case if you take Jn and half integer then W is 0 and bracket relation is given by this table and carton sub algebra is just linear span 3 dimensional L0 C is the center so actually it is 2 dimensional character of any equal to super conformal algebra representation are obtained from the character of admissible SL hat1 module by retic G1 is Z and G2 is minus C so this is the character of admissible representation so we put minus C then this is the character of any equal to highest rate representation so if we put children so modify to be extended to be modular modular form then we get extension modular function extension of characters and parameter domain of parameter is just given by this and central charge is given by this number and modular transformation of any equal to character is computed from the modular transformation formula of this function and we obtain such transformation formula and then n equal to highest rate cartons of algebra is 3 dimensional L0 and J0 and center so we put the iron value of L0 on the highest rate vector H and J0 is this means spin and central charge then HSC it is determined by 3 numbers and in this case n equal to representation parameterized by JK so in each case HJK, SJK central charge are given by this formula and in the case of m equal to 0 so m equal to 0 means L0 trivial representation of SLHAT1 so admissible representations correspond to trivial representation of SLHAT1 then the central charge just this part is equal to 0 and m minus 2 so it is well known central charge of minimal series of n equal to super conformal algebra but in our case we consider lm lambda0 for arbitrary m is arbitrary non-negative integer so in the case of m positive so in this case we need we get we need modification and we get modular properties and modular series then there are new representation of n equal to super conformal algebra and we hope that there should be exist the new n equal to conformal field theory corresponding to this moc modular series thank you very much why is that rational why is the corresponding CFP why should I be rational rational rational means the number of irreducible representation correspond to that central charge is the representation of vertex algebra vertex operator is finite so this is just fit to this case and vertex algebra representation is completely irreducible so just this fit this station so we hope but at present there are no theory so but we think there should be we hope that there should be thank you very much