[MUSIC] I would like to repeat that here we get the heat operator. And in this operator we see. A part which is equal to the differential operator D. We considered it this differential operator and its modular correction in the previous lecture. So from this point of view, the heat operator H is a generalization. So hear operator generalizes this differential operator which we used in the case of module form and one variable. And this additional part in the heat operator related to the latest part, to the all a billion variables of Jacobi form. Now let's analyze the action of the heat operator on Jacobi forms. We consider a Jacobi form. Of weight k for the latest a. This function has the following Fourier expansion. N is a natural number. And a is a vector from the. Moreover, we know, then the indices satisfy this condition. On the hyperbolic known of the index. The hyperbolic known of the index is greater or equal to 0. If I apply the hit operator H on this function. We get the additional coefficient 1/2 the summation and now, I have to use the fact which we discuss in the beginning of this lecture. So, in our case, the lattice m, this the lattice u plus l minus one. This is our lattice L, and this fact gives us the hyperbolic known As coefficient In front of any Fourier coefficient a m n. Certainly, this is not a Jacobi form in general but nevertheless, after the action of. We had a similar Forier expansion. It means then, we still have this condition. Moreover, we have now a stronger condition than the hyperbolic known strict because we cancel by the action operator. HO for [INAUDIBLE] coefficient with 0 null. In particular, let's assume that pi is a Jacobi form of. The minimum possible weight for the lattice weight for the [INAUDIBLE]. The rank if L = n0. For example, we constructed such Jacobi forms for the latest D8, E8 and so on. We know then if The Fourier coefficient within this n, l of this function is not equal to 0, then the hyperbolic null of its index is equal to 0. Therefore, Such Jacobi form of singular way belongs to the kernel of the [INAUDIBLE] rate. In this case, certainly the image of the Jacobi form is Jacobi form but this is true Jacobi form 0. But nevertheless, you see that if we compare H, the [UNKNOW] rate with differential operator D which we use as a usual modular form, then we can compare Jacobi forms of single weight with constant, because constant belong to the kernel of the differential operator D. And now, I would like to construct a modular correction of this differential operator H. Because the modular correction of the modular correction of d is the modular operator dk equal to d plus 2k times G2, separative multiplication by G2. And these modular differential operators transforms modular form of weight k into the space of modular form of weight k+2. I would like to construct a similar modular differential operator for the heat operator. But, to construct this operator, we need some transformation formulas. Now, we are ready to discuss the formula of the transformation of variables in the heat operator. I would like to consider the heat operator in the new variables. Tau tilda and zed tilda where as usual abcd as an element from the modular group, Sl2. And then in the blue color, I remind you the formula for the transformation of variable if you would like to change and if you would like to consider our numerator. Tilde, and zed tilde in terms of the old variables, you have to use the Jacobian matix to the power -1. So this formula gives us the following relation. This is simply the new differential, the new covectors are written in the term of the old covectors, delta over delta tau and d over delta, delta over delta z e. This elementary formula you can get without any problems, I hope. So, only I have to add one addition bracket right here. Now using resistance formation formula, we can write down the heat operator. H, a new coordinates. Tau tilde and zed tilde, so these formulas, in principle you can find, and it's very similar to the formulas in this integral model. So, and if you change all the variables, you get the additional term, which comes from this additional term. The additional coefficient 1 over 2 pi i comes from the fact that we take, This part in the heat operator with coefficient 1 over 2 pi i. Now, using this formula, I can prove the following proposition This formula is true for any metrics. M from SL sub 2 (Z). In particular if pi is a Jacobi form of weight k and index a example weak form. Then this action, the action gives us The original function. Because this is exactly the modular functional equation of Jacobi form. So. If you analyze now this proposition, this proposition follows from this formula. This is direct corollary from this formula. And you see that in the action of the heat operator on Jacobi form, we have the second term closed to the modular transformation of the quasi modular Eisenstein series. Therefore, if we act on Jacobi form by the heat operator, we get very correct action, and we have additional term related to the quasi-modular correction. But now let's assume that n0-2k = 0. This is equivalent to the fact that n0 is single weight. It's better to write down this as k = n0 over 2. Therefore, for the singular weight, the base formula is true. You can compare this argument with the arguments we used for modular form, modular function of [INAUDIBLE] zero. In this case, also we can consider meromorphic function, a function which will not allow holomorphic at infinity. And for the single weight, the heat operator is modular operator. Let me fix this conclusion. For the singular weight. The heat operator H is modular differential operator. Because this operator commutes with as a modular action. [MUSIC]