 You don't want to get what the action of L is, what do I do? Well, I find what the action of t is inside the coefficient function. Then I multiply this action of t by z to the pi n plus 1, and I integrate dz to the value of pi. If I isolate the part of the correlation function, then the A is like one of the z to the pi n plus 2. And there's a long way to do that using consciousness. Is this clear? Remember we said that the action of L might depend, you know, is given by this kind of path integral, this kind of integral, but changes, you know, depends on whether you can't just go through operator's actions. Yes, so we're in the state, right? We take a slice of constant time. A is some operator defined in some time. And if I wanted to insert L at that time, what do I do? I use the definition. I insert t everywhere on the circle, and do the following contour integral on correlation functions of t. That would give me correlation functions of n. Yeah. So the statement about the operator's actions was that if you insert this thing here, or you insert this thing here, you shouldn't get something different. Why is that? Why is that? Yeah, because correlation functions in a path integral are always time limit. So if you put something before or something else, you get one order. If you put something after something else, you get second order. And unless these two operators come in, you should get a different state. So we say that in terms of this picture. You see, my torche is there. Provided this t here, the function in question, is completely analytic. It does kind of work. What's on your main set? Is this clear? It doesn't sound good. That's it. Suppose we have, you know, every statement I'm making is a statement within a correlation function, inside a path integral. So suppose I come from some operator in the set. I wanted to put L at some time. So I do this. What makes the path integral that it needs a correlation function of t with all these other operators? Thought it was a function of the insertion point of z. And then you'd be written in this way. By the answer, the correlation function is analytic in z. In z for which it's analytic, I get freely formed along the door. It won't change my answer. That's what you said. For any analytic function, prenatally form your conflict in regions of analytics. But you can't go through regions of non-analytistic and reach to singularities with this. And we know that in general, t with other operators inserted will have singularities. So if you insert your head operator there, it will not be the same as inserting your head operator here. Because that will be a multiplication of singularities that you're not taking without. And that is the mathematical analog of the physical statement that operators, the insertion of time order operators, depend, you know, putting something just before or just after, is different unless the operators communicate. Take this intuation and use it as an activation device. So that will be a function of the head. To the rest of you. And it should also be provided. Commutate that. Okay, so let's actually use this as a nice calculation. You know, in order to calculate something. What we have to do is to take the definition we had of the LNs, which defines our operators on our open space. And I want to compute the commutation relations between these operators. So what I want to do is to compute what I get LN minus LNl. It is of various conformal transformations. You see that, right? We had a formula for the generators of conformal transformations last time. Okay, it was, the current was multiply epsilon of z with t of z. That was current. And if you wanted to charge, then you have to take this thing and integrate it over space, which is over the surface. Okay, so now, you remember we parameterized conformal transformations instead of taking arbitrary epsilon z and it empowers them. So it's like some power times t of z. Okay, integrate it over the surface. Now, if you perform z to the power k times t of z and integrate it over the surface, but t of z was z to the power k times LN divided by z to the power plus 2 sum over N. We do this integral over the surface if k is only when k is equal to N plus 1. Okay, so you get actually LN plus 1. So these LN plus 1 are the generators of conformal transformations corresponding to the conformal transformation in our k. Some link between k and N. These objects, because we've got a huge symmetry in our theory. Symmetry of that conformal transformation. And we want to see what symmetric group the commutation relation of the symmetry algebra generate. So what we're doing is conceptual equivalent to computing the combination relations of the generators in a direct relation. Let's go on. This is what we want to compute. Well, let's see. Let's write down the expressions for LN and LN. The first is to ask, how do you compute LN times LN inside a path integral? Well, remember, inside a path integral, anything that is inserted at an earlier time goes to the left. Things inserted at a later time go to the left. The left times, in z-lay, is smaller in this. So if I want to compute LN times LN, I should do the following. Remember, LN was equal to integral d z z to the pi n plus 1 t of z divided by 2 by n. LN was equal to, let's take w, w to the power n plus 1 by 2 pi i d of z. If I want to compute LN times LN, what I do is insertion with two t's, 1 and w and 1 and z to two contouring tables where the outer contour integral is z and inner contour integral. Let's make a final statement. The insertion of LN times LN is equal to the product of these two things. So d w, d z by 2 pi i, 2 pi i, z to pi n plus 1, w to pi n plus 1, t of z on this contour. Right? On this one. d w, d z to pi i, the whole thing squared. And then the same thing, z to pi n plus 1, w to pi n plus 1, t of z and t of w. But I move inside. Doesn't matter. Because unless these two t's approach each other, the correlation becomes fanatic. The question is there of what contour do you choose? What contour do you choose? Unless, you know, you pass through a point of insertion of t, it doesn't matter. There is inside and outside whatever v d shape you take, it doesn't matter. In-choose circles, but it doesn't matter. If you take a little more space in front of you, you want it to be in-choose circles. Once you've chosen that, you may deform it however you want. So let's deform it in a way that is of a boundary choice. Let's see. w and there are two integral values. I can do the w integral and the z. Let me first do the w and let it first do the z. So let me freeze w to a particular value. So let me freeze w to a particular value here. I imagine that the w integral is always at a particular contour and the z thing goes outside inside. Let me freeze w to a particular value here. First, we can form a contour. We can form this contour to here. I've got to go to w. Minus this. I'm just doing the integral. w is w. And the z integral is just in this. I have to do dw divided by 2 pi i. Let's use the pole because I've got a model of 2 pi i there. So it's just a residue of the pole. z to the pi plus 1. w to the pi plus 1. t to the z. t to the w. So it's a residue of the pole in value of the z. Because I've got so nicely replaced that the top case is no pressure. So let's move on. I'll be with the values. So all I can do is compute the residue of the pole. But frankly, who people supposed to have it at? I don't know. But where does the... I'm so sorry. Can we just have it now? Why not? We have to wind this up. Yeah, yeah, yeah. Okay, let's wind. Let me leave you as an exercise for you to compute this residue. I'll ask somebody in terms of the pole. Okay, so we stop for a bit. The reason for that string is technically complicated.