discuss a connection between infinite-dimensional algebra and conformal link to conformal field theory on probability. So I think a good place to look for such connection is field theory, because in some way, if you think about field theory, at least

Euclidean field theory in terms of statistical mechanics, Euclidean field theory is more or less a way to define or try to define probability measure on infinite-dimensional space, configuration space. That's what statistical mechanics does. So on the other hand, we all know, since Victor's work and many others,

that infinite-dimensional algebra is useful to try to make sense of some field theory and conformal field theory. So one simple example you surely know is that when you look at the Boolean motions, which are just a way to describe quantum function with some property, this may

be represented in terms of free fields. And the measure, the Boolean measure, is simply the data of the vacuum for the free field. So that's one of the basic examples. But the one I'm going to talk is a measure which are linked to random geometry, which

try to give measure on an extended object, which arise in statistical mechanics or field theory. On one hand, there is a stochastic linear evaluation, which gives measure on curves, which are planar curves in a simply connected domain.

Or a Boolean loop soup, which gives measures on loops. So a Boolean loop soup is a Poisson process with target space loops. So for each realization, we have a set of loops, which are overlapping. And if you look at the boundary of the cluster made by the overlapping loops, you get

the new loops, which may be notified with loops with a boundary of cluster in statistical mechanics. SLE gives, as I said, lead to measure of curves. And here is a sample of what SLE can be. This is taken from a realization of the Ising model

in two dimension at the critical temperature. So in the Ising model, at each point on the space, we have one degrees of freedom, which take two values, either blue or white or black. And if you look at the interface between black and white region, you have a curve, which is random because it

depends on the samples. And this curve has many properties. And the measure of this curve is given by SLE. OK, so what I want to try to explain is some algebra property linked to a virus or algebra and these measures.

So I have to tell you what SLE is. So since I'm of today's just to play with some algebra, I will more or less forget all about the random geometrical aspect of SLE, which is probably not a good idea. But that will be the whole of the game today.

So the way SLE works is try to define the measure of the curve by working along the curve and trying to decipher the properties of the curve when you let it go, when you work along it. So you code the curve by some conformal map. And this map is a map which uniformize the domain

where you draw the curve, here as a half plane. So that domain minus the curve is conformally equivalent to the domain without the curve. So that is a uniformizing conformal map, which could, if you normalize properly, completely the curve. So if you can give a measure on that conformal map, then you give a measure on the curve.

And SLE defines the measure on that conformal map by solving a stochastic differential equation whose source is just a Brownian motion. So from the measure on the Brownian motion, you induce a measure on the conformal map. So you induce a measure on the curve. That's SLE. So the way it works, if you just want to do it

purely algebraically, you look at the curve on the upper half plane. When you look at this conformal map, which I call Y, and you normalize the conformal map so that it fixes infinity, and that its derivative at infinity, so that it starts with Z. This gives two conditions.

And then you want that the tip of the curve is mapped, let's say, back to the point where I started. So this gives three conditions. It fixes the conformal map completely. So you may expand the conformal map into power of 1 over Z, which gives you a set of coefficients, where the first one is fixed to 1.

The second one is fixed to the Brownian motion. It's proportioned to the Brownian motion. And then you solve SLE, the linear equation. And to solve the linear equation, what you have to do, you just take your series, compute the inverse, which can be also expanded into power of 1 over Z.

This gives coefficients, which I call PG, which are functions of the A that you can define recursively. P1, so I will give two examples. P2 is 1, P3 is minus A1, and P4, I don't remember. And then you solve this series of infinite equations, which are just the derivative of ai is pi, or 2pi.

And this defines all the coefficients aj as integral of the Brownian motion, see, recursively. Now, everything is coordinate to this coefficient. So if you want to know property of the curve, geometrical property of the curve,

you have to ask for, look for function, which we call this function of the aji, a, a coefficient, which we code for this property. So all the observable in this game, in the SLE observable linked to the curve, are coded into function, which are aj.

And what I want to play with today is try to answer the question, or another answer, so answer the question, what is the structure of the space of martingales, which are polynomial in this coefficient? So when you have a process,

martingale is something which has a few properties, but more or less physically, it means that it's conserved in mean. So in the observable system, you have many conservation laws. Here, it will be asking for conservation law is too much for each realization. So you only act for conservation law in means. So martingale is a bit more, but it's more or less the same thing.

So we want to know the structure of the space of martingales, which are probabilistic objects. And why we want to know that is because from Boltzmann's basic rule of statistical mechanics, we know that correlation function of statistical mechanics, which come from statistical mechanics,

all observable in statistical mechanics, are going to be martingales. So if we know the space of martingales for our process, we will know the, more or less, all the correlation function, which are linked to the correlation function of the statistical model, which

are linked to the properties of the curve. And this is how you make the contact between SLE. You make a corresponding between SLE on the corresponding field theory, which is associated to the statistical model, which is a conformal field theory. So that was the motivation for looking at martingales. But now we forget about martingales. The motivation, I just looked at algebraic property.

So if you want to, if you ask for a martingale, you have to ask that its time derivative, or time derivative of its mean, is 0. This leads to a differential equation, which is just written here, which is a second differential equation.

And this second order differential operator is linked to the second order differential operator, which is associated to the stochastic process, which generates the curve. Every time you have a Markov process, you have a second order differential operators. So what I want to look at is that the property of the space of polynomial, which

are in the kernel of the differential operators. So you can look, you can give a degree to the variable a. The degree corresponds to the indices. And you may compute, degree by degree, what is this kernel. You find some polynomial. You can look at this from the polynomial, which

are function of the two first variables, a1 and a2. So if you solve the differential equation, which define SLE, you find that a2 is t, and a1 is buoyant motion. And the solution of this differential equation

is just, so here there should not be a 2, is this function, exponential of xi minus alpha square t, which is a generating function of Brownian martingale, which are located in time. That's the exponential, OK.

So that's one example. Now if you want to find the space of all polynomials, what I'm going to claim is that this space is a virus or module, irreducible for generating kappa. Kappa is just a proprietary coefficient here. What's characterized given here? So in a way, you characterize some space

of some martingales, which are functional of integral of the Brownian motion, and this space is a virus or module. So this comes from the connection between SLE and CFT. But I use this as a pretext to play it a bit with representation of the virus or algebra. And some of them are inspired by the work of Victor

on Victor Katz on Waki-Moto. Is the sentence just 1? No, less than 1, and depend on kappa. I will give you. The fact that there is this character means that there is a new vector in the module. So there is many ways to do it.

I remember one of the papers by Victor on E8 was, I don't remember, 118 ways of constructing E8 representation of alpha in E8 at level 1. So he has probably less way of proving this correspondence, but I knew at least 6, something like that.

So the way we do it is by using some kind of representation of the virus or algebra, which is inspired by the Borel-Weyl construction. So the Borel-Weyl construction is that if you look at section of this question of complex group

by the Borel subgroup, then you have an action of the group on the space of section, and it gives you a representation of the group as first-order differential operators acting on this function. So if you do it for R-finely algebra, you get something which is close to what Victor on Waki-Moto

did, which is, in physics, the free-field representation of the Vase-Domingo-Witten model. So here, I will do it for virus or algebra. So you look at, it's a bit formal, but it's probably algebraic. So you can do it even if you don't have a virus or group, you don't have gross decomposition, but you can nevertheless play all the algebraic rule, which

are involved in the Borel-Weyl construction. So the Borel, you have two Borel subalgebras. You have the same decomposition as we talk about. You have the two triangular subalgebras, n minus 1 and n plus. You have the Cartan subalgebras, which is made of the dilatation plus the central element.

And the n plus and n minus correspond to conform map, which fix either 0 or infinity. Now, if you look at the question, let's say, question on the left, by the Borel subgroup, this is more or less identify, at least locally, with n minus 1. And I can view my element, y, of the coefficient aj

as coordinate on n minus 1. So I am looking for the action of the virus or group, whatever it means, of n plus h1 minus 1 on function of variable a1, a2, all the ag, with some property of the conformal transformation. And if you apply the machinery, this

gives you representation of the virus or algebra on that function, on this operation, in terms of first order differential operators, differential operators in this infinite number of variables. These are different from the level 1 representation that you will get using free field.

And this representation depends on two numbers, which enters the way you define section, because to define section, you have to use some characters of the Cartan subalgebra. Cartan subalgebra has two elements, the dilatation and the central element. So representation will depend on two numbers, central charge and the weight, the Hagen value of the weight

of the highest weight vectors. This is formal. This is just complex. This is conformal map with complex coefficient.

You don't have group. I will tell you about what could be the group. You don't have group, but you nevertheless do all the manipulations, which are purely algebraic, which are involved in the Borel-Weyl construction, because you have this go-slide decomposition. So you can look at the two questions, either on the left

or on the right, and they give you two different sets of differential operators, which both satisfy the virus or algebra. And they do not compute, but they act on the same kind of function. Now, if I go back to my problem,

and I look at the differential operator I was interested in, which was the second differential operator, this one. And I can notify these differential operators in terms of the virus or generators I constructed using the Borel-Weyl construction. Because if you do the Borel-Weyl construction, you find that L1 is just a differentiation with respect

to A1, the first coefficient, and L2 is a formula. So now what I'm interested in is looking at the kernel of A on the function of S1. I can use now, so to define A, I use one of the constructions, the left one.

Now I can use a second construction with the right question. Then what you can check is that if you choose C on H in an appropriate way, the second virus or generator will act on the kernel of A.

We know that they form a representation, but it's only when you fix C on H that they act on the kernel. And why they act on the kernel? Because if you choose that value of C on H, then the commutator between these differential operators on Ln is proportional to the kernel with some polynomial.

So the Ln-white, which are first-order differential operators, unsatisfied the virus or algebra, act on the kernel. So the virus or the kernel is a module for the virus or algebra. It's a module with this value of central charge,

which is less than 1, on H, which is this value. And what you can check, so it means that you can check that it's irreducible module. You get all the elements of the kernel by acting with the Ln. You act with the Ln of the constant function, with L minus 1, L minus 2, L minus 3, and so on. You repeat, and you get all the function.

And for these values, a module has a new vector at level 2. It has a linear combination of operators of virus or algebra, of the universal algebra of degree 2, which annihilates the constant function. So it's new vectors.

So then it's by Catt's data-luminant formula for the beginner form in the Catt's module, the virus or module, we know that it's irreducible for generic kappa, not for all kappa, for generic kappa. So that's the proof of the claim. The kernel of A is a irreducible module

of the virus or algebra. And this has a simple interpretation in terms of conformal field theory, because this kernel is linked to the creation function of some field in the statistical model in presence of the interface. And from this number, we know what is the central charge of the corresponding conformal field

theory or the corresponding statistical model. And from this value of h, we can identify the operator which creates a curve, which creates a curve in the model. So that was one thing I wanted to say. I have five more minutes. So then you can go on and on with this relation

between CF conformal field theory and SLE. And so it's a way to see what's the relation between probability, because SLE is purely in term of probability, and conformal field theory, which was usually formulated in term of algebraic data.

But I want to use the construction I mentioned to point another connection between conformal field theory and the Boolean loop soup. Once again, before, you probably mean one particular conformal block, not correlation part. It's holomorphic. This thing is holomorphic, right?

So what you have in mind, you have a domain on which you define your statistical model.

And let's say when I talk about the Ising, I fix the boundary condition plus here, minus here, minus here, minus here. So for each realization, I will have an interface here, something like that, with plus, plus, plus, and minus, minus.

So since I change the condition here, I have to insert some operators, which I call psi, which has this dimension here at the position x0, which is that point. OK? And I have the same operator here, psi, if I call it x infinity.

So all the coefficient function you are dealing with is for this statistical model with this boundary condition. So you have to encode the fact that you have fixed the boundary condition. So all the statistical coefficient function, if you have some observable which corresponds to the intersection of local operators, say, or whatever,

that will be O, of the statistical model, in the domain with the boundary condition, they are given by you have to encode the fact that you change the boundary condition. So you have to insert these operators. So this is the CFT.

This is not normalized, so you have to normalize it. But that operator can be anything in the bulk. It can be whatever you want. OK, so the next point is connection

with a polynomial loop, I want to mention, a connection which I am not sure to really understand. So if you look at the co-formal map, which fits, let's say, infinity, you can view it as an element of n minus 1.

And then to any IS-Y TRC model, you can associate an element of the universal enveloping algebra of the n minus 1 of the negative part of the variational algebra, which implements this co-formal map. It means by adroit action on the primary, on the vertex, on the operators,

it will implement the co-formal transformation of that operator under that co-formal map. And you can do it for any operator on this G minus 1 will be infinity, which will converge in most cases. So then that's work without problem

if co-formal map is in one of the Borel subalgebra, fixed infinity. You can do the same thing if it's fixed 0. If the co-formal map is in n plus, you will get another operator, which we implement by adroit action, that co-formal map. So to define the group, you need to make product of such object.

So there is no problem to make the product in one order. If you want to act, if you think about not formally as this product, but as this product acting on some IS-wide module, then you have no problem to define the product of G plus times G minus 1, because this one is in the Borel subalgebra.

So if you, and this one is G minus. So if you take a matrix element of that one in a IS-wide module, only a fine number of term will be involved. Well, this operator will make sense. So it makes sense at least in a weak sense, just matrix element. So then the question, so that will be a way to define the n element of the virus or group,

whatever it needs. I don't know exactly, but the simplest and that will be that one. So I forget this part, which is link of H, you can deal with. OK, so you have to insert something in the middle. So now if you want to make product of such object, at some point you will encounter product of G in the wrong order with G minus 1 times G plus.

And this doesn't make sense apriori in IS-wide module, because it will involve infinite series of terms. So you have to know how to reorder this product. I mean, if you have this product, you want to write it as a product of something in the plus subalgebra on the right

and minus subalgebra on the left. Well, that's what we do when we do VIC, when we use VIC theorem or when we play with vertex operators in conformal physics or other things. So the way to do it is to look at some commutative jaguar, which is tell you how you uniformize some domain.

So think about the two conformal maps linked to uniformization of the upper half plan minus sum all, all. So that will be, there is an L associated to FA, which is in plus, and an L associated to the element, which is N minus 1, I call FP.

So that tell me what, if I fix some condition, what FA and FB are, then when you want to, on this product, will be linked to the uniformization of the two Ls. But there is two way to uniformize the domain, mainly these two Ls. Either you first erase B or you first erase A.

So if you first erase B, you use FB. Then A is transformed into some new L that you can uniformize with another FA tilde. Or you can do the first uniformize A. Then you get a BD form that you uniformize

with a D form map. If you look at this map, you can choose a map that is commutative. And the fact that it's commutative means that when you do the product of the element which are set to AB and A tilde, you have two way to do it. And this will correspond to the reorder of the operators. OK?

But here we are dealing with a VR, so algebra, which has a central extension. And in quantum mechanics, we can use central extension because we are interested in projective representation. Because we always use adenoid action. So this means that when you do this product of reordering, it's up to a central element.

So what you get is that if you do the product in the wrong order of that one, then you will get the product in the right order up to some matrix element here, which is what log is proportional to C. So that you can compute explicitly. And the miracle, which I don't understand, is that now this function has some probabilistic meaning.

It's a probability that no loop of the Brownian loop loop intersect both A and B. OK, that's a question for you. So if you know, if you can answer and tell me why a good explanation for it, except that probably it's

the only possible answer, because by conformal invariance. But I think it's not a good enough explanation. So if there is a better explanation, that would be nice. OK, so I stop here. I will just conclude by quoting a mail I got from Jean Thierrymy, who sent me a mail just last Sunday.

We made for Victor, so that's for him. And thank you. There are more questions, since we already had one.

Victor, comment. In the rest of the group, the commutator is not necessarily central element. Why do you hear from the central element? It has to be commutated. The commutator. But this is not the same element here.

It's different. So when I took two l's to conformal map, which I indexed by B and A, here I got A tilde and F tilde. So I have to change the map, which corresponds to the commutation in the virasal algebra. So if there was not these things, that would be just the width algebra.

But then there is a term which comes from the central charge, which is this extra term. And you see, that's exactly what you take for when you do vertex operators. You have the creation part, the annihilation part. You commute, you get the exponential of the log.

That's the analog of the exponential of the log. But the commutation are hidden here.