 et on va discuter de la connexion entre l'algebrae de l'infini-démotion et le lien conformal de la théorie et la probabilité. Donc, je pense que un bon endroit pour chercher la connexion est la théorie de la théorie, parce que si vous pensez à la théorie de la théorie, à la fin de la théorie clé, en termes de mécaniques, la théorie de la théorie est plus ou moins un moyen de définir, ou essayer de définir la probabilité de mesurer l'espace de l'infini-démotion, l'espace de configuration. C'est ce que la théorie de la théorie fait. Donc, d'autre côté, on le sait tous, depuis que Victor a travaillé et beaucoup d'autres, que l'algebrae de l'infini-démotion est utile pour essayer de faire le sens de la théorie et de la théorie conformale. Donc, un exemple simple que vous savez sûrement, c'est que quand vous regardez la motion de l'abond, c'est un moyen de décrire la fonction de l'abond de l'infini-démotion avec des propriétés, cela peut être représenté en termes de frein, et la mesure, la mesure de l'abond de l'infini-démotion, c'est simplement la date de l'accueil pour le frein. C'est un exemple basique. Mais l'un que je vais parler est la mesure qui est liée à la géométrie qui essaie de donner une mesure sur un objet externe qui augmente dans des mécanismes statistiques, ou de la théorie. En un an, il y a la révolution stochastique, qui donne une mesure sur les cartes, qui sont des cartes planères qui sont connectées ou simplement connectées au domaine, ou une soupe brunienne qui donne mesures sur les loûts. La soupe brunienne est un processus poisson avec des loûts de target space. Donc pour chaque réalisation, vous avez un set de loûts qui sont overlapés, et si vous regardez la bonderie de la cluster faite par les loûts overlapés, vous avez des nouveaux loûts qui peuvent être identifiés avec la bonderie de la cluster dans les mécanismes statistiques. SLE donne, comme je l'ai dit, une mesure de curve. Et ici, il y a un sample de ce que SLE peut être. C'est une réalisation de la modélisation de 2 dimensions à la température critique. Dans la modélisation de la modélisation, vous avez 2... À chaque point de l'espace, vous avez 1° de frein, avec 2 valeurs, soit bleu ou blanc ou blanche. Et si vous regardez l'interface entre blanc et blanc, vous avez une curve, qui dépend du sample, et cette curve a beaucoup de propriété et une mesure de curve est donnée par SLE. Ok, donc, ce que je veux essayer d'expliquer c'est des propriétés de l'algebra liées à l'algebra et à ces mesures. Donc, je dois vous dire ce que SLE est. Alors, depuis que aujourd'hui, c'est juste de jouer avec des algebras, je n'oublierai plus tout le plan de SLE, qui n'est probablement pas une bonne idée, mais ce sera tout le jeu d'aujourd'hui. Donc, la façon où SLE fonctionne, c'est de essayer de définir la mesure de la curve en regardant l'algebra et en essayant de vérifier la propriété de la curve quand vous le laisse, quand vous le travaillez. Donc, vous coudez la curve par une map conformale. Cette map est une map où vous vous entrez la curve. Ici, c'est le plan de l'algebra. Donc, ce domaine minus la curve est conformément équivalent à ce domaine sans la curve. Donc, il y a une uniformité conformale de la map conformale, si vous normalisez le correctement de la curve. Donc, si vous pouvez donner une mesure sur cette map conformale, vous donnez une mesure sur la curve. Et SLE définit la mesure sur cette map conformale avec juste une motion brunée. Donc, de la mesure sur la motion brunée, vous induisez une mesure sur la map conformale et vous induisez une mesure sur la curve. C'est SLE. Donc, la façon dont ça fonctionne, si vous voulez simplement faire l'algebra c'est que vous regardez la curve sur le plan de l'algebra, et vous regardez cette map conformale à cause de Y, et vous normalisez la map conformale, sur ce que c'est fixé à l'infinité, c'est que ça commence avec Z. Ça donne deux conditions. Et puis, vous voulez que la hauteur de la curve soit map, je dirais, au point où il a commencé. Donc, ça donne 3 conditions et fixe la map conformale complètement. Donc, vous pouvez exposer la map conformale en 2 pour 1 par Z, qui vous donne une série de coefficients, où la première est fixée à 1. La deuxième est fixée à la motion brunée. Et puis, vous étudiez l'équation de l'algebra. Et pour étudier l'équation de l'algebra, ce que vous avez à faire, c'est prendre votre série, compter l'inverse, qui peut aussi être expérimentée à 1 par Z. Ça donne une coefficient, qui s'appelle Pj, Pg, qui est la fonction de la A, que vous pouvez définir recursivement. P1, je vous donne deux exemples. P2 est 1, P3 est minus A1 je ne me souviens pas. Et puis, vous étudiez cette série d'équations infinites qui sont juste la dédévative de Ai est Pi, ou 2Pi. Et ça définit toute la coefficient Aj comme l'intégral de la motion brunée, recursivement. Maintenant, tout est quadré dans cette coefficient. Donc, si vous ne voulez pas savoir la propriété de la courbe, regardez pour la fonction qui coûte la fonction de l'Aji A coefficient, qui coûte cette propriété. Donc, tous les observables dans dans ce jeu, dans la SLE tous les observables liés à la courbe sont quadrés dans la fonction Aj. Et ce que je veux jouer avec aujourd'hui c'est essayer de répondre ce qu'il y a. Qu'est-ce que c'est la structure de Martin Gales, qui est polynomial dans cette coefficient ? Martin Gales quand vous avez un processus, Martin Gales c'est quelque chose qui est un peu plus plus physique, ça signifie que c'est conservé dans le milieu. Donc dans le système intégrable, vous avez beaucoup de conservation de l'auteure. Ici, il sera asking for conservation l'auteure too much for each realization, so you only act for conservation of the same thing. So we want to know the structure of the space of Martin Gales, 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 Martin Gales. So if we know the space of Martin Gales for our process, we will know the correlation function of statistical model which are linked to the property of the cap. And this is how you make the contact between SLE you make a component between SLE on the corresponding field theory for the which is associated to the statistical model which is a conformal field theory. So for looking at Martin Gales but now we forget about Martin Gales the motivation which I just look at algebraic property. So if you want to if you a polynomial q you ask for a polynomial q to be a martingale you have to ask that it's time derivative or time derivative of its mean is 0. This leads to differential equation which is just written here the operator is linked to the second order differential operator which is associated to the stochastic process which generates a curve. Every time you have a mark of process you have a second order differential operator. So what I want to look at is 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 and the degree correspond to if this is and you may compute degree by degree what is this kernel you find some polynomial you can look at this for example of the polynomial which have 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 born in motion and this differential equation just so it shouldn't be a2 is this function exponential of psi minus alpha square t which is generating function of born in martingale which are located in time that's the exponential ok so that's one example but now if you want to find the space of all polynomials what I'm going to claim is that this space is a VRSORO module irreducible for generate kappa kappa is just a propensity coefficient here what character is given here so in a way you characterize some space of some martingales which are functional of integral of the born in motion and this space is a VRSORO module so this comes from the connection between sle and cft but I use these as a pretext to play a little bit with representation of the VRSORO algebra and some of them are inspired by the work of Victor on Victor Katz on wakimoto no less than one and depend on kappa that will give you the fact that there is this character means that there is a new vector in the module ok so the way so there is many way to do it I remember one of the people on e8 I don't remember 118 ways of constructing e8 representation of affine e8 at level 1 so here there is 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 VRSORO algebra which is inspired by the Borel-Vail construction so the Borel-Vail construction is that if you look at section of this question of complex group by the Borel-Vail then you have an action of the group of the group on this space of section and this gives you representation of the group as the first order differential operators acting on this function so if you do it for affine e-algebra you get something which is close to what Victor on what Kimoto did which is in physics called the free field representation of the Vesumino written model so here I will do it for VRSORO algebra so you look at it's a bit format but it's purely algebraic so you can do it even if you don't have VRSORO group, you don't have gross decomposition but you can nevertheless play all the algebraic rules which are involved in the Borel-Vail construction so the Borel so you have the same decomposition as Conceviche talk about you have the two triangular sub-algebra n-100 plus you have the carton sub-algebra 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 the infinity now if you look at the question let's say question by the on the left by the Borel-Vail this is more or less identified with n minus 1 and I can view my element y or the coefficient aj as coordinate on n minus 1 so I am looking for the action of the VRSORO group whatever it means of n plus h1 minus 1 on function of of the function of variable a1, a2 or the aj with some property of the conformal transformation and if you apply the machinery this give you representation of the VRSORO algebra on on that function on this representation is in terms of first order differential operators differential operators in this infinite number of variables these are different from the level one representation that you will get using free free field and this representation depend on two numbers which enter the way you define section which because to define section you have to use some characters of the carton sub-algebra carton sub-algebra has two elements dilatation of the central element the representation will depend on two numbers, central charge and the weight the eigen value of the weight of the IS weight vectors this is formal GC just complex this 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 never let's do any all the manipulation which are purely algebraic which are involved in the model construction because you have this goes like decomposition so you can do you can look at at the two question on the left or on the right and this give you two different set of differential operators which both satisfy the VRSORO algebra ok? and they do not compute kind of function now if I go back to my problem and I look at the differential operator I was interested which was the second differential operator this one and I can identify these differential operators in terms of the of the VRSORO generators I constructed using the Borelville construction because if you do the Borelville construction you find that L1 is just a differential station with respect to A1 the first coefficient on L2 is a formula so now what I'm interested 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 the second construction with the right question and then what you can check is that if you use C on H in an appropriate way the second VRSORO generators will act on the kernel of A we know that they form a representation but it's the only one 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 white which are first order differential operators on satisfies the VRSORO algebra act on the kernel so the VRSORO kernel is a module for the VRSORO algebra it's a module with this value of S1 charge which is less than 1 on H with this value on what you can check so it means that you can check that it's a reducible module also element of the kernel by acting with LN you act with LN or the constant function with L-1, L-2, L-3 on you repeat and you get all the function and this is for this value a module has a new vector at level 2 that is a linear combination of operators of VRSORO algebra of the universal algebra of degree 2 which annihilates the constant function so it's a new vector so then it's by cut determinant formula for the for the billionaire form in the cut module VRSORO module we know that it's irreducible for generic kappa not for old kappa for generic kappa so that's the proof of the claim the kernel of A is irreducible module of the VRSORO algebra and this has a simple interpretation in term of conformal field theory because this kernel is linked to the correction 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 so that was one thing I wanted to say I have 5 more minutes so then so then you can go on on with this construction this relation between CF conformal field theory and SLE and so it's a way to see both the relation between probability because SLE is purely in term of probability conformal field theory which was usually formulated in term of algebraic data but I want to use the construction function I mention to point another connection between conformal field theory and the Boolean loop soup 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 icing fixable I condition plus here minus here, minus here, minus here so for each realization I will have an interface here something like that plus plus plus minus minus so since I change the condition here I have to insert some operators which I could 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 correction function you are doing is for this statistical model with this bond I condition so you have to encode the fact that you have fixed the bond I condition so all the statistical correction function if you have some observable which correspond to the section of local operators or whatever that will be o other statistical model in the domain with the bond I condition they are given by you have to encode the fact that you change the bond I condition so you have to insert these operators so this is the CFT this is not normalize so you have to normalize it but that operator can be anything in the bulk it can be whatever you want ok so next point is connection with a poignan loop soup I want to mention a connexion which I am not sure to really understand so if you look at a conformal map which fix let's say infinity you can view it as an element of N-1 and then to any ISWC receivable module you can associate an element of the universal universal overping algebra of the N-1 of the negative part of the VRSOR which implement this conformal map it means by adjoint action on the primary on the vertex on the operators it will implement the conformal transformation of that operator under that conformal map ok and you can do it for for any operator on this G-1 will be infinity which will converge in most cases so then that's work without problem if conformal map is in one of the Borel subalgebra fix infinity you can do the same thing if it fix 0 if the conformal map is in N plus you will get another operator which will implement by adjoint action that conformal 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 but as this product acting on some isoid module then you can 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 on this one is G minus so if you take a matrix element of that one in a isoid module only a fine number of term will be involved on this operator will make sense ok so it makes sense at least in a weak sense just matrix element so then the question what will be a way to define an element of the Virasso group whatever it means I don't know exactly but the simplest and that will be that one so I forget the part which is linked for H you can deal with ok so you have to insert something 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 G plus and this doesn't make sense in a isoid module because it will involve infine series of terms so you have to know how to reorder this product 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 on the left so that's what we do when we do when we use vixiorem or when we play with vertex operators in conformal physics with the other things so the way to do it is to look at some commutative diagram which is to tell you how you uniformise some domain so think about the two conformal map are linked to uniformisation of the upper half plane minus some all L so that will be like the L associated to FA which is N plus on the L associated to N minus when I call Fp if I fix some condition what FA and Fb are then when you want to on this product will be linked to the uniformisation of the two Ls but there is two way to uniformise the domain minus 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 so that you can uniformise with another FA tilde or you can do the first uniformise A then you get a B deform that you uniformise with a different map if you look at this map you can choose a map that is commutative and the fact is commutative means that when you do the product of the element which are associated to A, B and A tilde you have two way to do it and this will correspond to the reorder of the operators but here we have we are dealing with a VRS or Algebra which has a central extension in quantum mechanics we can use central extension because we are interested in projective representation because we always use adjoint 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 that you can compute explicitly and the miracle which I don't understand is that now this function has some probabilistic probabilistic meaning it's a probability that no loop of the Brunian loop intersect both A and B ok that's the question for you ok, if you can answer and tell me why a good explanation for it except that probably it's the only possible answer because a 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-Tier Rémi who sent me a mail just last Sunday we made for Victor so that's for him thank you ah, the commutator but this is not the same element here it's deformed so when I took 2 Ls to conformal map which I indexed by B and A E, I got A tilde and F tilde, so I have changed the map which correspond to the commutation in the vibratoalgebra so if I was noticing these things this would be just the width algebra so that ok but then there is a term which comes from the central charge which is this extra term and you see exactly what you take when you do vertex operators you have the creation part animation part you commut, you get the exponential of a log that's the analog of the exponential of the log ok but commutation are hidden here