 To je moja pravda in odličenja, da so tukaj vzvečili, vzvečenja, in zelo, da bomo povedali, organizacji Sasha, Samsono, Nikita, in institutom za toj posledanje. Ja postava scoop, da je v Vajdim, vzpravljena s njem, a ta razvorega vzpravljenje memjer. U rači memjeri, vzpravljeni, ..And particular memories are related to the way Vadim was sharing his remarkable ways ... ..of ...his remarkable new ways of looking at things. ..And not only in physics. Vse je to pravno delješ češčo se naščešči. Vi moramo se naščešči pravno. Vsi, da sajemšel prediv, da corrected vsi, da mu bo povedal kova češči. pa izgleda. Fizik se zelo, da ne zelo, da pa mi ne odliči. ... ... ... ... ... ... ... ... ... ... ... ...jo šta je jezik linjeva, kaj je z tem, da je menev izgleda bolje zelo. ...in nekaj da nečo, da je to zelo... ...kaj ti je dragged, da je to... ...zelo... ...zelo ... ...zelo... ...zelo... ...Zelo... ...zelo... ...zelo... ...zelo... ...zelo... ...zelo... So that my talk will be devoted to the subject, so it's a part of this talk would be some development of the subject that concerns conformal blocks. Most of the talk will be in the way of extended introduction, whatever the original part is based on the joint work with Alexei Litvinov, Sergei Lukyanov, Nikita. In, zelo, misljenje in misljenje je všeč mene. To je tajtelega klasične, konformalne bljok, in penlivega, siks. A tako, pričo stačno, nekaj nekaj dobro vzeličenih vzeličenih nomi, zelo je to najbolj obržena vzela. Prizeličenih vzeličenih, nekaj dobro nekaj dobro vzeličenih, nekaj dobro nekaj dobro nekaj dobro vzeličenih, Tako klasiklje vrštjev, je v srednjih počutji vyrsora, vyrštjev, kaj vrštjev se bariši, zelo počutje v rade. Vyrštjev se vrednje vrštjev, vrštjev se vrednje vrštjev, sa boljih pankčar, in modelija spasja. obž gelovnja surface, v punčer, ali počutim, da ima vzelo več sem bila, počutim, da je vzela vzela vzela vzela vzela vzela vzela vzela vzela vzela. Vzela vzela, da je vzela, da je vzela vzela vzela vzela vzela vzela. v tem vsem kompleksu koordinati, na svega svega. Svega, odlič, da se vse svega vsega bila svega naša punša. V nekaj različ, da vsega. Tako, nekaj punša. Zato, če je svega konformalne blok. Zelo, da je to svega koordinacija kajl primary vertex operators. The primary means the standard primary type of commutation relations with verisoral generators or operator product expansions with energy momentum tensor. But because these are chiral fields, galamorphic fields, they are not local fields. And so the notion of correlation function is ambiguous. And to give a meaning to the correlation function, one needs to specify what is known as a dual diagram or equivalently pendicomposition of this sphere and punctures here. And I will basically consider only this kind of hairbrush type of the dual diagrams. And in this case we can think of it as a correlation function of the, or expectation value of this vacuum-vacuum, expectation value of the product of this vertex operator with the instruction to put in between the operators the intermediate state decompositions in terms of the irreducible representations of the verisoral algebra with the dimensions with certain fixed conformal dimension delta of p. So this diagram is representing, or equivalently this is the instruction to use the operator product expansions in certain order. Well, in the order which is dictated by the dual diagram. Now delta of p is a specific parameterization of the conformal dimensions in terms of this parameter p alpha. At this moment this is not very important. This parameterization is motivated originally by the Fagan-Fuchs representation of the verisoral algebra and was very convenient in Liouville theory, although I will not actually refer in any systematic way to a Liouville theory, although I will mention it. All right, so coming back to this picture we see that there are n minus 3 intermediate legs here, so there are n minus 3 of these parameters. So the conformal block F depends on n minus 3 parameters p and it also depends on these positions, the alpha, and because of SL2 transformations the dependence is essentially up to projective transformations. So essentially we can just use SL2 transformations to send 3 of these points to predesign positions, usually 0, 1, or infinity, 0, 1, and infinity, and after that we are left with n minus 3 coordinates in this independent coordinates on this modular space. So it depends on n minus 3 coordinates in the modular space and also of course it depends on these external dimensions associated with the external legs in the dual diagram. But I will admit them in this notation because they will be regarded as fixed numbers, although of course the conformal block depends on them. So this is conformal block. I will be, just like I said, I will be dealing mostly with the four-point conformal block which now depends on a single intermediate state parameter p, a single coordinate x in the modular space and the notation will be f sub p of x. So the conformal blocks are fundamental objects in conformal field theory that were introduced in the 80s in that time in which we fondly remember when Vadim was with us and the basics of conformal field theory was originating. So it's a basic object in the sense that correlation functions of conformal field theory are built in terms of conformal blocks and through the coloromorphic factorization. So I will produce just an example how it looks in the alluvial conformal field theory where let's say four-point correlation function that's a simplest example where it enters in a trivially is expressed in terms of the absolute value squared of the conformal block. So you have two copies of conformal block depending of x and x bar and it's integrated over this parameter p with some coefficients which are known as the structure constant of alluvial theory. These are explicitly known and one of the reasons I write down this formula is to introduce this parameter b which is a sudden parameterization of the central chart which I will be using every now and then in the course of this discussion. Nikita, when I need to stop? It's not that I want to stop now, 12.50, OK, that sounds good. All right, so this is how... 50 minutes. 50 minutes, OK. So this is the way how originally conformal blocks appeared in conformal field theory but, I mean, it turned out that they are... the emergence seemed to be much more general in mathematical physics and lately they attract attention. In relation... Well, it's quite unexpected to me. I mean, people who are much smarter than myself probably expected that, to me, it came as a miracle that the same functions appeared in relation to the four-dimensional supersymmetric conformal field theories and, well, in particular, these conformal blocks appeared as... Well, they precisely coincided with the instanton parts in Nikrasov's partition function of the four-dimensional n equals to two supersymmetric gauge theory with certain... Well, if you were talking in Verozora conformal blocks, that's a supersymmetric gauge theory with certain content of product of SU2 gauge group and remarkably the parameters which originate from the modulate space on this Riemann surface plays the role of the gauge coupling constant relates to the gauge coupling constant in these gauge theories and the parameters p relate to the vacuum modulate in these gauge theories. So, to me, it looks sort of... Sort of, yeah, sort of remarkable, I would say. But nonetheless, it's so and I will... Now it's rigorously proven and I will mention that, but... So perhaps these functions play some sort of wide role in all these correspondences. But the point is that these functions we have some control on these functions. In particular, the conformal symmetry... Generally speaking, the conformal symmetry completely fixes these functions. Similarly, in the supersymmetric gauge theories, of course... Well, I mean, conformal symmetry fixes power series expansion and by implication, of course fixes these functions. If I write down power series expansion of this function in terms of the moduli, all the coefficients are fixed by conformal symmetry here. And similarly, this is an instanton expansion in the n equals to 2 supersymmetric gauge theory and these coefficients are given by Nekrasov's integrals of the moduli. Of the instantons, of course they are fixed. But that is a power series and we would like to have a better global analytic control over these functions and because they play this important role and that's one of the motivations of this work, which I am going to report is gaining some better analytic control. There are some things which are known or assumed about this. I will be speaking about a four-point conformal block but the generalization to the endpoint conformal blocks are more or less straightforward. So the conformal block is analytic on the universal cover of the moduli. Well, basically, yeah, on the universal cover, on the less space and as a function of this p parameter it's a metamorphic function and it obeys the crossing relay, what I call crossing relation. This is a relation when you interchange the position of the punctures by braiding transformations and when the conformal blocks are related through this integral transformation you transform a sort of conformal block through the conformal block in the cross channel with the integral with a certain kernel which is related. The kernel here is essentially a 6j symbol of continuous representations of q-deformed universal in the looping algebra of vessel 2. So this is part of the I think most essential part of the knowledge about the electricity of these functions. Now classical limit. I'm going to speak about classical conformal block that emerges when you send the central charge to infinity and interesting limit here appears when you also send before proceeding I will again write the central charge in this Liouville inspired or Fagan-Fuchs inspired actually because that appeared before parameterization of the central charge in terms of this B, this is not essential but I will use it because I like it. That's the most important reason. So I parameterize the central charge in terms of this B and the classical limit will be the limit of B going to zero. B is like a Planck's constant. Actually B squared will be one of the Planck's constant. So interesting limit emerges when you send the central charge to infinity in which case of course we know the Algebra converts reduces to plus one bracket Algebra but also you need to send the dimensions to infinity so that the ratios of the dimensions in the central charge remain fixed and are kept fixed and I will refer to this deltas, the ratios as the classical dimensions. This is not capital but lower case deltas as classical dimensions. And again I will use the parameterization of this classical dimension delta in terms of this parameter and new, depending of either I am speaking about the deltas associated with the external legs of the dual diagram or internal lines. And well, I mean the dimensions could be any so generally if they are real I could think of lambda as real or pure imaginary but generally I will think of this lambda as generic complex numbers. Now when I perform this limit the conformal blocks exponentially in this form that something which semi classical intuition makes us to expect but mathematical status is to me at least is not completely ultimately clear. So it exponentiates so that it it is exponential of 1 over b squared times f which is this f lower case f is called classical conformal block. Now there is no doubt that this is indeed correct statement and actually I think the proof is sort of exist. It's just my brain doesn't didn't yet incorporate all the details of it. It will happen eventually it probably consists of mixture of the results which come from different sides of this Al-Dai-Gaiyot or Tashikava correspondence which involves the form of nekrasov or form of nekrasov representation of the coefficients of the power like expansion which looks like a coefficients of the virial expansion of certain gas and then the classical limit is like a thermodynamic limit of this gas and then integration is usual thing which is and then there is now a sort of rigorous proof of this correspondence at the level of this 4-point function so I think the combination of this statement if we absorb it so it probably would lead to but I'm not particularly interested now there is a neat and well-known relation I think it's a game we all played 30 years ago a relation of this thing to the monodromy properties of second order differential equation so consider differential equation of this type which parameter with a variable z regarded as a coordinate on a Riemann sphere this is a differential equation with n regular singularities on the sphere and so to make it precise in the beginning so the potential term in this equation involves two kinds of parameters delta i and c i the delta i will be regarded as fixed numbers and actually I will take them equal to the classical dimensions which appeared in the previous transparencies and identify them with the parameters and c i are the celebrated accessory parameters and they so basically this differential equation is formatized by the positions of z i and c i well in fact there are only independent accessory parameters for obvious reasons which I sort of explained by this line if we don't want an additional singularity to height of infinity then there are three elementary relations between them so basically there are n minus three of these independent accessory parameters and also the positions z i because the form of the differential equation doesn't change under SL2 transformation again the z i's are defined model SL2 transformations so there are essentially again we can send three of them predesign favored positions and so there are n minus three parameters of this sort so this differential equation is is parametrized by two times n minus three complex parameters of course this differential equation defines a monodrami group which is a homomorphism fundamental group of a sphere with punctures into SL2 SL2 is a transformation of the basis of two independent solutions of this equation under the analytic continuation along contours representing the elements of the fundamental group and and so because I say that's lower case delta are fixed we are dealing with representation in which the conjugacy classes of the matrices m i which are representations of the basic elementary elements basic elements associated with the path elementary paths around individual punctures are fixed and actually related to these parameters lambda which is the parameterization of the delta and also there is of course ambiguity in the choice arbitrariness in the choices of basis, overall choice of basis and so we have this object which is a space of such homomorphisms with fixed conjugacy classes of the elementary of the elementary matrices and defined up to overall conjugation and that is roughly speaking the modulus space of the flat connections over the surface over the sphere with the end punctures well this is well known object it can be caught parameterized by various invariants like traces of this matrices with some relations between them it's well known that there are exactly two times three independent objects of this sort independent parameter independent invariants of this sort and that basically means that that the differential equation of this second order differential equation of this sort once the the parameters delta are fixed it doesn't have continuous isomandromic deformations or currently the parameters the positions of the punctures and accessory parameters I denote this n-3z and n-3 ci indicated by primes that are n-3 independent of them they can be regarded as local coordinates in this modular space of flat connections and and also it's well known that there is a natural symplectic form on this modular space due to idea I bought and these coordinates are Darbuk coordinates and this on this modular space alright so how the the classical conformal block is related to this is again very well known I'm still in the middle of extended introduction and that is related through the the the special cases of the conformal blocks when you have a degenerate representations with null vectors and there I will actually explicitly refer to to two of them in which null vectors appear on the level 2 and I will this I think is a standard notation by now these are delta 1, 2 and delta 2, 1 these are the associated conformal dimensions expressed in terms of this parameter b they are expressed in a very simple way that's why I like this parameterization and these are actually the coordinates of the null vectors in explicit form and from this it follows that the conformal block in which you I'm going to I went back to the quantum case where b is fine the conformal block which involves the the insertion of the degenerate vertex operator with the null vector in this stated representation I base the second order differential equation of this form and this is called null vector decoupling equation right and the second one I don't write down for the 2,1 it's the same equation which be replaced by 1 or for b there is this symmetry so I will it will appear shortly in this discussion all right so this is equation and now I want to send b to 0 again with this delta divided by b squared delta also going to infinity with lower case delta kept finite but the insertion dimension of this delta 1,2 doesn't go to infinity remains finite minus 1,5 and as usual in classical limit we expect by usual semi classical expectation we expect to have exponential factor and pre exponential factor the only effect of this of this additional insertion is going to be on the pre exponential factor and from that and from that the differential equation in the previous section of the previous slide this one transforms into the differential equation which appeared here with the accessory parameters being the derivatives of the classical conformal law this of course is very similar to the expression for the accessory parameters which appeared in the uniformization problem and this was conjectured by Polakov long ago again in this early 80s and then proven by Tektajan and Zograf in the but the associated monodrami problem is different as I am going to explain as I am going to explain now in the uniformization problem we are dealing with the kondition that the monodrami group is going to be fuchsian that means the monodrami group associated with this secondary differential equation is going to be we have to impose the condition that it is embedded into a real subgroup of SL2 that means I wanted to put SL2R but but it still see it must be SL2R here either SL2R or SL2 that's the the fault of doing this thing with mouse alright anyways it's either SL2R SL2 and in that case the accessory parameters become gradients in terms of Z of the classical action calculated on the solution of classical level equation which depend on Z not in galamorific way but it depends on Z in non-galamorific phase alright and this is going to be compared to what we have now this is the accessory parameters which we come up with are gradients of the classical conformal block and and of course these things are somehow related in particular the the liovil action liovil action which is related to the solution of the monotomy problem associated with the information problem can be expressed can be solved in terms of the conformal block I write down schematically the solution the liovil action is expressed in terms of a combination of conformal blocks plus some capital xi which is known function but you need to evaluate it at nu which in turn depends on Z and Z bar so it's not simply a sum of the functions of Z and Z bar because nu are determined by extremization of this function in terms of of nu so in fact the liovil action is related to to the conformal blocks by some sort of a genre transformation but what I want to say is that it looks like the conformal blocks are more basic objects these are galamorphic objects and how to solve the inverse problem how to express the conformal blocks in terms of liovil action this problem is not solved and I suspect it is impossible to solve this problem and so I am not sure I understand why people sometimes call problem of conformal blocks the complex liovil theory so I think it's it's quite different problem so we can we can ask now the questions F appears as a classic in the form of classical in the exponential so it looks like it is a classical action of some system of classical mechanics so natural question is what classical mechanical system is behind all that and and this is a this is a question which I I think I will produce maybe not entirely ultimately satisfactory answer but some answer too and also but let me first start with much simpler question which is easy to answer and this is what kind of monodromy problem this equation that accessory parameters gradients of the conformal block which kind of monodromy problem is solved by this equation that's pretty obvious because we remember that the classical conformal block well conformal block itself and classical conformal block is defined relative to the perpendicular dual diagram and just like I said one of the ways to interpret dual diagram is the way the suggestion in which operator product expansion are used and because we are dealing with the insertion of the degenerate vertex operator V12 we need the operator product expansion V12 and those this expansion be the well known fusion rules from which one immediately read out the answer and it basically says that if we fix the accessory parameter according to this gradient formula the accessory parameter are gradients of the derivatives of the that fixes the monodromy around the succession of contours which are exactly associated with the pan decomposition corresponding to this dual diagram that means if I take these contours and shrink them it would correspond to the degeneration of the sphere in which it splits into succession of three puncture spheres in which these three puncture spheres are exactly the vertices of this dual diagram so so, yeah, now let me sort of put this in the this thing in the neat form in general form which is due to Nikrasov, Rosli and Tashvili a couple of years ago and well, first of all one immediately absorbs because these contours can be this path here can be chosen to be non intersecting associated parameters new which also can be part of coordinates in this modular space they are post on commuting in the in the with respect to the idea about symplectic form and so one can one can define another set of Darbouk coordinates which I denote now new, this one of the parameters associated with this counter and conjugated parameters new which conjugated parameters new and then the classical conformal block becomes important part of the associated generating function of associated conformal canonical transformation so the new Darbouk coordinates are related to the C and Z which has also Darbouk coordinate which I already mentioned so we have a generating function actually it's convenient to add a certain term there is ambiguity of course defining this canonically conjugated variables and one can convenient to add certain term well which is convenient both from geometric construction of the canonically conjugated variables and also it has nice interpretations in terms of supersymmetric gauge theories ah ah I will come back a little bit so there is something interesting things to say about about canonical transformations between different Darbouk coordinates associated with different dual diagrams and that's closely related again to this work but I think the time is running out and I will better skip this part anyway let's try to move forward and try to explore the second nullvector equation which is related to to the nullvector which obtained from the generated representation which is obtained from that one by replacing b by y over b now this decoupling equation takes this form the delta now is becoming large in the classical limit and therefore the decoupling equation converts in the classical limit converts not in the in the second-order ordinary differential equation but instead it becomes a Hamiltonian Jacobi equation of this form this is of course very well known so it's it's still a partial differential equation but in fact if you have only four four points it's simply a Hamiltonian Jacobi equation for one-dimensional system so if you take four-point conformal block and take a limit classical limit here the equation which you have is a Hamiltonian Jacobi equation of one-dimensional system with a coordinate which I denote y that's where we insert the two one degenerate operator and just like I said three of the points I sent to certain positions that's my SL2 freedom and one of the moduli is still one of the points is still there that's the cross ratio independent cross ratio so I denote it t because it's going to play the role of time so then the classical limit generates this Hamiltonian Jacobi equation with this Hamiltonian it may look ugly but everyone who is familiar with the pin lever equations would recognize that this is a Hamiltonian associated with the pin lever 6 equation pin lever 6 of course is one most general and of the of the six equations which produce transcendental solutions and have this pin lever property that they have no no movable singularities that means there are no the solutions if you take a solution y of t as a regarded as a function of complex time there are no singularities which depend on the initial data all right so but simple poles but here because why is by construction why is better considered as a coordinate on a Riemann sphere simple poles are also not really singularities so we don't have a singularities movable singularities at all and of course there are fixed singularities at this points 1 0 and infinity which are some power like singularities which I will mention soon all right so this is I think this is also rather straight forward thing I think many people knew that but it turns out that this is quite useful in analysis of the conformity this is quite useful in actually creating the conformal block and basically I will not go into many details but it's based on very straight forward elementary observation that by definition of the classical action consider some trajectory, some solution y of t of the such that it passes through 1 and t2 it passes through 2 points then by definition of classical action we have this relation between the classical limits of these conformal blocks at y and t associated with these two points with the coefficient which is the classical action evaluated at this piece of the classical trajectory that means is to choose properly t1 and t2 to choose properly the solution and the initial point and the final point so the solution we take generic solution and the generic solution is known to behave like a power near any of the singular point and to be definite I choose 0 as the singularity like that with some power nu the nu will be identified with the parameter monodrmi parameter associated with this intermediate intermediate classical dimension and and so if I take this solution and then we see that as t goes to zero y also goes to zero and in this correlation function in this conformal block t goes to zero this point and this point merge with this point so three of the insertions merge and we have a three point conformal block which is a constant and so if we take t1 equals zero then the starting point here is just some constant ok so now if we look at the trajectory y of t then at certain complex generally complex t it hits one of other other points where other insertions sit and we take t2 the other t in this expression over what do I do? yeah, if I take t2 in such a way that t2 equals to some equals to some x at which y hits one of the other insertions here again for definiteness I take that y there hits the point infinity then this point this reduces to four point conformal block with some shifted dimension associated with the infinity which can also be computed using using the fusion rules standard fusion rules and so forth I skip the details but but basically the bottom line with this particular choice the solution certain solution of the of this equation defined by this initial conditions which eventually would hit infinity one of the other insertions basically interpolates between the three points conformal block and four point conformal block and basically we need to calculate the classical action on this solution there are some subtleties because although the solution itself is regular because of bin lever property is regular the action has some simple logarithmic singularities so that's I have to put this is something which I may put in the frames that's on the level of four point function that's the answer how the classical conformal block is expressed in terms of of the classical action of the evaluated on the evaluated on the classical on certain classical solution of bin lever six equation so in some sense it answers the question of what classical problem is behind this I'm not sure it's completely satisfactory I will say probably one word about that but at least it could be called some sort of lazy answer to this question there is a little bit to say about long standing of century old problem about the monodrame problem of basically the first non-trivial problem after the Riemann equation of the differential equation with regular singularity where you have four singularities and the monodrame problem then is is related to the through this is related to the connection problem of bin lever six the connection problem is again if you start with the solution given in terms of that cap and nu at the singular point then it eventually arrives at at certain point x where it at certain time x it says hits infinity or any other of these points but let's say infinity and you want to connect the initial data nu and kappa at one end to the parameters why not and if you manage to do that then the accessory parameter which solves the monodrame problem as I mentioned is explicitly related to that I don't want to enter the details here this is fairly straightforward well of course the one can compare that to known things about the power series this goes through to known expansions about the bin lever and this is just the just taking in grinding some power series expansion that's simple let me make some remarks here one is that well everyone who who looked at the bin lever and whatever all this kind of things would recognize close relation to the monodrame preserving deformations of flux systems in this case SL2 differential equation you have a now matrix differential equation where again with regular singularities where now AI are 2x2 matrices traceless matrices and now we have a monodrame preserving deformations which are described by this Schlozinger system or Schlozinger flow and the bin lever appears in the simplest non-trivial case when the number of regular singularities is 4 and and well it was as was demonstrated by Richard Eiching long ago this Schlozinger flow is just a classical limit of the system which bears Vadim Vadim's name and so perhaps the true understanding of all this thing and not only the classical limit but the quantum thing lays through this system and also the another kind of incarnation of another manifestation of bin lever was discovered not so long ago in the remarkable series of papers by this physicist from Kiev who discovered that this time not the classical action but a famous tau function which is more or less the same tau function which Stasher mentioned in the first part of the in the first talk to this morning appears as the generating function of the conformal blocks but not the classical conformal blocks but quantum conformal blocks at the central charge equals to 1 and just lately Nikita suggested that it might be closely related again through this Vadim's system alright so that's kind of what I wanted questions? I was I wrote some supporting letter for this guy who found this nice expansion for this so I know this paper and surprising really surprising that there we have sequel 1 and to here we have sequel infinity may it happen that if we expand in 1 over B square that the next term of expansion will coincide more or less with sequel 1 and probably with other C and other terms of this expansion I don't know about another terms of this expansion but but it might be so, yes it is a little bit miraculous it is a little bit miraculous but it might be that just like I mentioned that Nikita suggested the true Vadim's system that could be a sort of direct relation you consider the limit C goes to infinity will you have to understand so sure if C goes to minus infinity? it doesn't matter the limit from the point of view I am discussing the limit is regular it doesn't depend from which side you approach it so then the sequel minus infinity as an interpretation in terms of processing it's a small noise limit of SLE and we have an understanding of why they span way in SLE SLE C goes to minus infinity and second order differential equation is a focal point equation in the small noise limit of SLE so it means that time is somewhere in SLE processes it must be possible to fish it out from there there is a gazillion of questions about that emerging from this observation in particular in particular gain back related to Liouville theory then there is a question how come what is the meaning of this because it should show up in the Liouville theory and what is the meaning of it I mean it definitely is there and people notice that but what is the meaning of it that's interesting question I think no one raised it let's thank Sasha again