Classical conformal blocks and Painleve IV

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Classical conformal blocks and Painleve IV
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Zamolodchikov, Alexander
Belavin, Alexander (Organization)
Nekrasov, Nikita (Organization)
Shatashvili, Samson (Organization)
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Institut des Hautes Études Scientifiques (IHÉS)
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In the limit of large central charge and large conformal dimensions Virasoro conformal blocks exponentiate, leading to classical conformal blocks. The latter are of much interest, in particular they give solution to the monodromy problem of ordinary linear differential equations with regular singularities. It will be shown how to relate four-point classical conformal blocks to classical action evaluated on special solutions of Painleve VI equation.
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at the time and From the it's my pleasure and 1 of the 1st speaking at this remarkable occasion and I would like to the 1st to thank the organisers said fashion and that some people and keep them the institute for this opportunity and he II I had the privilege of working closely with redeem and hopefully collaborating with them and that this occasion brings memories a lot of memories formed members said members of obviously In and them particularly memories that are related to the To the way have was sharing his remarkable waves of of his remarkable new ways of looking at things and not only in physics and also His particular joy of doing physics lab will all enjoy doing physics but that I think everyone who knew him that would that would lead to denies that he had some Of particular flavor of jury of approaching his except that they will I will not forget and then well there would deem slice away from physics was shortened the clock time but this is obviously a very long in terms of impact and there he said and she was at the origin of many things including and particularly the and the conformal field theory and that so that will be
so this is now with the there are an important part of the footer medical physics and that and the pure mathematics and it's still developing and there are so that they might like block will be devoted to this subject so it's it's it's some part of the stock would be sound development of the subject that concerns conformal books most of the talk will be in the way of extended introduction whatever their original up 5 is based on the on the joint work with and it extended to enough said guilty enough them at the end of course all mistakes and misconceptions on the mind and there so this is the title of the work classical conformal blocks and and 6 and their sold before starting probably although conformal blocks I think it's there now that sort of well-established notion in the momentum article physics Maddox adults still I think I will need to hide my will use a few words in the way of fall often production so or classical conformal blocks appearance of me all 4 US Oregon formal blocks would news that sent the of central charge the Infiniti together with the conformal dimensions and the virus or generally the virus or a conformal blocks are associated with surfaces with that bunch and the modular space of those object 3 months of this with punches but I will limit the potential ended will make them exposition much simpler I will limit attention too adventure orchids I will limit the pension bill this year was punches here this year and that was the I will always mean that did note the positions of this sponges are in terms of some pomp export on this agreement here ,comma and actually I hear you will see that I will be in mostly talking about was up for punches but survived there for some time we will consider so consider and bunch so what is the conformal blogger after speaking that's some sort of correlation function of guile primary that's all operated as the primary means the standard primary type of on the station relations was generators or operate the plant expansions was energy momentum density but because they are so these bizarre but battlefield good Norfolk feels they have not local fields and so on the notion of correlation function is ambiguous and to to to give meaning to the correlation function well and needs to specify what is what is known as the world Bagram or a quarter a the Quran did Ben decompose and decomposition of this sphere and functions here and there I will be considered that I will consider only this kind of an brash out of the dual diagrams are and the interstates are we can think of it as a correlation function of that or expectation value of this way can break protection any of the products of this verdicts operator was the instruction to put in between the old operators are that the intermediate stayed the compositions or in terms of the irreducible representations of the algebra will use the dimensions with certain fixed the formal dimension built them off the so this is the this diagram is presenting a view that is the way it sure got the better of representing this ordeal or if you equivalently this is the instruction tool used to operate the products branch in certain quarters well in in the water which is which is dictated by the tool that no doubt over the Is that I don't know of peace at specific parameterization all of formal dimensions and In terms of this parameter of the this moment this is not very important this parameterization is motivated than originally by the failure folks in the present had a presentation of the all of algebra and and that was very convenient in Duval theory although I will not I will not actually I'm referring the incident to systematically to William Wilson also I will mention all right so becoming better too To this picture we see that there are in minus the intermediate legs here so the I minus 3 of these parameters so that informal walk depends on them minus parameters of the and it also depends on this position the alpha ended it because of the result to the transformations of the dependencies essentially up to projective transformations soul essentially there would be no just use itself to transformation the standard CEO of this points to preside position usually 0 1 hour Infinity has 0 1 and Infiniti and let alone after that we're left with and minus 30 coordinates in this independent coordinates with this on this model is based on their cell it depends on and minus-3 coordinates and the wireless space and also of course it depends on this external dimensions associated with the external legs and the dual that again the but there but then I will I will and nip them in this patients because they will be disregarded this fixed number although of course and for the will back so this is conformal I will be just like they say they will be dealing Wallstreet outweighs the 4 .period conformal block we just now depends on the single intermediate state parameter be a single at all in the model a space and that the station will be at the or fax the length of so the the the the father the conformal blocks of fundamental objects conformal field theory that were introduced in the 80's and at the time
when the which we all wanted a member of was possessed this conforms to basics and for field 53 was generating a sewer by a submits a basic objectivity in the sense that good relations functions of conformal field theory built Aruban are built in terms of conformal blocks the rules that govern market but there is a nation so I knew I will produce just an example of how it's being it looks and and you conformal field theory where that's a four-point coalition function that's the simplest example where the 10 percent daily are is expressed in terms of their after the to Suitland square of them conformal blocks so you have to these of of conformal block venue of exon digs by and it's integrated this parameter the Wii's some professions which are known as the stocks of constant all versions of Uriel theory of this Act explicitly known into 1 of the reasons I write down this followers 3 interviews this parameter B which is it's a student parameterization of the the sample chapter which I will be used the Everton Alan and then In that in the course of this discussion I that media there when they need to say Due to stop and it's not that they're going to stop now I just need to be filled yeah 1250 OK that good sound alright and so and so this is it this is how well 15 minutes soul this is it this is the way home genetic informal blocks appeared in conformal field theory but the but the timing it it turned out that they are more alike than they ever a modern system to be much more general came in in a in mathematical physics and a low lately there be there about the tension in relations well it's quite an expected it to me I mean people who are much smarter than myself probably expected that but I do need to be there it came as a miracle that the same functions that appear in relation to the four-dimensional supersymmetric conformal field and the well particularly this conformal blocks up in Europe as well the the besides the coincide with that of the incident on parts of the new grasses that partition function of the are over 4 dimensional and equals to 2 supersymmetric as exterior with certain uh well if you were talking their Sorokin for conformal blocks that's a but with that said supersymmetric gage theory was certain contempt of of product the 1st 2 gage group and and that is remarkably there the the parameters set which are which is originate from the moduli space on this agreement surface plays the role in of the gage coupling constant relates to the death of the huge topping constants in this city In the gage theories and the parameters p relate to the to the vacant model in this so it all up to meet looks to sort preferred of firm's honey market but I would think about the nonetheless it's a it's a it's a soul and and will it's now it's vigorously pool over there and I will mention there but there and but they're sold that is perhaps this dysfunction in place some sort of a wider role in In all this correspondence but the the point is that this functions and have some of which have some control on this phone the but they got there in the conformal invariant conformal symmetry 0 generals figure the conformal significant taken completed fixes this functions similarly on the in the the supersymmetric gage theories of course while I'm in conformal symmetry for excessive power series expansion and and by implication of course fixes this function solely by all right down power series expansion of this function in the office of the model and all the coefficients are fixed by the formal symmetry here are and similarly this is an instant on expansion in the in the chemicals the Semitic history and this and that this coefficients are given by Nicholas In integral over the bodies of the incentives of course there fixed but that is that power serious and we would like to have a better analytical global analytic control of over this functions and because they play this important role and it's 1 of them at evasions of this this and this work which is linked to the board is is gaining some better analytic control that some things which are known by air or water air or assumed about uh and I will be speaking about 4 conformal well but this is positioned to do the end .period conformal logs or respect for and so the builders of the conformal blockers sees his views 90 take it on the universal column of the of the of the model while basically yeah and on the gossip column 1 space and it as a function of this people around at the it's metamorphic function all and base the the the crossing the only what call crossing variation this is relations when you interchange that this change is in the position of the the the punctured by vibrating the confirmations and then the conformal blocks are related throws integral transformation transformed but sort of well through the conformal blocked in the cross-Channel and was the Indigo and was so to governor we choose it which is related with the
"quotation mark Cardinal here is essentially a six-digit symbol J. symbol of continuous presentation of food deformed in universal and loping algebra offical too so this is a part of the I think most essential part the knowledge this functions :colon now classical Limited unwilling to talk with about classical conformal moderate emerges when you send the central touched Finucane and interestingly refuted to appears when when you also send that I will well before before proceeding I will again parade the central challenge in this new deal inspired Figgins folks inspired attitude because the year before a parameterization all 4 the Sentul charges in Toronto this but I'm coupling parameter the This is not essential but there but there but I will use it because I like that's going with the story the the most important reason so so I prioritize the central tenets of the terms of visibility and the classical limit will be the limit of the going to 0 all these like of a man plants constant actually be square will be analyzed blocks ,comma so interesting limited marriages when you stand sentinel just Finnerty in which case of course where we know there are some algebra converts reduces the was on bracket algebra but also you need to send their dimensions feeding so that the ratios of the dimensions of central challenge remain fixed gadgets fixed and I will refer to this Delta's for ratios as the classical dance club has not kept folded but loyalties spelled as classical Indian area I will use the parameterization part of this surge of this classical dimension belt In terms of this parameter slammed new depending on the floor of either and speaking about the Dolphus associated with the external legs of the little diagram or internal internal lines and that well I mean they're dimensions could be any sold generating while I could think of a very all I could think of London as real appeal of Manginelli but generally I will think of the slammed Cisco genetic now when I perform this limit the conformal blocks exponentially it is for that said something which is in a classical intuition makes do you expect but mathematical status he is said to me at least is not is not completely if it ultimately clear so it expand initiates so that it it it's 6 financial so if 1 or be square at times ever and which is after all lowercase Afleet school classical conformal well now that there is no doubt that this is indeed correct statement and actually there said that I think the proper proof was sort of exist it's it's just my my brain doesn't didn't yet incorporate all the details of it and it will happen eventually it probably consist of a mixture of their results which come from different sides of this uh I'll daily at the should Omar crisp London's which involve involves the other in the form of an aggressive for formal the colossus of presentation of the coefficients of the of the 2 power like expansion which which looks like which look like a coefficients of the veal expansion of certain gasses and their than the classical images like that the economic limitations of desk and then exponentiation theses usual thing which is intended that there is now a and so rigorous as proof of this correspondents the level of this for functions so so think the combination of the statement you flip sorbet such property it would too but I know I'm not particularly interested in proved so it me just proceed now that is a nice then that is there is need and well-known relation I think it's a it's a again in dealing with we all played the 30 years ago In a way it relation of this set of this thing to do was to the monogamy properties of 2nd order differential equation so considered differential pressure of this died .period Chu was there was an apparent good barometer reserve variables he regarded as support on many months here this is a differential equation was an illegal Arrigo singularities on CIA endit and so on to to do to make precise from the very beginning I would regard soldiers there is the potential torrents of the from this equation involves the programs of parameters are built NCI belt that I will be right the fixed as fixed number of and actually I will I will take them equal to the classical dimensions which it appeared in the Paris transparencies and as identified them whose parameters and NCI harvest immigrated accessory parameters and the the ourselves so this basic this differential equations for a reprise by the positions of the CIA and the and year well in fact there are only and minus the independent accessory parameters for all this season which is which and I sort of explained by this line that city in the fleet I don't want that an additional singularity provided that Kennedy then there is the lamented relations between them so that basically there and minus city of this independent accessory products and also all the positions he eye because of the form of the differential equation doesn't change under mistletoe to transformation agenda is the eyes are defined
model it's 0 2 transformations votes so there essentially again we're concerned 3 of them too pre-designed favored positions and so there and minus the parameters of this sort so this is this differential equation is it is parametrized by 2 0 times and minus 3 "quotation mark to experiment Of course there is the rational equation defines someone under mingle and which is the homomorphism of fundamental group almost here wasn't punches into a cell tool this is a transformation of the basis of an offer to independent solutions of this equation on the the analytic continuation of Qantas of presenting the elements of the fundamental group In the end so because because of the idea I say that the that's what Laura Keyes built the fixed our way of dealing with the representation in which in which that Kaunda was a classes of the Madison spam I which are representations of the authors this basic elementary elements of basic elements associated with the with the past elementary passed that on individual bunch of are fixed and actually related to this parameter slammed which is the parameterization of the no-fly Delta on and also there is of course is a of ambiguity in ditches arbitrariness and juices of Britain's overall traditional bases of and so we have this object which is a space of of such a common more efficient was fixed quantities to classes a we the mentally Of the lamented the Marcoses and their defined up conjugation and that is office speaking at the end of the modules modulates space of the what connections all 4 of the agreement surface Over this furor was the was the and punches well this is well known objects it can be caught in the parametrized by various invariants like traces of the mattresses with some relations between them it's well known that said but there are exactly 2 on bench and minus independent objects of this sort are independent from the parameters independent variants of this sort and that laser going means that fewer the 3rd the differential equation of this sort 2nd row of differential equation of this sort once there the Department of Delta have fixed it doesn't have continuous among monarch formations or didn't play here the parameters for the positions of the polishes and accessory parameter saving about this and minus the disease and minus 30 C indicated by a branch of the reminded the independent of them they can be regarded as local coordinates this morning on the basis of the connections and the and also it's well known that there is an angel sympathetic for on this wanderlust based utility I bought a underscored in it's on that walk all over them and this wanderlust all right so how do do do classical conformal blockers so this is is again very well known I'm still In the middle of extended the production and that is related to the rule of version on special cases of the of them conformal conformal blocks when you have and degenerative presentations wasn't all that and ask that I will actually explicitly referred to 2 of them special today in which there no letters appear on this on the level of 2 and I will make this I think is suspended station by now there these Delta wants to work built 2 1 these are associated conformal dimensions expressed in terms of the sport on the there expressed in this very simple way that's why I like this sensation and these are actually at the ordinary itself that know that this explicit form and from this it follows that the court that the conformal block in which you I'm going to do but I I went back to the quantum case would be fine the the conformal block which involves 3 do insertion of his degenerative plastics operator and all that the in in this state that representation of a base the 6 2nd or differential equations all of this for are and that this is called nor that of the copying application for a part in it and the 2nd 1 I don't write down for them too ,comma 1 it's the same equation was be replaced for by 1 of 4 B image so I will be a big debate will appear shortly in this discussion so so this is this is equation and now I want to send me to 0 gain was this the Delta divided by the square and the Delta I'm also going to infinity with Wizard a lowercase Delta kept final but there has been such an the dimension of this Delta 1 tool that doesn't go to infinity but remains find minus 1 half of tends to 1 1 calf and above the as usual Indians in classical and limit we expect by my usual classical expectations were expected to have an exponential factor and that creates financial factors which depends not on the phone be effectiveness of this additional insertion is going to have to be on the league's finance factory and from that and from that the there differential equation in the previous section both the previous slide this 1
transforms into into the differential equation
which which appear here we used
that sets the parameters that being the very rigid Of the classical conformal this of course is very similar 2 of expression for the common could accessory parameters which appeared in the zation problem and that this was bad conjecture but political long again these are really good ladies and then brought on by the Fed John organ of power in that but there seceded monogamy problem is different as I'm going to explain it as an willing to explain and now in the uniforms Asian problem when dealing with the wizard is the condition that the monogamy group is going to be forks and that means the monogamy group associated to is this 2nd order differential equation is going to be in that we knew we'd have to impose the condition that it is embedded into a real subgroup offical tool that means that I wanted to put this to but but but it's still sees it must be useful you ourselves to in that's the game the the fault of doing this thing was now all right anyway there is a universal to our old DeSoto inductees Of those accessory parameters are becoming gradients in terms of the order of the Classical Action calculated on on the solution of classical Neuville equation which depend on the ones you not go more frequently but it depends on one's In longer Faso depends on the ends about all right and that this is going to be compared to what we have now this excessive parameters which will come up with these are gradients Seoul the classical conformal blocks In the end of course listings on some call related but in particular there there is a new deal the action that action which is the only real action which is related to the salute to the solution of the memorandum a problem this season was for musician problem can be expressed and that can be solved in terms of the conformal but I II III down schematically the solution that the removal action is expressed in form of a combination of conformal blocks brought plus some capital oxide which is now known function but you need to do to evaluate it at a new which is in turn depends on GM's about so it does it's not simply as some of the functions of zillions of but because you are determined by extreme musicians of this function in the south of New the in fact that there is a new election is related to 2 of the conformal blocks but by some sort of resigned confirmation but what I want to say is that is that it looks like a conformal blocks are more basic logic is anamorphic In order to send them and how to sort out how to solve the inverse problem how to express their I conformal locks in torrents of real action that 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 called the problem of conformal blocks the complex to the so I think it's it's quite different from so many weekend we can ask knowledge questions with after Pierce a classic in the form of classical index dimensional so it looks like it is a classical action of some system of classical mechanics sold natural question is what classical mechanics system is behind all this and this and this is this is a question which I I think I will produce maybe not entirely ultimately satisfactory answer but but some amounts of and also about this let me 1st stop was much simpler question which is easy to answer and this is what kind of monogamy problem this equation that the accessory parameters the radiance of all the informal block which kind of monogamy problem is solved by this equation that then that's it that's pretty obvious because we remember that the conformal classical conformal blocked well good formal about itself and classical conformal but is defined the relative to to the bond bend the composition of this dual Dagenham ended just like they said 1 of the ways to interpret dual diagrams to concede the dialog is the way the succession in which operates a product expansion are used and because we are dealing with the insertion of word the verdicts of the general looked except that the lead you want to we need there were peripheral products especially around town in those in this is that this is fashionable the well-known funerals from which 1 immediately about the onset ended basically say is that if we fix that but accessory parameter according to the Canadian Formula One the successfully by gradient so they have derivatives of the classical conformal blood that fixes them around the succession of Quantum which had exactly associated with his depend decomposition corresponding to this
that that means if I
take this and then showing them that would correspond to the generational was field in which it splits into a succession of of 3 punctuates in which these 3 bunch of fears that rejected the vote the sizzle of this duel .period since so so yeah now let me this sort of food producers in the this thing in the need for a and then the general form which is the standard you to do the worker nickel from across a frozen and the state of couple of years ago and then and there well 1st of all 1 immediately observes becomes because these quantities can be this bad this past year can be chosen to be known intersecting associated and parameters new which also can be part of coordinates into this model is based they are Paulson commuting in in the quality of book In the end it was suspected that the ability to Blackduck form and so 1 can no 1 can define another set of bearable coordinates at which I do not now knew this was the parameters associated business that was discovered in the kitchen and the conjugated parameters I was going to give brand new and the the classical conformal becomes important part of the associated to the generating function of assisted them for all of us the canonical transformation so that the new global coordinates are related to the assumes the which has also been on the books of literature which ability mentioned our so we have a generating function actually it's a it's with it's convenient put them at a certain this there is ambiguity of course that defining money contributed to the variables and and 1 can convenient periods of time ah well which is convenient was from dramatic construction of the canonically conjugate variables and also it was nice interpretations of the soul supersymmetric gage theaters I cholera and I will benefit comeback in little bit so I know I have do something interesting things to say about that and about that but canonical transformations being between different to say yes that will coordinate associated with different dual diagrams and that's closes related again to this work of food refers to the she with I think good things too at the time is running out and that will but the skip this
party anyway that the 3rd to move forward In the 3rd to explore the 2nd know that the creation which is related to order there to the to the no letter which obtained from the the Norton the degenerative presentation which is obtained from that 1 when replacing the dealt by well over the mountains the coupling Gration fix this the Delta now is that it becoming a lot in the classical limits and therefore the decapitated equation converts into classical elements convert it's not in in the 2nd or ordinary differential equation but instead is that it becomes a and don't report will look like equation of this form this is of course very well known and so it it's a it's still a partial differential equation but but in fact if you have only 4 4 plants it's simply and upon Jacoby Commission for one-dimensional systems so if you take a floor .period conformal block and take agreement classical limit here but the equation which you have is a organized on Jacoby creation of one-dimensional system it was a good was according to which I do you know why you would swear this so we accept it to 2 1 degenerative operator and just like I saved 3 of a point I sent to a certain position that's my soul to freedom and are 1 of the more delay still is still 1 of the points is aware that some of the cross independent restoration so I denoted because is willing to play the role of the so I then and there a classical image generates this demagogic the Gration was this committee got hit by a local look ugly but everyone who is familiar with and recreational would recognize that this is a notorious associated with 6 the coalition I want to know the 6 of course it is they're 1 of the most general and and also of the 6 equations which produced transcendental solutions and have suspended their property that they have no city no movable singularities that means and that the the solutions if you take a solution while he is the president is regarded as a function of complex time there are no singularities which depend on the initial data right so I so bad about this simple but simple but here because while you like but but construction was but the considered that as a as a coordinator from Lehman's few simple polls are also not really singularity so we don't have a singularity of small wobbles in the and of course there are fixed singularities and this point said 1 0 infinity and which are from politics and which and I will mention all right so
this is the I think users also will rather than rather straightforward saying will I think many people knew that but there but there are a total of and this is quite useful in its analysis of the of the conform with this is this is quite useful actually reading the portable walked on and that and I will basically I will not go into many details with but it's based on very straightforward .period mentally observation about that bite by definition of the classical action that will consider consider some trajectories absolution alive he of the peninsula 6 equations that passes through that sometimes the 1 and to it passes through 2 points away to them that way definition of Classical action we have this relation between the classical leaders of this conformal blocks Ed White in the Associated this With the coefficient which is glass collection everything this piece of the classic suggested so what remains is to do to chose proper rated G 1 and the 2 Dutroux's properties a solution the initial point and the flight so the solutions well would take genetics and moved right and this generic solution is known to be like a power near any of the singular appointment to be deafened I choose a 0 as the singularity of indicators like that without power and you all new wood will be identified with the parameter and did mother parameters associated to release this intermediate intermediate and classical dimensions 10 but in Seoul earlier this year in so and so if I take this solution and then and then we see that the 2 as goes to 0 White also accused pizzeria and inverse correlation function in this in this kind of formal block it he goes to 0 this wind and this and this .period members which can we use this himself sealed the insufficient knowledge and where we have a three-point conformal block which is a course all right so it would take the 1 equals 0 than the starting point here is just is just from "quotation mark OK so far knowledge look at it objectively while 50 than at certain complex generally Quantic state the kits and 1 of this other points where our instructions so if we take the told the other key in this expression of it would be yeah yeah if I take
but in such a way that the moon and the 2 equals to some goes some accident which is why he it's 1 of the other instructions again for that month the site II I take it that while that hits the .period infinity then there disappointed that this reduces before .period conformal blocked with some shifted dimensions stated with the Infinity which can be also was the shift can be computed using them using the the diffusional offender funerals and so forth and ideas give details but but there are basically the bottom line is that is in the air over this particular choice the dissolution of certain solution of the Glacier defied the defined by by this initial conditions which eventually would hit what hit infinity 1 of the other sections basically interpolate between 3 . Milwaukee and 4 . informal look and the lucidity we need to call late on the classical action on this notion that some subtleties because although the than that of the solution itself is a regular job because of their properties regularly the action has some simple getting picked singularity is 1 to make decision and so forth so that's what I have to to put this is something which I knew I I'm a footed the frames that when the level 4 .period function that found some Hall the classical conformal look it as expressed in himself and the classical Action all of them on the Adriatic turned and related on the classical on certain classical solutions to the nearest excoriation so in some sense accounts of the question of what classical program is behind this so I'm not sure it's the its factory I will say it was probably 1 worried about that but then is still it could be called some sort of collegiate answer to this question there is a little bit to say about that the Mile long-standing century-old problems about the monogamy problem of well basically the 1st month real the problem after the human inflation also the differential equation which I don't think worry except for singularity and there and the mother the problem years L His related to the for this is related to the connection problems often 6 the connection problem the game if you start with the solution you in advance of the gap and the single required then it eventually arrives at the at some point acts with everybody but that suit at certain time exited through it's the 6 hits Infiniti or any other all of this of this points but gutsy infinity and you want to connect the initial David Newton cap on 1 end to the To the parameters that explains why not in the field but managed to do do that then the accessory parameter which solves another problem as I mentioned is that this citizens related to them and want to go into the details here this is fairly straightforward
well of course there is was the 1 can compare the 2 known things about the power series that this goes through the unknown expansion said about the Nunavut ended this is just just taking it in drinking some of status expansion that simple let make some
remarks that she 1 is that well everyone who who my hope that the Pinevale and whatever all those kind of things but over the recognize closer relations to their momentum preserving the formation of Fox's systems in this case are still too differential equation you have 1 now Maddox differential equation when they was a regular as well now AI I have to wait to mattresses traces mother and I now would travel from another and preserving the formations which are described by the stewards into a system of citizens full and that of the dependability of said the simplest month relocate when the number of 4 games this is for ended in there well it was as was demonstrated by should taking longer to go this is Angela Florida just the classical limit offer of the system which was but the misnamed and sold that is the effects toward understanding will fall this thing and not only the classical limit but the quantum playing ladies the rule this this system and also then there was another man you kind of Nacional another manifestation of was discovered that not so long ago but this is a very remarkable serious of papers might this this physicists from Kiev who discovered that the that the new this time they're not the glossy collection but famous style function which is more or less the same tell function with special mention in the in the 1st part of the the book to this morning but we feel this is the generating function of the conformal blocks but not the classical conformal looks but quantum conformal looks at them of the central challenge he goes to 1 and the and the poet just blatantly on suggested that the might be closely related to for all
this system operates so that's the kind of what I want to this was the most I also wanted to work with him in the world along with mistakes all this so long as they don't surprising but it is surprising that the CIA along and you will see more of this the nature of the universe is expanding while on his way Over the next several things to coincide with the list is well and probably will I think the Government and but I don't know about another torrents of this expansion but but but it might be so yes because the miracle it is a little bit Morocco's but debt but that it might do that just just like I mentioned that political suggested the rule but the system of it could be a sort of direct relation if you he was using the lessons and specialist who might thinking that of it doesn't matter and that the delimitation of the limit and from what we want from the point of view I am discussing the the limit to the regular it doesn't depend from which side you're part of the sequence when he hears that the definition of some for the things that the learned as the facilities and the funding of the flight they face faith in the city as they seek my thinking the other side of the question forgotten equation is loosely yes yes we that the somewhere in face was not being mugged .period money it might be possible to fish topping yeah that that is that there is a good of questions about the emerging from the from from From this observation that and in particular in particular gained back related to the little thing then that that is a question of how come the isn't what is the meaning of this kind of question because it should show up at the interval city and what is the meaning but I mean it did but different things there and the people noticed that the time and what is and what is the meaning that's interesting but I think no 1 raised differently you would think that the thank them


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