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Integrable deformations of sineLiouville model and duality
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Title  Integrable deformations of sineLiouville model and duality 
Title of Series  Vadim Knizhnik Memorial Conference 
Number of Parts  12 
Author 
Fateev, Vladimir

Contributors 
Belavin, Alexander (Organization)

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/16413 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  We show that sineLioville CFT, which is dual to Witten 2d black hole model, has three different types of integrable perturbations. The corresponding theories also possess non trivial duality properties. The first two models give the examples of weakstrong coupling dualities between charged fermions and bosons. The third type gives two different field theories, which are dual to massive and massless nonlinear sigma models. 
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as it turned on in and then I want to say thank you very much to organizer and this is what
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I after the 1st of this conference we cannot say we can see our old friends the people close to why do you it was over the young changes could during the In any case I think this conference is for people was here is reviving Boston and the memory of courses the deal was also reported so the I want to discuss with you yesterday some some say that all models and terrible models can be derived from an equal doses this is the Gish serious progress not all so that they have a lot to talk about some different diabetes in detectable the verdict you terrible models Byron Boston the 1st because many terrible models beside son the feels theory the Phoenician Lagrangian definition have some other definition parents of asthmatics riches rather simple object you the theory is intolerable this is metrics is contained several functions of but and they have all the information about the cheery the all of this function is enough to reconstruct all the fear it gives us the possibilities sometimes the show this theory has some of the the litter presentations In some sense of can show that there is the MEPs between strong call police To the call blinker region really gives us the possibility to study the serious rather write the range of the probably because it would result in durability in injured by different madam but commissions theorists and was so but before I should say probably about signed routine .period missions also poses some problems over the so some eluded the Orioles as I said I should say that some of the results which I will talk about where derived during the it's saturated by nurses isn't much different in the corner so I what is the sign theory it this summer conformal fill theater which is normal To the Clinton Blake hold Richard some concept model it means that at some point in the Munich tender it also describes some signal model the end of this evening model is well known c . little cigars but even inside this model is some of them do it someday to it is the dynamic to between 1 month and winding command bursts of this series it is some of the dreaded to this model to the site kind of surface which is called the home but it's going to be completely embedded to twodimensional space but In threedimensional but the visible part can distinguish what doors the conjecture which was done Sasser understrength means many years ago in 96 with this model is has the dual representation the the has the same in parameterize it contains on of screen and the Richard elated by 1 immigration so but it with 4 1 mandatory life the central chartists buried them exactly as this 1 in the parameterize drives them by Parliament case it is the instrument to expose a central charge both this concept model for or the cigarette now White this power this conjecture or why it was based at the age real parts so what I have with this theory is but some continuing conformal field fit this part of the notes are noncallable assigned to 1 1 the rooms which can very simple representation in terms of too few and it is possible is in approved the this part of federal loans commute in the strong sales is with of hills fish form signed union the provisions of In this Our which is the potential of such analysis it's going to use in the strong sense it tho it means that that Colonials locally or that this feels odd hello Moravcik feels after the 5th permits in principle it gives the possibility to construct all Walgebra all all all always local feels this feels a lot local but there operator product expansion contend the local fields which for infinite dimensional WR algebra with station number of the comments of spinning 3 4 and so for example the Spin City home Orchard current fears this 4
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so far no newmont ideas this bias over primary feels here Is isn't alone describing it themselves the fields since signing theory it depends on people mom were spent them then them in their can be considerate 1 is the moment number and what the other winding command by the end of momentum for cigar is gone soft and the not wined and 9 were not consulted principle around here will introduce some ball field which related this simple but relations officials of serve the CIA the editor for example detained Strong bed between cigar and prompt that it is deleted which changed the number M and M it the widening number is concerned because they have no nowhere 2 most cigar outside this don't buy under another important characteristics of this theory riches important for Australia but the behavior of the theory or in the debate is the buoyant function all the reflection amplitude it knew the take for example this field and will build the police formation all but 1 that which does not change all quantum number of all quantum numbers it is the number selected through they can of W. feels then those of course after this transformation signed only some numerical coefficient can appear for this field and this medical coefficient is an into the sink and important objects can colorcoordinated and was the the limit was also calculated the important functions in this period indeed it is enough From our from at least from my point of view if good theorists have the same kind of senator algebra and due to buoyant threepoint functions it is enough to shoulder the win but there's still a lot of other papers which contained this warrant more rigorous broke but will stop about the spoils instant now will consider possible intolerable distorted bass defined the possible integral perturbation it is To find different viruses which How would we condemn it can consider the enveloping culture profile Walgebra and to see them some Carbone said Bulger wrote or the sub of intangible objects which can be cold this isn't the girls of motion and the song on his shoulder that's at level 3 it can be only 1 fuels which can the density of the conserved stars of course proportional to W the field England but it is impossible to find another field which continued its level for which continue to its it and of course this object of defiant up the David because only the integral of this charge in the current users advertisers and continue so as it news of the possibility To show what is the best perturbation so that the better perturbation so in the leg irons and Richard Pearce should also commuters this In the girls of motion not in the strong sales at its was before but that we as charges were Figaro so the 1st perturbation is can be found in the 1st Baron can be found is indicated by this directive its then it is possible to find a 2nd truck it will generated by the localbased so it will not have this thing the garlic themselves :colon but it will have been for and of course all spinning for on different the first one just because there is no longer feels just biding his career categorized bottled the status and thousands so this theory contained more senior today it is symmetrical with respect to this transformation and that could respond to perturbation operators of this kind the there is this sort and probably the most interesting but probation which there is semantic with spectral the With respect this exchange into it is it could respond to that signed the 1 that model which crippled by this that they're so good there is an inprinciple several the database and what they want to saying How the duality works in this no longer so let's consider them the 1st one probably the simplest 1 so it is but the total Lagrangian here Wikinews because they're too rich will the slugger arms and for the prevention favorite and then there is the usual corn among among the stunning to the coast of India b a small but it lots more the square some Mosquera physical difference is equal to onehalf cent to prepare our Kansas please to build the 1st the use delighted to have him on the stand and
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calling for an end to prior to this Lagrangian is relying on I think model cowpoke with some respondents fulfilled the some others Integra will failed here are the 2nd explanation heroes as the going to there to work in the preparation of the summit a small companies now we can use the perturbation theory and with Wendy's small and this the rewards sweater will enter Canada concluded that the spectrum consists of 2 Charge at Feather Maryland there is also some particle is equal to that which disappears from the spectrum because it is not stable with respect to respect decay but also what is interesting here with care there is a song charge generated by the current post in 3 and reach doesn't period the reflection of a serious this Maddox's Wataru with those of the phase then we know that the particle it's monster disappears from spent on so it should be it there I'd here among physical state Close to give people buy into can ride there the result In 1 of them general form their only the is it was shown below it is to compare and then To find the dependence of his but on it so it is possible he reconsider the asymptotic all the ground state television in the external fuel and it is Is it would In from like handson intuition legislator from bloodhounds and from there skating theory and standard they are the same and tipped as possible From my grounds alone and politics has this aura the All My American take out so it will be the American model in directing career and OK and images of this model is well known to the from other point of view mechanical collated from the scattering theory promise medics approach then this initial to share original sold some integral equation some of them undermining their plans out the creation but in principle it is enough I don't know not a lot about all this kitchen so for example the same but can be concluded that there's no the Fourier Transform will follow Campbell to the tools you so comparing this delirious you'll find that this relation between the 2 them fix power scattering theory and then let us see what happens when we moved to from small tho big and will find and that will introduce Parliament that Gumbel rich in office power be this Maddox begin transformed into this method which is proportional but the table wine as egomaniacal 0 it that there is some other but apart for a coupla include presentation the Boulder presentation of the city defined the Lagrange American calculated the senators in this league and it is not that simple down the From now on the point of view we will see this theory now it is completely was earning it the other but under we have some of them also thinks which was that the fairness of the buzz only for example the became aware of this image of new official to have been called to them so intrusive that now our theory is described by the action which contained for the Bulls loans and to support this week and coca led this manual and from the graves semantics that that is of bound to see the visit coincides bank details that we see that it we had some interesting Ed small call opening we have Chavez at federal loans and to indeed in the weak opening could have discharge at federal and and the strong call we have the judge free bosons ending In my point of view this as a little bit None deal because we understand that strongly interacting bosons can be considered lifethreatening but relies strongly interacting betterknown candidates they become the walls that there's a little bit this looks a little bit strange all I care about another bit of probation I will not talk a lot about it it can also but in this case referred asthmatics which is more or less 3 it should be the solution of some special aggression is methods Is it expected to be the solution of Foucault and Botsford that repression and the solution is the only known solution with the unity What is this Maddox both Simon Gordon model up to multiple litigation to Sun City defied and viability discussed below 2 0 calculations so only we'll say it is also available it does the example not the numeral but some city the fact that appears not manual solution of but that is a creation bellowed that it is possible go "quotation mark total against all possible calculations and to find the
23:45
Lagrangian for this we should solve some integral creation it can sold by the Ministry of funneled to break functional is given parameter :colon damn good it's exactly Qatarsponsored too so this Lagrangian the meaning of this like I wasn't visit because it is on a quandary can use the classical Theater and the minimal gives us exactly the early this like Guen's what is a very interesting phenomena which appears in this series and in my opinion a little bit strange is the following not only numbers on intelligence but also probably I forgot to say before that difficulty at the beginning and family alike around In terms of perturbation theory you will find that it has 2 particles charged particles to Chavez variant into and was which can be a concert there their balance that but when we go to the strong call playing a bound state disappears at some point and from other point of view we can consider going from small gun the big government here and for small drama this perturbation looks like that directive the there's no In bound states but for all of the big gun if you tho C When model the nature of this that bit of this term we feel the bound state but I peers some balance that you but the potential looks like it directive so now I To say said probably the most interesting good it is In this historic models the 1st is all of them it is not unlike the sausage more so this model has nevertheless this this Inc for the perturbation theory and this that is initial Michelangelo radicalization assess possibilities to find the semantics In this matrix it is now also it is taken by young Boxster a question and it coincided to visit this method extrusion plant recession many years ago and we didn't know which model with correspond and call themselves the model With denies this metaphor but the 15 years that following this model and merit it was in the sausage models which have local legend so again again collect parameter In "quotation mark laggard aims estimate Texas but I'm the brain In In our stated Inc series in the icons and to see that was increased by about in the consider this is metrics for small when being is that it be reveals soon that this this standstill this matter so of the signal model so it means that the power the area is you on CNN tripled the formation of our signal more or 3 signal in this differential to say about this deformation and we should say it is denies that they need said for several words about this equation socalled 3 so but consider this but it should flow or is your unsanitary and today twodimensional thousands space endit you may consider problems at space neurons that is there is only 1 solution riches Cecil property of protection essentially ended has the parameter you endit annual is a delay that is effective curvature of pollen a few years and is the monarchs all as dynamic in but liberalization broke it really independent someone Parliament that you more of the list riches this product and that it looks in the beginning for small awareness of you it looks like sphere then it becomes it'll be it's like a lips and then it becomes like sausage ended nearly a week of the sources by parts will find that the store vitamin C god Michigan together a number of so this is from 1 point to refer the solution of optimization question from other the solution of from mineralization Group a question a very cheerful so we should now we want to prove that they the respond to the same model which is the goal for the field theory Richard considerate in the beginning :colon goes to infinity so there are many Of course the possibility this exactly terrible kitchen knows many recited the site's Ivan talk about them but for example it introduce the external field we will find that the relation between but I'm telling you and find very well parliament the future of skated and then we can find something modernistic functional advantages in Toronto will but up 1 approximation
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"quotation mark understand and now we want could what happens visit power theory when we introduce to see how this parameterization group a question of we should introduce some scale and consider series in the winter Lila provisions for example we can consider our theory of the circle the if we consider this and the rival like you new multiplied by Logan fire them better in Ronald all our results should be exact stay reconsidering for example the central charge which is related to the ground state energy of power series of the circle In it can be a that In the socalled Minnesota space approximation which is a the approximation which is exactly in the this agreement in the state including the can obtain some the differential creation told the Times it is interesting that exactly the same aggression it appears later in the war itself I think it is a better tomorrow element of the 0 angular value of the "quotation mark it will play the role of some entropy which has the known decreasing property and what it is in some sense from the physics thus supposedly them while went in that it really is affiliated to the central charge will buys a logic of terrorism should increase yeah when are Goldston this is so low when skilled in coastal was provided end so this immigration going to start in this case and other details the there were missing can expressed only in terms of environmentalists parameter you and the Commission which appears for example for the GAO state then the we'll good bundled various often equal to 0 and 2 in this case is this the failure of Ibsen's 0 of couple which is up to some constants the multiplicative costs of respond to the central task there will coincide to that some of the most symmetric accessory coefficient about recession yesterday it is when all hails from the same various all them parliament the prepared close to pool work to slow the it can be Duncan that we can find it asymptotic for small you and for large you endit just from this aggression it is 1st it will be same so get a creation in the potential In inside the is a small you it will be however the memory of the and all the vendors of the spherical 1 but when we will go 2 on the violent here we will have the potential Mitchell look alike gallstones almost everywhere but in the end it will be some walls and it is isn't the poker later this and put it into giving also there the reflection coefficient in this quantum mechanical fault and compare it to with the reflection coefficients Inc which can be calculated was calculated exactly incendiary theory because this would discover that was thrown property feared from the left and from the right you to cover total base the derive their US and politics with you is that big and now there is this week compared analysts increases like we can cope with it where is like trousers which was in the beginning With the wall like rounds at because calculated all this videos what exactly would 73 but this possible also to collated them From there creations socalled stem enabling bedtimes and secretion which span of their own coupla system of integral equation it is not failed but now it is possible I'm completely exactly exactly concluded more than a reserve at the accuracy of the video this function corresponding to ground state television wanted to play in society as the corresponding cleared which follows from there reflections on
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plates utensils she died for the people who knows this can be a technique she is a victim the Isabel's diagram for
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this integral Christian the first one critics
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point to the sausage system the the 2nd 1 critics point of some of the model which took so long Bender How much you don't know
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15 minutes later so know I want to talk about some of the models which can be derived from the sausage model by someone analytical consideration In the continuation in the EU and find wife in this case will have the metric signal model with the medic which will be singled out it will be element to it as defined Poland 4 why more than minus the younger but all in many case at all jet basic the instances on the surface of fine it means that we can regain it apply this equation Minnesota space aggression for analysis of this known but How will look that the 90 Macoby surfaces In the 1st the case for the sausage model with had the surgery both to goddess 3 consecutive disorders at all good figure it is the pseudonym of where is the from end of this figure looks like this there now where can consider you like in in infirmary in deep in front from deep in front 3rd St deep australia From you go stimulus minus infinity 2 going to the Boston Phoenix and at the beginning when you use very negative out figure looks like well then In the accounts the a warrant more along Inc and so on unfortunately in the presence of the it is impossible to find something corresponding name for this model but they say it has no some but this is the Nymex is but other interesting and you consider the social in this case it it is possible to show that the In proper way the soldier is possible only when King a number of flour Lagrangian around the forms and is equal to the integer surveys some kind of and there is some kind of composition of the upandcoming and now can begin to see this accreditation it will change now a little bit but it can this the 2nd offenders can be expressed and someone threw away from them it is the ground state can be expressed through various 1 where it may in the buoyant when you why is close to minus when our figure is looks like the bell it may also that imaginative the goes to infinity and in this limited it as possible to solve the sequential exactly to find that this correspondent conformal field theory we cannot obtain all spectrum of this come home with filthy and central charge from the secretion and the calculations show
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that's then central charge end corresponding spectral dimensions corresponds exactly to so cold as but afternoon a conformal field theory this the models described the critical be favored over was at the end analysis of Pfizer Inc model so that was that there wasn't model more interior is almost exactly the spectrum of more although of course we fit in the in Fort exact it will have fewer of possible but we our approximation works in the case where there is goes is a rather big so that it it would consider now the perturbation theory near this exact solutions them but their mission will be in terms of models of elliptic function spot on the region quantum In exact case will have also some corrections but this what is interesting about this but perturbation serious convergence From the old originally from this equal to 0 principle on which correspond to the ultraviolet it condos and but for ground state is very beautiful we so in In information that is in health the Kompong what will be in the world tour 1 of vision in this case will have the potential of this time so now Our potential will be really look like the wall from 1 side and from the side to Peru length the directive but there's and in this case we have some also Newfoundland their peers some bound states in this equation it means that in quantum case there are some models of this surface is related to his this effective part of the potential I which which appears for m more than 0 here it is believed that the ice and politics when you will see the Infiniti it is for GAO stated that this the same as for the sausage but from underground station that different belittled and here receive the Bshare were all the view level it is order of 10 minutes have but of many different levels here we from infrared from conformal field the then other conformal field theory corresponding to the humor something quite the bell is at the 1 of the Shearer fed up directive a somewhat attractive part but have the series on the long cylinder where is the sequel to the anger that we see that the levels all levels have the same monopolistic for properties although decrease or and do it's issue because it's minus with minus if there the sense of list of later its central chart and share with him all levels discouraged labels here and there collector told them but stolen contingent spectrum like should be converted a long season so it meant I want to say that now Ivica locally that right and left metrics there will be a differently but if this Armatrix can be derived you get from other sites the perturbation of the fallen except here and his begins signed governing political base but we had the ball fields so altogether this potential will look like this part of this model With singular meant end but this still towns I mutually local their local with 1 respect other only if I square His integer or cable for is integer so we refer in our lives the coupling constant .period moved I consider several lost more about something provision fearing new at this In fact Poland but where it's possible to call marvelous the sink but
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I don't think it is very important for people not working for this tradable models so
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that they can progress on come concluding could remarks the it declined to fill theater which can be considerate of the perturbation integral part of the convention Camden lessefficient immensely and they have some rather interesting appropriate like federally along the edited and directed to rid signal models all this picture can be generalized to really signed museum model candidates to the assigned other model which also has some interesting properties of doing Bellendir the sausage model also can be generalized to probably the most solution an exact solution of this the Christian both played down the career In with threedimensional start its space and it also this theory is also the Dole still to parameter family also intangibles series which contained almost all all almost all intelligible quantum field theory and thank you I hope that worst thing work about this signal model with the men in the single whose name shall not you name it if you have a lot of good possibly because of the strength of the world will in the future you will you know what got him the lower not here yet here will not be you say that this is the place where some of these like this 1 so that the working on it does not mind at this mathematician integers this worth a dash but don't know how a but but but by the when this is all over the world in this considerable work Hawaii industries so the such as among the new EU what but the problem is that it nature wave function when you're ready when along 1 is is the wave function of prompt and demand is the wave function of cigars and he'll this surgery can be done without With some honest as it is who but it is only half of them in this way you is proportional to the influence it did little let the breadth for the number over for example landings bent momentum number they should be but I the Kommersant and this I appears only 40 integer villages that's what it is not possible for them without going through this time as to yeah but they haven't been able to take some you consider the the but like I wasn't formulation which correspond to this Ahmeti gives exactly the enmity which it described the polls was derided properties of this series that right then it will be not sausage but some different from address the dated to between a winding current momentum and then you will find that this theory also is welldefined goal for this villages last year let me do you want to be with you in this way it cannot be conclude that the thinking of the system and the use of the but from another point of view the serious falls in the infrared told the paraphernalia theory and this also if it is not the intention of the spectrum we'll be a fined up to 2 half infinite number of Primary Fuels until Monday the vessel field theory so the fear is that good to many of this parameters in treasure other questions what is home of and what is known but it is all something in some sense it can be a letter related to observe expressive of the blocks of the news some of them but exactly sounds and quake it is a duty relations it is not known at the moment but it's still but the last problem because salute it is the simplest model probably after you the so sometimes it will be solved by insurance the other versions the world
00:00
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Conformal field theory
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07:48
Group action
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23:41
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31:23
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39:18
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44:35
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Grothendieck topology
Summation
Database normalization
Length
Geometric quantization
Elliptische Funktion
Infinity
Parameter (computer programming)
Perturbation theory
Term (mathematics)
Trigonometric functions
Metric tensor
Category of being
Cylinder (geometry)
Eigenvalues and eigenvectors
Hausdorff dimension
Order (biology)
Conformal field theory
Equation
Quantum
Right angle
Sequel
Sigmaalgebra
Modulform
Exponentiation
Perturbation theory
Limit (category theory)
Infinity
Vector potential
Frequency
Centralizer and normalizer
Numerical analysis
Term (mathematics)
Energy level
Integer
Equation
Field (mathematics)
Series (mathematics)
Mathematical model
Sine
Surface
Classical physics
State of matter
Mathematical analysis
Theory
Operator (mathematics)
Exponential function
Mathematical model
Group action
Approximation
Vector potential
Computer animation
Function (mathematics)
Field (mathematics)
Integer
Spectrum (functional analysis)
Series (mathematics)
51:23
Point (geometry)
Threedimensional space
Momentum
INTEGRAL
Conformal field theory
Multiplication sign
Modulform
Feldtheorie
Parameter (computer programming)
Surgery
Mereology
Theory
Number
Duality (mathematics)
Natural number
Quantum field theory
Integer
Length
Physical system
Series (mathematics)
Spacetime
Theory of relativity
Block (periodic table)
Wave function
Moment (mathematics)
Infinity
Perturbation theory
Mathematical model
Category of being
Negative number
Quantum field theory
Mathematician
Family
Spectrum (functional analysis)