 Okay, thank you Slava, thanks for the invitation. I'm very happy to be here in this very interesting meeting. Unfortunately, I cannot say long, but anyway, I tried to benefit myself the best I can. So, what I'm going to tell you today is a very disturbing fact of quantum field theory, and actually how little we know in dozvori, da je bilo všeč, da se pošliče, tkaj je teori, obočen je o nekompatno pošliče. Vseč, da tez vseče, da pošliče, da pošliče, da se o te zelo... ...ga vseče vseč otebo. Zelo, da se se vseče, da se otebo ... z njega generaličnih. Zdaj z njega generaličnih, kaj bi teželj, je več izveč, ali je več, več, da je več vse, da je in tegrabilit, da je vse, da je vse, da je vse, da je vse, da je vse, vzvečenih za vzvečenih problemov, ki je kar vizivno. Ten model vzvečen je tudi vzvečen in vzvečen. Tukaj, je to, kaj vzvečen je njegma vzvečen. Zato zelo svojnju vsakaj, kaj si zelo počekno odvori, kaj smo izvečili, da je to teori, je momovore in normlijizacija vša stavne oborje, da je ovo v kljav in minimal, komformalno teori, zanoje posledno na kotik, bo, da je taj in ambigustnih tjusov, in vzaj biti vzaj vzaj. So all the root consist exactly in this problem. And u v property. And, finally, I am going to present you the problematic formula, which are exact formula, but nevertheless very problematic. So the exact mass formula of the model and the exact formula of the vacuum respetiation value of many, in in in in in many operators. In then we discuss the problematic aspect of this model, both from Monte Carlo point of view and truncated conformal space approach. Ok, so let me first present the model. The model is very simple. It's a bosonic field, a scalar bosonic field with this kind of interaction. So this seems really the best field theory you can think of. Why? Well, several reasons. There is a unique vacuum. So it means that you are not worried about hidden sector, topological sector, solitons, whatever. Whatever is in the theory seems to be under your eyes. Second, there is really fast growing potential. Then these lead you to think that you can simulate this theory very easily. Then is the simplest symmetry, just z2. And moreover, if you expand to lowest order, it's five to the fourth, which is repulsive field theory. So you have not even to bother about bound states. So the theory, maybe the only content is whatever, once again, is under your eyes alias one particle, massive particle created by the field itself interacting. And that's it, nothing else. Now, let's dig more in detail. So if we expand the cosine, we get infinite number of potential term, power law. So the theory looks like an infinite type Landau-Ginsburg theory. Remember that the field in one plus one dimension is zero dimension, so there is no problem with renormalization. All theory is trivially normalized if you just put normal order. So you have infinite number of coupling, even coupling of that type. However, the fact that there are infinite number of these vertices change radically the nature of the theory and make it integrable. Now, this is very, very interesting. You can do really a back envelope calculation. It's extremely attractive because what you can compute is imagine I assign you a Lagrangian with one bosonic field and arbitrary set of even coupling. And then you ask yourself, what are relations between this coupling constant such that there are no production? No, production means you have n particle, m particle, n different from m. And then you want that on-shell these amplitude are zero. All for any n and any m at all order in coupling constant. Well, you can do the systematic way of doing was pioneer by Patrick Dory, and then he did let me just first do the three levels, and then when I find that condition, the three level I will dress with vertex and this and that. And the story is extremely attractive. So let me just show you the simplest calculation. So simplest calculation is two going to four. So imagine I have just five to the fourth. So two to four consists in the following graph. If you have five to the fourth, so you can compute it. And the result is m square lambda four square divide 32 i minus i m square lambda four square divide 96. And then if you have five to the fourth, definitely the sum is non-zero. So you can have a production two going to four as far as you have enough energy to produce it. Now, if you want to kill this term, you imagine that you have the possibility to add a sixth term here. So you can add a sixth term here, you see, two going to four. But this is five, six term. And the result is if you compute it with the proper normalization of doing this is 48. And all these result end up to be zero. So what I mean is at any given order you can compute some graph which involve the lowest order of vertex in propagator and the result is not zero. But then if you have possibility to include exactly the vertex which make n goes to m. So here there are no propagators. You can kill it. So if you do this... Why are you not putting any loop diagram? No, it's just at the three level. You are perfectly right. I'm just saying strategically you can find some condition, simple condition, just look in the three levels. Imagine you find this, then you elaborate further. That's all. I will just make a few comments on that. So if you do at the three level you can do at any order of this you find an amazing beautiful recursive equation that as far as the coupling constant of the two n vertex is proportional lambda to the four in this way all these theory are integrable and at the three level there are no production. And then if you elaborate further you see that with Z2 theory so only even in vertex you have only the ambiguity if this term is going to be negative or positive. So at the end of the day the three level calculation leave as only possible theory made of single bosons just the sine Gordon or the sinh Gordon. So here is Kosh, G phi and then there is a relative sign and the Kosh. Now imagine I fix this at the three level I told you before that the theory is trivially normalizable so you can, it's non-trivial but you can check that all the loops essentially normalize uniformly the mass so doesn't scale differently the coupling. Somehow what changes the overall scaling here. It's not trivial. I'm not claiming is something that you can see it but if you do all the calculation properly what happens is that you have a finite or infinite so the loop for instance are finite here in this theory you make a finite shift but always the same so doesn't change anything. So say differently I can put in a different way. It's known that sine Gordon and sinh Gordon is classical integrable because there are inverse scattering method you can compute all the lux pair and then classically they are integrable. Then you have infinite number of conservation law that you can derive from this method so you have to check that the normal order expression this current doesn't get anomaly when you quantize it. So this is remain finite. So to make the story short sinh Gordon is the simplest integrable model which is Z2 even consists of only one particle in the spectrum all together I'm going to derive in minutes and therefore looks absolutely ideal to try to understand basic things of quantum filtering. So if you want epistemologically is the ideal laboratory playground where you can test something. But as I said there are very disturbing features this coming later. OK. So let's assume therefore is quantum integrable as I say I can prove it. So this means that the as matrix is elastic and factorizable so any n go to n things I can write in term of two body problem pictorially something like this so I can concentrate only in two particle scattering then I parameterize my dispersion relation in term of rapidities. I think it's one problem theory or family no no it's one theory. What you call one family? Sure I mean there is a coupling constant. No no it's going to play an important role. OK. If you want from this point of view there is an honest coupling constant. So I parameterize my dispersion energy of my particle in this way you see they automatically satisfy Lorentz dispersion relation under Lorentz transformation theta just additively change therefore the as matrix which has to be Lorentz invariance should depend on rapidity differences. So I define my as matrix like this two particle I have two particle or rapidity theta one theta two and then if I interchange them the amplitude in front is called as matrix. So this is a convention also Gabor was mentioning before theta one is order larger than theta two and this one is order differently means the two particle just have scattered together and the amplitude is the s matrix so s matrix depend on the difference of rapidities and they satisfy unitarity and crossing now where this equation come from come from the following in s matrix you have a Mandelstein variable s which is p one plus p two square if you write down is and t the Mandelstein variable t is p one minus p two this is essentially changing theta one theta two in i pi minus theta because then this become minus sign so in the Mandelstein variable plane the analytic structure s matrix is like this you have a branch cut at m one plus m two square and m one minus m two square is the t cut so unitarity means the value of the s matrix on top of the lip and below has to be one and t channel is this analytic continuation so if you say this map as analytic map in the theta plane so let me call the upper lip of this branch cut like this so e is mapped here e two is mapped here e three here and e four here you can check this is the analytic map I am using so you see that the unitarity is s s minus theta s to be one so this is the and the crossing is the same value of the s matrix here and there so these are the general condition that s matrix has to satisfied and for the s for the sinh Gordon we have an exact solution of them so the exact s matrix of sinh Gordon is the following s theta is equal sinh theta minus i sinh pb mb is a function of the coupling constant s theta now I am going to convince you that all the trouble of the theory come from this expression alias from the duality let me explain what I mean ok so first of all this is really very very simple function so it is periodic 2 pi periodic 2 pi i periodic and it is zeroes than poles so here if there are poles in this physical strip physical strip is between zero and i pi any poles here correspond to bound state there are no poles there there are just zeroes ok so where are the zeroes use this one so let me show you the analytic structure of the s matrix so there are two zeroes located in i pi b and i pi 1 minus p ok you can see immediately if there is a zero somewhere by this relation there also should be a zero i pi minus the same ok now this s matrix much perfectly the perturbation theory what I mean is you can take you can take this Lagrangian this Lagrangian compute the Feynman diagram corresponding to the scattering this is very easy actually at least the lowest order for instance you have something like this plus this plus this plus you can compute all this they are finite so this guy for instance give rise to 1 pi minus 1 over sin theta something like this and then you can expand this expression in term of coupling constant here and matching order by order ok clear what I'm doing I have an s matrix which is an exact expression of the coupling this formula is exact so to all order in the coupling constant so I can expand order by order in g, g square and comparing with the corresponding Feynman diagram which come from the Lagrangian and the match is one to one ok now you will see why this is kind of mysterious things now notice that this matching works for both the sinh yeah yeah now I'm going to tell you yeah now you can ask indeed thanks for the question so you can ask where the hell you come with this expression I mean you see I mean it's like NP complete problem in mathematics you don't know where they come from but if you know the solution it's easy to check so this is the same it's easy to check you can do all Feynman diagram but it's very difficult to do vice versa ok so where the hell come this expression from to start with and so this is precisely the point that this theory is very much related to sinh Gordon theory just by analytic continuation so the analytic continuation so sinh Gordon you have let me use lambda phi minus one now this theory is profoundly different from the other because here the corresponding vertex operator the exponential are compact while there the corresponding vertex operator are uncompact ok so here the theory is very very well studied since long mathematics was known consist of kink and anti kink the spectrum consist of kink and anti kink and bound state thereof and the first bound state is the breeder so kink anti kink bound state and that's matrix of this breeder was sinh theta plus i sinh let me call c sinh theta minus i sinh c and in this case as a pole because breeder create a bound state to themselves ok so at this point this is the famous Kolemann things which as a validity as far as lambda square is less than 8 pi this is the famous Kolemann bound where the bounds come from irrelevant or this vertex operator when lambda square is bigger than 8 pi this guy is irrelevant so at least from renormalization point of view the theory strictly speaking is free theory although I mean one can discuss but the think is no longer there is no longer a mass gap so if you want to make sense sinh Gordon as theory of kink bound state mass gap lambda square has to be strictly than 8 pi but when we make this analytic continuation analytic continuation in lambda goes in i lambda which I call g this become minus g square and this become plus so what was the pole here become my zeros and what was a bound there now become completely invisible bound ok so this is where this matters come from ok then once you have it you can check it ok now these things as it is very nice but is very very positive all the enigma of the model is here so let me explain why the think is so let me cancel now the sinh Gordon let me concentrate here so you see that imagine I'm moving g I'm making g1 bigger than j2 so this guy are gonna move one toward the other ok so I increase g and they move in the complex plane if I increase g then I can plot this function what happen if you send g in 8 pi over g if I make weak strong duality what happen is that b simply go in 1 minus b so what happen is that the two zeros swap just the position and the analytic structure is the same ok so if you make weak strong duality b going to 1 minus g b goes to 1 minus b but for what the analytic structure is concerned you have just swap the position of the zeros so is exactly the same so the theory is self dual for what the arithmetic are concerned but where the L is written in the Lagrangian if in the Lagrangian you substitute g1 over g completely different things there is no 1 over g in the Lagrangian nevertheless the calculation that you get from duality matching with the Feynman diagram is exact what I mean is once you have a function like this the dependence is not g2 the dependence is g2 divided 1 over g2 therefore imagine that you expand this which is already a sinus you expand in coupling constant already at the fourth order term the contribution is both from sinus of g2 but some term which come from downstairs ok so what I want to say is even at the level g4 and further the number precise number is a combination which come from the function of sinus but of the function also this function a much perfectly the perturbation theory of this Lagrangian written only in exponential of g there is no 1 over g there ok so this is a very puzzling thing but it is the same for all strong weak strong cutting dualities no you mean for other theories phenomenon that the duality is not not ever, yeah but you see what I found here at least I mean is that the theory looks so simple there are no hidden sectors you see in other theory the structure is so rich you have monopoles you have kings even a sin order is not self-dual so you have other degrees of freedom you see what I mean is here since the theory consists only one bosel and it is able to do all these things ok ok now let me also mention something which make the theory particularly appealing also from an application point of view what is really appealing is that imagine you restore the velocity of light in all the Lagrangian in all the formalism and then you make the double limit c goes to infinity c is the velocity of light and g goes to zero such that c time g is equal lambda is equal finite what you can prove is that the sinh Gordon reduce to the Libliniker model whose s matrix so the Libliniker model is the one which consists just of three particles no relativistic because I am taking c goes to infinity with delta function interaction this is really the easy model of cold atoms is how you can discuss the property of one dimensional bosons and actually you can use all the technology of sinh Gordon for instance the form factor bed answers I am going to describe in a minutes to compute property directly in atomic physics so in particular you can derive the combination rate you can derive correlation function this and that so this has been checked also in lab so I want to say that apart of being model in itself there is a payoff a byproduct that goes directly in experimental physics atomic physics if you just do the proper no relativistic limit now let me introduce new tools to study the theory more detail and the first tool is the thermodynamic bed answers how did you end up with this as I said you have to restore the velocity of light so I usually in field theory with this regard c equal 1, h equal 1 you have to do all this exercise and then you take c goes to infinity limit this is how you realize no relativistic one and then there is a bed interesting trick so you have to disentangle fast mode from slow mode and then the fast mode goes to zero and then what is left out is non-linear Schrodinger equation and this matrix much perfectly with the Liplinear one and then once you do all the other formula of that you just make this trick simply like that you can recover three all the quantity of Liplinear which is pretty nice for what for which reason because in field theory usually you have more constraint that in non-relativistic one for instance in field theory you have duality this equation which doesn't hold in non-relativistic one so the theory is much more constrained and this is the reason why people working in atomic physics were unable to compute easily correlation function Liplinear while if you come from and you need the result ok, so let me tell you other aspect of the theory which appear in finite volume through the thermodynamic better answers so let me sketch the idea what it is thermodynamic better answers you take large volume L you compactified in temperature one over r and you compute the trace of your theory so here the space is made of particle which scatter all the way around but the number of particle is conserved because it is integrable so you can do exactly trace ok, so I make the story short you can compute the ground state energy is a function of the temperature r and this is parameterized in this way in term of what is called effective central charge so this is standard normalization of conformal field theory and then there are set of equation integral equation which compute the ground state energy gonna write it and then I will comment so the formula has a very very clear interpretation I am going to spell out for you so look this formula here the formula here remind very very much the formula of a free fermion particle ok, so it's an integral if you write like this you will recognize immediately logaritmic 1 plus e to the minus beta e right this is just so the course here adjust change of variable in the rapidity I am doing the only thing which change is that instead of having the energy of the particle we have a function which is called pseudo energy which is self consistently determined by itself so these things satisfy an integral equation which involve the free part that will be this one but then bootstrapping through the interaction with all the other particles with the same distributions so this equation is very very nice because you are enforcing the most you can the free quality of the theorem because integrable theory is essentially the closest of free theory you can think of but the only difference is that what played the role of energy is not the energy of single particle but is determined self consistently by the all other particle present in the system through a kernel which is just derivative of this matrix so the message is here is that for what the finite volume is concerned this matrix determine everything even the ultraviolet this is the message so you can in particular study what happens for r goes to zero so you can study what happen to the central charge the effective central charge for r goes to zero so you see you can solve numerically this equation as a function of r and then plot c versus mr and the plot is like this so is one minus some coefficient of logarithmic square mr plus so you get the central charge of a free boson theory ok, now I will come back to this because once again is very disquising seems very simple, elementary but then you will see what is behind this so now let me introduce another part of the story which is the phone factor and it will illuminate a significant very significant very significant absolutely this is absolutely significant because in other theory for instance in theory of the minimal models if you take Pots model or if you take easy magnetic field the correction is always power law never logarithmic and this is the signal the theory is I mean relate to the wheel is what I am going to tell you in a minute ok, phone factor provide another window another view on the model because give you access to the exactest matrix of local operators so phone factors are defined of a field I call psi by definition is this matrix element now is I am not losing generality because any matrix element in which I have any number of particle here or even the point at x so the field at x I can shift it with the momentum and this particle are against the momentum so this matrix element if I have x differ from this just by phases on the other end I can cross any particle on the left hand side I can cross here by cross in symmetry so what I want to say is that if I know this function precisely this order vacuum this I have access to the full glory matrix element in any configurations now for the singed Gordon this calculation has been done in full glory and let me just sketch what it is is like this ok, so let me tell you what it is so an s particle is like this a phone factor is like this here is the vacuum here is the field here is the particle so this matrix element has a certain very strong constraint for instance when I cross this this is going to be the s matrix so this means if I take f theta i theta i plus 1 and the rest untouch is going to be related to the same function when I cross the two of them ok because each time I interchange for the relation s matrix so this property of this function is taking care by this function f min which satisfy this functional equation it's easy to solve it this function has a mean free number of pole and zeroes in the complex plane but I mean there's precise expression we don't need it I just want to say that everything once again depends on s matrix I can find explicitly the solution it's very simple by re-transform imagine that this matrix is exponential i t over t ft sinh t over pi is a phases I can always write like this f and f which satisfy that equation is simple given by these for re-transform immediately and then if you make the infinite product representation you find infinite number of pole and zeroes and so on so forth so I want to say this function is well known and determined by this then what is this factor here I told you that sinh gordon do not have poles so you can never have a configuration like this in which you take two particles and go on shell but what can have is I can take three particles and make this configuration where here is the transmission of the s matrix now you see to do that two particles has to be head to head so this is really the s matrix so this means that the only pole this amplitude can have is when all the rapidity differen by any other by i pi ok so this kind of amplitude shall have necessarily pole each time the rapidity difference end up to be i pi and this is taking care by this term here in all possible channel so this is product of this is elementarism ok so what is left out is a generic symmetric polynomial in x which is not determined at all by the analytic structure this matrix this is what genuinely depend on the operators because so far I never tell you which operator I am considering so summary the form factor has a structure x by the s matrix alone you cannot do anything about it what is left out is a symmetric polynomial in x and difference in the polynomial characterize different operators ok and in the past I have been computing it exactly and I have been found together with kubek an infinite number of solution on that because there are infinite number of operators with very remarkable result in term of determinant of symmetric polynomial I am not going to write for you and just tell you that this symmetric polynomial are known you ask me any operator you want you say can I give me the form factor of phi square yes I can I just go take the formula and give for any number of particle and for any couplings to all order ok, now this is the things now let's go in more detail sorry, is that a constraint or yeah, yeah, indeed I am going to describe exactly now indeed how you determine this symmetric polynomial you determine exactly through the recursive equation that the form factor has to satisfy so this n particle form factor is related to n minus 2 precisely by residue equation which involve this matrix, so here is a pole you see there is one particle here and there is a pole so this determinant are a cursive equation not going to write for you something like minus xx x1, xn has to be a very well definite polynomial in symmetric polynomial in x, qn minus 2 x1, xn you have to find solution of this infinite number of recursive equation at once so this is why this form of determinant works because in a way encoded this recursive structure in the field so far once again the field you have to fix me some condition for is some phi square I define as the field which create is different from 0 at level 2 phi4 normal order I define the field which has 0 matrix element up to 4 so I can classify the operator which is called self clustering alias I can take the exponential by definition if you take the river then this and that always remain the same so is a special operator after all I can do it now let me not enter in the things now what I want to point out leave out all the detail the thing is noticed I have a cursive equation which jump in 2 so let me denoted by this the number of particles n equal n for me is the number of external particles so you see I have a way of relating form factor in this way or in this other way but I'm never able to have an operator which link even with odd number of particles they are completely coupled so let's talk about stress energy tensor which is probably one of the most important field of the theory for the stress energy tensor we know 2 things by basic stuff we know it's vacuum so so is the simplest form factor and this come from better answers because I can compute the free energy from that and the free energy is the and the expectation is by m square the renormalized mass which will be my concern later sinus the same function b and then what you know is the 2 particle form factor on a general basis simply because the stress energy tensor if you integrate is the energy of the theory this is fixed to be uniquely this one the stress energy tensor the trace the stress energy tensor all together since satisfy conservation law if I know the matrix element only the trace I can reconstruct all the other component just simply by conservation so is enough to have that so you see general theory allow me to fix zero expectation value and 2 particle never no no this is the point the point is stress energy tensor you can alter with charge at infinity so this is the point so if you have a theory with T mu nu I can alter the theory with T mu nu with a parameters which I call charge at infinity I can have always this ambiguity to add the divergence ok how about phi squared phi squared at least there is no problem no, but there is no problem phi squared I mean you have to tell me what you are calling phi squared phi squared is the solution of this equation which has on the 2 particle states is normalized to 1 and then on go to phi 4 but am I correct to say that phi squared is not going to couple to 3 yeah sure no no no no this is the point the only things where the theory has this ambiguity is the stress energy tensor all the other there is no way you can do it also here when you add the odd term all the chain which come from the odd term goes in its own way there is no way of relating the the 2 chains ok but when you do that the UV ultraviolet the theory change completely because once you add this tt 2 point function become 1 plus 6 q squared divide z minus z2 4 you can check using form factor of course and then the vertex operator become a different with delta of v alpha delta of alpha so v alpha for me is exponential alpha t now is alpha q minus alpha ok so what is the important lesson the important lesson is that Lagrangian and S matrix UV ok this is what I mean is this much with the content of the theory this is the problem so I have to take once again the Lagrangian but you said the center charge is 1 starting from the estimator wait wait wait this is really wait actually I have to be precise this is the effective so let's take once again the Lagrangian and now let me write like this and now I want to interpret this theory as deformation of some UV field theory but then I have a big ambiguity because I can take this theory as Gaussian deformed by this symmetric Z2 combination the vertex operator this make a very precise commitment of the stress and the tens and choosing central charge equals 0 or I can take this as UV this is UV but to make sense of UV as conformal field theory I have to add charge at infinity because this operator now is dimension 1 deformed by this one so the stress energy tensor will change the central charge or I can take this as UV and deformed by that so there is an intrinsic ambiguity what you are doing you can have infinity many other possibilities like add some term subtracted and so which of these gives rise to the S matrix no this matter is untouched untouched because this term is completely invisible this is total derivative this is the point this is precisely the point this matrix which is infrared data is completely invisible to any perturbation theory ok it's not obvious maybe some of these non perturbative definitions could give rise not to an integrable theory but no no no this preserve as I say this preserve any integrable this is built up because it just admitting the presence of all term in the stress energy tensor simple like that this is completely compatible doesn't affect all the chain of recarces equation everything is fine now unfortunately I'm running out of time so let me tell you two amazing things which coming out from it so from Liouville you know that Liouville is a non compact theory and has a lot of is a very complicated theory to build to discuss it in particular there are no correspondence between operator and states as it happens in any other conformal field theory in particular there are continuous set of states that correspond to vertex operator alpha but alpha has to be q half plus e the momentum p now here is the point central charge effective is remember that the Hamiltonian is L0 plus L0 bar minus c over 12 and then the dimension 1 over r in the so you see that if this L0 again values it's non zero but effective central charge which doesn't consider central charge but there are minus 24 delta min ok so if you consider the theory as sinh Gordon so c is 1 and this guy is 0 so you get exactly this one minus this one but if you consider Liouville you get exactly the same result q this alpha remember the formula delta was alpha q minus alpha so if you insert this as a minimum states in the theory you compute c effective c effective is 1 minus 24 p square the momentum of this particle and the momentum is quantized let's call the Liouville reflection wall so you can take sinh Gordon like 2 Liouville one far apart of the other so particle leaving here momentum p arrive here reflect back and so you get a quantization like sp square equal to 1 there is factor in r here which determine the story so you can compute p as a function of r and when you substitute you get exactly the same so what i mean is even from the final formula you cannot decide it's completely compatible with both interpretation even sinh Gordon has z2 theory everything or as Liouville uncompact with hidden sector with a lot of stuff behind is a completely the same but since I promised you some mystery I have to tell you the mystery unfortunately I am running out of time the mystery is you can compute for this theory sinh Gordon the exact mass formula is an exact expression I am going to write for you is 4 pi gamma 1 plus 2 plus g square gamma 1 plus what it is I cannot do much you have to be pleased that exist an exact formula to all order so this is the formula once again you can check it perturbatively but what is the mystery of this formula what does this formula define define the exact mass gap of the theory there is a function Decatov which you put to define the theory and there is a function of the coupling to all order in the coupling ok now what is the starbing of this formula what is the starbing is that if you plot it to fix mu vanished at the self dual point vanished ok and so this is pretty disturbing because you don't know what the theory is below that I mean beyond that moreover this formula is the only formula of the theory which is not self dual doesn't respect self duality so here is the real problem what is the theory here for g square larger than 8 pi what it is what is behind the column what is behind it is a theory which is still massive or here become massless now I have to tell you little part of the story and then I close you remember that there was these two zero that move together when I move the coupling when I write to the self dual point I can put them in the complex plane I can just make analytic continuation I can compute the central charge effectively so I'm doing b equal 1f plus i theta zero so I'm putting here theta zero now what happen is that if I plot the central charge like this the central charge start to get staring case behavior with quantize value which match all the minimal models so the value is 1 minus 6 p plus 1 with p integers so the theories knows all this scale behind amazing moreover if I make the plot real and complex coupling here is the sine Gordon line so here the theory is massless here is the self dual point 8 pi now if you do the massless roaming which is massless the trajectory is just a semicircle like this where the theory is massless as well so you see what is the puzzling guess here the theory is massless here is massless well I mean in the scenario that all together as a generic function the coupling the theory is massless wherever even here and the only massive case is here unfortunately we have no way of checking this is a technical point it implied TSS once again the origin is what kind of conformal field is taking in the UV because you have to choose a basis to make the calculation and the fact that you have non-compact bosons affect avely the calculation so we have been trying a couple of approach all of them failed in a way or another one failed trivially in the sense that we choose the basis of compact bosons to compute the matrix element of the energy level were pretty pretty bad so the energy level were something like these are the lowest one and there is always some highest line which cross all the other and make the story completely out of then we have using zero mode technique alias to select out that zero mode and construct excitation on it it work pretty well for small value of g when you go g square bigger than one half why that because you have to unknowledge the following things operator of 2 g on two points is kind of trigonometric expression you get something which goes minus 2 g square and this is fine but then you get cos 2 g phi cos 2 2 g square since the theory has expectation value this term become marginally irrelevant as soon as g square become bigger than one half so you have to take this term in but once you have this you have to add the third term, the fourth term the fifth term so the theory just explode under your eyes and so it's kind of problematic and Monte Carlo doesn't help either for the simple fact that if you want to decide what the master here is you know in Monte Carlo you have to do the simulation but then you have to scale the lattice side to zero you have to go to the critical point of the theory to get the scale but here means that the mass is already zero or you have to play with the cutoff to be finite because the easy answer from quantum field theory point is that look the mass is not zero as far as you take mu goes to infinity but in Monte Carlo you have no you define mu and then the theory get you what it is so it is a question and moreover the fact that the theory grows so fast make the Monte Carlo worst and worst and worst because you can never span more than the vacuum larger g you got you stuck to the vacuum so I mean it's amazing frustration that you have a theory which you know essentially everything but there are some basic disturbing things which which are really at the moment impossible theory to figure out so this work has been done with Robert I mean we are working on it since long actually and as I say it is really challenging and so this is what I can offer at the moment so just puzzling question and the poor understanding we have so far okay thank you so the basic picture is that the naive duality that you had in the asthmatic is not correct you are saying no no no so you can work it out from duality for what asthmatic is concerned okay so everything I mean you are never sensitive to the issue now you start be sensitive to the issue when you pose the problem can I compute the actual mass gap for the theory as a function on the cutoff and the coupling and the answer is yes this formula come once again from better answers you have on one side the energy expressed in term of mass physical mass on the other term expressed in term of perturbation theory you match the two and you extract the mass gap but once you write to this formula you can check once again perturbative that is working you just compute the loops and this and that but when you go to strong coupling I mean the theory vanish there you don't know at least I never be able to go to high order so you see this and the formula is not self-dual is the only formula which is not self-dual now the way out this is what Alyosha Zamologigov which actually computed first in this formula is way out to this he said well at the end of the day nu is an arbitrary parameter so if I go g goes in one over g I will define some nu mu so you see I can write this as mu sum function g this function is not self-dual but then I can make this guy self-dual just define nu tilde such that he hold like this after all he say these are arbitrary parameters I'm not very satisfied with that I mean because seems at a certain point you are changing the rule of the game and moreover the question remain is this really a critical point of the theory I mean critical point defined must be at zero or is something else and from renormalization I mean from roaming trajectory this is the picture this point is directly connected to this by roaming trajectory I don't know I mean it's really puzzling I really don't know honestly I have no idea so it's fair to say that for g less than 8 pi we are pretty confident that your is massive everything works to the best you can do you have out of nothing this self-duality because it just pop up this self-duality by this analytic continuation sign Gordon you bite seems that in any formula workout except ask the right question what is the mass gap mass gap is not self-duality and then has all this property is there a lack dispersion in the field that preserves no, no I mean in the sense that once again or better there is but it's not efficient as well because once again the problem is that the boss is not self is uncompact this causes a lot of problems this is the real final question is there a boundary version of the story that could be sure there is you work it out exactly I didn't talk about the web the web is exactly the same phenomenology everything is self-dual except term which depends on the mass in the boundary there is similar story there are web as well everything the mass once again pop up there so you you got exactly the same problem there is no way of opening a window and making progress on that exactly so for the reason the web any expectation value any of any operator as mass the same mass time and exact function of the coupling this guy is self-dual function so if I do g go in 1 over g this function is self-dual but this term is not because it's the exact same formula and this web satisfies amazing property if I call ga ga satisfy the reflection matrices of q which depend on q times g q minus 8 and then you might say but where the hell I am telling you are using Ljubil I never because you see the anomalous dimension is gaussian very positive really very very mysterious very very mysterious is it fair to rephrase this problem is saying that you are looking for a renormalization scheme that preserves this self-duality because all these problems you are seeing is related to the the cutoff you are imposing it's not really a purely quantum field theory it's not a continuum question how the mass in the continuum is related with the cutoff What do you mean by this? I mean that all the formulas that only talk about quantum field theory observables they are self-dual only when you express things in terms of a UV cutoff you are seen to have some problems so there is no UV cutoff there is no no no no this is a no cutoff I mean is mu the one you define the bare mass if you want bare parameter nothing else nothing else but it's not physical observable indeed there is a formula itself give me your mu I will define the masses to be expressed in terms of mu question is that formula does it define the theory or not it does which formula the first formula on the platform is it a definition of UV complete theory I mean this is it just some platonic equation because there are some formulas of that type for example if the interaction is not exponential but polynomial then provided the normal order it's a definition of UV complete theory in a mathematical sense that formula goes out of that framework so perhaps there is a problem with that formula that formula doesn't make sense I think you can rephrase like this as I said if I give you this you don't know what to interpret because if you interpret like you will deform it by the way that's a separate question let's interpret in the simplest possible way simply possible way is that this theory defined like this a central charge equals 0 because you are treating them any expansion in G is Z2 even so essentially from the if you want my main message is theory Lagrangian probably in this cases particular CV or other is simpler are unable to fix the UV is somehow a parameters free even though the formula you can derive are not contradictory is the same expression that you can interpret in two different way both because this matter doesn't know anything about UV so it's a basic thing it's not simply to find contradiction you see what I mean you say well if I did this then the formula doesn't match means no no no everything work perfectly the only way where you start thinking that some alarming something is going on weird is this commitment of this mass actually I probably have to tell you how this mass formula come out