 Я очень рада быть в such a nice place on a nice occasion of Samson's 60th birthday. Nobody believes it happened. Actually the number is small. But when I try to recall to remember how we met for the first time, I cannot even remember if I first met you or I heard about you. Anyway it was the same. A very sharp impression about a young researcher, a valuable member of Fadiev school. And we started to, I wouldn't say collaborate, but a lot of scientific communication on various occasions in that country. Actually it was long ago, and to give you understanding of how long ago it was, it was the time when Saint Petersburg Russia was laying right in Soviet Union. And it was the time when the string theory was not yet science number one. And I was going to say that by that time everybody was young, except Brezhnev of course. Actually it was nice time. But since I am staying in front of a great storyteller, as everybody knows. So I won't bother you with stories. And also I have to mention that I guess among those who are present here only the third oldest friend of Samson's. So I switched to my talk. But I promise there will be small birthday gift for you inside the talk. So it will be about conformal properties of a field theory called self-dual Young Mills theory. And my talk will be based on the joint work with Andrei Losev and Igor Polyubin. And also unfinished work with Igor Polyubin. This is about a theory which is described by the following Lagrangian, probably first considered by Chalmers and Diegel, and which is described as follows. First of all we have in this theory two fields, gauge field, a mu with arbitrary gauge group. And another field is anti-symmetric tensor field p mu nu equals minus p nu mu, which is kinematically constrained to be anti-self-dual. Then we can form the following Lagrangian or action function. Yes, and the anti-symmetric tensor field is in the joint representation of the gauge group. So we can write pf plus the topological term. The coefficient I will denote by tau over 8p squared. Here I wrote the scalar product of tensors, but because p's anti-self-dual are up to a sign, it's the same as writing which product. So the equations of motion for this theory are immediately understood. Our variation in respect to p gives the equation of self-duality for the gauge field. And the other equation is the covariant divergence of p equals zero. Again you can understand it as a divergence or exterior derivative, because p is anti-self-dual, it's the same. Now we can regard this equation as follows. It is like we would take the Young-Mills equations and make its anti-self-dual sector linearized. This is a linear equation with respect to p, and leaving the self-dual sector nonlinear. I can make these words more precise by passing to the light-con gauge. So in the light-con gauge in the standard Young-Mills theory we have only two physical degrees of freedom, and therefore in certain reformulation called by this name, we are left only with two fields, phi plus and phi minus. Phi plus is responsible to the right polarized, gluins and phi minus is responsible for left polarized gluins. And the Lagrangian in these terms is written symbolically in the following way. Phi minus Laplace phi plus, and then interaction terms, cubic and quartica Young-Mills, which are of the following type. I will not write them explicitly because I don't need this. There is a cubic term, which is quadratic with respect to phi plus and linear with respect to phi minus. There is, of course, by CPT the opposite one with two phi minus and one phi plus, and there is a quartic term to phi minus to phi plus. So this is the action, like on gauge action for the standard Young-Mills. Now if we just drop last two terms and leave only these two, this will correspond to the light-con gauge formulation of the self-dual Young-Mills. And here we see exactly that we leave only the terms linear with respect to phi minus and nonlinear with respect to phi plus. The equations which come from here with respect to equations for phi plus, which are nonlinear, are exactly this self-duality equation. Therefore, the words, that this theory comes as linearization with respect to half of degrees of freedom, make sense. Okay, now I should mention, of course, that the theory is not left-right symmetric, which is clear in this formulation because we imposed such constraint. And also in this formulation, because it is not symmetric with respect to exchange of minus and plus. Therefore, it's not CPT invariant. And you can immediately observe that this Lagrangian cannot be formulated in terms of real fields in Minkowski space, only in Euclidean signature or in signature to plus, to minus. But for the questions I am going to discuss, this doesn't matter, we regard this field theory as just a playground for understanding some hopefully useful features of quantum field theory. But what are these features and why it's a good playground? It's a good playground, first of all, because this theory is very simple in the following sense. So, if we consider correlation functions, there are very few. The perturbation theory stops at one loop. So, let me write this in the following way. I list the only non-vanishing correlation functions. All other will vanish. So, the non-vanishing are the following. If you consider correlation function of one field A and several fields P, then we have and take connected path and three-level Feynman digraphs. This can be non-zero. Another non-vanishing case is when we consider several fields P. I mean inserted in different points. And again consider connected path and take one loop. This is non-zero. All others vanish. This makes the theory similar to some extent similar to the string theory, which is called, I guess, N equals 2 super string considered by Augurian Waffa. In this theory, one also has on the level of effective field theory the equations look similar to self-duality equations. And it also manifests the feature that it is real in the signature to plus to minus. And also has no amplitudes beyond one loop. As concerns amplitudes, it is difficult to speak in this theory. Difficult to speak about amplitudes in the sense of physics because there is no chance for uniterity. It's again because in Minkowski space Lagrangian cannot be made real. But formally speaking, we can take correlation function, apply that formula, reduction formula, which means amputate the legs and put them on shell and obtain expressions which can be called amplitudes. What we shall find here. So, on the level of amplitudes these three-level correlation functions will give zero after applying the reduction formula because otherwise they would coincide with amplitudes in the standard young mills which have only one plus and several minus. And this vanish as it is well known from this vanish. So, three amplitudes vanish. And one loop amplitudes coming from this correlation function correspond to one loop amplitude with all plus with all positive helicities at the ends and this is done vanish and coincide with standard young mills. This is a distinguished amplitude in the standard question, in the standard young mills because it is given by a rational function which is easy to see when you cut this diagram then the halves are three amplitudes which vanish in young mills. There is no imaginary part therefore it is given by rational function and it is regarded often as sort of anomaly it vanishes in any supersymmetric version of the young mill theory but in standard young mills it is non-vanishing and it is the only amplitude which can appear for this standard young mill theory. So this was about I was saying this to say that this theory is quite simple therefore I still very hope that everything can be solved exactly but it did not happen. By the way the formula for this amplitude is known since the 90's it was computed by Malone it is very explicit formula for arbitrary number of legs. Now what should I say this would be nice to solve this theory exactly but since we cannot do this right now we still are trying to investigate various properties and in particular with most attention to conformal properties but to say in advance Several months ago I was much more enthusiastic about conformal properties of this theory but since that we found that it's unfortunately not that simple as we expected so therefore my conclusions will be which I will give at the end will be not that definite but you will see. Now what about quantum properties of this theory? First of all there are ultraviolet divergences I should have said again that if we consider the supersymmetric version of this theory then the biggest change will be that the one loop correlation function will vanish and there will be no divergence at all and the theory would be essentially classical may be yes but it will be about if you pass to the amplitudes ah no no the answer is no because the issue about infrared is applicable to the case of amplitudes but all the amplitudes in supersymmetric version will vanish the three amplitudes vanish as I told you and the one loop vanish for supersymmetry because these one loop diagrams correspond to in super field notation they correspond to one loop with all chiral fields at the end of the same chirality and this vanish this f term so but in bosonic theory which I am talking about there are actual ultraviolet divergences I will describe them shortly but as we shall see these ultraviolet divergences result only in renormalization of the field of the yes of the field but no running coupling constant the fact that there is no running coupling constant is actually very simple it is immediately understood because any coupling constant would be coefficient in front of this part of the action and it can be always absorbed into the field p therefore there is essentially no coupling constant and consequently no running of coupling constant into I will discuss possible other terms if we add p w h p it would be yes if we add p squared or p w h p which is the same this would make the theory equivalent to the standard can wheels if we perform Gaussian integration with respect to p we get f squared sure but I am asking how do you know that it is not generated because of computation the result of computing ultraviolet divergences in the one loop over there gives no p squared there must be some way to understand why you mean conceptually because I don't know I cannot say this simply the question that this is not I cannot I have no good explanation it will be still metric independent because this is the conformal who with this term classically conformal you cannot acquire metric independence because you already pose a self duality condition yes it is metric dependent I will come to this issue in a moment still of course it depends only conformal class of the metric on the classical level and if it will be p squared then it will depend on metric not only conformal on the quantum level yes of course for the function which I see yeah yeah yeah but it is still not an argument that it is not generated I mean I know this only as a result of computation but even before computation when we write the final rules you understand that ok I tell you I tell you let us discuss this in seconds ok I come to this this is a sort of a list of features which I will describe in slightly more detail but as a result of cancelling divergences or introducing contraviolet counter terms we get not only field renormalization but also renormalization of the topological coupling so we have running topological coupling tau another feature is that this theory is classically conformally invariant which is obvious on the quantum level what can be immediately said that the one loop amplitude not the correlation function but the amplitude is conformally invariant another thing one can prove on a certain formal level that this theory is almost a free theory if I have time I will explain this it is a free theory up to some anomaly and as concerns the precise statement of conformal invariance of the theory this is left without answer but the answer is not so easy what would it mean that this theory is conformal it would mean that we are able to construct a sufficient number of conformal operators whose relation functions are conformally covariant so for this I have no definite answer so I put this with the question mark is the theory conformal or conformally invariant so let me answer this give more details answer your question so first before describing more precisely let us discuss what could be the possible other terms in such an action so I mean local terms of the same dimension as you suggested there could be a p squared term but this would lead to standard Young Mills by the way it's gives us a possibility to study the standard Young Mills theory with respect to as a perturbation theory with respect to inverse coupling constant if we add p squared here and consider it's in perturbation theory but we don't do this but in principle it's a good probably point to investigate what would happen if we add f squared term then the theory remains the same because f squared can be written as let me see like I guess one half f squared when I write square I mean scalar product as in field theory we usually write so this is sorry precisely this is what I'm going to say so it can be I guess I should have one half here f minus squared plus f times f dual wave means the dual of f where f plus minus are are cell dual and anti-cell dual part of f which means one half f plus or minus f dual so if we add this term f squared it is the same as writing this appearance of f times f wave simply changes tau and f minus can be absorbed as you anticipated f minus squared can be absorbed by field definition with some coefficient since p is anti-cell dual it's okay so this is essentially doesn't change the theory now this is a good point to discuss the stress tensor for such a Lagrangian the stress tensor is similar in structure to the stress tensor of young mill theory is given by the following expression plus symmetrization minus one half p scalar product is f minus one half not one fourth because here I have two terms so t is t is traceless t is traceless but of course there is some degree of arbitrariness in definition of the stress tensor in particular the coefficient of this term in front of this term is at our hand because this means choosing a special conformal thank you because this particular choice means that p is not this standard two form but the two form with certain conformal weight it is additionally I guess minus one half density then it is traceless now this is a scalar product contraction with respect to indices okay now notice that what would happen if I apply such a function here so if I write t menu as a function of p and f then consider shifted p then it is easy to see that nothing changes they are equal okay because if we consider such an expression where p equals f minus it is zero it is easy to see but now a puzzle for interested listeners which is I give you as a birthday present that was my promise now let us do in other way let us first shift the field stress tensor then we would have the Lagrangian which is pf minus plus c f minus squared okay plus topological term and f minus is the same up to the definition here is the same as f squared right and then the stress tensor will be p f plus f minus p this symbolic for such structure minus g over 2 pf minus this is the old result plus the contribution from here so it is not the same what happened this is a puzzle I define stress tensor there are several ways of defining it if you define it as a generating function just variation with respect to the metric I define it definition will fail here because your shift as we discussed the metric dependence on shell it is difficult to implement on shell does not matter here you are on the right way but mentioning on shell is not needed I define it you can do this but I define it as a variation with respect to metric okay we discussed it later yeah it's not examination I give you this gift that you solve it and then you can give it to your students to your students I teach students that stress tensor is the same the same thing will happen here up to the terms which conserve trivially these two definitions are the same you know okay let's discuss it later alright now let us pass to the issue of ultraviolet divergencies there are only three diagrams which diverge it's one loop with two legs with three legs the same as young mills otherwise the theory would not be renormalizable now as I mentioned there are only non-trivial correlation functions with all p's after amputation p becomes a because the propagator is diagonal therefore in terms of amputated legs I put a field a here so we can consider this as a functional we compute something in the external field a gauge field a therefore there are only divergencies which the only counter terms will be local functional of a p squared is not generated did I answer your question such a counter term would appear if we had a one loop divergence where p is on the legs when they are amputated but if we continue to non amputated case it would be field a here this would correspond to one loop function of two fields a but there are no such diagrams at all you simply cannot draw such a diagram p a sits here but let us try to draw actually you should have asked your question when I wrote the statement about that those are the only non vanishing relation functions but actually that statement is a simple consequence of the fact that there is no coupling constant that it can be absorbed you immediately derive that statement but in simple terms if I want to draw such a diagram let me start from here there should be the only vertex we have is p a a right then there are following possibilities I can send a here, p here and a here then it will be similar to this to have p here I should send both a to the loop then they become to p here and there is no such a vertex that's all yeah therefore p squared does not generate ok now if we compute this divergences of these diagrams we get two counter terms because of lack of time I don't try them in detail which generate f minus squared and logarithmic divergences and f times f dual which renormalize the field p this makes field p renormalized and this makes tau renormalized and for tau we get the following renormalization group running remarkably or maybe not surprisingly I don't know for the group SUN we have the same exactly the same running as tau runs in young mills up to one loop and here it is exact result but the law is the same sorry yeah yeah in standard young mills in instant on computations within it is convenient to write tau something like this plus i theta to separated real and imaginary part the real part in standard young mills has the meaning of the coupling constant of the genuine coupling constant not like here and theta is the logical term and this guy runs, this doesn't run we have the same but for different reason ok now as concerns are the normalization of the field p let me explain it in the following way first let me mention the general shape of the renormalization group equation it will have the following shape something like this are mu over d mu of some correlation function equals here we have two terms one corresponding to anomalous dimensions and one to the beta function the one with the beta function here vanishes so we have only the contribution from anomalous dimensions and let me symbolically write it like this where by gamma hat I understand some operator acting on the entries here operator of anomalous dimensions in our particular case let us consider the correlation function of two p's was it important to fix some gauge which gauge I am fixing yeah of course I should have mentioned I assume that are the way introducing performing Fadiya Popov procedure introducing gauge fixing term and ghost terms the only thing I prefer in this area I prefer to deal not with I prefer to write the gauge fixing with help of the Lagrangian multiplier so the extra terms added to that action will be B because everything is in the joint times d mu A mu plus ghost terms ghost Lagrangian usually we integrate over B and get the divergence of A squared because the kinetic term is second order but here the kinetic term is first order therefore it is more convenient to work with such formulation ok if we consider this correlation function as a particular case differentiate with respect to logarithm of mu we should get as I mentioned the only contribution will correspond to the renormalization of the field p which is renormalized by the following law gamma is some numeric efficient which is computed exactly logarithm of regulator mass and mu times f minus as a result it will be the insertion of f minus here so it will be plus the same terms where f minus inserted in the second place I don't spend time then note that f minus vanishes on the equations of motion and therefore it is a pure contact term so f minus let me leave this space f minus in the sense of Dyson-Schwinger equations f minus can be written as say i d over dp inserting this means that we get in the second term is the same actually we shall get e gamma times delta function of x minus y four dimensional delta function so this correlation function depends on the scale in such a simple way and this is confirmed by computation it is correction constant and to get some central charge what sense I mean this partition functions partition function is not invariant and rescaling but multiplied by some constant constant how do I say are you mean because of this no I mean ea these things is counter term get kind of one dependent on this but first of all yeah maybe yes something like this but actually I would prefer to say that if we change the transformation law of the fields that a the field a is transformed as usual under dilatation that p is transformed in a modified way it is transformed as a tensor or density tensor plus additional term minus then the whole thing remains invariant this is in principle my goal would be to prove this and this is almost proof but the complicated part is when we consider transformations with respect to conformal boosts then everything becomes more complicated and I cannot give definite yeah oh yeah okay okay I am going to finish alright so just for curiosity let me let me mention what is the explicit form for done так yes so the correlation function which I mentioned of 2p if we compute it for separated points then it is the following the tensor structure is given by the operator of the projector to the antisocial dual forms dependent on the points is very simple but now this needs regularization or renormalization in terms of in the usual terms and this becomes as follows so I mean the numerator sorry I simply write the following way such a 2-point function for coinciding points it has has a bad singularity should be replaced by the following expression Laplace applied to a ratio logarithm of x-y squared times some scale mu squared divided by x-y squared this is already normalized 2-point function which correspond to the standard renormalization when we do everything in terms of momentum space integration anyway the result is as written now if we differentiate with respect to logarithm of mu we obtain if we differentiate 2 times and this is our dealt function ok, so it's confirmed now if we pass to but usually people don't call this beta function this phenomenon is not usually we don't think about it as RG flow we think about it as just a red definition of how things transform it's the same as RG beta function you need to do it's not in this case it's not about a better function it's about filtering definition renormalization but it can appear in the shape of better function it could come from the vertices right, but it's not a physically running company it is ok it's precisely that dependence it's another way this description is another way which was developed by Dan Friedman with collaborators but here you need to go to contact terms if you don't look at contact terms it does not always have to be a contact term in this particular case it is it's just a particular case I tend to agree with Zohar if there is a renormalization you have to see logs at separated points there are no logs at separated points there is no RG flow at separated points it's not physical it's just a kind of choice it's a particular case but the procedure is the same as used in all other renormalizations I know there is a formalism it looks like a better function but I'm saying it's not a better function in my case it's also not a better function if renormalization is by a term which vanishes on shell it should be a contact term it's confirmed so we agree okay now as I said the question when we do conformal boosts I don't try anything it would be much more complicated to check even in the two point case but in three point case it's even more so I have no definite result in this case but what would be good for me a good thing for me would be if okay this transformation law already on the level of dilatations means that the trace of the stress tensor doesn't vanish now the question is what is exactly the answer for this conformal anomaly and what we did not decide because of we were lost in some computations not because it's a solvable question but we did not solve maybe to your next jubilium I will give you more definite results here okay there are two possibilities either it is f squared or it is f minus squared if it is f minus squared it is the same as writing f minus this operator is the same as this one and this would mean that as concern the generator of dilatations or the current or the divergence the divergence of the current let me rate it in this way of dilatations is the stress tensor the trace of the stress tensor and it would equal to this one in the second case this would mean that I can redefine the transformation law to make everything invariant and also this will also apply to boost because the divergence of the current corresponding to conformal boosts labeled by the index lambda equals x lambda times t mu mu and would be also good for my case but in this case it would mean the real breakdown of boost invariance with respect to boosts so it looks to finish to summarize the situation looks as follows that in the first possibility we would probably have an example of a theory which is dilatation invariant but is not invariant to conformal boosts which is in principle academically speaking also interesting thing to have such an example but in the second case as far as I understand we would get a fully conformal theory okay thank you I have to finish time for a couple more questions at the classical level does it depend on the metric it does depend on the metric conformal class so why does the union vanish classically the trace of the stress tensor just the full T mu nu doesn't vanish but f minus is 0 on the shell on shell it vanishes so there are no solutions that carry energy yes that's right but you know that also in standard theory stress tensor of cell dual fields is vanishing because it's stress tensor in euclidean space I'm just asking something general if classically no solution carries energy we say that the theory is topological no there is no notion of energy density it is not an energy the notion of the energy in euclidean space has no such meaning as energy it is not positive definite fine but the definition of classical topological invariance is that T mu nu vanishes on every solution no topological invariance means T mu nu vanishes identically I think no there is no distinction between the two notions because you can always add things that vanish on shell I don't know okay let it be topological I don't know I think it's topological classically by any reasonable definition if classically T mu nu vanishes yeah but then there should be some selection rule for saying which correlation functions are topological something like BRST I don't think that all of them all the gaugin variant correlation functions are topological no what is topological about this expression this one but in the bilian case it is in the bilian even in the bilian case it is not topological I don't think that such a straight argument about stress standard can work I'm sorry do you understand that if things will make sense if we need a complex space time to be some kind of natural measurements on finite dimensional manifold to do connections and you just calculate finite dimensional interval that's another paper of the master so to be some kind of natural density yeah it's of course we should think about this but we didn't if you solve the one of the question motion if you solve and plug in the point you solve you plug in then you have more integral 4 dimensional 4 dimensional 4 dimensional 4 dimensional when you are analog of the it's an integral it's an integral of some depends on what you calculate and as I said that was another paper almost here there is a degree of freedom p which has a meaning of something like normal bundle to the submanifold of instantons inside all solutions integrate over p, f equals 0 you get the function of plug in so yeah the integral will be over instantons right of course but if you consider some source for p but we'll say that it localizes on the modular space of the instantons it doesn't take questions of motion in this case it does in particular this is reflected by the fact that there is only one loop it's because it's Lagrange multiplier it's a Lagrange multiplier you are having a delta function so it's a finite dimensional integral it's a finite I actually called it the 4 dimensional avatars of 2 dimensional conformal yeah yeah all right