 P blood, sorry for the little delay. We had a technical issue.ilateral is going to talk about the boost up quantum field theory. This is actually not a technical talk, it's a colloquium style presentation. I will take the opportunity to continue my lecture during it z neko je boje, kaj sem bilo zelo odličen, da je neko se različen, ga je zelo odličen na to, a pa se bo začeli, ko mi je začeljno šličen. Na minulosti, se je to zelo, da se pridušajte in zelo, da se so se zelo pojeliti na vse, ker je začelj, in ne so pojeliti na vse. Zelo, da zelo pojelim izličenom, kako smo se vse zelo se vse zelo, da so se zelo pojelim zeno, vun claiming of quantum field theory. So we have local quantum fields, in the context of fundamental physics are functions of coordinates in space time. So this is not important for this audience, but since I have the opportunity, I will give you a mini introduction to quantum field theory in three slides. It's a bit difficult to do it in the context do agricultural physics is because it's more abstract and I will instead do it in the contour of statistical mechanics, but first I'm just going to start with one slide that summarizes what you all know. Quantific theory is the language of particle physics for each particle species you have a field and when you go to the quantum theory, excitation, I mean Achosne predanie do vseh pih delarje iz forjivovine in je odstavile, kako na jesetu na zelo imamo našelje pendantu. Tristim pa, kaj ker poč Evn GUY, tudij se zaradi zašlič natičnji. Sve tako je zašloysnno, ta je, da se začem bilo, z njehovo težnje, zgodno, posačo. A poča, kako je to. thega težena je obučenja v Čutom, zelo začevna, fondamentalno-physics, ali zelo da sonto začaj z vsi kičen kvantunč, in da se malo poživno vsega zelo in začaj začaj delo in zelo da je zelo. Svoje tudi je nekaj neštradega vsega od nekaj fondamentalnega, ali zelo da je vsega pogrešnja izgleda na vsega nekaj ste začaj zrednjenja naredno v 1970-ju. Zdaj so boj, da se zelo kaj je vsega Kaj je to velik? Ko lahko modljamo, s kratikom vizivu takovom antukov, da je ovo, da je ovo pomembnij. S modljami, staro, kaj je to autokop, bo ne, da se kratiko po vse. V zelo, da zelo pomembnjem ovo klasiko, satisiko, mechaniko modlj, da se vse zelija vladje površenje in zegma-i, Sred in sred almost sure, whether this actually work. Yes. The text values plus and minus one and you write in the area snp type of Hamiltonian. And so in this context we are just in the equilibrium statistical mechanics. So we are time is out of the window and we just write down this effective description of the system near the face sensation and the system starts as a discretized statistical mechanical model. kot izgleda, je zelo da je však koralizacija, je, da je zelo, da je zelo vse zelo, in so vičnim z vrkodem, in izgleda se s koncijutnem vkoradom z tudi, vzelo, da je vse, tako, v to, in so, vse je, kako je vse, začnega, z vrkodem, na naša boljna vrkod, in, kako je v Landau, smo se počeškel, nekaj nekaj zevrčenj, in nabjič nekaj druga kot se vnit. Rada v željih pustenju, se zelo pošljeno in zelo ušlično za jeljeno energične zelo, zelo vzivno očastveni možni danes in zelo ušlično zelo obrženih vstavne obrženih možničnih. To je na vse, ne možno, tako, da ne bolj zelo ima, kaj je neko inberazno. Se, da se vse pošljeno independent of space, you look at the potential and then famously there are two distinct phases. If this parameter m square is positive, then you have a unique vacuum and if it is negative, then you have this famous phenomenon of spontaneous symmetry breaking. And so we should identify m square with the deviation from the critical temperature. Above the critical temperature we are in the order phase and below the critical temperature, although the original Hamiltonian is of course symmetric under exchanging the sign of each spin under up and down transformation, below the critical temperature the symmetry is spontaneously broken and you have a spontaneous magnetization. Very familiar story. So you may immediately question and surely you have a question and probably thought about why this approach should be trusted and should be useful given, of course, the enormous uncertainty that there is in this additional dot, dot, dot term that haven't written down. And so these are examples of higher operators that you could, I mean, I use the language of, OK, I'm sorry. To for the audience, I don't have to make too many caveats, but we are really in the context of classical statistical mechanics, so phi is the local monetization and these are higher powers of monetization. So I'm using the language operators because it's standard, but forget. There's nothing, there's no Hilbert space, there's no operator statement anywhere, it's just a name. It's just a name to denote these monomials of the field and various gradient combinations of them. And of course the answer to why this is an OK thing to do is that these dot, dot, dots are irrelevant if you just want to describe the universal large distance physics near the phase transition. And irrelevant is in this context is both a ordinary use of the word in English and also as a technical term. And the rule of thumb is that we are entitled to neglect these operators if their scaling weight is greater than the dimension of space. Remember, in this context time is out of the window, so we are in three dimensions and these additional terms have scaling greater than three if one follows these assignment rules where the elementary field is assigned weight one half in dimension three and then coordinates are assigned weight minus one which means that the derivative are assigned weight plus one and so you can easily check that the terms have been neglected are irrelevant according to this criterion with one small cheat that I could have added phi to the sixth and then, of course as you probably know, higher order calculations will show that that actually is truly irrelevant when those higher calculations are taken into account. And so there's a way to make this very precise which is the Wilsonian approach to second order phase transitions and this criterion, this becomes the criterion that the universal physics near the phase transition is only captured by fine number of terms which are the relevant terms in the action, the one with scaling smaller than d. And those universal properties are famously, for example, the values of various critical exponents such as how the monetization depends on the deviation from the critical temperature that would be a spawn in beta, for example, et cetera, et cetera. The critical exponents famously do not depend on any of the microscopic details of the original Hamiltonian that only depend on very coarse properties of the system such as the dimensional space and the symmetry. And Wilsonian theory gives you a picture for how this happens. This happens because once you understand what is happening, you are really attracted to this finite dimensional space of interactions regardless of what is the original detailed microscopic description of the model. And that is the first goal at the phenomenon of... And so people have had observed in the 60s and 70s that widely different systems such as boiling water or uniaxia ferromagnet exhibited with good accuracy, the same critical exponents despite having a very different... I think it's a very different microscopic origin. And this Wilsonian theory is the first goal at that. It's pretty sure we'll come back to it. And so this counts as a crash course in quantum field theory because if you are confused by continuum fields, you can always go back to this lattice version of the story where you have a finite number of variables for each point of the lattice and then you have to imagine taking this continuum limit where you send the lattice space into zero while keeping the large distance properties fixed. That is a procedure that by the standards of rigor in physics I think is well understood and it's called the renormalization group. And so this picture I've given you is a traditional one where we formulate quantum field theory as a theory of local fluctuating variables. So in the example above, I was either a function of spacetime if I want to do particle physics or a function of space if I want to do statistical physics and there is a certain measure, a certain ensemble which is defined by this functional of the field phi which has two different interpretations according to whether I do particle physics or statistical mechanics. In particle physics would be the action. In statistical mechanics it would be the Hamiltonian and also this coupling g would have two very different mathematically equivalent formulations that would interpret these fluctuations either as quantum or thermal fluctuations. So it's clear that this is powerful but it's also limited because it's very hard to compute this object and you have to take this difficult continuum limit and surely we have no controlled way to do this when the coupling is not small. When the coupling is small you can do a saddle point approximation which famously leads to Feynman diagrams, et cetera, et cetera. However, the viewpoint that I would like to advocate is that apart from a calculational difficulty there are more conceptual reasons why you may want to look for a different framework and let me start with the left. So a famous story that you have probably seen in several examples of at least some of you is the phenomenon of duality. What is duality? Duality is the idea that you can have two very different looking formulations of the same theory in terms of fluctuating fields such that this is very schematic of course but this is meant to indicate that when the first description in terms of the set of fields phi with coupling constant g is good meaning that g is small the alternative description in terms of phi prime and g prime is hard because g prime is large. This is a schematic way in which the two couplets might be related. And so you can have a theory of weakly coupled fish down here or which up here would look like a theory of strongly coupled fishes but you are better off describing in terms of a theory of weakly coupled birds. But what is it? Is it fish or is it birds? Well, it's clearly either both or neither. The fundamental description of the theory is one in which one should be able to talk about both at the same time and once you are here in the middle it's not clear how you make the distinction anyway and I want to emphasize that the theories are truly the same as either statistical mechanical theory that would have the same set of correlation functions or as quantum theories the same Hilbert space, the same set of observables the same everything but they have very different looking descriptions using these two sets of fluctuating fields. So perhaps there is a more fundamental way to think about quantum field theory without ever bothering with fluctuating fields. And this truly becomes a necessity because we have discovered and I already mentioned this this morning in my talk that our theory is that as far as we can tell truly have no description in terms of elementary fluctuating fields and the paradigm that I will use in this talk is the one of a somewhat exotic if you haven't seen it before six dimensional theory with maximum super symmetry for technical reason it's called 2,0 of course this is not something you stumble upon if you don't have a reason for but it's actually very natural in string theory it's the theory that arises as a factory description a large distance on a stack of these mysterious five dimensional brain like objects in amp theory, n of them and we have seen and I'll come back to it that this is actually a and this is also a theme that Nida Slom and I will really talk much about so you're gonna get in this talk this theory appears to be a very good organizing principle for a huge number of quantum field theories in lower dimensions and there are reasons to which I will come back later on to think that such a theory will never have a description in terms of elementary fields and Lagrangians and so the vision here is to graduate from the standard approach to quantum field theory in terms of fluctuating fields and embrace instead a more abstract viewpoint where the first thing you want to do is to enlarge the view from studying an individual model such as the Eisen model I had before the 2.0 theory to studying all at once the entire set of possible theories you want to study theory space this is already a viewpoint that was very much implicit or even explicit in the Wilsonian paradigm so Wilson told you write down the most general action where you keep this infinite set of higher terms and you may be attracted to a specific subset of this infinite set of couplings in particular regions and so there was already a notion of enlarging the view to the entire set of theories but this is even larger because we are not committing ourselves to any choice of specific elementary set of fluctuating fields and then the slogan is that you like to carve out out of this infinite dimension of space they are still infinite dimensional but much smaller subspace which is consistent and what are we going to use we are going to use the constraints of quantum mechanics such as positive probability and symmetries of the problem spacetime symmetries or whatever additional symmetries the problem might have and the surprising news is that although you don't think that such a generic slogan would lead anywhere in fact it does lead to rigorous prediction that short cut this issue of having to do semi-classical approximation where g is small and so the alternate catalog which is of course very ambitious is to compile a complete catalog of the consistent quantum field theories and so this vision is called the bootstrap it's an old idea it started in the 50s and then later in the 60s in a different context it was in the context of the S-matrix when people were despairing of writing down a local Lagrangian that described the strong interaction so they said okay well let's just focus on universal properties of scattering of this complicated zoo of resonances and let's try to see whether imposing consistent condition we could nail it of course that program didn't quite succeed but among other things it led to the discovery of string theory and there are many reasons why it didn't succeed but one of them was of course it was too ambitious you are really trying to use this general set of principles to zoom in on a unique theory that's not what we think is happening now you really should think of carving out the full theory space rather than aiming for a single theory and this program also then had a second life starting in the early 70s where it was applied to conformal field theories and it led to this very beautiful success in two dimensions with the work of BPZ in the early 80s that led to the solutions of many models and now we are continuing it in higher dimensional conformal field theories vikipedia entry always look at vikipedia when in doubt the vikipedia entry says that the bootstrap is a self-starting process supposed to proceed without external input so this is a perfect quote the idea of this bootstrap actually in the original book is the Baron of Munchasin was pulling himself out of the mud by pulling his own air in the more modern version you are trying to pull yourself up by your own bootstraps and the idea itself starting is clear but also the fact that it is supposed to proceed is something one should keep in mind this doggone as formulated is too simplistic there is a lot of fine print that goes into precisely identifying which input you need to get consistent output and so going back to the idea of theory space which was already very much part of the Wilsonian paradigm this is how theory space looks like very approximately and famously there are ok so here of course this is just a joke it's just a landscape of theories but what it is supposed to impress upon you is that there are stationary points in this landscape height is stationary and the height is some sort of measure of the number of degrees of freedom this can be made rigorous in two-dimension where the height is the c function and in the Wilsonian approach the distinguished points are the stationary points those are scaling very unfixed point of the normalization group and as you move away a little bit from a local maximum you will follow in the normalization group trajectory so in this audience I don't really have to explain these terms because you have seen them before but the idea is again you do this coarse graining at the level of the Wilsonian action and that generates a trajectory in this infinite space of couplings and what we will be focusing on for reasons that we will explain momentarily are the stationary points the fixed points of the normalization group that are points that are invariant under scale transformations and so conformal field theories are singled out in theory space because they are sort of special mark points and you are supposed to be able to reach any other point in theory space or think about any other point in theory space as a normalization group trajectory between two conformal fixed points and the main theme of the rest of the talk will be that at least for conformal field theories and we have also high hopes of this landscape you don't need to think in terms of explicit fluctuating fields and lagrangents you can define them more algebraically in terms of an abstract algebra of operators so this is to continue on that point before so we are familiar with the fact that physics simplifies when the intrinsic length or mass scale so the problem can be neglected and this can happen in quantum field theory either at very high energy or at very low energy and in statistical mechanics this happens near the phase transition because then you develop an infinite correlation length which if you recall from my second slide was the identify with this m-square parameter and so m-square is going to zero the system becomes scaling variant and and genetically we have learned although there is no universal proof but many partial results that under a reasonable set of physical assumptions that certainly include positive probability so the theory is a unitary quantum field theory discrete spectrum etc barring some rather trivial counter examples that you can spot with your own eyes because they are free or pathological it seems to be rather universally true that scaling variance is always enhanced to this additional bigger group of symmetry which is the conformal symmetries and here as I already said in the morning in higher dimensions in two the conformal group is finite dimension in two dimension it's an infinite dimension group and using that infinite dimension group do some beautiful art I'm illustrated here the idea of conformal invariance from using this famous drawing of Escher it's a boy in the gallery the boy looks down the gallery and the gallery contains a painting that depicts the city that contains a gallery that contains a boy etc etc so actually apparently didn't know any math but he drew this grid intuitively this is the grid that is responsible for this drawing and this grid is actually some whatever some tallest covering of the plane and the main idea here as you can see is that although distances are not preserved from if you go from the rectangle from the standard straight grid to this curve grid angles are preserved and so conformal transformations are transformations that act locally as a rotation and as a dilatation ok, so now we get to the part of the lecture that I was going to give tomorrow and of course I will have to repeat the more technical aspects of it but the main point I hope to impress already today and the idea is that we can define a conformal field theory not by giving any sort of elementary field but rather by enumerating a set of local operators again you have to keep in mind that here I'm using the notion of operator in the same sense I was using it before it's just a local disturbance of my statistical system going back to statistical mechanics perhaps again the most intuitive picture in the example I had before with a single scalar field phi this infinite set of OKs would be various monomials of phi and various gradients and various derivatives and so in that special case you get to build this entire infinite tower of local disturbances from a single object phi but that's a lecture that you don't typically have you should think that in the most general case set of local disturbance an abstractly defined set where k takes an infinite set of values it's just a label and going back to the previous example we could have for example in the I's in safety we could have this elementary spin operator sigma which is just a field phi or we could have the square phi square which measures the local energy and of course infinity many more and the most interesting property of each of these fluctuating of these local operators is the fact that the theory is getting the scaling variance so it must transform covariantly under scale transformations and the way this works out is that two point function of these local operators have homogeneous scaling with what is called the scaling dimension delta I and using the additional conformal transformation you can also prove that operators with different scaling weights are orthogonal to each other and these scaling weights are what you can directly relate to the critical exponents that I had earlier so the defining property in this abstract framework of a conformal field theory is that these operators obey an algebraic structure which is known as the operator prior to a sponge you take a disturbance I at point X a disturbance J at point Y and that can be expanded into an infinite series where you will recognize that these prefectors of the distance of the differences the difference of the distances is they are to take into account the properties under scaling with some coefficient C i J K and it's an infinite sum which however when inserted in a correlation function so the way you should view this relation is an operator relation so it holds when you replace this product of operators inside a correlation function with other operators and this sum will in fact converge with the radius of convergence which is dictated by the nearest additional operator in the correlation function so there will be a final radius of convergence that this sum will sub-converge once one of the two operators reaches the radius where at the same radius and other disturbances presence and so this is really what makes conformal field theories very special you wouldn't expect a similar algebraic structure at least now with the same nice properties to hold in a generic quantum filter but now the idea is I mean it's sort of obvious but if you know the scaling dimensions of the operators those fix the two-point functions because I told you that this is the universal formula two-point function if the operators are different this is zero and then it's clear that by taking successive products of the operators go from an endpoint correlation function to an m-1 and you keep going and everything is completely fixed in terms of the scaling weights and in terms of the structure constants cijk in the operator product's function all the way down till you reach the correlation function of a single operator but under the assumption that conformal invariance is also a property of the vacuum and not just of the operator or algebra the expectation value of an operator must be zero unless it's the identity operator because under a scaling transformation you'd find a this one-point function changes and that's forbidden and so if conformal symmetry is not spontaneously broken if you are in a situation where the vacuum respects it then you can completely process this endpoint function by taking successive opc you reach the identity and so you have completely determined the correlation function just in terms of this data and so this is a famous story so this data determines the theory and that's the old aspiration of the conformal booster which was I think actually most clearly formulated by Ferrara Gattogrid and then Paulicov had some beautiful insight in the early 70s and this story then became really powerful in two dimensions in the 80s and the idea is that this data of course determines the theory but they're surely not arbitrary they're not arbitrary for many reasons but a rather obvious one is that this procedure of fusing operators together and reducing the number of operators in the correlation function if it is to make any sense it must be independent of the order of these multiplications in terms this operator process function is associative and so I could fuse operator 1 and 2 together and 3 and 4 together and I told you that if you get different operators they are orthogonal so at the end of the day there's a single sum or I could perform the operation in a different order where I fuse 2 and 3 together and 1 and 4 and I get a different sum external operators are identical it's actually the same sum that appears on both sides it's a different sum but nevertheless this looks like an infinite set of equations that must be obeyed by this infinite set of data and that also is explained in the logo that I had in the beginning of my slides it's a girl and a boy that pulled themselves up by their boost subs and they are either in DS channel or in the T channel and so this looks like a very constrained algebraic problem in its broad outlines it's actually not too different from the problem of classifying say le algebra simple le algebra is also fixed by a set of structure constants and with suitable assumptions you can come up with a complicated catalog and so that's the dream that with suitable assumptions we could actually classify the entire set of consistent conformal field series so to remind you of the just the tip of the iceberg of this two dimensional success that I mentioned earlier in two dimensions if you assume that this height function this C central charge is more than one you get a complete catalog of unitary conformal field series these are the famous minimal models which are labeled by a single integer in which you can identify with the critical behavior of various famous statistical mechanical models so this is sort of the the blueprint the golden standard for what we mean by accomplishing this program under a small elegance set of assumptions you come up with a complete catalog and solve everything so the model booster program had a huge boost by now it's been a while more than ten years ago this gentleman wrote a beautiful paper where they pointed out that rather than trying to be too greedy and trying to nail everything at once perhaps also aided by modern computational methods what you can try to do rather than fixing the theory you can try to impose a set of inequalities on this set of data and the idea is actually rather simple because we are insisting that in having positive probability which means that these two-point functions have to have positive coefficients and it means that there is a basis you can choose where these OP coefficients C, I, J, K are real so that their squares are positive if you insist on that then this infinite set of crossing equations is an infinite set of equalities but you control the signs and so if you are clever you can tronkate them to a finite set of inequalities there is a way to organize this infinite set of inequalities by neglecting positive terms so that although you are now reduced the infinite system to just a five-dimensional system so it's less powerful and you have lost the fact that you had an inequality nevertheless with no loss of rigor you have found a finite set of perfectly true inequalities so that's the idea we control the signs and so we can turn this infinite system of equalities into a finite set of inequalities that we can now with modern computational methods make larger and larger and see what that implies and so the flagship result of this approach is a product perhaps you have seen before and it's truly a beautifully powerful and also conceptual relevant story so let's go back to the izning example so what do we know about this phase diagram of say in axial mannets of water at various values of the temperature and the pressure but we know that in order to tune ourselves to a critical point we have to tune two experimental knobs that's clear just a statement that the phase diagram is two dimensional one experimental knob is the temperature and the other in the case of the mannets would be the magnetic field we have to set the magnetic field to zero if you are to find a critical point which has infinite correlation length and so in the language I was using earlier this means that the theory has only two operators with dimension greater than three because those are the two parameters I need to control to tune my critical point I need to set their coefficients to zero if I am to be at the critical point and one of them is even under this Z2 transformation that flips spin up and down and the other is odd and so under just these assumptions we are in three dimension there is a Z2 symmetry one operator is odd one operator is even apart from the identity and they are the only two relevant operators and then of course it's an infinite set of irrelevant operators you find that this set of inequality as I mentioned earlier nail you on a tiny little island of allowed values for the dimension of these two relevant operators in fact although there's no proof and people are still discussing whether this is eventually going to shrink to zero and it does really seem that with increasing the set of these inequalities we can systematically do on a computer the island wants to shrink to zero size and so this is truly a beautiful conceptual explanation of inversality why is it the critical exponents of the universe? because it's a unique conformal field theory with the relevant symmetries and with the two dimensional phase diagram and moreover given that these inequalities are rigorous you get to compute these exponents with rigorous error bars and this method beats Monte Carlo by a very large amount by the way my colleagues tell me that the work is in progress now for the A2 model the A2 model is harder you need to consider a more complicated set of correlators and the A2 model is famous or in some circles at least because it's the only good thing that ever came out of the International Space Station was the most accurate measurement of the critical exponents in Helium superfluid which you must do in the absence of gravity and so one of the most accurate experiments ever done in statistical mechanics is this experiment done on the International Space Station which however is spectacularly falsified by Monte Carlo or vice versa so the Monte Carlo results are in spectacular disagreement about this experiment that was done on the space station and so stay tuned for what the bootstrap is going to tell you but I'm told that actually Monte Carlo is right and so this experiment done on the space station is actually probably wrong so so then we can clearly this is a very flexible tool we can use it various numbers per centimeter the kind of symmetry we want and of course my prejudices that you may have guessed from the lectures that I'm giving here are to try and do something along these lines with super symmetry so I don't really have to apologize for it there are a variety of things you can do combining these abstract bootstrap ideas with the power of super symmetry one of them is all fashion numerical results that I had that I have outlined so far you can do them in super symmetry models and we are kind of on our way to give a sort of non perturbative bootstrap solution a strong coupling for something like n-equal force-sprave you can really do that put it on a computer and derive rigorous bounds but of course it would also be nice to do is to combine this kind of abstract operator algebraic approach with the tractability of super symmetry and try to find exact results in this context and and so that's part of what we are doing so one comment is that the unlike the case of conformity which a priori can exist in arbitrarily high dimension just from symmetry reasons but as I will describe in a minute the standard will turn a viewpoint suggest that there shouldn't really be conformity theories in high dimensions for super conformity theory argument is sharper because the relevant super algebra only exist in dimension smaller or equal than 6 that's a classification result from num so already the number of possible symmetry structures and if you further impose that that you have a single stress tensor and no pathologies then you really just find a finite set of possible super algebras in dimension between 2 and 6 and and the classification here is felicitated by the existence of solvial subsetter which is a subject of my lecture such as this vertex operator algebra and so we can now look forward to to use these methods to address some of the conceptual questions I had first of all should be clear that from this abstract viewpoint duality is not an issue so duality is in the eye of the beholder you choose to take a certain semi-classical limit in which certain variables are more useful in my guess but the abstract theory does not depend on any choice of explicit description in terms of elementary fields there's one quantum theory that emits multiple semi-classical limit that's perfectly fine the abstract theory and the abstract operator algebra and the abstract operator products functions are the same in some sense duality does nothing duality in this sense is a gauge symmetry copies of the same thing and so the other the other challenge was the one posed by few that have nolagranjan descriptions such as 2,0 theory but again from this viewpoint nobody stops us from considering it as an abstract conformal field theory with the relevant symmetries and so this theory as I was saying earlier is this language dynamics on m5 brains and it is singular by being the maximally supersymmetrical quantity in the highest possible dimension which is 6 and the reason why you may be skeptical of ever finding a explicit microscopic filterated description for it is just power counting so all the explicit lagrangians that we know that are useful to describe conformal fixed points have the property that you start from in the uv from some free theory the Gaussian fixed point and you flow the infrared so all sorts of exotic things could happen but if you use the usual rules in which you must perturb the uv fixed point by a relevant operator then you cannot win this game in high dimensions so in particular if I assume that I just have a scalar field in 6 dimension this will have dimension 2 and so the simplest interaction that does not lead to pathological behavior of b5 to the 4 which is irrelevant and so you are not going to be able so easy to describe the theory in terms of some local Lagrangian that will turn a paradigm of course that could be all sorts of caveats perhaps there is a dangerous irrelevant operator perhaps your arg trajectory looks like he wants to start with an operator dimension greater than 6 but then eventually you flow down perhaps you need to relax locality at some level the game is still on but surely there are very simple power counting arguments that tell you that nothing like a standard Lagrangian description should be possible and then the other point is that by taking this theory and compactifying some of the dimensions some of the 6 dimensions to smaller space such as the Riemann surface is already what Slomo and I were discussing a little bit earlier we find a huge cave of lower dimensional quantum field theories so so the other reason why this is interesting and this is a comment that applies rather generally to many of these super-symmetric theories is that they often have large and limits the limit in which a number of these m5 brain goes to infinity in which they can be surprisingly described in terms of a theory of quantum gravity so, again, this is a separate colloquium I'm sure you have seen it at some level, so this is the most surprising dualities of all a standard quantum field theory a local quantum field theory in the sense you have a local system so you have local operators etc in flat space becomes a theory of quantum gravity in anti-desidious space and the good news is that for large n the bulk theory is weakly coupled you have just super-graviton scattering and you can think of the correlation functions so this is another picture of anti-desidious space which famously has a bound and you can think, for example, as a four-point function as some sort of a scattering amplitude in ADS7 and using this approach, actually rather recently we made some progress in which we could fix a large n, this correlation function just from symmetry without having to deal with all the messy details of super-gravity and so here is now the blueprint for how you may want to solve the 2,0 theory by booster methods but obviously you want to input the symmetry structure that we know is this maximally supersymmetric algebra in sixth dimension which is known as 2,0 algebra hence the name and then you want to input a little bit more and what you can input because we know this is indirectly from the n-theory construction from the brain construction and from a variety of other considerations is that there is a tower of operators with exactly known integer dimensions that go in fact from 4 to 2n and there are lines special short-term representation of the super-algebra and it turns out as you will see in the rest of my lectures in the special version of the same story then there is in fact a larger set of operators which I am denoted here by okai which contain this tower as a very small special subset that have meromorphic correlation function so you fix a plane R2 inside the sixth dimensional space and put coordinate z and z bar on that plane and it turns out that this special subset of operators, although a priori and both on z and z bar their correlation function only depend on z and this is the idea that you can carve out the left moving set of the 2dCFT or a vertex operator algebra inside the 2,0 theory and then now using the fact that we know that this must be there and in fact we also know that they must generate the algebra one very quickly arrives and then the compelling guess which under certain assumptions is actually theorem that this subsetter of the theory this vertex operator algebra that we have found inside the 2,0 theory must in fact be just a WN algebra which is a generalization of the Virasoro algebra where you have generators of allomorphic dimension 2,3 et cetera up to n and we can compute the structure constant WN algebra just from by knowing how the algebra acts and so we can compute exactly the correlation function of this subsetter and compare them with supergravity and so rather amazingly you find that the nonlinear structure of supergravity is captured at the cubic level is captured entirely by the structure constants of W infinity and then you can keep going down, there are a few of our colleagues who are doing this and can relate one of our n corrections of the structure constant of this WN algebra as n goes to infinity with quantum and theory correction about which of course field needle is not and then once you have accumulated this very large set of data for this protected subsetter you can really roll up your sleeves and find good collaborators and put the rest of the theory that is much more complicated because no exact methods are known on a computer and find bounds analogous to the one that I was showing earlier for the 3d ison model in this case we do not find a small island we found a golden cube or it's a smoothed out cuboid whatever you wanna call it and so what are shown here on this three axis is delta2, delta4 and delta0 those are the dimensions of the lightest spin0, spin2 and spin4 nonprotected operators that you they one that lie in long representational super conformal algebra and we have some very good theoretical arguments that show that the true dimensions of the operators are actually the one that accomplishes this corner and so rather amazingly you get to approximate numerically the conformal dimension of this nonprotected sector of this seemingly mysterious intractable 2-zero theory just by using symmetry and nothing else, symmetry and positivity of the of the norms and so this is a blueprint for a nonperturbative solution 2,0 theory, of course this is a numerical result and eventually you really like to show that this is can be done analytically also for this nonprotected sector so we really have strong indication to think that the method is working one way to argue for it we can put a numerical bound on the central charge of the theory in some units where the free theory that corresponds with single M5 brain has central charge normalized to 1 the simplest intract in 2,0 the one that corresponds to N equal to 2 brain has central charge equal to 25 and numerically we find with rather good accuracy that that is indeed the minimal allowed value and anytime that this happens, anytime that your your booster bounds are saturated then you can rather convincingly argue that the entire spectrum of the theory is determined uniquely by this solution of the busap equations and so by this numerical solution of the busap equation you can in principle determine the entire spectrum of the theory and moreover, although we don't quite find 25, but some numerical approximation of 25 this is surely an indication that somebody clever can take this set of equations and find a more analytic approach that will give you precisely 25 in the term in the entire rest of the spectrum ok, so so this I should probably really skip because it's going to be the subject of my lectures we can play a similar game in four dimensions where we will extract this kind of algebra and and finally let me conclude with a a more different idea that is in spirit within the busap realm but a busap in a different sense so this is something as I've been mentioned couple of time this morning we can get this very large class of four dimensional theories by taking the 2,0 theory on a remand surface with special points special blue punk shirts here and what as you flow to the infrared in this two dimensions you find an interesting four dimensional an interesting class of four dimensional theories which are then now labeled by these geometric choices you have made and so using the language of this morning we now get to identify the shape of this physical curve which really is a piece of space time with the conformal manifold of the four dimensional theory in fact it turns out that in this process you lose memory of the careless structure of the remand surface but the memory of the conformal moduli is kept and so the conformal moduli of this surface are identified with the conformal manifold with the space of couplings of the n-equal to super conformal field theory and now you can play a game which is kind of in the busap spirit but in an interesting way rather than boost trapping a single theory we can boost up the whole space of theories by deriving consistency conditions in theory space what do I mean by that well it means that I could for example consider various decoupling limits in which various of these remand surfaces develop long necks and I can do this in different ways and certain quantities one can argue do not depend on the way you perform this violet operation of stretching your remand surface and decoupling the blocks and for example the chiral algebra that I'm going to discuss in these lectures does not depend on the details of the conformal moduli of the surface only depends on the topology and so you must get the same chiral algebra if you consider various limiting procedures in which you stretch the surface and so in some more mathematical kind of language this means that you must have a topological quantum field theory valued in vertex operator algebra because to the topology of the remand surface associated vertex operator algebra and different ways of the composing the surface in different pairs of pencil composition must give the same vertex operator algebra and that's the idea of topological quantum field theory and so let me conclude so I've emphasized two principles one is the idea that rather than being stubborn and try to solve one theory should really enlarge your view to the space of quantum field theories and the other idea is this boost up approach using internal rules consistency rules and symmetries to charge this theory space rather than solving detail macroscopic models as the Hamiltonian of the Isome model and of course concrete models are useful still we have been trained in some cases to build the dynamical models with whatever springs and spins and whatnot where the degrees of freedom are represented by some set of frustrating fields and write a model and solve it but the other viewpoint is more general you really want to ask this meta question which is what are the allowed theories and in some cases even famous cases of experimental relevance such as the 3D Isome model this is so concerning to actually give you very concrete results and ideally again dream is to find the entire catalog of consistent theories and that's for the future thank you so questions so what was the experiment that you were referring to at the beginning in the space station oh that's the liquid helium experiment it's the O2 the transition these are the critical in statistical mechanical language these will be the critical exponents of the O2 model and my colleagues tell me that actually the experiment is probably correct but the error bars were generous sorry too small they should have put larger error bars so I was wondering when you discussed Lagrangian theories in 6 dimensions you said that 5, 4 theories and so on they were sort of irrelevant but I was wondering why you excluded 5 cube theories because sometimes that's used as a textbook example of asymptotically free theory in 6 dimensions which is also indirect I mean of course the reason of course 5 cube ok so who knows maybe one can make sense of it but of course 5 cube is unstable so nonpertor, but if you don't expect the theory to make sense there's a famous story about 6 minus epsilon dimensions which if you give a purely imaginary cap into 5 cube then this is this describes the universality class of the Liya model which is non-unitary so you can actually get they're not very good because just starting from the upper critical dimension is 6 but you can get reasonable estimates for the critical exponents of the Liya model in 4 dimensions by doing this 6 minus epsilon expansion of the 5 cube theory with the imaginary coupling just from this very naive viewpoint just from this very naive point you would not expect interacting quantum field theories above 4 dimensions so I think it was one of the surprises of the string dualities and quantum field dualities of the 90s interetim models were discovered and of course the interetim models that we can trust at the moment are the ones that have sufficient amount of super symmetry so we can embed in string theory so that we are relatively confident that they really exist the good question still very much open whether there are non-super symmetric quantum field theories in interetim quantum field theories in the sense that I've described full-fledged consistent conformal field theories that you can then perturb in 5 and 6 dimensions and it's another excellent question whether there are any necessarily non-super symmetric because of the non-classification that I mentioned conformal field theory dimension higher than 6 this is an open question that the booster should one day solve whether conformal field theory exist in high dimension we don't know naiv rg counting would tell you that you're going to have a hard time Is there a classical limit for theories with non-lograngia? Sometimes if you have some parameters you may some of these series come in with continuous parameters in some cases you can you can tune it like the picture I had this morning where you have this conformal manifold and you have to a cusp and perhaps that point has a weekly couple description which is a semi-classical description or you can have a discrete family the canonical examples are these large n dualities where in the large n limit the theory has a weekly couple description in terms of the sft that is also a semi-classical limit but in general no why would you so I will mention later this week in 4 dimension he is 6 global symmetry no tunable couplings that theory shouldn't have and does not have any semi-classical limit ok good maybe we are in order not to run out of time so let's thank Leonardo again