 Tako, počekaj smo pričočati. To je najbolj, da je najbolj by Leonardo Rastelli, kaj je pričočiti o karalalgebra programu za 4-dimensionalne, superkonformalne teorijs. Tako, počekaj. Vseč nekaj, da je 4-dimensionalne, konformalne teorijs, nekaj, cft, in 2-dimensionalne karalgebra. Or, as mathematicians would prefer to call them vertex operator algebras, I will use the two names interchangeably. So, it's a by now rather rich and complicated story and it will be impossible for me to completely do adjusting for lectures. And although I was told that this is supposed to be a very advanced audience, I didn't quite, perhaps we should wait, people are still coming in. I didn't quite had the courage to assume that you already know everything about n equal to two field theories. And so I will start today with a rather pedagogic and elementary overview of n equal to two super conformal field theories in four dimensions, partly as a way of motivation. So, our work, so I can say it in a slogan, so we have this rich landscape of four dimensional theories with n equal to two super symmetry, it's a growing list. It's in a way much richer than the maximum super symmetric case, which is n equals to four, which is more constrained, but still has sufficient structure to allow remarkable analytic progress. And much progress has been done, but what we will describe is a particular sub-sector of these complicated theories, which turns out to be isomorphic to something very tractable. So, you're familiar with chiral algebras, chiral algebras are the purely holomorphic left moving sector of a two dimensional conformal field theory, so meromorph is very powerful. And this correspondence has many uses, as always often is the case with this kind of correspondence or dualities, the error both is powerful both ways. We can use the fact that the bottom line is tractable and has infinite dimensional amount of symmetry to find interesting results, interesting result for dimensional theories. And we can use physical intuition about four dimensional theory in the other direction to formulate mathematical results about two dimensional chiral algebras, which you can then hand to mathematicians and they will prove theorems. And perhaps the most ambitious goal, which really is my current focus and this will be, ambitiously, the punchline of these lectures is to use this correspondence as an organizing principle for the whole landscape of, so the idea is to use this as an organizing principle and perhaps even as the beginning of a classification program. Ok, so, this is all great, but I think it would be a little premature for me to jump into the structure without any motivation. As a way of motivation, I will describe today in a really lightening way the basic structure of n-equal 2-2 theories in four dimensions and I will do it from, so the outline for today will be a review of n-equal 2-2 super conformal field theories and optimistically I will start with basic stuff, such as we will start with baby steps, familiar vanilla type of n-equal 2-2 theories, I have a Lagrangian formulation and then I will promote a more abstract viewpoint based on super conformal representation theory and I will tell you how the algebra of local operators is useful, that will be my basic organizing principle and try to connect that also with the geometry of vacuum branches. Ok, so that's a little ambitious, but we need to start somewhere. Ok, so, I assume the basis of supersymmetry, I just want to remind you of the basic elements, so there are two types of supersymmetric multiplets that we will consider, we will consider the n-equal 2-2 vector multiplet, which in familiar component language consists of complex scalar phi and two gauginos lambdas, so phi is a complex scalar, so I will also use the notation lambda i alpha where i goes from 1 to 2, so, just to, ok, so, again, I need to assume you know the basics of supersymmetry, let me write the supersymmetry algebra in this. So, we have two copies of a Susie algebra, so the index i is an SU2R index, so this is the art symmetry of the theory, the full art symmetry of the theory is really a U2, which we will decompose into an SU2 times a U1R, and my notation throughout these lectures will be very consistent, I will always denote with capital R the non-habilian art symmetry and with small R the abelian one, and also by abuser notation I will often denote by R the carton of SU2R and by little R the generator of U1R, and those of you who are familiar with n-equal 1 supersymmetry in terms of n-equal 1 multiplets, we can decompose n-equal 2 vector multiplets into n-equal 1 vector and into n-equal 1 chiral super fields. Ok, so, it's clear that if you have not seen this before, this is going to be meaningless, so you are just doing it as a way of recap notation and also because I will need some of this notation to write down a few examples. So, this is the vector multiplet contains the gauge fields, which I am writing here by spinorial notation, and so necessarily by the general principle of quantum field theory the moment you have a massless spin one object you must have gauge symmetry, this is a famous result from the old days, and so necessarily the n-equal 2 vector multiplet has in this app now written down, it transforms in the a joint representation of some group G, which is going to be the gauge group. The other multiplet I am going to write is going to, well, let me do it in a slightly more precise way than it is often done, I am going to write the n-equal 2 to half hypermultiplet, which will be a, let me try to be precise, it's going to be again a complex scalar q vile fermia, I am going to write psi alpha, and cpt conjugation does not leave this multiplet invariant, there is a whole discussion that I need to shortcut a bit, but if you look at the representations of one particle states, which hopefully you are familiar with that story where you construct, you know, you go to the, you look at massless representations, you construct this cliffer vacuum and combine different representations in such a way that they are cpt self conjugate, it almost looks like this one is cpt self conjugate, but that's not quite the case, and the reason for it is that the, necessarily we will find a pseudo real representation of SU2R, the fundamental representation of SU2R, so in order for this object to make sense, really one must consider a whole pseudo real representation under some other group, let me call it G, it doesn't have to be necessarily the same as the gauge group, there has to be some additional, so this is forces, this guy to be in a pseudo real representation, in some pseudo real representation are pseudo real of G tilde and then the whole thing can be made, cpt self conjugate. Ok, so for those of you who are slightly puzzled by this, so the more, the most familiar case of this situation is one where we consider a full hyper, which will consist of two half hyper, a half hyper you will have recognized, it's really the same thing as any whole one, kind of super filled, and we are going to consider a full hyper where we consider this in representation R and this in representation R star, and then in the R times R star there is a natural pseudo real, but the most general situation is the one where I align there, where we are not necessarily assuming that we have two different copies in two complex conjugate representation, we may have a situation where we have a honest pseudo real representation, which cannot be written as a direct sum of two complex conjugate representations. Ok, so, with this data you can now start constructing Lagrangians, and again, I'm not going to really be able to do, I mean truth be told, what people usually do is write even n equal to Lagrangian n equal to one super space, and then the covariant ties with respect to the non abelian arts symmetry, that's still the most efficient way to do it, because n equal to two super space is a bit cumbersome. There's a nice simple construction, so this is a side note, n equal to two super space is naturally and easy for the n equal to two vector, but it's a bit hard for the n equal to two hyper or half hyper, because it requires an infinite number of auxiliary fields. For the vector there's a nice simple way of doing it that I just want to mention briefly. You introduce Grasman coordinates theta i alpha, where i again is one to two, and alpha is my index, is my vile index, and then you write down the full super field, again by abuser notation, you write down a full chiral super field in this n equal to two super space, so it depends only on the thesis and not on the theta bars, and the expansion of this guy will contain the gauginos that we saw earlier, and then the theta square component you will find the gauge field, and then there will also be a piece that contains a triplet, an s2 triplet of auxiliary fields. I mentioned this briefly because it will allow me to save a little bit of time later. In the more familiar n equal to one notation, this triplet of auxiliary fields will be written, so in n equal to one notation be even by the familiar f, f bar, and the auxiliary fields of the chiral super field, and of the n equal to one vector super field. Yes, of course, I'm sorry. I was just a typo, sorry. Ok, so, again, I'm sure cutting this whole discussion, but let me just assert the following that if we, at the classical level, at the level of writing down classical Lagrangians, if we insist that they are classically scaling variant, then if you fix the matter content, which means you fix the gauge group, and you fix the representation under which the half hyper transform, then the Lagrangian is unique up to choice of the gauge coupling. Ok, so, the basic data of n n equal to two Lagrangian are choice of g, which a priori can be a product, where each gi is a simple vector, or at this classical level it could also be a u1, and then a choice of a representation r for g, which by the quick, not really argument, but the quick assertion I had earlier, must be should be real. Ok. So, the simplest, and just to belabor the point I made earlier, that the simplest and cheapest way to construct a shoulderier representation of g is to take r to be the direct sum of, let's call them, well, let's call them r prime of two representations r prime in its complex condition r prime star, but more general possibilities are allowed. And so, these are the discrete data, and then it turns out that n equal to two supersymmetry fixes the Lagrangian uniquely. For the vector multiple, that's very easy to believe, because you just take the familiar Lagrangian for n equal one vector multiple, it's in your covariantize with respect to SU2, remember we had two lambdas, lambda one and lambda two, you covariantize the n equal to one Lagrangian, you get the right answer. In other way of writing the Lagrangian for the vector multiple is to write it using this simple superspace, which you can do easily. The construction of the full Lagrangian including matter is a little bit more involved, but a cheap way of saying what you need to do, you do minimal coupling. OK, so you have the familiar story in where you want to couple the gauge field minimally to the nether current under the symmetry, and you want to write down an n equal to two SUSE version of minimal coupling. And that can be done, although the details in superspace are a little bit involved, so I'm not gonna write it there, write it down, but what you are doing in simple terms, you are, and we will come back to that in a minute, because it will, right now I'm unfortunately painful, it's slowly just a view in basic Lagrangian territory, and I will graduate soon to a more abstract presentation. The way to do it is to recognize that the conserved current really belongs to a full super field, so we're gonna view the conserved current j mu, or in the notation I had earlier, j alpha alpha dot, as a member of a super field, and this super field, again, you take my word and later we will do this in a little bit more detail, as top component, or according to how you count, bottom component, a element, which has dimension two, and it's an SU2R triplet, and then there will be some fermions, which I'm not writing down, and in the next level you will find, when you add with two supercharges, you will find the current and also a complex scalar. So in n equal to 2 theory you write j alpha alpha dot as the action of two supercharges, two Poincaré supercharging, acting on this bottom component of the super field, which for reasons that come from geometry and that perhaps will become mere to some of you and later on we will quickly review what is also known as the moment map operator. So once you have recognized that the current you want to count minimally to the gauge field has this nice n equal to covalentization, what you basically want to write down is some souped up version of this where you couple the full n equal to 2 vector to this full fledge, let's call it mu, to conserve current superfield, and this fixes uniquely the coupling of the matter to the gauge field. Under which assumption, so whether on the assumption that the theory is classically scale invariant, so I'm not allowing for mass terms, and as always is the case, each simple factor of the gauge group comes with the choice of complexified gauge coupling in the standard. So to summarize, to write down a classically scale invariant n equal to 2 Lagrangian, you choose a product gauge group, a should be a representation of this product gauge group, and for each simple factor, a priori one factor, you choose a complex parameter tau, which lives in the complex upper half plane, because the coupling is assumed to be, of course, real. Now, as we graduate to quantum field theory, various choices are possible, and it is certainly allowed to consider general, much more general type of structures if you are interested in viewing these Lagrangians as low energy effect field theories, but my focus here right now in writing down UV-complete models that make sense as you stipulate them to short distance. And so, you will recall that you won famously as a positive beta function, and so it's not well defined in the UV, so we will insist. So in writing n equal to 2 quantum super conformal field theories, we will dismiss the U1s, and so then these are the data, the same discrete data as before, and then a set of continuous gauge couplings. Now, it turns out that nonperturbative dualities will carve out generically a smaller subset of these complexified gauge couplings because there will be discrete identification in this space, so that theories related by the discrete identification are in fact physically equivalent. So the complexified gauge couplings a priori should be modded out by some duality group. Let me give you an example. N equal to 4 superion mills is a special case of N equal to 2 theory, and there we have a single non-abelian gauge group G, and we take R to be the a giant representation of G. We have a single complexified gauge coupling, but famously there is a SL2z invariance that restricts the physically distinct values of tau to the fundamental domain of SL2z. In that case, duality group is SL2z. Since I need to pace myself, I want a quick raise of hand. Who among you already knew everything about this? It's actually not... It's less than half, so I'm glad I'm not completely wasting your time. OK, so the next step if you are interested in conformal... OK, so I already made an assumption that I want to exclude the U1 factors because they lead to inconsistency in the FIUV and I'm trying to write down something which is a fundamental Lagrangian description, not some Lagrangian description. Of course, those of you who are familiar would say it's a Beewittian theory. You know that there's, of course, very much beautiful use of U1 gauge fields. If you view your Lagrangian as in a derivative expansion, as in a Lagrangian effective action, and then, of course, a Lagrangian is a lot more complicated than what I wrote down. I should have also emphasized. When I said N equal to 2 super similarly fixed Lagrangian uniquely, it fixes the... There are various ways of saying this. It fixes the renormalizable Lagrangian uniquely, or if you assume that you have classical scaling variance, of course, all the higher irrelevant operators are forbidden, and so you have a unique structure. But if you don't make this assumption, then you can have a complicated derivative expansion and then N equal to super similarly only relate some terms, but it does not fix the Lagrangian uniquely. If you are now interested in promoting the story to a full quantum story, we need to make sure that the scaling variance is preserved. Remember, our goal is conformal field theories. We are in particular scaling variance, so we need to make sure that the beta function is zero, and there is a beautiful implication that happens here. It is sufficient to check the vanishing of the one loop beta function. So a quick heuristic argument for the vanishing of the higher perturbative terms comes from holomorfi. You know that N equal to one theories have a holomorfi scheme where only the running is one loop. That holomorfi scheme, however, introduces wave function normalizations for the terms that appear in the Keter potential, and so it's not the usual physical scheme where the Kira superfields are chronically normalized. But thankfully N equal to two supersymmetry relates that piece of the Lagrangian with the better piece of the Lagrangian, so you know that that is actually the physical scheme in which simultaneous you can have holomorphic and absence of the wave function normalization, and then holomorphic, remember tau was this combination for pi i over g square plus theta over 2 pi, holomorphic allows you to quickly conclude that you have loop correction beyond one loop because they would have to depend on the imaginary part of tau, and so they would not be holomorphic. So we simply need to impose the vanishing of the one loop beta function, which, of course, for the u1 factors, that could never be possible, and that's why we discarded them. And then for each simple factor g i in my product gauge group, so let me not put the index i to avoid cluttering, there will be a negative contribution that comes from the better multiplet. A-check is the dual-coxater number of the particular... I'm writing the full gauge group as a product, and then this is the dual-coxater number of the particular factor g i for each i I'm doing this, and then you get a contribution from the half-hypers, which turns out to be the index of the representation heart. This is the notation, which is proper for half. So, for example, if I take g to be SUn, the number is n, and I'm using conventions where the index of the fundamental representation is one-half, and the index of the adjoint representation is n. And so now we can check familiar examples of n equal to two Lagrangian field theories. So, I wrote that formula for half-hypers, and I'm sorry for being a little bit pedantic about half-hypers, but I'll tell you a little secret. So people claim to have classified all Lagrangian n equal to superconform of field theories, and it turns out they had made a little mistake. And this is the last one, because I had not been careful about this basis of half-hypers, so it's actually something that one should always keep in mind. So I wrote this for half-hypers, of course for full-hypers, which is the simplest situation, then we would just put it twice there, obviously. And so if I take g equal SUn, there are two familiar ways in which you can satisfy this condition. You can either have nf equal to n fundamental representation, you can just check that this works out, minus two little n plus two times two n times one-half gives you zero, or you could have one adjoint, and this is what we would call n equal to two superqcd, and this is the case in which, in fact, the super symmetry enhances to the full n equals to four. OK. So another little exercise you can do is to convince yourself, which can be done by specific calculation, that the one loop coefficient of the beta function also controls the anomaly of the U1r. OK. So remember, classically we have this SU2r times U1r, if I don't break the classical conformal invariant by my terms, this is the true symmetry of the classical Lagrangian, and this is manifestly preserved by quantum effects, but the U1r is broken by the anomaly, but the anomaly is actually proportional to the beta function. This is something you can quickly do if you recall that under a chiral rotation the measure for vile fermions transforms under something which is this times k, where k is the instanton. And so if you look back at the matter content of the n equals to vector multiplied, there were two gaugenas, who are careful about little arc symmetries. Well, I didn't say that, but let me say that. So remember we had this, we are going to assign r equal minus one, r equal minus a half, and r equal to zero, whereas for the half hyper, this is our assignment, and with these assignments you can convince yourself that the anomaly precisely cancels when the beta function is here. So in n equals to two theory with vanishing one loop beta function, first of all the beta function vanishes to all orders, so the theory is truly scaling variant, you have an exact SU2r times U1r symmetry, but then let me assert without further ado that so we have scaling variance, d, we have arc symmetry SU2r times U1r, and then of course we have Poincare, and as is familiar, the scaling variance and the Poincare invariance will enhance to the conformal algebra, so I should perhaps write it here, the Poincare symmetry, so we have Lorentz generators, and we have momenta, and these will enhance to the full SU4,2 or SU2 slash 2 conformal algebra, and in fact we have super symmetry, okay, so let me just write it, and then I will comment on it, there will be an enhancement of the superalgebra in the full-fledged superconformal algebra, which almost each of the same as I make would require a separate lecture, so I need to take it for granted that you know what the conformal algebra is, so the additional generators that you find here are the momenta, now come with partners, which are the special conformal generators in spacetime dimension d, special conformal generators, and then if you have Poincare supercharges, they come accompanied by their conformal analogs, which are conventionally denoted by s, and furthermore the full structure will close on the right hand side of the commutator of q and s, you will also find generators of the full art symmetry so that the whole superalgebra is, in fact, s2,2 slash 2, okay, I'm going to discuss the structure superalgebra in a little bit more in detail in a minute, but that's the story, so questions about this. Okay, I need to be ruthless, because otherwise I'm never gonna get to the heart of the matter. So what are the features that I'd like to emphasize? First of all, you can now solve a fun combinatorial hazard size, of which many aspects were known a long time ago, but there was a nice paper by Tashi Kava and collaborator who did a systematic analysis missing one example, where you can take this discrete data, which is the choice of the product gauge group and the choice of the pseudo-rear representation of the gauge group and this condition that the one-lubita function must vanish and classify all possibilities for n equals to superconformal Lagrangians. That's a fun hazard size, I'm not gonna go into it, but let me just mention some prominent examples. One thing you can do just for the fun of it is to assume that the gauge group is a product of SUN factors and further assume that the matter comes into bifundamental representation of any two of these SUN gauge groups, and then you will probably have encountered this fun quiver notation where we have, sorry, I missed your lecture, perhaps you had quivers, where we have a little blob with an N1 here and then we have an N2 here and this means that we have an SUN1 and this means that we have an SUN2 gauge group. It's a way of encoding the structure of the gauge group and that we are connecting them by a line means that we have a full hyper in the bifundamental representation of the SUN1 times SUN2. And then the question is what kind of quiver diagrams you can draw that are conformal. And the answer turns out to be very beautiful. First of all, okay, I don't have, it's a fun hazard size to do in detail but I really don't have the time. One thing you can do, well, you can easily convince yourself that you need to ensure the vanishing of the beta function at each node. So the beta function for this node gets, remember the formula, it was minus 2hv plus 2c2r in the notations where I'm using notation for full hyper multiplets and you will remember that the SU2 of the fundamental is equal to 1hv. So from here we get for, if I focus on the contribution of the function from this gadget, I will find minus 2n plus 1hv times 2 and then I need to count how many hyper multiplets I have but I clearly have n from this side, n from this side because from the point of view of the symbol symmetry and n from the other side so I get 2n and this is happily zero. And so clearly I can't keep going except that, of course, I don't quite know what to do at the end and the simplest way to make this work is to just connect it back and write down a sort of circular river diagram with all SUn groups of equal rank. The same would be to end this story here. OK, let me call this 2n would be at the end in this way and then there are a couple of three more exotic possibilities but the statement is that if you assume that there are no flavored symmetries and if you assume that the gauge group takes a product of SUn and factors, all the possibilities are classified by drawing the affine dinking diagrams for ADE. So there's this ADE affine classification. OK, so this was a bit quick but perhaps you've seen this before. Another fun example I could do would be the... Well, let me not draw it. So this is the A k hat where k is the number of nodes. This would be the dk and then you can go to our colleague Wikipedia and check the dinking diagrams for E and... Anyway, these are all very nice examples of n equal to 2 super conformative theories in four dimensions. So this is also a little bit more elaborate where you assume we can improve our set of possibilities. We can enlarge our set of possibilities by assuming that we can also have hypermultiple which are charged so this would be a hypermultiplet which is in the bifondamental of this gauge group SUn but it also transform the mental representation of some additional symmetry group SUk which, however, is not gauged. So the standard notation is that we use circles to denote gauge groups and boxes to denote flavor groups and so you can now enlarge your set of possibilities by decorating these quivers with these kind of boxes and I will leave you with the answer and then you can have some fun trying to argue for it so if I have let me call this v if I consider a quiver of this type with... in principle now I can have I can decorate it with additional flavor symmetries of this kind then the condition that this be conformal at each node is this one where Cij is the Hartan matrix so ok, so this is just a very quick list of possibilities just impress upon you that there is a huge class of just Lagrangian theories and the once you have spent 10 seconds to be impressed by that I'm gonna graduate and tell you that this set of Lagrangian theories of which I just enumerated a few examples by drawing quivers are in fact just a tiny subset of what we now believe to be the full set of any kultutu super conformal filled theories and how do we know that? Well we know that a bit indirectly the simplest and perhaps from what I've told you so far the easiest way to get to that conclusion is to try to take the strong coupling limit of any one of these Lagrangian models and to realize that there is a duality transformation that will require the existence of another super conformal field theory that cannot be written in the language that I just described so far another way of doing this of course which is also related is by constructing more mysterious that cannot be written in this language from various decoupling limits of string theory or amp theory so the emerging picture is the following so now we are getting into the more conceptual territory where I want to eventually go for the rest of the lectures which is to try to delineate the full space of any quantum super conformal field theories and the emerging picture is the following so here we had a nice separation between what you may want to call matter which were the hyper multiplets or the half hyper multiplets and gauge fields and the continuous parameter of the theory the compensified gauge coupling were purely carried by the gauge field so I'm going to speculate that this is so the conjecture the full space of n equal to 2 d equal to 4 super conformal field theories arises by gauging of isolated super conformal field theories of which the free hyper multiplets is a simple example one can make it more precise but the idea is that you have exotic type of matter you have ordinary matter which is the hyper multiplets those we understand because you can write down a free field Lagrangian for it and then you have exotic matter which is characterized by the idea that there are no continuous parameters this parameter tau that I had earlier is from the point of view of conformal field theory is what would be called an exactly marginal deformation it's a deformation that preserves in this case the full super symmetry but in fact more to the point the full super conformal symmetry in some sense call, exotic matter a theory that has no exactly marginal deformation it's a isolated fixed point of the renormalization group and the only such theory for which we can write down a Lagrangian model are the free hyper multiplets and these other more mysterious type of matter theories should be understood as some strongly coupled isolated fixed point that we must describe more abstractly but the idea is that we can then treat as an abstract beast and introduce parameters by gauging some subalgebra of the global symmetry algebra of these theories and this is the only way in which you can get exactly marginal deformations or to say differently the picture of the space of exactly marginal couplings of a general n equal to 2 super conformal field theories looks a little bit like this where this cas point corresponds to points where the gauge couplings at least one of the gauge couplings in my private gauge group is going to zero and at this point you will decouple a piece that has no marginal deformations and this is clearly a special point Am I making sense? So there are familiar examples of this story where one point is perfectly described by Lagrangian tools you have a bunch of hyper multiplets coupled to gauge fields and then you go to some other point where you recognize that in that limit some other gauge group is becoming with the cup and what you are left with is a product of perhaps familiar stuff like hyper but also unfamiliar stuff that has no obvious Lagrangian description and the full set of theories is obtained by gluing together these elementary pieces making sure that the beta function is zero and so really one of the main goals of our abstract considerations when we got the way to this chiral algebra correspondence is to be an understanding of these strongly coupled pieces Ok, so what next? So since I mention this let me also put a little bit of more equations so remember I wrote down this abstract I wrote down this abstract the way in which I was describing the minimal coupling of the gauge field to some conserve current and you recognize that if my picture is correct I get to treat the gauge fields using a conventional type of Lagrangian but it might be that the what the gauge field coupled to is something that does not have a simple Lagrangian description nevertheless I can think of this moment multiple that contains a conserve current abstractly and so we'll be coupling the gauge field to some conserve current of my strongly coupled piece and then I can rephrase the vanishing of the beta function in this purely abstract way by declaring that my conserve so this will become a little bit more clear once we graduate to confirm a representation theory but the idea is that the strongly coupled theory may have as a certain symmetry algebra and corresponding there is a conserve current associated with the symmetry algebra and I can define the level abstractly in some normalizations that I need to be write rather quickly because I don't have time to get into all these details there's a way to normalize the currents that so a conserve current has dimension 3 and so you see that dimensionally this works and the non-abelian structure constant fix uniquely this normalization so I have then meaningful data is contained in the two point function and this is the level of my four dimensional current algebra and then in those conventions you can write down the condition for the vanishing of the data function in this more abstract way making sense this of course we reproduce the condition that I had earlier for in the case where I can think of the current as a composite field of free hyper multiplets but the more general subscondition is this one and so this is the first baby step towards this sort of more abstract view point that we want to take we want to think of this theories as define abstractly in terms of a set of local operators and their operator prowess functions and then even something as pedestrian as a vanishing of the beta function can be set in that pure lepser language ok, so how am I doing with time? ok, so ok, so this is about half of what I want to cover but I will have to be a little bit rootless next time