 Okay, let's start. So the first speaker today is Kenny Trilligator from UC San Diego. And he's going to talk about aspect of five-dimensional, six-dimensional super conformal theories. Thank you. I'd like to thank the organizers for the invitation to come back here. I think this is my seventh time to TRIES, so it's always great to be here. And I'd also like to thank my collaborators, Clay Cordova, Thomas Dimitrescu, Nadie Seiberg, Dave Morrison, and I'll apologize in advance for admitting many references. So a lot of the talk will be kind of following somewhat things that I've worked on myself. Okay, so for today's lecture plan, I'll start off with slides and then switch to the chalkboard. So I'll discuss some motivation and overview, some tangential details. And in the first lecture today, I'll introduce some unitarity bounds and group theory for conformal field theories and super conformal field theories. One didn't get very far using just group theory, so I thought in the first lecture I would discuss some of the group theory. And then in lecture two this afternoon, I'll introduce some of the strength theory and brain constructions of 5D and 60 theories. Okay, for some motivation, we can ask what is quantum field theory? These are some nearsighted physicists trying to study quantum field theory. And when we first meet quantum field theory, we study it using perturbation theory around free field Lagrangians. And so this works very well, for instance, for the standard model. But in that way, you can only explore part of the space of quantum field theories. There are other theories that we can study. For instance, we could start with conformal field theories or super conformal field theories, which might not be close to any perturbative description. And nevertheless, they can often be analyzed by using, for instance, bootstrap methods or other kinds of techniques. And then once we understand the conformal field theory, we can perturb the conformal field theory to explore more of the space of quantum field theories. So for example, the topics of these lectures, 5D and 60 super conformal field theories, these theories aren't in any way perturbations around free field Lagrangians. If you start off with free field theories in 5D and 60 dimensions, you just get infrared free theories. Nevertheless, they're interacting 5D and 60 super conformal field theories. I put this F in capital letters, but not for F theory necessarily, but because these are field theories, despite being somewhat exotic, and sometimes they're referred to as non-Lagrangian because they're not close to any free field description. And it's possible that some of these non-Lagrangian theories or maybe something else unexplored will be crucial for the future. Okay, so just to give some kind of picture of renormalization group flows, the idea is that we can start off with some ultraviolet conformal field theory and then perturb it by some relevant operator and then flow down to some infrared conformal field theory. This reminds me a little bit of this game that I used to play before as a kid called shoots and ladders, where the irrelevant operators are like ladders that you can use to try to climb up, and then the relevant operators are like shoots that take you down to lower energies. So for example in the standard model there are two relevant operators, the identity operator and the Higgs mass, and so the fine-tuning those operators are the two fine-tuning problems. And then these irrelevant operators we can try to use to explore things beyond the standard model. This is some picture of the renormalization group flows as kind of flowing down. It's flowing down in the number of degrees of freedom. So we start off with some ultraviolet conformal field theory and then we add some relevant deformation and then do some coarse-graining and get some infrared conformal field theory plus some irrelevant operators if we want to go a little bit away from the infrared fixed point. And these kinds of deformations could be for instance various operators in the theory. And here I put delta L in quotes because this is okay even if the theory is not Lagrangian. So the idea is that if we can understand what are these operators we can start off with the conformal field theory. We could look at for instance what are all the relevant operators. Once we know what are the relevant operators then we can turn those operators on and try to see where we go and then we can explore more of the space of conformal field theory, more of the space of quantum field theories. So the conformal field theory is the starting point and then we move away from there by these kinds of deformations. There are other deformations that will be useful to discuss. One is that's especially useful in supersymmetric theories is moving on the modular space of supersymmetric vacua. This is one of the aspects of supersymmetric quantum field theories that makes them so attractable is that they have this space of vacua and you could move around on this space of vacua and sometimes connect it to some limit that you understand better and then try to go back and gain some insights. Another way to modify the theory is to gauge a global symmetry. Okay so just to give an example in four dimensions we could start off with non supersymmetric SU engage theory with F massless flavors and then depending on F there are various different possibilities. So if F is bigger than 11 halves times the number of colors then the beta function is positive near recoupling at least so the theory is not asymptotically free. Not asymptotically free in the ultraviolet means it's infrared free. So if we start off with this theory and we go to low energies so we have g something like that and the theory wants to flow in the infrared to zero coupling. Okay and in the ultraviolets we need we need either some kind of cutoff or we could UV complete it in terms of some other theory or in some cases the theory is ultraviolet safe like if this beta function eventually comes back down to zero then there could be an ultraviolet fixed point. There are some examples of that. So that's if the number of flavors is bigger than 11 halves times the number of colors. If the number of flavors is less than that but bigger than some n star let's call it this is what's called the conformal window. In this range what happens instead is that the the beta function as a function of g looks like that and the beta function can have a zero and so that's some conformal field theory in the infrared with some non-zero g star and the lower limit of this is is some subject of interest that that people have studied using various methods and then if the number of flavors is less than that critical value then there are ultraviolet so in the ultraviolet we still have free quarks and gluons but the infrared instead there's confinement chiral symmetry breaking and infrared free goldstone boson pions. So this is kind of an example of a duality where the theory the ultraviolet is free and the theory the infrared is free but it's a different theories. There was some recent progress in this paper by Galletto, Cyberg and Kormogatsky for recent insights including connections to 3d quantum field theory dynamics and dualities on the domain wells. Part of why I wanted to talk about this is that it's just to illustrate that in general we're unable to follow actually this renormalization group flow in detail like here we're unable to follow it from the ultraviolet to the infrared of showing that there's confinement this is one of the million dollar clay prizes to show that and so no one's collected the million dollars yet but nevertheless it's often possible to use basically to guess the answer and to use symmetry constraints to check to do some non-trivial cross checks and to gain some confidence if the answer is right. Okay so there are various constraints on the renormalization group flows. One is that if the theory has non-trivial etouffe anomalies so I'll be discussing various examples with etouffe anomalies and discuss more in detail what they are then there's a powerful constraint which is that they have to match they have to be constant on renormalization group flows so this is the condition of etouffe anomaly matching. So that's one constraint another constraint is the intuition that renormalization group flows reduce the number of degrees of freedom so one count of the degrees of freedom in two four and I believe also in six dimensions is the conformal anomaly a so if we look at a conformal field theory the trace of the stress tensor should be zero but if the theory is put on a curved background it could be non-zero and these coefficients are the conformal anomalies these were these were first studied by Duff and he he has a nice review about these conformal anomalies where he talks about how he was a student of salam and some of the the controversy originally surrounding these conformal anomalies anyway the conformal anomaly a is this coefficient of the Euler density so this is something that you can build out of the curvature tensor in even space time dimensions and in odd dimensions there's an analog of this using the sphere partition function and the entanglement entropy and then there's additional power from supersymmetry because for instance operators have to form super multiplets and the anomalies have to form super multiplets and this can relate for instance ethof anomalies to conformal anomalies that's something that I'll discuss in six dimensions in the third lecture is the connection between ethof anomalies and the conformal anomaly okay so I thought I would discuss a little bit first some of the uniterity bounds and the structure of conformal field theories so the idea in a conformal field theory is that we have a bigger symmetry than space time than the usual Lorentz symmetry so if we're indeed in space time dimensions if we think about the theory in Euclidean space we could say that there's an SOD symmetry which is the Euclidean rotations and Lorentz transformations and then we could combine that with the translation symmetries and with the dilatation symmetry and the special conformal symmetry to get a bigger symmetry group which is SOD comma two so in d space time dimensions there's an conformal field theories have an SOD comma two symmetry and the operators form representations of this symmetry and the way that the representations look is that there's some operator at the bottom of the multiplet which so here the raising operators are the momentum generators which are like derivative operators and then the lowering operators are the special conformal generators k mu and so this primary operator is annihilated by k mu at the origin and then we form descendants by acting with p mu and if the theory is unitary there's a condition which is that well by using the operator state correspondence we can form an inner product of these so the inner product is basically related to the two point function of two operators and that if the theory is unitary that's inner product has to be positive definite so the norm of the operator has to be positive and it can only be zero if the operator is zero and so then we can impose that condition not just on the operator but on its various descendants so like the p mu acting on the operator has to satisfy this condition that it's bigger than or equal to zero and it's equal to zero if and only if this operator here is zero and then using the algebra this leads to unitary constraints so here d is the dilatation operator and mu nu is the rotations and so you can relate you can put bounds on the dimensions of operators and related to their spin and when these inequalities are saturated then some operator is zero and so those those operators are referred to as short representations so examples of that are conserved currents if we have a conserved current operator then some descend into zero and and so that's an example of a short representation and these representations are protected in various ways okay so just to mention some of these unitarity bounds for scalar operators in d spacetime dimensions this is the bound so it has to be bigger than or equal to d minus two over two and if this inequality is saturated then the operators of free operator the scalar operator satisfies it has a level two descendant which is basically just like saying it satisfies a free field clangorten equation and such operators have to have for instance boring operator product expansions because it's a free field operator another example is if we look at an operator which has s lorenz indices that are symmetrized and traceless then the dimension of that operator has to be bigger than or equal to d plus s minus two and these are conserved currents so for instance if s is one then this is just like a usual flavor current so its dimension if we saturate this its dimension is d minus one and it satisfies a conservation law if s is two this is the energy momentum tensor so the dimension is d and it satisfies the conservation of energy momentum for s bigger than two such conserved operators are believed not to exist unless the theory is kind of a free theory and so usually we only have these operators in interesting theory for s equals one or two let's see so just to write down in five dimensions what these inequalities look like in five dimensions the lorenz group is s o five and i could write down representations of s o five using this this notation of two quantum numbers j i'll call them j one and j two so like in four dimensions we have two quantum numbers like j and j bar and in five dimensions because those are like the s u two left and s u two right quantum numbers in five dimensions we have quantum numbers lorenz quantum numbers j one and j two and so if an operator has and j one is just just to give some examples if j one is one that's a spinner and if j two is one that's a vector those those are the two main representations of s o five so if we have a representation with j two bigger than one and any j one then the then the uniterity bound is that its dimension has to be bounded by this thing so it's a half j one plus two j two plus three i put a little start by that three just to remind myself that um so this is if j two is bigger than or equal to one if j two is zero instead of three it's two and so there are various things like this where for small quantum numbers some of these bounds are slightly different but anyway if we look at where this is saturated uh for instance this is this is saturated if um if this operator which is a level one descendant has zero norm okay so so just to give an example of this a flavor current is in the vector representation so the vector representation or i write a zero four or sorry zero one and its dimension is four so that that's a conserved flavor current this is a conserved energy momentum tensor and we could check that that these are the right quantum numbers of the divergence and the supercurrent is in the representation one comma one so it's like a spinner it has a spinner index and a vector index and its dimension is four and a half so that's what we get from this uh this is this is another short multiple well so this is another bound that if the quantum number j two is is zero and uh and if we have general j one then well for instance for j one equals one then the inequality is that the dimension has to be bigger than or equal to two and this is saturated for a free fermion okay in um six dimensions i thought i would also just briefly mention some of the uniterity bounds and the group theory so in six dimensions we can write if we go to euclidean space the lorenz group is s o six which we could think of as su four and so um so since su four has ranked three we need three labels so j one j two and j three the three labels for representations so we can write operators and their lorenz quantum numbers is with these labels j one j two and j three and then this deltas the dimension of the operator and so um if if j one is anything and j two is bigger than one and j three is anything then this this is the bound on the dimension and this is saturated for conserved currents so examples are for instance a conserved vector current j mu has these quantum numbers um if you think about the young tableaus j one is is the number of columns with just one box j two is the number of columns with two boxes and j three is the number of columns with three boxes so um so so for instance the vector if as a representation of su four is that representation and so that's uh zero one zero it has two boxes and according to this inequality it has dimension five which is the right when it's saturated which is the right dimension for a conserved current uh the stress tensor uh stress energy tensor is in this representation so it has um two columns with two boxes so this is j two is two and its dimension according to this inequality is six and the conserved super occurrence um are in this representation has one spinner index and one vector index and its dimension is five and a half and uh so when its dimension is five and a half it's conserved another example is if we take this j two quantum number to be zero and j one and and j three to be non-zero then this is the inequality for unitarity and an example of where that's saturated is if we look at a representation where j one is is one and j three is one so that's something with three and one boxes and that's the adjoint of su four which which um we could we could think about and if we write these as pure mu and new or so six lorenz indices it's a two index antisymmetric quantity and so this is like a um two form this is a conserved two form so if if we take these quantum numbers and if the dimension is four it satisfies this conservation law and if if we have such an operator then we have uh generalized global symmetry with the two form current yeah yeah so so this is yeah this is uh right so the question is where where do these these things come from um yeah basically one has to just work out the the commutator and with the algebra to see where this thing comes from so it's a bit of a long story yeah to actually to actually show where these offsets come from it's a bit of a long story to to work through the commutator and um maybe just take my word for it and then I can I can point out some references that that work through it uh huh well they are definitely from the representation point of view no yeah so from the representation point of view this this is fine I mean definitely studying the theory kind of the the properties of correlation functions in euclidean versus lorenzine this can be quite different but in terms of just the counting of the operators and this representation theory this this works yeah so so the question was about which have protected dimensions so anytime this this inequality is is saturated this this is like a short operator and so that's a protected operator and so so for instance um the statement that conserved currents can't have anomalous dimension is because it's in this short representation and so if we have some kinds of continuous parameters that we can dial yeah thank you for the the question I wanted I should have emphasized that more whenever these inequalities are saturated one of the interesting things about them is is that these are um operators that if as you vary kind of continuous things like coupling constants or moving on spaces of aqua and things like that um these are protected operators the only way that they can become non protected is if sometimes um there can be some recombination rule where like like for instance if we have a conserved current that's ends up being not conserved like if we turn on some operator that violates some symmetry then what happens is that in that case the conserved current can get an anomalous dimension and the way that that that can happen is by pairing up with another operator so sometimes we can have a situation where two short operators can form together to form a long operator like a conserved current plus a scalar operator can become a non conserved current but um by themselves all of these operators are are protected operators and if there's nothing that they can pair up with sometimes you can argue that they have to remain conserved always thanks yeah i just i realized that i was talking probably too fast and so i should i should remember to ask people if there are other questions also okay um so just just to give a few other examples of six-dimensional uh operators where where we saturate these inequalities um if if we have any j one and if j two and j three are are zero so this is like a representation of of su four like that then this is the inequality that we get and and these inequalities when they're saturated the the fields are free fields so the simplest example of that is if this j one is one this is a free fermion so a six-dimensional fermion is in the four-dimensional representation of the lorenz group the lorenz group is su four it's in the it could be either in the four or the four bar so this is the four this is the four bar and when this inequality is saturated its dimension is two and a half which is the right dimension for a free fermion if we if we write down like psi bar d slash psi in six dimensions this thing will have dimension six if if psi has dimension two and a half and um another example of an operator of this type is to to take instead of uh being the firm instead of the fermion where this is one we can take something which is like a something that has two spinner indices so we can take two indices here so if we just have two boxes and that's something which we could also write so these are two symmetrized su four fundamental indices and we could also write that as two anti-symmetrized lorenz indices this is just like in four dimensions where we can write f mu nu for instance as as like a by spinner and so in general we can write down like a commutator of of gamma matrices to get something which is goes between being a by spinner and being a two form so this is a two form operator and just like this one is a two form operator but this is a two form operator that we get from one chiral and one anti chiral spinner index this is a two form where both of them are chiral so this is a non chiral two form and this is a chiral two form and this this this is a chiral boson and the way that its chirality shows up is that if you write down the the if we take d of this two form so we form a three form by taking dx then that three form has to be either self-dual or anti self-dual so if it's this one it's self-dual if it's this one it's anti self-dual and these operators play a big role in the six dimensional theories that's part of why i wanted also to kind of go through this group theory was to to introduce some of the things that will come up later these chiral two form operators play a big role in six dimensional super conformal theories and we can see from here that they have to have dimension three if they are if they saturate this inequality so if if j three so if either j one is two or j three is two we get three here uh-huh oh yeah yeah sorry yeah that's a typo yeah that should be zero zero two thank you thanks okay so um when we get to to super conformal field theories the algebra is uh some super algebra version that should that should contain also s o d comma two so if we want to have a super conformal theory the algebra should have s o d comma two plus some super charges and the super charges have to be in the spinner representation of s o d comma two and that turns out to be something that's not possible to do in general the super algebras were classified by cats and if you just look at these different super algebras and look for the condition of having a s o d comma two subgroup and having uh fermionic operators that are in the spinner representation in general it's it's not possible it's only possible in six dimensions and less so there could be no super conformal field theories above six dimensions just because the algebra doesn't doesn't exist and in cases where it does exist it always uses special properties of spinner representations like for instance in six dimensions it exists thanks to the fact that so the the bosonic piece again includes this s o d comma two which is like s o six comma two and so this is if we think about it's in a euclidean version this is like s o eight and s o eight has triality between the spinner the vector and the other spinner representation and the only reason why this super algebra exists is thanks to the triality because instead of thinking about it as a spinner we could think about it as a vector and so so in six dimensions this is the name of the super group it's the osp six comma two n the algebra exists for any n and it contains the conformal group and then it contains this r symmetry which is sp and r and so this can take this is written as n comma zero supersymmetry the supersymmetry in this case has to be chiral so super conformal symmetries in in six dimensions only exists for with chiral supersymmetry all of the supercharges have to have the same chirality and so there are eight n supercharges where so so the minimal supersymmetry would be eight supercharges that would be one zero okay in in five dimensions the supergroup is kind of a supergroup version of f four it's an exceptional group and so it only exists for one it only exists for like n equals one supersymmetry so it contains as a subgroup this conformal group so five comma two and its r symmetry is sp one which is the same as su two so there's an su two r symmetry and there are eight supercharges and there's no unlike six dimensions where it could exist for any and here it only exists for n equals one and then in four dimensions it can be written down for any n three dimensions it could be written down for any n and also in two dimensions okay when we look at the unitary unitary conditions in super conformal field theories in addition to requiring that so we have this inner product just like in the non supersymmetric case so there's a positive definite inner product and all of the descendants have to have non negative norm but now we can form descendants with both the momentum and also with the supercharges so here we can form descendants with both and so if we look at the representation there's a super conformal primary at the bottom which is annihilated so basically what happens is just like p mu can be written as the anti-commutator of q's k mu which is the special conformal symmetry in general can be written as the anti-commutator of some some operators called s so p mu raises the dimension by by one k mu lowers it by one and q raises the dimension by a half and s lowers it by a half so there's an operator at the bottom which has the smallest dimension and that's annihilated by the q by the s's rather so s annihilates this operator at the bottom and then we could fill out the multipletive operators by acting with q's and so this is the conformal analog the conformal field theory analog of a of like a super field there's some operator in the super field and then we form the other operators by acting with the q's so here there's the super primary at the bottom and then we form all of the other ones by acting with the q's and if we look modulo conformal descendants so the q's anti-commute up to to p mu so if we look modulo conformal descendants we could think about the q's as basically like anti-commuting with each other and so this is a grassman algebra so if we look at what are these operators what do they look like in general we can act with an anti-symmetric product of q's up to l times where l has to be at most by Fermi statistics at most the number of super charges and so if all of these operators are non-zero then the multiplet is called long so if this l max is the number of super charges then it's referred to as a long multiplet and if the number maximum number that we can get to here is something less than that it's called a short multiplet so for instance half bps operators satisfy that the condition that this l q is half the number of super charges so um and if if an operator is short then that means that if we act with one more super charge on that on that operator we get an operator which is zero so at the top of the multiplet we get something where if we act with another q we get zero and that means that there's some null operator okay so if we look at the form of these uniterity constraints what it looks like in general is like this so there's some the dimension is bounded by some lower value and operators above this are called long operators and then the short operators with the largest dimension are called a type short operators and these are right at the threshold of uniterity and so these are operators where they could for instance get an anomalous dimension and pair up with some other operator and become a long operator then then there's a gap where no operators can exist and then there can be some new short operators which that are called for instance b multiplets and then there could be another gap and then there could be more short operators that could that's called them c with some specific specific dimension delta c um in five dimensions we have short multiplets of this type a b and c and in six dimensions we have short multiplets one more which is called d so the difference between these is that if we look at the uniterity bounds um there's some function of the Lorentz quantum numbers then there's something that depends on the r symmetry quantum numbers and then there are these shifts like this three and four that we saw also in the in the bosonic case and these shifts depend on whether it's this a b c or d multiplet so for instance in six d one zero um this is the dependence on the Lorentz quantum numbers this is the dependence on the r symmetry quantum numbers so for one zero the r symmetry is an su2 r symmetry and this r is just the number of boxes if we think about it is a su2 young tableau so the dimensions of the operators are bounded by this thing that depends on the Lorentz quantum numbers this thing that's the r symmetry and then there are these shifts which for the a multiplet it's six for the b multiplets it's four c it's two and d it's zero so the interest one interesting thing about these shifts is for instance like this six here uh means that the the smallest dimension of an a multiplet even if all of these quantum numbers are zero would be six and six is already like the dimension of the operator for being marginal in six dimensions and so we can have a an a multiplet whose bottom operator is marginal but actually if we wanted to have something that's supersymmetric this the supersymmetric operators are at the top of the multiplet because those are the operators that we can't act with any more cues if we act with any more cues we'll get something that's um zero up to total derivatives that's the condition that supersymmetry is preserved and so what we see is that for instance the an a multiplet even the bottom is is just marginal and by the time we get to the top it'll be irrelevant and so what what we can show from these shifts is that um actually no supersymmetry preserving operators are relevant in a one zero super conformal field theory every operator is irrelevant anyway um these short operators also play a big role in super conformal indices which count these different short representations okay so so that's all i wanted to say with the slides and then i'll transition switch over to the chalkboard maybe are there any questions yeah yeah yeah that's that's okay the um yeah so the the condition that we preserve supersymmetry is is that um um like we want the supercharge acting on like so if we have some deformation then then the super charge acting on this deformation should be zero up to total derivatives so it could be like p mu of something and then and then we'll still have um the supersymmetry algebra being preserved the um these s operators don't satisfy this condition and and that's fine that's it still preserves supersymmetry what that means is that it breaks the super conformal symmetries which um when we turn on these operators in general we break conformal symmetry and so we break the the extra the extra supersymmetries yeah the s's the s's are are the aren't like the q bars they're they're some extra supersymmetry generators that that are special to the fact that it's super conformal so yeah so in the theory that's super conformal you get an extra you kind of double the number of supercharges but really it's better to think about them as like raising and lowering so just like um you have this p mu which is the raising operator then the special conformal operator that's k mu and so like for instance if if we want um yeah so so in general like for instance um we can get something that's lorenzen variant by writing down something that's an integral of some local thing and then we preserve this thing but we don't preserve in general these k mus because we break we break conformal symmetry yeah in general in general there there are many different unitary representations and so there there are many in general there are many ways to deform that for instance it's supersymmetric theory by adding irrelevant operators for instance and um yeah so like like for instance in in 4d n equals 1 you know that we can write down something which in superspace well you should write it as like d4 theta times k and so so this is this is a kind of deformation that we would call a d term or we can write an integral over half of the superspace and so so we can think about this the way that i'll think about this uh in terms of this description is that this is of the form like q to the fourth acting on a long operator and so if we act with with all of the supercharges on a long operator then we get something that's supersymmetric and that's that's a d term and then this would be acting with q squared on some short operator and in this case q squared acting on the short operator is also the top of the multiplet and so if we act with any more supercharges we'll get zero and so so these are examples of supersymmetry preserving deformations and in general we'll have some very we'll have lots of operators and we can look at this for any of these operators i'm not sure if that was that the question yeah if if we want to preserve supersymmetry then this is how we could deform the theory we could deform the Lagrangian like delta l or deformations that preserve supersymmetry uh if they preserve supersymmetry if they're at the top of the multiplet so in this case what we would do is we would say that there's an operator o short at the bottom we can act with so we can act with with like q and q to get two more operators and then when we act with q squared we get to the operator here which is at the top so here we're increasing the dimension as we move up and so so this is and this is this q squared so this would be an example of like an f term deformation in supergravity yeah we could we could uh we could do something also including like the graviton multiplet and and all of this but yeah yeah i mean part of what i'm saying is just the structure of of supersymmetry really but at various places i'll be discussing also like conformal symmetry but yeah yeah basically this is also we could apply this also in supergravity okay um yeah so if we look at the different super multiplets we could ask what are the super multiplets of the conserved currents and so so one example of a conserved current multiplied is a flavor current j some current let's say j mu whose dimension is d minus one so that if we integrate this over uh space we'll get some dimensionless charge and this thing respects supersymmetry let's say that it's uh should commute modulo something that's like p mu total derivatives in order to satisfy the algebra because we want the the charge associated with this current this is the supercharge we want the the algebra should be that the supercharge commutes with this j mu unless it's an r symmetry current so the case where this doesn't happen is for an r symmetry current so for an r symmetry current q is charged and so then in that case this thing is non-zero but it's something that measures the charge of of q under that r symmetry so these these are two kinds of symmetries that we'll often talk about are flavor symmetries or r symmetries and um in in the case of the flavor current this condition that q annihilates it up up to total derivatives means that this j mu has to be at the top of the multiplet so anything that's a flavor current has to be at the top of its multiplet whereas um this for r symmetries the condition that q is charged means that um j mu if it's an r symmetry is one the supercurrent so the the supercurrent is the operator whose charge is j mu so basically the structure in in this case is that we have j mu r if we act with this supercharge let's put here alpha for some spinner index so if we look in the multiplet what it looks like is that we have the r symmetry current we have um this supercurrent let's call it q mu alpha so this is the thing where if we integrate um so here q alpha is the integral over space the time component so so this is the the density associated with the supercharge and then if we act with one more q alpha we have to get the stress energy tensor so this is in order for to have the right commutation relations where two q's anticommute to be p mu of the condition that q q anticommutes to p mu tells us that q the anticommutator of q with this q um beta mu is the energy momentum tensor yes there's some twos and then the energy momentum tensor has to be the top of its multiplet if we act with any more supercharges on this because this is already an anticommutator of two q's or because this is already an anticommutator with q we'll see that if we act with another supercharge we have to get zero up to total derivatives so this thing also has to be the top of the multiplet so this the stress tensor is at the top of its multiplet and flavor currents are at the top of their multiplet and actually this condition that these things are at the top of the multiplet tells you tells us that they can only exist if there aren't too many supersymmetries like this condition that that this j mu is at the top of the super multiplet this implies that the number of supercharges has to be at most eight in order to have a conserved flavor current so theories with more than eight supercharges um if we just look at the condition to have the the current be at the top of the multiplet it's just not possible with more than eight supercharges yes yes yeah for any dimension that that we have to have at most eight supercharges um yeah so if you if you just look at the representations um yeah be it's a little bit of work to to show where this thing where this condition comes from but if we just look at this condition that um that we should have a operator which is like a Lorentz vector at the top of the multiplet with and by the way since it's a conserved current its dimension has to have this special value what what you see is that these conditions um are just not compatible with the representation theory if the number of super in any dimension if the number of supercharges is is bigger than eight so like like an example an example of this is if we look at 4d n equals 2 this has eight supercharges and there we can have flavor currents whereas if we look at 4d n equals 4 then we have 16 supercharges and there we can't have flavor currents also in 4d there's n equals 3 supersymmetry which also doesn't admit flavor currents yeah so so it just comes from it's a little bit of work to show where it comes from but this condition that's at the top of the multiplet is very restrictive and likewise this condition that the stress tensor is at the top of its multiplet is very restrictive and actually this condition this this condition implies that um if d is bigger than three the number of supercharges can be at most 16 in order to have the stress tensor be at the top of its multiplet so like for instance in in six dimensions in six dimensions we can have one zero supersymmetry which has eight supercharges or we can have two zero which has 16 supercharges and we can write down an algebra with three zero symmetry so this would have 24 supercharges but but this one and and higher ones aren't allowed because even though the the super super conformal algebra exists it's impossible to write down these charges as being integrals of local currents in that case because you don't get a stress tensor which is at the top of the multiplet so so this thing is not possible because the t mu nu is not at the at the top all right put uh if this thing about the dimension having to be bigger than three d equals three is an exception and d equals three we can have any nq is possible but um and if nq is bigger than 16 it's a necessarily a free theory we can write down a free field theory with arbitrary numbers of supersymmetries in three dimensions but we can't write down an interacting version yeah so the so the condition that the team you knew was at the top of the multiplet is because um it follows from this condition that if we look at a commutator of q alpha with t mu nu we get zero mod p mu like if if we just look at the algebra and we use something which is like a jacobi identity we can see that this q has to commute with t mu nu modulo p mu just just to have the right algebra and so this is this is exactly the condition that it's at the top of the multiplet like q qx with these commutators or anti commutators and zero mod p mu means that it's at the top of its multiplet that's right that's right yeah so we can write down uh three comma zero algebra but it's it doesn't have as it can't have it possibly a stress tensor with the right properties so that so that rules out it's like three zero supersymmetry so so the biggest supersymmetry um yeah so like in fact in the next lecture i'll be focusing on this two zero theory in six dimensions which is the most number of supersymmetries in the highest dimension and so this this kind of thing about the group just the group theory tells us that even though the charge algebra would exist for three zero there's no way to write it down in terms of a local stress energy tensor that integrates to give those charges yeah so there there's uh i guess there are other things one could say about the group theory but i think i think um yeah i think in the next lecture i'll i'll kind of just skip ahead to properties of like the two zero theory