 Tako dobro za invitacij. To je odličen, da se počnem v sku. Prišliš, da je nekaj moj prav. To je OK. OK, zato. Zatim si, da ne možeš odličen. In, kaj sem počnem, da sem počnem, da sem počnem, da sem počnem, da sem počnem, da sem počnem. as as in said ask any question during my talk. So what I will talk about in my series of lectures is supersymmetric theories on curved spaces. So let me start by giving some introduction and motivation to the topic. To je skul o lokalizacijenem. Zato nekaj, lokalizacija je več potrebačna tehnika, v kajštji se potrebač nekajštih nekajštih nekajštih nekajštih nekajštih nekajštih nekajštih nekajštih nekajštih nekajštih nekajštih nekajštih, vizations ka이� evolike ide talbe in tehnika reka se pravam symmetry. Cdaj je izreizveno ko sem nekajšte počilersi k militarno zo completivnji priklade, spoj bath, bo s otroril아러 starsim ekstitutom, tak takene na spke nasrval zd announced. To, da se prikoneče, je zbora vzbeni supersimetri, in vzbeni vzbeni vzbeni vzbeni vzbeni vzbeni, ali imač sami kompakt češnje, prezervili sedje supersimetri. A z dvej, da je tudi čestno izgledaj, da je kaj vzbeni na vzbeni vzbeni. Tukaj vse je ozbunil je ovo pošličenje, kako je to izgledan z praktynima, potrebno sem ne vzelo, da je inak do vse zelo. V mojo priplenju je naj neko vsi vsočenje, neko vsi vsočenje. Vse najmaj spolito, da si počnem, je vznikala v terせて na nekaj supersymmetričen ljudi in mizila vstajnika tudi z časem ovoj nešto Petron, kako z n انemi dvorinih dokrotov. Kdaj poėljamo tracje v tle ljudi o T-3, od "-1 p.m". Zrča je vstajnja, kako vrstala in izgleda, zato ta je maybe, zelo je, da je prejvične zelo, da je včistil teori iz vrstva placpe, načo moraš različiti na Toroz. Idov nekaj zelo, da je izgleda, je, da je vzelična teori n-2 v d.Of for our choosing. Another interesting deformation that one can start in the context of N equal to theories is the so-called omega background. And this was very successfully used by Nek gidasov to compute instanton partition functions. OK, so now that's so, and then it can be drawn instanton partition funkcija. OK. So, these are somewhat older examples, but then more recently there has been like a lot of interesting developments that make use of localization techniques in supersymmetric field theory and they rely on placing supersymmetric field theory, deforming supersymmetric field theory from flat space. So, let me start with another famous example. So, one can take an n equal to theory and place it on the force sphere. So, this was done by Pestun. And then this allows to use localization techniques to compute the partition function of the theory or even the expectation value of certain supersymmetric Wilson loops along the equator of the sphere. And motivated by these results, there has been, I mean, people have looked at theories in higher and lower dimensions, so there has been a lot of work on n equal to theories in 3D with a u1r symmetry, which can be placed supersymmetically on the 3 sphere. So, let me just remark about the previous example that you could say, OK, well, I already have a way to place an n equal to field theory on S4, I can just do the topological twist. So, the theory of Pestun is indeed different from the topologically twisted theory. In particular, it allows, so it has eight supercharges, which is much more than what you have in the topological twisted theory. So, similarly, one can work in three dimensions with n equal to theories and put them on the 3 sphere and again compute partition functions and other observable. So, this was initiated actually for n equal for theories by Kapustin, Willett and Diakov, and then there was further work by Amos and Michenli, and by Jeff Ferris in 2D up a Francesco Benini, who is here, and Kremonesi, considered n equal to 2,2 theories on the 2 sphere. So, I suspect you will hear much more about this example in Francesco's lectures. And, OK, so, and also people looked at theories in five dimensions. Yes, then there is, as I said, I was not gonna, mine is not an exhaustive list. So, if I miss one example that you worked on, like, you should not feel slighted. OK, so then there are also examples in five dimensions, where, like, starting with work of Schellen's, Abzin and Q, and also Michi's song in Terashima, and also Jeff Ferris and Puffel, people looked at n equal one field theories in five dimension on the five sphere. Let me also mention another class of geometries, so that it's not always a sphere. So, one can take an n equal one field theory in 4D with a one-arsymmetry, and it can be placed preserving all of the four supercharges on S3 times S1. And then one can define an index by taking the trace over the Hilbert space on S3 of minus 1 to the f times e to the minus beta h plus decorations. And this index can be computed exactly. So, this was started by Romesberger, even if the super symmetric field theory on S3 times S1 was actually written longer, much longer before by Dipti Manson. OK, so many of the geometries and theories that I've placed on the blackboard actually allow for interesting deformations that preserve some amount of supersymmetry. So, for instance, one can start squashing the sphere so that they are no longer round. And so, let's put here, and the various observables that you can compute can then depend on the deformations that you apply to the geometry in different ways. Sometimes there are deformations that do not change supersymmetric observables, or it could be that you find a deformation that actually does change the value of a supersymmetric observable. So, then there is a question of understanding, like, why is that the case, and what geometrical information do the supersymmetric observables that you can compute the localization or with some other technique actually depend on? OK, so, this was just a list that is supposed to motivate, like, the following three questions that I will address during my lectures. So, the first question, which I guess is quite natural, is, like, so, given some su, given some suzy theory on which geometries can it be placed preserving some supersymmetry. So, here I'm being glib about what geometry means. So, this geometrical information can include, like, the metric or any other structure that we will find relevant. So, the second question that we would like to shed some light on is, what is the structure that was of the resulting theory? And finally, as I was saying before, we would like to understand how supersymmetric observables depend on the geometry. So, what do you mean by what is the structure, more concretely? Well, for instance, you would like to know, say, I take an nipple one field theory in 4D, and I placed on some curved manifolds. So, suppose this theory is a Lagrangian description, then what are the couplings that I have to write down on some specific manifolds? That's one question that you might want to. So, yes, these are pretty general, but any other question? OK, so, if there are no other questions, I can proceed. So, let me think about what to do with the blackboard. Maybe I'll just erase this one. OK, but I don't think I need this one, so I'll just erase it and then I'll have to come to space later. So, now, during the first lecture, I would actually like to introduce a topic, which is perhaps not completely familiar to many in the audience, which is that of supercurrence. So, let me motivate a little while I want to talk about this to start. So, let's consider some theory in flat space. Then you can imagine deforming the metric to be slightly different than flat. So, we can take some theory, then take the metric of flat space and introduce some deformation. So, the way that the theory reacts to such deformation is via its energy momentum tensor. So, the linearized metric deformation couples to the energy momentum tensor of the theory. Now, in supersymmetric field theory, the energy momentum tensor does, first of all, there is an energy momentum tensor and it's part of a multiplet with other currents and other operators. So, it's tends to reason that in studying how to place supersymmetric field theories in curved space, we will need to know what is the structure of this multiplet. So, this multiplet is called the supercurrent. So, the references that you can look at for this part of the talk, so, there is a paper by Seiberd and Dumitreskoy. That's 11 or 6, it's 0031. And also a paper by Komarotovsk and Seiberd, which one is this, 1, 002, 228. No, I messed it up. No, no, it's correct. And, well, I mean, this is by no way an exhaustive reference, but if you look at these two papers and that the list of references therein, I think you'll be in a good shape. So, in particular, during today's lecture, I will talk about the supercurrent multiplets, which are appropriate for n equal 1 field theories in four dimensions. So, at this point, I would like to make some comments about notations. So, I will follow Wesson-Bagger notations. So, I hope you are familiar with that, but if you're not, we can maybe discuss this during the exercise sessions. Or, if you have a question about some notational issue, you can ask me during the lecture, and I will try to answer directly. So, let me remind you about the Susi algebra for these theories. So, we have the anticomitator between q alpha and q bar alpha dot. So, that's even in terms of the four momentum. So, actually, as we will see, we will modify this in a short while. And then, two supercharges of the same chirality commutes, anticommute. So, we would like to study the supermultiplet, where the energy momentum tensor resides for this particular theories. And, as we will see, by dimensional reduction, then one can get also information about other theories with four supercharges, namely n equal to in three dimensions and n equal to comma two in two. So, oh, it's okay, but I think you might be, I mean, is there a problem because of the shade? Okay, so, what do we want our multiplet to contain? So, well, as we said, we want the multiplet to contain the energy momentum tensor. So, I remind you that in a local field theory, which is sponkare invariant, there exists a conserved real symmetric energy momentum tensor. So, it's important here that the theory is sponkare invariant, so that, like, you can show that you can always improve the energy momentum tensor to be symmetric. So, if you haven't seen this already, I think one of the possible exercises for this afternoon will be to actually show that this is the case. Okay, so, the four momentum p mu is, as usual, obtained by taking the integral over space of the mu zero component of the energy momentum tensor. Now, the energy momentum tensor, the symmetric real energy momentum tensor is not unique because it can be changed by improvements. So, in particular, we will focus in these lectures on improvements of the following form. Of some scalar, let's call it small u. So, this is not the most general improvement, which keeps the energy momentum tensor symmetric. But you can check that with this, what, missing a piece. So, with this definition, the second piece that we added is automatically conserved. And also, if you try to compute the momentum corresponding to the improved energy momentum tensor, you see that it won't change. So, this improvement of the energy momentum tensor does not spoil the conservation and does not change the corresponding charge. So, this is one object that our supercurrent has to contain. And then, because we are dealing with supersymmetric field theories, there will be supercharges, q alpha and q bar alpha dot. And by applying another procedure to the supersymmetry, you can find that there are currents, conserved currents which correspond to these supercharges. So, in particular, we will have conserved currents s mu s alpha mu s bar alpha dot mu. So, we are conserved. They give, when integrated, they can give rise to the supercharges. So, for instance, q alpha is the integral in the 3x of s alpha zero. And the same is for q bar. Again, the supercurrent is not unique, but it can be changed by improvements. And we will not write the most general improvement but just the ones which will be relevant for later. So, I can take my supercurrent s alpha mu and shift it by a term of this form. And again, it's immediate to check that this piece is automatically conserved due to the anti-symmetry of sigma mu nu and that it doesn't change the supercharge. And there is an equivalent expression for s bar. So, we certainly want these two objects in our supercurrent multiplet. But we also want to impose other requirements. So, another requirement that we want is that s bar and t are the only operators in the multiplet which have spin greater than one. So, this is because we eventually want to couple this theory to gravity and in supergravity in 4D we only have degrees of freedom with spin greater than one are the ones corresponding to the metric and the gravitino. And the metric, as we know, couples to the energy momentum tensor and the gravitino will couple to the supercurrent. So, that means that if you want to embed our supercurrent multiplet in some super field, then its highest spin component is the theta-teta-bar component of the super field. So, we would imagine that our super field would be something that's called s mu. It would be real and its theta-sigma nu-teta-bar component will contain the energy momentum tensor plus lower spin stuff. Why does it have to be in this component? What I'm saying is that if we want to take embed our supercurrent in a super field, then it's the top spin, the operator with larger spin in the multiplet we reside in the theta-teta-bar component of the super field. So, that means that because we want to have some energy momentum tensor is the operator with larger spin then it will have to reside in the theta-teta-bar component of some super field s mu. Then we will see what the constraints are. So, that's the name of the game is to find what constraints does this super field satisfy. Okay. So, another requirement that we will impose is that the multiplet is in the composable, which means that we cannot separate it into two multiplets, two separate multiplets. Okay. So, we want a minimal multiplet. This, by the way, does not mean that inside the multiplet there aren't smaller sub-multiplets, which are still multiplets under n equal 1 supersymmetry. However, these sub-multiplets cannot be separated from the total multiplet without spoiling Susie. Okay. And finally, there is another very important requirement is that... Why is this important? Well, it's just because we would like to have the minimal structure which contains the energy momentum tensor and other currents. But is there an obstruction when you need it in some vital way in future or you're just looking for the simplicity then? Well, I mean the point is that this will contain the minimal structure which is indispensable. Now, when, for instance, you couple this theory to gravity, like the the supercurrent multiplet will tell you which supergravity to couple to. And it's true that you could imagine adding more fields to your supergravity and then maybe this will couple to a larger supermultiplet. But somehow those fields can be separated. And the minimal part which contains the metric and the gravitino will be coupling to this in the composable multiplet. So, the other requirement that we want to have is that all operators in the multiplet are well defined. So, for instance, they have to be gauge invariant if there is some gauge symmetry. But we will see other examples of constraints that follow from this requirement. Are there any other questions so far? So, now we come to the point where, like, there is a very long computation to be done to understand what is the well, either what are the constraints that need to be imposed on this S such that it contains these objects or in component notations how do you construct the minimal amount of operators which will be closed under supersymmetry and have these properties. So, this is a computation that I will not show but I will tell you what the result is. And to my defense, if you look at the references, they don't show the computation either. So, we will call this object the S-multiplet and in the following I will show super field expressions but always translate them in component notations for people who are maybe less familiar with super field manipulations. Yes? So, you are saying S is the super field that contains T mu nu T mu nu, so S is the multiple which contains T mu nu and the then the other fields will be in the lower components. Ok, so, unfortunately, I don't know if I can show this blackboard instead of this I don't know if you can see over there but ok. Or maybe we can put this Ok, so, so let me tell you what are the components, the operators that appear in this multiplet. So, as we know, there is T mu nu which is a conserved symmetric tensor. So, it has ten components in the symmetric tensor but then it's conserved, so we have tracked four, so we are left with six degrees of freedom. Then it turns out that this multiplet contains a T-form, F mu nu which is closed so this gives rise to another three degrees of freedom then it also contains so this, as we know, is real this is also real now there is a complex one form which is also closed so because this is complex it has two degrees of freedom then we have a real scalar we call it A so that's one degree of freedom so let's count missing something, yes then there is a current which is not generically conserved call it J mu and that's four degrees of freedom so if we sum this we get six plus four ten plus four plus two sixteen bosonic degrees of freedom and then we have fermions so there is the super current S mu alpha and S bar mu alpha dot so these are conserved so instead of having eight degrees of freedom each they have eight minus two so it's six plus six and then in the multiplet there are also spin-a-half fermions psi alpha and psi bar alpha dot so that gives two and two so again the sum is sixteen so we have a multiplet which constane sixteen bosonic degrees of freedom and sixteen fermionic degrees of freedom so excuse me what do you mean by this complex one form has two degrees of freedom well okay so let's forget about the fact that instead of taking this to be closed let's take it to be exact then it will correspond to a complex scalar or you can just count constraints you have four components minus three constraints but it's complex does that answer the question? I didn't understand what you mean by this complex means it is always decomposed into so there is a real part and imaginary part the real part is one form so it has four components and there are three constraints so one and then times two because there is also the imaginary part okay so these are the components which are contained into this multiplet now I can also give the super field expression so which shows which constraint it has to satisfy alpha s alpha alpha dot equals chi alpha plus y alpha and d bar alpha dot chi alpha is equal to zero d alpha chi alpha alpha dot chi bar alpha dot and finally alpha y beta plus d beta y alpha is equal to zero with d bar square y alpha is equal to zero sometimes in the literature you find this last field y alpha replaced by d alpha of a carrow field however it's not always the case this carrow field x is well defined so one has to be careful so this is the most the most general one and you can find component expression for the super fields in the papers that I cited I'm not gonna write it down because it's gonna take completely because it's gonna take a long time but I just give a flavor for it so this super field s mu is gonna start with this non conserved current J mu and then in the theta component you will find the super current plus derivative of psi which I'm not put then there is also a theta bar component which contains a spar plus other stuff then the theta square component contains this complex one form and then finally as advertised and well it's gonna be here the theta theta bar component contains the energy momentum so but it also contains this scalar field A and something which is proportional to this close to form f mu nu and finally a derivative of the current ok and in the papers you can also find expression for kaj alpha and y but I'm not gonna spend time writing those down just to make sure the statement is that if you source the conditions you have before then that fixes the s mu to this one so if it isn't decomposable it only has operators it up to spin 2 it contains a symmetric energy momentum tensor and the super current and what does dot stand for here is it lazy but for instance here you have minus i over square root of 2 sigma mu of psi and some similar term over there but they are uniquely fixed yes, all of it is fixed uniquely so for exercise I mean if you really want you could take this and show that so you can write down the most general real super field s mu it will contain a lot of terms then you can impose these constraints and you will find out you will find out that these constraints impose that t mu nu is conserved that s mu and s bar mu are conserved and this f has to be closed and this y and y bar have to be closed as well and what will be the start of the argument in the other way so how would you really start arguing from essentially the requirements you made to this set of equations basically you have to use the supersymmetry algebra so one way you can do it is by using super fields you start with the most general real super field and then you want to find constraints which impose the fact that t mu nu is conserved so that's one way the other way is that you start with t mu nu and with s and then use the supersymmetry algebra so this you will see in a second maybe it will be more apparent ok so before I get to the improvements let me I have another question so you are expressing s mu and there is s mu in this question oh sorry this is bad so this is the super field let's call it s with a squiggle net and this is the super current so this is actually spinor indices contracted in theta so this is the super current this is just the super multiplet sorry for the any other question this structure of the super current can you understand from higher dimension I am not sure because this is n equal 1 so I am sure for instance if you are considering n equal 2 in 4D you could presumably understand it from n equal 1 in 5D and so on so so now as I argued that this multiplet contains all these various operators and in particular you could ask yourself can I see these operators in the supersymmetry algebra and the answer is yes but it requires to extend a little bit what we usually write for the suzy algebra so let me maybe write here can everybody see if I write here so one thing one can do is by using the super field expression and the supersymmetry transformation rules we can build out what the anti-commutator between a super charge and say the super current is so if I were to integrate the left hand side of this expression with respect to the 3x I will get the anti-commutator between q bar and q so if I use the super field and this is 2 sigma nu alpha alpha dot then there is a piece that you imagine has to be there because when I integrate over the 3x I want to recover the 4 momentum on the right hand side of the super algebra but you also get another piece call it semi nu where semi nu is not symmetric it's anti-symmetric and also there can be schringer terms the schringer terms as well are completely fixed by the super field maybe can you say one more word what are the schringer terms well we can even write them down well first I want to say one more word about this and then we can go on to the one more word about the schringer terms so semi nu you don't see semi nu in this list so what is it okay semi nu is actually built out of this f this is a bad place to be so semi nu is written in this way in terms of f and from this definition and the fact that f is closed you recover immediately that semi nu is conserved so semi nu is a conserved current which is anti-symmetric in the two indices so it's a string current so in particular if I were to integrate with respect to d3x the zero component of s0 of s mu I will get from this that q bar alpha dot q alpha is equal to two sigma nu alpha dot p nu plus z nu where p nu is as defined over there and z mu is similarly defined as the integral in d3x of c mu zero the way you do it is by just so it's q bar which is acting on s mu alpha so you can take your super field then you know how the super charge acts on the super field you can compute your derivatives and that's what you get on the other side vice versa the other way to find this multiplet would be to start from q bar with s is equal to two sigma t and then say okay so then I should get something proportional to t so that tells you what the relation between s and t is and so on and so forth try to build it component by component which is however somewhat harder working in super field this is easier Is this the string current related to this equation self-dual equation or something like that? Yes so this string current that appears here in terms of component of the s multiplet is related to f so there is no it's just the dual of f and because f is closed then it's automatically conserved okay so like the reason why you don't usually see the supersymmetry algebra written in this way is that this string charge coming from the string current is actually infinite any string object which would carry it like it's an infinite long string it will have an infinite charge so in the same way as for the q's we can also do the repeat this for the other commutation relations which are q alpha with s mu beta so what do we find here well so with the correct realizations so we get sigma alpha beta nu rho times c mu nu rho and what is c mu nu rho that okay so c mu nu rho is antisymmetric and it is equal it's proportional to epsilon mu nu rho lambda y bar lambda so again because this y is closed then it's automatically conserved and it corresponds to a domain wall current again for any domain wall you can find in an n equal 1 field theory this would be infinite so the corresponding charge will be infinite so we can write here what the commutator between q alpha and q beta is it's gonna be sigma mu nu alpha beta times z mu nu where again z mu nu is defined as the integral in d3x of c mu 0 sorry is it a general statement that conserved antisymmetric tensors correspond to higher strain currents of domain wall currents yeah I mean those are objects that like are like naturally carried by extended object then in any theory you can actually check if there are extended objects that do carry this this charge so actually we will see examples if I get to it those don't appear in the algebra so you can see for instance that this current j mu would not appear in this commutator but it's not even conserved so there is no corresponding charge but besides I'm only considering these anticommutators if you consider anticommutators with other components then you might find these other operators here I'm just looking at the anticommutators which result in the suzi algebra but I mean you could consider others like for instance you could consider the commutator between j mu and q yeah and that would be proportional to s and so on no this one actually does not so right and now I remember that they should comment on the shingar terms of terms which do not contribute to the charge at least under the assumption that things behave nicely enough at infinity and in this particular case they are determined by the structure of the s-multiple so for instance those shingar terms over there will be equal to sigma nu alpha alpha dot which multiplies so you can convince yourself that indeed this gives like total derivatives that do not contribute to the super charge so what is domain wall here so it depends so given some tier you can check if it has or not domain walls which carry this charge so for instance if you consider some vestumino model with some cubic super potential like it might have two supersymmetric vacuum and then there could be a bps domain wall correlating between the two and then what is the domain here why you write those that's why I'm saying this come from so from taking the anti-commutators of quvitas you discover this current and then you interpret it as some conserved current which when integrated would give the current which would be carried by some extended object so this is the bps charge you are saying it would be well the object can or cannot be bps but if it saturates some bound then it could be bps now I have to decide what to erase so you said usually the string charge and the domain wall charge are infinite that's because of the infinite variable yeah the infinite volume of the string yeah yeah if you were to compact but right then it would be different okay so running late but that's fine I guess so any other question on this on this so far okay so if there are no other questions let me proceed with talking about the possible improvements to this super current multiplet so as we saw before the various components that enter into this multiplet can be improved for instance the energy momentum tensor can be improved the super current might be improved so there is a way to encode all these improvements in a supersymmetric fashion so I guess I'm gonna erase this top so the improvements can be encoded inside a real super field u so as it's familiar from people who have looked at Wesson-Bagger these are some component expansion like so et cetera et cetera and then in terms of this super field the improvement of the super current works in the following way so s changes this way and chi alpha goes into chi alpha plus s follows so actually this u need not be well defined it can be changed by a constant and nothing will be a miss because the various improvements will not change if you shift u by a constant u is well defined up to a constant okay so this looks a little bit esoteric but we can check what these improvements are on the various components well there is a reason why I used little u over there so the improvement of the energy momentum tensor is just the same as I wrote there but with u being the bottom component of the super field and that's also the reason why I've chosen this family factor of 2 in that expression because the eta over there is the same eta which appears in the super field u okay so that tells us what are the improvements for the energy momentum tensor but there are also improvements for other operators which enter into the super multiplet and in particular the improvement for the 2 form f mu nu is as follows f mu nu goes into f mu nu minus an exact 2 form which as you would imagine is related to this v and the one complex 1 form y mu get shifted by an exact 1 form which is related to the derivative of could you motivate a little bit why you need this? okay so yes we will get exactly there but the idea I can tell you what the idea is in a second so basically we said that this multiplet is in the composable so this is true as long as there are no further no further constraints that are correct but you can see that if for instance this 2 form being closed is exact then by doing this improvement I could set it to 0 so that means that there are cases in which I can expect to be able to shorten the multiplet even more than it is already short but this requires extra constraints to be put in so we will understand what these extra requirements are and what are the different structures look at these expressions and very easily you see that if it were true that there exists a well defined u such that chi alpha is equal to minus 3 half d bar d alpha u then I can set all chi alpha to 0 and on the other hand if there is a well defined u such that y alpha is minus a half d alpha d bar alpha u then I can set y alpha to 0 so this is in super field language so this will correspond to two different shortenings that the multiplets can undergo other questions so given the fact that there is been a lot of quite formal stuff going on it's a good idea to give an example so the example I will consider is that Vesumino models so Vesumino models are depend on two datas one is a killer potential which depends on the carol field and anti carol fields in your theory now the killer potential in a Vesumino model need not be well defined indeed the theory only depends on the killer metric so that means that I can shift the killer potential by killer transformations so k can be shifted by an holomorphic function of the carol field plus an anti-holomorphic function of the anti carol fields and indeed this does not change the killer metric which is given by taking the derivative of the killer potential and because there is an holomorphic and an anti-holomorphic derivative this shift does not change the other piece of data that you need to define your Vesumino model is a superpotential which is an holomorphic function of the carol fields phi i and again the superpotential need not be well defined indeed you can always shift it by a constant and nothing changes because the theory only depends on derivatives of the superpotentials what do you mean by not well defined for instance you could have carol field so let's consider a carol a theory which has canonical killer potential and there is one carol field phi which is identified with radius 1 so phi is the same as phi plus 1 it's more than 1 this carol field is identified with phi plus 1 so then I can consider a theory which has superpotential equal to phi and this theory will be super symmetric even if the superpotential is not well defined because it changes by a constant as I go around phi it's defined on the circle it by 1 phi lives on the circle but w is phi so phi w is not well defined on the circle it's more of a cylinder because phi is complex for instance so again when you write the Lagrangian of a sumino model you only have derivatives of superpotential so shift by a constant is not problematic ok so then we can write down what the various operator in the super symmetry multiplet in the super current multiplet r so we can start it so we can just write the entire multiplet so s alpha alpha dot is q times the killer metric so there are very explicit expression for all these objects and to your art content you can check that they do satisfy the definition over there so that's d bar square d alpha k and y alpha is 4 times d alpha of w so one thing that you can notice is that they are all well defined when k undergoes keller transformation of w those shift by a constant ok so you probably all know what the expression for the energy momentum tensor and the super current is for these models but it would be interesting to check what these other operators in the super current multiplets actually look like so for instance the two form f mu nu is given by the following expression so it's i times g i j bar times d mu phi i d mu phi bar j bar ok so if you stare at this you recognize that this is nothing else but the pullback of the keller form so if you consider the keller form corresponding to the keller metric k that's i g i j bar d phi i wedge d phi bar j bar ok so this is on target space then you can pull it back to spacetime and what you get is this f mu nu ok so this is already enough for us to make some interesting comments because it's well known that at least locally the keller form can be written is d of something where this a is given by keller connection is given by this so now you can check that this connection a is not well defined under keller transformations so it does shift so in particular that means that this f mu nu whenever so whenever this connection is not well defined will be closed but not exact ok so when does this happen so for instance whenever the target space of your vasomeno model is compact then the keller form cannot be exact because some power of it is going to give the volume form and then in those cases the f mu nu which as we saw is related to the string current will not be exact but it will just be closed so a simple example of this occurrence is the cp1 model where you take the keller potential to be f squared times the log of 1 plus phi phi bar so you have a single chiral field with this keller potential then the target space is cp1 which is compact and the keller and the keller metric is just the fubini study metric so this model will have an f mu nu which is closed but not exact and then it means that there is no improvement transformation which can be used to set this f to zero because f improvement transformation can only shift away the exact part of f but cannot shift away f if it's closed but not exact so the other object of interest is this one form y mu which is very easy is just di of the superpotential times d mu of ii so again you could say this is just the derivative of the superpotential but that's so that's fine but the superpotential itself is not well defined because it can shift by a constant so in particular if we consider the case this model I was talking about before where we have a scalar field with shifts by a constant is identified with itself plus a constant then y mu will be closed one form but it's not gonna be exact because the object that it will be the differential of is not well defined and the superpotential is shifts by a constant the superpotential is shifted by constant derivative is well defined why mu is well defined the superpotential but it's not exact yeah it's not exact derivative is well defined yeah it's closed but not that's the point so in particular that means that in this particular model we will not be able to use these improvements to shift away why mu because the only way to do it would be to use n proportional to the superpotential but that is not a well defined operator ok so I have nine minutes correct so let's see what ok so I will talk about two special cases and then I will leave the rest for the next lecture unfortunately I'm going a little bit slower than expected sorry I have a question I'm confused how you've labeled all of these fields in the end of the one but I'm confused what energy momentum tensor is clear how do you get that mu mu in how do you get this one this non conserved current they have to be there in order for the multiple to be consistent with the supersymmetry algebra but is there some definition somebody giving me a theory and is there a prescription how I get each of these so team you know there is we know how to do that the supercurrent as well and then the others you get by doing suzy transformations so they are all related by supersymmetry ok, I see well, if you add an expression for the lowest component then it would be pretty easy to find all the other ones by just varying that but is there a physical interpretation of them yes there is well, I've given two interpretations here those are the domain wall current and the string current but in a second there are also nice interpretations for other objects ok, other questions ok, if not let's see what I can erase probably I will erase this one so now we look at special cases where the s-multiplet can be made shorter by using improvements ok, so there are two main special cases so I don't know if I'll be able to talk about both of them but let's start with the first one so the first one happens whenever there is a well-defined u such that you can shift away chi alpha so if this operator chi alpha can be written as d-bar square of d-alpha u minus 3 halves then by using this improvement transformation I can decouple chi alpha or I can set chi alpha to 0 ok yes this one? it's better maybe even that we can set chi alpha to be equal to 0 so I'll tell you what this implies for the various components so it implies that f mu nu is equal to 0 so this form is actually equal to 0 so in the improvement it had to be exact so that I can shift it away using v so this requires f mu nu before the improvement to be exact then the other thing that implies that this scalar field A is equal proportional to the trace of the energy momentum tensor and in the similar way is proportional to the contraction of sigma mu with s-bar so when f mu nu is exact it is totally equal to 0 no no no no if you start with f mu nu which is exact then you can use the improvement to make it 0 even if it's exact it's the definition of the curvature tensor if you get it's exact it doesn't come from supersymmetric I'm not saying that this comes from I'm saying if you have an f mu nu which is exact then it's d of a one form so then you can improve it away I'm not saying this is ok so ok so what does this mean so this means that in any such theory where the s-multiple can be improved in this way such theory cannot have the string charge because the string charge is related to f so in particular in this theory there will not be bps strings so this multiplet which one obtains by setting kaj alpha to 0 as a name it is called the ferrarazumino supermultiple and we can count the number of degrees on freedom in the ferrarazumino supermultiple it's 16 plus 16 but then we take away the 3 degrees of freedom in f mu nu and 1 degree of freedom in a so that means that we are left with 12 and we are also left with 12 degrees of freedom for the fermions because the psi's are related to the s's so these are 12 plus 12 degrees of freedom 12 ozonic and 12 thermionic so that's the first special case the second special case is maybe more interesting but I don't know if I really have time for it 5 minutes then I have time so the other special case let me just erase the name here so let me say some other comments so from this it's clear for instance that if you have a vasumino model where the target space is compact then the killer form cannot be exact therefore this model cannot have an fc-multiple so that's already a statement that we can make and there will be other examples of models which do not allow an fc-multiple so the other case is let's suppose that y alpha can be written as minus a half of the alpha d bar squared of u for a well defined u then we can improve y mu so then we can improve y alpha to zero so what does this mean on the component is actually quite interesting so it sets the scalar a to zero the one form y mu can be set to zero again this is using improvement so that means that the one form has to be exact and not just closed in order for this to be done the fermion psi alpha is also set to zero and there is a further constraint that needs to be satisfied which is quite interesting and that's the fact that the non conserved current j mu is actually conserved so now what is conserved current so it's a u1 current which is conserved and appears in the supersymmetry algebra it's a conserved r-current so what this means is that if I have a model which has a conserved r-current then actually I can revert this logic and I can start from the conserved r-current put it at the bottom of the multiplet and then obtain all the rest so any model which has a conserved r-current allows for this particular kind of super multiplet in particular you can see that this model here with the scalar field which shifts by a constant does not allow an r-current if there were an r-symmetry the superpotential would have to have r-charge 2 in some units which means that phi would have r-charge 2 and f plus 1 it is only consistent with having r-charge 0 so then that means that this model cannot have an r-current and therefore it cannot have this multiplet which is called the r-multiplet ok, I will say more things about fz-multiplet and r-multiplet in the next lecture it is usually in the n-proplete series r-current always suffers from a normal no there are cases where it does not if you have an anomalous r-current if you have an anomalous r-current then the theory does not have an r-multiplet so for instance, if you take pure superior means that has an anomalous r-current therefore it has no r-multiplet it turns out it has an fz-multiplet and indeed there are domain walls but no supersymmetric strings