 Tako, zelo sem izgledati nekaj, kaj smo zelo izgledali v prejnega lekova. Tako, smo izgledali o superkarantih multipletih. In spesivno, smo zelo skupnili na različenju n-1 filteoriji v 4D. Supercurrant multiplet je multiplet, kaj je vse supercurrant, kaj je vse supercharge in energiomomentom tansor v teori, in ki smo videli, ta multiplet vse je vse vse operatori. Vse vse je vse vse multiplet, kaj je vse teori poses, kaj je začel je na nekaj naprej S-tiblet. Kaj je, kaj je, kaj je zelo energijomentom tenser in superkarant. Kaj je začel je varajitva občasov. Vom, kaj je vse strin karant, ki je vzouto v tem, da je vse vse vse vse vse. then there is also a domain wall current, which is given in terms of one form y, which is complex and also closed. And finally there is an arc current, which is not conserved, generikali, konečnji, več značen, konečno, nekonečnji, ko je ali bolj filetir, in po veselji, nete nekonečnji za zdaj, po takosanju R-sefetri. Danes je tudi včetno, ki so tamo 16 vsoženje, vse i izgledaj izgledaj vsoženje, in tudi nebezgledaj, kod tudi, nebezgledaj, tako ti so 16 vsoženje, vse i izgledaj izgledaj vsoženje, in tudi se 16 vsoženje, vse i izgledaj izgledaj vsoženje, tudi vsaj superkurrent, jih ima se fermion, psi alpha in psi bar, alpha dot. In as we saw this string current and the brain current, the domain wall current will like appear, so the corresponding charges will appear in the superalgebra and we wrote down the resulting superalgebra where we have the usual momentum appearing and then this string charge which is obtained by integrating the string current and similarly in the q alpha, q beta, anti-commutator you find the appearance of the domain wall charge. So this is we also looked at what kind of improvements the supercurrent can be subjected to and so we found out that like also the improvements can be treated in a super symmetric fashion and we have started describing possible particular instances in which this supercurrent can be improved to a shorter multiplet. Ok, so there are so for instance so we discussed two different scenarios so the first scenario happens when this form f instead of being just close it is also exact and then that means that there is no that the string current can be can be improved away and one is left with a smaller multiplet which is called the Ferrara-Zumino supercurrent multiplet and this Ferrara-Zumino supercurrent multiplet contains less than 16 bosonic and 16 fermionic degrees of freedom indeed there are only 12 plus 12 degrees of freedom so what happens is that there is no so there is no string current so there is no f and also like there is a relation between a and and the trace of the of the energy momentum tensor so so in particular we saw that there are cases there are examples of field theories which do not have a Ferrara-Zumino supercurrent namely if you consider a Vesumino model with target space is compact this cannot have a Ferrara-Zumino supercurrent because the two form f is going to be proportional to the kater form and it's going to be therefore close but not exact and so in general whenever the kater form of some Vesumino model is close but not exact then you have you cannot improve the s-multiplet to the Ferrara-Zumino multiplet okay there is also another example of a field theory which does not allow for a Ferrara-Zumino supercurrent and that is obtained by just taking the case of a u1 so it's got another example so if you consider some u1 gauge theory with an f i term so then your Lagrangian is going to include the usual field strength part plus complex conjugate and there is also the f i term okay this is in usual super field terminology then one can show that for this theory one can write down one can write down the supercurrent multiplet and in particular the the two form f which appears in the string current is proportional to the f i parameter times the field strength of the gauge field ends because the gauge field so if it were possible to improve away the this this f mu nu then it would mean that it has to be it has to be exact but it's not because a by itself is not the gauge invariant so this is another example of a theory which cannot have a supercurrent which is improbable to the fz fz kind and indeed in one of the exercises you could check that you could couple this you could consider sqd so the u1 gauge field couple to matter and then you can check that this theory does indeed have string configurations which carry this string charge okay so the other case that we took into consideration is the case where instead of being able to improve away the string current you are able to improve away the domain wall current and in particular we found that there is another multiplet which exists whenever the theory has conserved r symmetry so when this current which appears in the multiplet is conserved then again the multiplet shortens and becomes includes only 12 pozonic degrees of freedom and 12 fermionic degrees of freedom so this theories clearly you cannot have such a multiplet if your theory does not have conserved r symmetry so for instance one example could be if I take some vasuminum model with some generical cubic or even non cubic superpotential like this will not have an r symmetry therefore it will not have an r multiplet another example which is maybe more interesting is that of purium mills so if you take supersymmetricium mills like the r symmetry would be anomalous and then this theory would not have an r multiplet so in both these cases then when a theory of an fc multiplet then there is no string charge in the algebra and vice versa in the case that the theory has an r multiplet then there would be no domain wall charge in the supersymmetric algebra we can also think about theories which have both an fc multiplet and an r multiplet so certainly there are such theories so in these theories you would not have either domain wall charges or string charges and then it may be that the improvement that you need to make the s multiplets the fc multiplet and the improvement that you need to make the s multiplet the r multiplet actually do coincide and then it is possible to get rid of short and even further the multiplet so that's the third possibility so this happens when you can get rid of the both this field kai, super field kai alpha and y alpha in the definition of the s multiplet and this happens for theories which are super conformal so indeed if you look at the conditions that needed to be satisfied in order to have an fc multiplet and you combine them with those that need to be satisfied in order to have an r multiplet then you will discover that the clearly the r symmetry has to be conserved but also so to have an fc multiplet this scalar field a had to be proportional to the trace of the energy momentum tensor but to have a nr multiplet this field a has to be zero so that means that the trace of the energy momentum tensor is equal to a and is also equal to zero then you don't have you don't have a string current nor you have a domain wall current so fd nu is equal to zero y mu is equal to zero and there are also conditions on the on the fermions namely what you find is that psi alpha is equal to zero and sigma mu s bar is equal to zero and similarly for psi bar and s so these are indeed the conditions that need to be satisfied for like the super current to actually be as If I do a short rating of I am in masses like what we had yesterday from birthday in which month should I keep up? Well that depends on which multiplet you want so suppose that just for short breaking no so if you do a soft breaking then like the trace of the energy momentum tensor won't be zero and this guys will not and this guy in particular will not stay zero so you will have to then you will have to fall into one of the larger multiplets ok so this is for super conformal theories Is it possible to get rid of just the domain wall charge for example while keeping the J mu for example? What is J mu? J mu is the you and I are current Yeah so that's the R multiplet No sorry I am keeping that not to be non conserved but just eliminate say domain wall charge I mean there are many ingredients so you can there seems to be many options of which one to eliminate Right so as discussed so there is there are two possible way to do it consistently with supersymmetry one reduces to the ferrazumino supercurrent and the other one reduces to the to the R multiplet Ok but those are the only possibilities? Yeah those are the only two shortings I see Well and the SCFT shorting which includes both so if you introduce superpotential mass then that will break R symmetry Right so for instance if you let's take this let's take this theory so this theory is the one gauge theory with f i term like ok it has an R symmetry so indeed you can write for this particular theory you can write down some R multiplet indeed I think the R multiplet is gonna be just given by the following expression ok but then suppose you couple this theory to matter and you take some generic superpotential so that the R symmetry is broken then this theory will not have an R multiplet anymore but it will only have an S multiplet so indeed the most generic the most generic theory will have an S multiplet and then there are more special theories that allow for Ferrara-Zumino super multiplets or for R multiplets and then in the intersection there are some more even special theories that are conformal What happens if this conservation condition is broken by memory? that's the same so for instance you could consider the case of n equal 1 superium means pure superium means so classically it has an R symmetry but it's broken by the anomaly so that means that this theory does not have an R multiplet I think what we use is the super-satellar distance no but this is ok so let me comment on this in a second ok, so these are the three possible possible shortnakes What happens if the F5 parameter is dynamic power in the support of the xamod here or some other field for example it's a conclusion state the same I'm asking partly because people discussed for example dynamic and context of detail integration I'm cutting super graphically and we seem to say that so if you have a theory which generates between the infrared you want gauge field with F5 parameters that means that this theory cannot have cannot have an fc multiplet so this cannot happen if you start, ok so let me just make this comment now because it already rose twice so I should I should address it so one application of this of these ideas is that to put constraints on the RG flows of some theory that you are interested in or in general so let me talk about constraints on RG flows so the idea is that the supercurrent multiplet of a theory is some kind of short multiplet or alternatively you can say that the supercurrent is embedded in a super field which needs to satisfy some constraints so then this means that the structure of the supercurrent multiplet is preserved along the RG flow so it does so if you start from the nearly UV with the theory which has an fc multiplet then this theory will have an fc multiplet all along the RG flow and the same is true for a theory which has an R multiplet so when you get UV so the structure is conserved along the RG flow so there is a comment here that so first of all like if you start in the actual UV then suppose you have some asymptotically free theory then there you will have a free theory which is also conformal and then you might think well this does not really follow these rules but actually what you have to think about is like the theory at any high energy very high energy but not infinite and then like the statement applies and the other caveat is what happens in the extreme IR so in the extreme IR like your theory could like be some SCFT so in that case what happens is that some of the operators in the supercurrent might become redundant so they just a couple they don't have correlators at separated points but in between just up to when you get to the deep IR like your theory will have the structure of the multiplet and it will stay the same so in particular that means that you can determine what the structure of the multiplet is by just doing some computation in the UV so now this computation does not just have to be a classical computation it might include quantum effects of non-posterbative effects but those must be a possible computation in the UV so for instance for the case of superium meals like classically it would have an arm multiplet because it has conserved dark current but if you include quantum effects you discover that there is no are symmetry current and then the theory does not have an arm multiplet it however does have an FC multiplet and indeed in the the theory has like so if you consider superium meals it will have an distinct vector and there will be supersymmetric domain walls that interpolate between the various vector so that means that the theory does allow for a domain wall charge and then but not for a string charge because it has no a because it does not have an arm multiplet ok so between case 1 and case 2 is there a difference between how the theories can be coupled to super graphically so that is the other so that is the first comment but let me just say some more things about this so that this can be used to constrain the behavior of theories like quite a lot for instance if you start with a theory which has a Ferrara-Zumino supercurrent then you know that it will not develop FI terms for u1 so in the in the AR there could be some emergent u1 some emergent u1 and then you could say oh maybe this emergent u1 will have FI terms but that cannot happen because then the theory will lose its and the same thing can be said for like possible suppose that the AR description of this theory is given by some Vesumino model then you could ask about like it's a target space and well what we just said if the theory is an FZ multiplet then like the target space of this Vesumino model that you have in the AR cannot have cannot have a compact cannot be compact because if it were compact then the the color form would not be exact and that cannot happen if the theory is an FZ multiplet so there are all sorts of statements that you can make by just using the structure of the multiplet on the behavior of the theory along the arg flow and then the other comment that we want to make is that indeed eventually we are interested in some n-1 supersymmetric theory to supergravity and then the structure of the multiplet actually dictates which supergravity you have to couple to so in particular there is the most common supergravity for n-1 field theories this is called old minimal supergravity so in old minimal supergravity if you count the bosonic degrees of freedom there are 12 of them and indeed this couples to theories which have an FZ multiplet so this is appropriate for theories with an FZ multiplet and in particular you cannot couple to old minimal supergravity any vasumino model which has a compact target space also you cannot couple to old minimal supergravity u1 gauge theory with an FITER then there is another version of supergravity which is called new minimal supergravity I guess it's new because it was newer when it was invented and the so this this version of supergravity which we will describe more we will describe a little bit of them in more detail this couples to the r multiplet so you can use it whenever you have a theory which has a conserved r symmetry now both the old minimal supergravity and new minimal supergravity have 12 bosonic and 12 fermionic degrees of freedom so they can couple to these shorter multiplets and then the question arises of what happens for a theory which doesn't have either a fc multiplet nor an r multiplet as for instance u1 gauge theory with f i terms and generic superpotential so then in that case in order to couple to supergravity you have to use a new minimal supergravity which has more degrees of freedom in it so this is called 1616 supergravity and this couples to the s multiplet so the price you pay is that you have a longer multiplet and therefore your supergravity contains more fields and you can also go in the other direction and if you have an scft then you can couple it to conformal supergravity which has less fields now one important point is that if you look at old minimal supergravity and new minimal supergravity then it is true that they are different but the difference only arises in auxiliary fields so if you integrate out the auxiliary fields they are actually equivalent so on shell they are equivalent these two supergravity are equivalent on shell but because we are interested in like we will be interested in like backgrounds so in considering the fields of supergravity as like fixed backgrounds then we will work off shell and therefore the two formalism of the old minimal supergravity and new minimal supergravity are actually not equivalent okay so are there any questions there is some similar story in 4 dimensional n equals 2 okay so in 4 and then you have a super current current field of mass spread and I think it has anomali mass spread and for super conformal theory the anomali mass spread becomes without yes so also for n equals so you could repeat this story for like theories with different amount of supersymmetries and then it would be it would be similar well clear the details would change but so you can also have super current multiplets for n equal 2 field theories the most common one is called the sonius multiplet and it contains like the conserved SU2R symmetry in it but then like if the theory becomes super conformal then the multiplets shortens and indeed like you lose the trace of the enjoyment on 10 so you lose you also acquire another conserved d1 because there is also then the r symmetry becomes SU2R then SU1R and so on so but in 16-16 super we are not just adding auxiliary field we are adding so in the 16 so one way you can think about 16-16 supergravity is that you take some you basically you have one of you take new minimal supergravity and then like you add an extra chiral field that you use as compensator so indeed yes there are there is more propagating stuff 16-16 super is not irreducible irreducible 16-16 super is not irreducible 16-16 super is not irreducible irreducible or reducible it couples to this multiplet that as we discussed is generically not not reducible supergravity looks like reducible again this might be something which has to do with like imposing the equation of motion I am not sure and also there is a constraint in column super multiply in which you can replace one of real field in terms of tivine and fermions up to a fork of multiple mm-hmm ok, maybe I can ask we can talk about that later mm-hmm I mean, are you talking about ok, let's talk about that later ok, so so are there any more questions about this different supercurrent multiplets? sorry, I don't understand there is versions of the super current multiplets dictate how we can couple the theory to supergravity, why is that? just because like in the structure of the super multiple you have to choose a different supergravity to couple to so for instance at the linear level like you would have some coupling of the so if you do linearized supergravity the light so then like you added some theory in flat space then you want to couple it to linearized supergravity so that like at first order you would add a coupling of the supercurrent multiplet s mu to some superfield h mu which contains the linearized metric and then depending on which constraints s mu satisfies that changes the nature of the metric superfield so you see the difference at linearized order then I mean if you want to do the full non linear theory you have to work further just to make sure I understand your statement that the structure is preserved along the rg flow if there exists a shortening condition on the s multiple at any point on the rg flow then it must exist all along the flow is that the equivalent statement? yeah so it as I said with the caveats of what might happen in the extreme where we are with some operator might couple could you repeat the argument for why why that is? it's the usual argument that it's in a short multiplet so it cannot the nature of the multiplet will not change sometimes you can make it more than one short multiplet they can combine yeah so so you want to combine them in a longer multiplet but I think then the other multiplet should already be there so I think so if there is already another multiplet which it can combine then I think you are not in this in this case okay so so now I can jump to something different so what we want to put to use is let me erase something first so now we want to put to use this structure that we have uncovered to like to explain how to address the questions that we talked about in last lectures that is like given some supersymmetric field theory in flat space we would like to understand on which manifolds it can be placed preserving some supersymmetry and what kind of properties the resulting theory will have so the references for this part of the lectures so there is this paper 11 of 5 of 689 by cyber than myself and there is also a nice review by Dumitrescu 160802957 so let me start by talking about something somewhat trivial but which I think gives an idea of what is that we want to do in supersymmetric theory so we'll just give an example which is without supersymmetry so let's consider some theory in flat space now we can you can take your fabric theory maybe just some scalar field and couple it to gravity so once you couple it to gravity the metric becomes dynamical and your theory will be different variant so now the metric can be whatever you wish so in particular you can you can imagine your theory being on some manifold of your choosing with some particular metric and you can couple gravity by sending the Planck mass to infinity so now we can fix the metric and on some manifold M and the couple the fluctuations of the metric by sending the Planck mass to infinity so what I have done is that I have basically took my theory in flat space and I have coupled it to this like on this manifold with some particular metric so now I can ask what happened to the different variants of the theory with gravity while clearly fixing the metric breaks the different variants of the original theory but there are some different morphism that stay in particular all the different morphism which do not change the metric that you have chosen as background will be actual symmetries of the resulting theory so and what are these so if you look for a different morphism for which the change in the metric is zero that means that the infinitesimal parameter for the different morphism which is epsilon mu has to satisfy the killing vector equation these are just isometries of your Romanian manifold so this is a somewhat trivial example you don't need to go through this series of steps in order to figure out that if you have a theory and you put it on some manifold with some isometry then like it will have a symmetry corresponding to that isometry but so it's a way which explains what we want to do in the more complicated supersymmetric setting any questions so one topic which will play a very important role in the following is that of coupling some conserved currents in your theory to background fields so for instance in the example above once you fix the metric that the metric becomes a background field and if you look at the deformation of the theory from flat space then the the formation of the metric couples to the energy momentum tensor which is one of the conserved currents in your theory so this is recur during the next few lectures so I'd like to make some comments about this what happens with background metric under normalization under what normalization how do you plot so metric is a kind of couple collection of couples so I'm fixing the metric to be some background you turn on the group effects the logic might get normal upgrade is a couple so first of all stress on the tensor but I found the terms so it is true that in in general there are lots of you can always change so the coupling to the metric is not universal you can always add terms which are suppressed by 1 over r where r is some scale in the manifold so in principle this could be generated during the RG flow so the metric itself maybe not even observable the metric is the curvature the metric is not unobservable we take it to be a background field which we fix with couples to in your theory so if I change the metric so if I take my theory coupled to background metric and other background fields then I can compute its partition function on some manifold then taking functional derivatives of the result with respect to the background metric this will give me information about correlators of virus operator in the theory so what's the difference between metric and let's say the gauge coupling in Yang-Mills theory so you could say Yang-Mills coupling is also a kind of background field but we know it's not a parameter you cannot compute the function the function of the gauge coupling it becomes a scale if your theory is not conformed so how is the metric different for example could it be that as we go down in the RG flow the background metric has to become flat or inflective so it eventually becomes just a flat spot conversely there is some concentration of curvature and becomes similar this is the choice of scheme what they are doing they fix the curvature and they cannot change but then other quantities the way they normalize will be affected and you can really think of it as a choice of scheme of the way you calculate your loops it's like the opposite of true to pole mass scheme it's somehow the opposite of that when you say that it's a curvature that's presumed as constant curvature metric constant so that's a very special class of metrics one feature of you I think of what general for many fold has many features here it's one curvature and there it's another curvature but usually they would do sphere so they are very simple things that's a good discussion so it's a general discussion so let's go ask the general question ok, so let me continue but I'll go back to this to this question ok so now where was I left so let's I wanted to make some comments about the two background fields so as an example again without supersymmetry we can consider some theory with a u1 with some u1 conserved sorry with some u1 symmetry then this will mean that your theory will have some conserved current jmu and I can imagine conserved current to a background to a gauge field and then to make this gauge field non-binanical so ok, so then the structure of your Lagrangian will be left the usual the Lagrangian you started with and then you add the coupling of the background gauge field the mu to your conserved current but in general this will not be enough and so up to here this is invariant under gauge transformation of a mu because jmu is conserved but this is true at first order like at higher order you might have to add terms of other a squared which are usually called single terms so again these terms are here to preserve gauge invariance if you do this for a theory where say you have some theory of scalars with u1 symmetry then like these are the usual single terms in scalar qd another example is that that we talked a little bit about is that of the energy momentum tensor so let's take some theory let's take some conserved energy momentum tensor which is symmetric then we can couple this to fluctuations of the metric so if you take jmu mu to be flat space plus linearized correction then again the structure of your theory would be the theory you start with plus the coupling of linearized metric to the energy momentum tensor but again in order to preserve diff invariance at higher order you might in general have to add more terms and here we can also discuss what happens to this under improvement so as we said in the last lecture team mu nu is even when we consider just the symmetric and still be changed by improvement transformations so for instance it can be shifted by something of this form this is not the most general improvement but it is some improvement where u is some other scalar operator in the theory and then what this does to this variation is that it will add it changes t here but you can reinterpret the change in t by pulling the derivatives on h and then this is just working as with the old t but adding a coupling of the linearized richer curvature as a function of h to this scalar operator u makes the general point that by improvements I can shift the coupling to the metric by terms which scale down like one over the radius of the manifold to some power so the richer curvature of the manifold scales like one over r squared and indeed so the coupling of a theory to some space is not universal but there are all sorts of one over r terms that can be that can be changed so for instance they can be changed by improvements of the conserve currents so sorry in the s-max rate in the s-max set you have a real scalar a can you consider a similar I think you can consider similar linearized coupling in principle ok is the physical meaning clear in that crystal? ok so to be fair I don't remember what a exactly couples to in 16-16 superglality but in principle there is so a will couple to some operator in your theory sorry a is your operator in your theory it will couple to some field in the supergravity which one? but other than that if there be any Cisco meaning other than ok there is a coupling so there is some coupling to some field so that coupling if you do background 16-16 supergravity that coupling would be required to preserve supersymmetry so if it weren't there you wouldn't have supersymmetric field theory for one field like see me see me then the physical meanings are clear they are brain current for a 16-16 supergravity is a little bit more complicated anyway because there are more propagating degrees of freedom in particular both in the fc multiplet and the r multiplet a is determined in terms of the it's either 0 or it's determined by the trace of t ok so the other wow so the other example that I wanted to make before getting to the more complicated supergravity case is that of the so let's now add supersymmetry and again consider some supersymmetric field theory which has a u1 flavor symmetry some u1 global symmetry then that means that this theory will have a conserved current let's call it jmu corresponding to this symmetry but because the theory supersymmetric the current will be part of a multiplet exactly in the same way as the energy momentum tensor was part of a multiplet also just global symmetry will be part of a multiplet maybe this is actually an example I should have given before so this multiplet is a linear multiplet and it contains other operators beside the conserved the conserved current so it can be embedded in a super field j which satisfies the following constraints so it's a real super field which satisfies this constraint so we can write j in components so there is a bottom component which I will call j again which is a scalar real scalar then there are fermions little j and little j bar and finally there is in the top in the component with that and that bar there is the conserved the conserved current plus other stuff so again there are 4 pozonic degrees of freedom and 4 fermionic degrees of freedom and then this fields can be coupled to gauge multiplet preserving supersymmetry so again so the gauge multiplet contains a u1 gauge field a mu then there is an auxiliary field d which is a real scalar and then there are gauginos lambda alpha and lambda bar a dot alpha dot and well also these you can in a super field and you can write down supersymmetric interaction between the fields in the gauge multiplet and the fields in the linear multiplet so your Lagrangian will change by something which will contain the bozonic term we discussed above the coupling of the conserved current to the gauge field and we also contain the coupling of the auxiliary field d to the scalar at the bottom of the multiplet and finally it will contain like fermions and possibly there will be seagul terms so now we can we can think of making this gauge multiplet non dynamical so we can set fermions in the gauge multiplet to zero and give some value to the a mu and d but these in general will break supersymmetry so when is supersymmetry not broken so supersymmetry is not broken when the supersymmetry variation of my chosen background is zero and what that means is that delta of a mu must be zero but that's fine because if you look at the supersymmetry transformations the variation of a mu is always proportional to the gauge genus so because we set them to zero this will not give any constraint the same thing for delta of the auxiliary field it's also going to be zero because it's proportional to derivatives of the gauge genus so we only have to check that the variation of the gauge genus themselves is zero and that will give rise to some interesting constraint so for instance if we look at the variation of lambda then we get that so there are two terms so you get that the variation of the gauge is zero provided that these holds at least for some of the supersymmetry variation parameters so that means that if you choose a background which means that you choose some f mu nu and some d such that this combination is zero at least for some spin or zeta then that spin or zeta will generate supersymmetry transformation resulting theory and ok there is a similar equation for the variation of lambda bar ok is there any questions so far? If we don't go to the original gauge we have a little bit more more auxiliary fields can this help us a little bit or to have a bigger freedom for the spin or zeta that we have to pick no I don't think going I mean this you always want to write gauge invariant interactions so whatever thing brings you out of the sumino gauge I think could be irrelevant So when you say and you get another equation from lambda out of that bar Is that essentially going to be the complex time together where you get any new actual constraints? Well ok so if you work in Koski space then they are related by complex conjugation Ok so you don't get essentially any new constraints on d in the gauge field from the closing that the other gauge unit has finished the equation Who said that the background should be real? Again So if you work in Lorentzian spacetime with usual reality conditions then the background will be real But the background is just some cut way If you want some unitary field theory Ok but indeed like if when we go to Euclidean spacetime there is no reason where the background should be real So then and besides in Euclidean spacetime when you take the complex conjugate of zeta you don't get zeta bar So then like the background can be complex and indeed we will see it's important So there are cases where it's useful to think about complexified background Talking about Euclidean Koski makes much sense because it could be complex If it's just the perturbation of flat space as an engine I could have complex coefficients that's most important What I'm saying is that in relation functions of the operators to which my background couples I should allow all possible couplings to be able to really differentiate in respect to them But in a unitary field theory in Lorentzian flat space like the various operators can couple to these currents will satisfy reality conditions Unitary is a bonus for special slides If you want to go beyond that then you can consider also complexified deformations a good So then you should write another equation because they are truly independent I agree especially we will write both equations So now we can go to the supergravity case There it is That's much better So next let's consider the case with supergravity So what we want to do is to couple some field theory with some supercurrent multiplet to supergravity So we take some n equal 1 field theory or some suzy field theory in general and we couple it to the appropriate supergravity So at linearized level this coupling can be described in quite some detail and it will depend on the structure of the supercurrent multiplet that the theory has So there will be the original theory you started with and then there will be couplings of the supergravity fields to the operators that appear in the current multiplet So for instance you will have the coupling of the linearized metric to the energy momentum tensor but then there will be various bosonic fields in the supergravity and this will couple to various objects in the supercurrent multiplet which we will call ji So again in specific cases this could be the string current or the domain wall current or the arc current and so on Then I will have also couplings of the fermions So there will be the gravitino which couples to the supercurrent the similar coupling for the other gravitino and then there will also be other fermionic fields in your supergravity so those will couple to some fermionic operators in the current multiplet besides s And finally there can be Siegel terms which are there and you can discover supergravity Lagrangian Ok So some comments So apart from the Siegel terms which in order to work this out you actually need to work out the entire supergravity Lagrangian All these terms here are completely dictated just by the currents in your multiplet And in particular they can be described in theories which don't have Lagrangian So if you have some theory it has some supersymmetry it will have some supercurrent multiplet in the supercurrent multiplet these operators and they will couple in this way to the various fields in the supergravity So now So this theory it's like a supergravity theory so it has it is invariant under the transformations And these are parameterized by some spears zeta alpha which depend on the position you are in your manifold and its friend zeta bar alpha dot of x So now we want to proceed exactly in the same way as in the trivial example we discussed in the beginning So we will set our background to whatever background of our choice we will also set the in our background we will set the fermions to zero so we will set the gravitinos to zero And then so we choose a background so for the metric we choose bosonic fields in the supergravity background we will set the fermions in the supergravity to zero And then we send the Planck mass to infinity to the couple of the fluctuations of the supergravity fields So what we are left with is some theory on some manifold with some metric and there is going to be also various couplings of these fields that appear In general, this procedure breaks all the local supersymmetry except as we discussed in the previous cases if there are some of the local supersymmetries that keep the background invariant then these local supersymmetries will remain in the theory once with the frozen supergravity fields So in order to figure out which supersymmetries remain, we need to figure out what are the conditions for this background to preserve some of the local supersymmetries But luckily enough this is not a very daunting task what we have to check is the variations of all the various fields to be zero but the variation of the bosonic fields in the supergravity are always proportional to the fermionic fields in the supergravity so those are not going to give us any equation to solve so we only have to check that the variation of the fermions is zero and in particular for the case of the minimal supergravity that we will be concerned with we just have to check that the variation of the gravitino is zero and its friend So now this equation has a general structure so that we will see born out in examples so it starts with the covariant derivative of the spinor parameter z and that has to be equal to some matrix m which depends on the metric and the values other background supergravity fields which acts on z and then there can also be another piece let's call it m tilde acting on zeta bar and there is a similar equation which comes out of the variation of the gravitino with bars so basically the task of finding which manifolds allow for some super symmetry just becomes the task of finding for which values of the metric and various auxiliary fields you can solve these equations or you can find some solutions to these equations So now I would like to make some comments on this equation and the general structure of them so one important comment is that this equation does not does not really depend very much on which theory you started with because what these objects depend on are just the background supergravity fields so there are no fields there are no matter fields inside these matrices because if you work off-shell in the supergravity variation of the gravitino you only have fields in the supergravity multiplet appear you don't have fields in the supergravity so you can solve this equation and then the results that you have will apply to many different theories not just one so now this is so in some sense finding super symmetric backgrounds is independent of the theory you start with except that this is not a completely correct statement because as we saw depending on the theory you start working and the supergravities are not equivalent of shell so depending on the theory you have if the theory is an fz-multiplet you will use all minimal supergravity and then you have one equation and you can find a certain set of backgrounds which are super symmetric if the theory is an r-multiplet then you will have to use new minimal supergravity and you will find a different set of equations and a different set of solutions ok any questions on this you could decide to just work with conformal supergravity if you wanted to couple conformal field theories but I think if you just compensated you can also couple non-conformal yeah I think that's equivalent to working with the non-conformal supergravities so but I think it's actually important to use the non-conformal supergravities in actual first of all you might be interested in like coupling non-conformal theories but if you are just interested in conformal theory many times I mean to do some computation you might have to introduce some regulator and then this regulator will like introduce dependence on the will like break I mean usually breaks conformal invariance so then in order to so then I think it's more appropriate to use the non-conformal couplings so for instance if you compute some partition function using some regulator, some regulator procedure like there will be this will depend on the on the background fields but the possible dependence on the background fields will be like will have to follow from like some there will be the respect gauging invariance so they will be dictated by some supergravity and the appropriate supergravity I think is the non-conformal one so it's true that you can choose the scheme carefully so that like you will be left only with couplings which preserve which are conformal but that might be difficult in any specific scheme so in the open physical questions are asked for conformal conformal theory and for such questions that it makes sense to try to without answer in terms of conformal supergravity somehow people don't not many people don't seem to many people seem to do this I don't understand why is there a good reason no but as I said I mean if you are just concerned with conformal field theories then you can use conformal supergravity up to this comment that like when you actually do some specific computation you might have to introduce a regulator and this may break conformality so Is there a more supersymmetric way of analyzing this? Yes there is so supersymmetric you mean can you work in superspace? Yes so there is a way which to encode all this in some superspace formalism and I think you can actually find some of it in this book by Kuzenko which is called a walk through superspace so I don't remember but it has some comments about about how to get supercurrents and how to yeah so I have five minutes so I don't know what can I do in five minutes let's see well ok so I can give you I guess an example so that this does not seem to dry so let's see what happens for a theory which has an fz-multiplet so then as we said this theory will couple to naturally to old minimal supergravity so then we can be a little bit more specific so first of all let me say that I'm going to write the couplings so you have some theory with an fz-multiplet so the fz-multiplet that we discussed in the lectures we had this super field y alpha but we said that in many cases y alpha can be written as d alpha of some chiral super field x so this is the case I'm going to consider so then the couplings that we've wrote in the previous blackboard are going to be the coupling of the linearized metric to the energy momentum tensor but then you have some coupling of some auxiliary vector field in the supergravity which we call b mu to the non conserved arcurrent which appears in the fz-multiplet so this is a supergravity field which maybe I can write the supergravity fields in a different color so that's a field in the supergravity which is an auxiliary field then we also have couplings of some other complex scalar auxiliary field in the supergravity to the bottom component of the multiplet x and in this particular case you can also write down what are the equations that come from setting the variations of the gravitino to zero so here they are maybe the best idea and there is another one for the right-handed you know so here they are so as you see the general structure that I advertised like is indeed borne out by this example this equation they only depend on the supergravity fields so now let me make some general comments which actually do continue to be true in other examples so if we look at some specific manifold and you solve these equations then you can indeed check that there is a scaling all these fields all the auxiliary fields in the supergravity multiplet m and b in bar in this case scale like 1 over r so that is just given by their dimensions so and again this same field m and b will also appear in the transformation laws of matter as we will see in the next lecture so what that means is that in the uv the theory that we've wrote approaches a susi theory in flat space if you want the original susi theory in flat space so in by using this formalism you are not gonna obtain the formations of the theory that you started with which are present even in flat space so there are the formations of the supersymmetry of some theory that might be there even in flat space and we are not gonna obtain those in this way and the other thing is that again as we discussed before if you improve the fz-multiplet then this will change the body's currents which couple to these fields and this will introduce the difference is gonna be in the couplings which scale as the radius of the manifold so that's exactly as in the bosonic case and the final comment that I wanted to make but which I've basically already discussed is that it might be important to consider cases in which like the auxiliary fields or whatever the fields in the background supergravity are not real especially when you consider the Euclidean case where like the spinors are not related by complex conjugation now however to be fair I never consider like the case of a complex metric so that's might be something that some of you can think about that's it