 Yeah, now it's now it's on. Okay. Very good. Yep. Okay. Great. So we are happy to have Stefan van der Rijn the next speaker Thank you. Welcome everybody I'm still a bit sleepy because I arrived in the middle of the night. So I'll try to be as sharp as possible and To make me even be awake further. Please do allow a lot of her or please do ask a lot of questions So that makes it more lively and I get more feedback Very good. I see my lectures a little bit as Serving to the other lectures in the sense of building up and introducing certain topics that will make certain following lectures more accessible and so I thought to Really give some basic introduction also to aspects of supergravity and also black holes Of course when we talk about black holes and supergravity, we first have to learn a little bit about supergravity So I thought I spent some time teaching you the basics of Supergravity with and without matter couplings and then also discuss some black hole solutions in it and so I tried to do this in a kind of like sort of Little bit of a from a new point of view or a different point of view as you might usually see also to to well To make it not too technical Probably everybody has seen once this book and you can just lecture 500 pages or thousand pages about supergravity. So I kind of like distilled what I needed from it and Well, let's see how it goes So let's start with something really simple before we go to n equals to supergravity I want to go to just minimal supergravity and in fact even let's forget about supersymmetry for the moment Let's think about the gauge principle where the simplest set up is that we have a gauge field and a gauge transformation which involves a gauge function epsilon which is a scalar function and Of course, we know how to make gauge invariant quantities by constructing field strengths The action is then just f mu nu squared and this is all of course well known so what you can think about now is is is to Change this principle of a u1 gauge symmetry where you promote this epsilon this gauge scalar function into a spinorial quantity I want to change this into a spinorial quantity by Trying to come up with a gauge principle that has a spinner Epsilon is a spinorial quantity and so if you do this, of course, you have to change something on the left hand side as well and so well the most kind of obvious thing to do is to add here make of this thing a spinner as well and We give it of course a different notation It's going to be the gravity no later on but for the for the present remark I don't even need to be in there has nothing to do with gravity at all. So This spin zero became spin one-half and this spin one became a spin three-half and of course because this is spinorial it has to be anti commuting and and This field here is a spin three-half field and you can make similarly a gauge invariant combination Now I suppress the spinner indices This here is clearly invariant under this spinorial gauge transformation and of course you cannot repeat the same trick in making an action say this is F mu nu squared This we cannot do F mu or if you call this for the moment that this this you can call this psi mu nu You cannot you cannot make an action with psi mu nu. We know that for spinors. We need first order actions and with first order actions They only need one derivative and so the action that We write down for it is the Rita Schwinger action. It was I think written down maybe even in the 30s or something You can do this in arbitrary dimensions and so the new Siro This is a Rita Schwinger and it is gauge invariant because for the notation If you want to look up the pluses and the minuses I followed this book here by Friedmann and von Poehen which I recommend for both students and staff and This is of course on the symmetric so that means that here only the field strength for this spin three-half particle is Is is there? So this is a spin three-half particle And so people have been playing with this action before the days of supergravity even Either in Schwinger And so if you count the number of degrees of freedom of shell degrees of freedom then Yes, because This is invariant under this transformation and here only the anti-symmetric part is Is present because this this gamma mu nu is essentially the product of three gammas totally anti-symmetric on the three indices Yes, but you can do partial integration here You get a d mu you partial integrate and then it's this and then it's the d mu d nu that that makes it invariant. Yes, very good Very good. Please do ask. Yeah Even if you know the answer very good host choke number one Very good, let's count the degrees of freedom a spinner has of course Dimensions two to the power d over two the even part or the integer part of it. That's the number of components But then this in this this this has a this on top of it It's a Lorentz vector, so it has d degrees of freedom extra So you would multiply this by d, but you have to subtract one because you have one spinorial One spinorial gauge degree of freedom that you can subtract so that the spinner Is again two to the d minus one So you have d minus one times two to the d over two off shell degrees of freedom and then d equals four We have 12 so an off shell Gravitino has 12 degrees of freedom This is my arena. That's also important Indeed in n equals although this is not yet supergravity, but in n equals one supergravity We're going to have one gravitino, which is my arena in an n equals two. We're going to have two my oranas or one dirac Fermion Okay, so now I can couple to gravity and we know of course that if you couple to gravity We're going to introduce wheelbinds And these wheelbinds from these wheelbinds you can construct a metric Can use this here and so the metric is constructed out of a wheelbind just to set a notation and you can try to Assign a transformation rule because we know that this is going to couple and so now I'm going to Enter supergravity And so the wheelbinder on the supersymmetry has the following form Well, you can ask how was this derived that that is a little bit more difficult to explain Essentially how it historically was derived was by trial and error and Here's the spinor Super supersymmetry transformation gamma matrices and the gravitino so the graviton transforms into the gravitino And so that means the delta g mu nu Is just epsilon bar gamma mu Sign u symmetrized And so we have a metric here and the off-shell degrees of freedom of a metric is We have a symmetric tensor. So that's one half d d times d plus one But this metric also has general coordinate transformations just like a general relativity. These are d So you can impose d gauge conditions so then You subtract d and that is then of course one half d times d minus one and in d equals four We have that is just six So if you want to build the supergravity theory Having equal number of bosons and fermions you see immediately at least if you want to do that off-shell You see there's a mismatch here. We have six bosons and we have 12 fermions. So that doesn't match But of course, this is an off-shell counting and if you just want to do on-shell supergravity It's sufficient We know that off-shell This six goes into the two polarizations of the graviton on-shell And this 12, this is an exercise that you can do this 12 on-shell Also goes into two That I won't do here in this class because it's a little bit playing around with the equation of motion for the Rarita-Schwinger field It's standard techniques that you also do for the Dirac field except you have more indices So you you do more than with Dirac just happening the degrees of freedom Um And so you can look up the argumentation in this book or ask me separately if you If you want to know how to count the degrees of freedom of a spin the on-shell degrees of freedom of a spin three half Then the answer is two try it first yourself if it doesn't work, please ask me So on-shell we do have a matching And of course the first supergravity theories were constructed by well Friedman Ferrara for newheisen and so I'm going to write down the answer and restrict myself now to four dimensions So n equals one supergravity is a minimal supergravity. It's not coupled to anything yet There is no matter But I'm just writing this down also for the purpose of setting the stage explaining the conventions This here is a covariant derivative is no longer a flat derivative And the covariant derivative also involves the spin connection and the spin connection is not a new Is not a new field it's determined in terms of the wheel binds By formula that probably I guess you all have once seen and so The variation of the Because this is no longer a flat derivative The variation on the supersymmetry of the gravity, you know is something like And here you have the spin connection epsilon So that is this d new effect And of course, uh, this is going to be important when we discuss also bps solutions because bps solutions Are solutions to the equations of motion that preserve some supersymmetry. That means it's annihilated by um, a particular epsilon Um, on the right hand side here Now this is n equals one supergravity It contains just the ordinary gr plus the spin three half field If you're interested in black hole solutions that have no fermions excited Then you look for solutions that have the fermions set to zero in this case There's only one the spin three half you back to gr The schwarzschild solution and the curve solutions are solutions, uh of this, uh equation of motion of this theory And uh, but they don't preserve any supersymmetry. This there is no epsilon for which, uh, uh, it leaves The solution invariant also even the the extreme curve Solution is not a bps solution of um, uh of this theory You can also, uh, add additional terms and construct again with the same number of fields can construct a Supergravity theory that doesn't live in In asymptotically flat space, uh, but it lives in anti the sitter Uh space so you can extend this So let's call it ads supergravity Or if you don't yeah, well ads supergravity Is it just a deformation later on? We will see this deformation is a gauging And ads supergravity is let's call this as as zero Then s equals s zero plus one over two kappa squared integral d four x Square root of g or the determinant of the wheel binds also sometimes used and then we have One over l is a deformation parameter. This looks like a mass term for the gravitino And then there is a cosmological constant And if you write down the equation of motion, then, uh, you will see that until the sitter space is a solution of this With a cosmological constant cosmological constant so r mu nu Equals uh minus three over l G mu nu so we say that lambda is equal to minus three over l Being negative Okay, of course when you deform the action with a parameter l here that that Well, this isn't very important for on the supersymmetry for any l, but you have to adapt the Transformation rules and for instance the the transformation for the View bind remains the same but we get for instance additional terms like this here d mu Minus one over two l gamma mu epsilon And that is the supersymmetry transformation rule that leaves this this action invariant, okay Good are there any questions about this? good, um, if not, so this is the story basically that Uh, is a story of the of the 70s When supergravity was first discovered and I did hear an off-shell counting and off-shell matching of the degrees of freedom But what happens if you want to construct off-shell methods? This is useful for both Well applications to do with localization. We're going to need an off-shell formulation Uh, in fact also in field theory in the localization program You often need the gravitational background as a as a background not not propagating, but you all you often need off-shell versions of of supergravity And so this will be important also for the black hole story and the lectures of Of some near murky in particular So, um, you see with the with the off-shell we have a problem because we have six Uh, we have six bosons and 12 fermions degrees of freedom. So there's a mismatch. So how to restore this Um, I want to explain this Uh, of course later in n equals two, but at this moment, uh, also in n equals one And this makes use of a program that is called the super conformal calculus Now this is something very technical many people would run away. So maybe we'll lock the doors um, and um I'm not going to give all the full details. I want to give well the ingredients of the mechanism and um Emphasize the most important points So the off-shell methods We're going to use super conformal methods or calculus also called So of course conformal symmetry is something very important in our, um field of research also for field theories So some of these aspects you have even might have even seen or learned about just in the study of super conformal field theories without gravity And so what is the idea here? So well first of all before let me repeat that the problem was that we had That we had six bosons and 12 fermions. So there's a mismatch. We need to introduce auxiliary fields and so, um Um Historically people tried by trial and error to add these auxiliary fields and so we need to add essentially Six auxiliary fields six bosons And that can be something like a gauge field a massive gauge field It would have four degrees of freedom and then a complex auxiliary scalar field I will I will I will make that more explicit in a moment Uh, or you can add a a gauge field with a gauge symmetry and An tensor field And if you do the counting then you get an off shell matching, but let's see how that comes out So I want to explain here what the super conformal method is the idea is to construct super gravity theories That first have a much larger symmetry namely super conformal symmetry The hope is that when you have more symmetry symmetry is more restrictive. It's more constraining So it should be easier to write down actions for it and and How then to construct ordinary? Super gravity well, that is by gauge fixing procedure gauge fixing all the generators That are in this algebra for instance the dilatation and later on the r symmetries that are going to be Um Part of this algebra, but not of the Poincare, but we'll we'll we'll do that in detail Um, good. So let's first get the main idea here forget about supersymmetry. Just focus on gravity by itself So we can consider consider the action The action based on the Lagrangian L is square root of g or minus g And then we have one half d mu phi d mu phi Plus one 12th r Phi squared and if you look up the conventions about the metric here, this has the wrong The wrong sign kinetic term, but uh, that's not important for the present purpose or not too much This action here is gravity coupled to a scalar Hitchy scalar times the scalar field here and it has a dilatation or a scale symmetry And the symmetry is delta of Phi equals lambda is the parameter from the dilatations times phi and delta g mu nu Is minus two lambda d G mu nu you could check explicitly that this action is invariant under this scale symmetry And so whenever we have a gauge symmetry, we're allowed to do gauge fixing And it's kind of clear how we should gauge fix to get general relativity out We should fix phi to a constant then this term disappears And if phi is a constant then this becomes Newton constant Of course, we know this also from string theory where we uh, often This is called the dilaton field then or the exponent of it So we gauge fix since we have a local symmetry this this parameter is allowed to be dependent on x as we can gauge fix And then phi is equal to square root of six over kappa Newton's coupling constant And so, uh, then of course we get that l equals one over two kappa squared times square root Of g r Alternatively, you can do something else You can start with gravity Einstein general relativity and introduce a scalar By saying, um, well, we're going to take grab or alternatively we can take gravity and introduce um Or introduce g mu nu prime by definition kappa squared over six g mu nu times phi squared This combination here Is invariant under dilatations because here the weight this has weight minus two This has weight one So if we put phi square here, this is invariant under dilatation And if you define this combination you rewrite the action You'll figure out that you can rewrite the action in terms of g mu nu prime And then you just get Einstein-Hilbert term on the nose That's not a surprise because this is how it was constructed. You can start from Einstein-Hilbert action Write this metric in in this form By introducing a scalar field and rewriting it such that you introduce now dilatation symmetry given by this If you plug it in in the Einstein-Hilbert action, you get back this You can see it reversely by introducing a scalar field that compensates for the scale transformation that Einstein-Hilbert does not have That's why this phi field is called a compensator Yeah, so this is a simple trick to to make Invari actions invariant under under scale transformations If you had a theory that was not invariant on the scale transformation You could make it invariant on the scale transformations at least at the classical level by introducing compensators that account for The right scaling weight in a scale invariant theory I'm not doing anything quantum here. I'm also not talking about conformal gravity in the form of Well, you can call this conformal gravity, but usually People call conformal gravity based on the wild tensor. The wild tensor square is a higher derivative theory That is also part of the story once you include higher derivatives into this game You will get the wild tensor square. I don't know if you will talk about higher derivative terms Maybe touch it. Yeah So I will not do this. I will not include any higher derivatives in my lectures here Good. This is the main idea of the super conformal method. Just add some fields to compensate for The lack of scale invariant or start with theories that are conformally invariant and gauge fix all the all the transformations that Or the symmetries that are not part of coordinate transformations We gauge fix them such as you end up with just coordinate invariant theory Now we have to just repeat this game in supersymmetry The idea is still as simple But the technicalities become a little bit more involved Because everything has to be well sitting in multiplets in off-shell multiplets So this compensator needs to be part of a multiplet itself And in n equals excuse me In n equals 1 supergravity that is still quite simple But in n equals 2 supergravity that is a little bit more Involved So let me say a few methods about or a few let me make a few remarks about super conformal symmetry Because that's also useful perhaps for Your general knowledge So there's a super conformal algebra and the generators Of super conformal algebra Kind of can be depicted as this It's a super algebra. So it contains bosonic and fermionic generators Here it contains here Here there is the conformal algebra Which in four dimensions is just SU 2 2 which is SO 4 2 Up to double coverings but these double coverings are important when you discuss fermions and we are discussing fermions And here are also bosonic generators and there are symmetries There can be one or there can be many And here are the supersymmetries we call q But the supersymmetry that's not the only fermionic generator The supersymmetry the supersymmetries are kind of the super partners of the general coordinate transformations But the conformal algebra also has all their generators even apart from the dilatations also has special conformal transformations These special conformal transformations have their own super partners. They are called s supersymmetries And these generators here are on the off diagonal because they transform if this acts on bosons fermions Then a q and s transform bosons into fermions. Whereas these They transform bosons into bosons So in n equals 1 d equals 4 This super algebra well has the notation su 2 comma 2 slash 2 And it contains as a bosonic super algebra contains the conformal algebra Which is isomorphic to s of 4 2 Times u 1 And that is the r symmetry called the r symmetry 1 yes i'm already in a In this n equals 2 setting very good in my notes. It's fine So the conformal algebra Contains of course the translations. There's four the Lorentz transformations that six Then there are special conformal transformations. That's another four and then there's dilatations that one So these are the generators are pa for translations the Lorentz transformations m a b the dilatations Well called a d and then we have special conformal transformations k a And of course only the 4 plus 6 is part of a Poincare theory So we have to gauge fix away both the um special conformal transformations and the dilatations And also the r symmetry r symmetry is not is a symmetry of the algebra But it's not a symmetry of a Lagrangian or at least not necessarily a symmetry of the Lagrangian But in a super conformal theory these are symmetries of the Lagrangian That's why we can gauge fix them if we gauge fix the super conformal theory We obtain Poincare symmetry where of course these symmetries are gauge fixed. So they're no longer symmetries because they're gauge fixed So r symmetry is not is not not necessarily a symmetry of a Poincare super gravity theory So now you have to make this is the algebra Maybe you can write down a few commutators to make that a little bit more visible For instance the ones you're least familiar with I guess is the ones that involve the s super symmetry So if you take for instance, well a special conformal transformation with a super symmetry I'm not writing spinorial indices here. Then this is gamma a gamma matrices times s For instance, if we do translations commutator with special super symmetry Then that's gamma aq Um, what else did I write down? Oh, we can have the s alpha s beta Now there are spinorial indices That's minus one-half gamma a alpha beta They anticommute into the special conformal. So that's very different from the qq anticommutator Anticommute into the translations these Objects here anticommute into the special conformal transformations And then perhaps one more q with s Anticommutes into the dilatations. That's a generator. That's a symmetry that sits in here Minus one-quarter gamma ab alpha beta times the Lorentz generators And then plus i over 2 gamma 5 alpha beta times t and this t is just the u1r The generator of u1r. So you see how everything mixes and forms one big algebra called the super conformal algebra So what we want to do now is to construct a multiplet That on which this super conformal algebra acts In other words, we want to find a realization of this algebra in terms of fields The next step after we construct the multiplet is to construct actions with it. But first the multiplet So we have a lot of gauge Symmetry is here And we know from poincare or from the gauge principle that once you want to make a symmetry local You have to introduce gauge fields And so, uh, let me make a list Let me make a list these these gauge fields they form a multiplet called a while multiplet and um So the gauge fields form the while multiplet Um, and let me make a table So we have the translations pa mab Dilatations special conformal transformation r symmetry And Then we make q and s Let me not make these sorry. I like to get the cosmetics, right? These are the bosons here and these are the fermions So, uh, what are the gauge fields here? Well for translations, we introduce a metric or or the field bind emu a For the Lorentz transformations while we introduce the spin connection But you know that the spin connection. It's actually not a independent field. You know the spin connection Uh, if you you can impose a constraint. It's called the tate hat postulate That's already in the absence of supersymmetry spin connection Unless you introduce torsion, uh, the spin connection is not an independent field. It has no dynamical, uh, physical degrees of freedom by itself They're completely determined in terms of emu a but we'll give them these names The dilatation It's one, uh, symmetry So the gauge field is just one vector field just like a u one And then here we have special conformal transformation. We call the gauge field f mu a and um, we call this here a mu And this here is of course going to be the gravitino And then there's a gauge field for S supersymmetry, which is also spinorial field psi mu So the degrees of freedom off shell Can be counted as follows now, um When we do an off shell count and we don't only count the number of components of fields We impose the gauge symmetry as well So we can use the gauge symmetry to subtract or to gauge fix certain components away We do not use the equations of motion. That means on shell So I already told you that, um, uh, uh, gravitone has six degrees of freedom But if I want to I can use the dilatations already to impose a I can I can make various gauge choices and later on I'm going to do another gauge choice But I can use the gauge the gauge transformation to reduce the six degrees of freedom to five And here I'll write down what I have gauge fix general coordinate transformations Plus dilatations then we go from six to five This year was not an independent field It's not an independent field. So it has no degrees of freedom Be yes Yes, but in supergravity indeed, there is a torsion Maybe I didn't say that the very precise enough, but the torsion is induced by fermion bilinears It's still the is still not an independent field the torsion component and equals one supergravity Is bilinear in the gravitinos and so You can still call that torsion But but my main statement is that there is still it's still in Uh, uh dependent field the the the the spin connection here Yes, yes, yes Yes, it can be written there's a formula that Expresses this in terms of the spin connection and in terms of fermi gravitino bilinears I could give you the formula in the break. Yeah Yeah, that is torsion. Yes. Yes, you could also the reason why I brought up torsion is that Uh already in general relativity you can introduce torsion in the game And of course the counting works differently in the bosonic part of it And here there's torsion, but it sits in the gravitino bilinear Yeah, uh for this one here Um Well, this is 16 So it's symmetric. So it's 10 In let's do it in four dimensions. We have 10 10 minus four coordinate transformation. That's that's six and that dilatation that brings it to five b mu here has d or four component fields But special conformal transformations are also a gate symmetry. I can use them actually to gauge away This b mu. So I'm very often one one picks a gauge where this b mu is zero So then there is no off-shell degrees of freedom here because I fixed here The special conformal transformations now A little bit of a longer story here, uh to see that f mu a is also not an independent gauge field There's lots of gauge fields here that are Not independent well the spin connection you all know but these this one and this one turn out also to be dependent on the other fields How can you notice well, you cannot really notice in advance what people do is you can you you construct actions and then you see that The equation of motion for f mu is algebraic It's like an auxiliary field almost It's algebraic and when you eliminate the equation of motion Then you find that f mu is a function of the other fields You can do in fact the same with the spin connection if you would not impose a tetrad formulaism and Postulate this one you essentially figure out that you can Use the equation of motion start with two independent fields and then use the equation of motion the equation of motion Will put omega as a function of e so that happens for this one And this one as well if you want to understand a little bit more in detail Then uh, you should really consult this book. Otherwise, I will spend too much time on this. Yes Yes Correct. Yes. So these different constraints I had exactly the same question when I was preparing this lecture and luckily better. David was still in utrecht So I could ask him like why should I choose this constraint? I don't want to choose anything What what happens if I choose something different? And he says, okay It's still the same principle. There will be the equation of motion, but it will be late Your our two choices will be related by field field redefinitions Well, you have to make sure that your algebra is of course holds of shell And then these constraints are usual curvature constraints And At least the way I understand the topic is that there's there's no Different choices that lead to different theories the different choices that you will make lead to Physically equivalent theories and the relation are just field redefinitions. They can look Complicated depending on what precisely we choose, but if you have a concrete example, maybe we can discuss afterwards Yes, very good How shall I phrase this? I can be of shell, but we still don't count this as dynamical degrees of freedom if you are an n equals 1 Well, let me Yeah That's right. Yes. Yes. Yeah, but that's still an equation of motion That's right. Yeah, that's the language in which we usually we impose a constraint Which can later on also be understood as an equation of motion But I agree that's a little bit confusing So I would like to think a little bit about this to answer this more more sharply. Yes Here we have a gauge field and that's three and Here this here Would have 12 degrees of freedom of shell, but we have more supersymmetry. I have engaged fix the s supersymmetry. That's the spinor So you can reduce four components out of the gravitino Further down to from 12 to 8. So here I fixed Both q and and s okay Yeah, I think the answers now Let me not tempt because I might give a wrong answer that these equations of motion are so special that even if you impose them You are still off shell if you you don't need the equation of motion in the To close the super algebra for these fields. I think that is the answer and so You're kind of both imposing question of motion But you're still off shell from the point of view of the closure of the super algebra But I will double check that to you Good. So now you see we have a matching because we have 8 here and 8 here Now this is not the end of the story here because With this this is a multiplet. It's called the wild multiplet But with this multiplet by itself you cannot make an action It's not because you have a multiplet that you can write down an action We've seen this already in this toy model for gravity Because we needed to in order to make the theory invariant We needed this compensator this compensator was a scalar field. There's no scalar field in here So to make or you can also say phrase that in different ways And this is maybe a chiral multiplet and in order to make a full density you need to Multiply it with another Anti chiral to have a full density I'm not going to go into too much detail At this moment. I think from the toy model Gravity you see that that we need a compensator the compensator is not part of the wild multiplet And so it is a multiplet by itself And that multiplet because it has a scalar field the most canonical thing of course to do is to Introduce a chiral multiplet compensator There's there's different choices for the compensator one is a chiral multiplet if you go to the gauge fixing You get old minimal supergravity if you use a tensor multiplet and you get to new minimal supergravity And let me for the moment just do the chiral multiplet So a chiral multiplet contains a complex scalar A spinner two components spinner wild spinner And an auxiliary field complex field f And so now i'm going to make this exercise again Now including Including this compensator and now the counting was somewhat differently Yes, uh for the dilatations. Yeah Well, it will do the entire job So, um So let's add this compensator here and do this exercise again We've seen in the in this toy model that in fact, I use the dilatation symmetry to to fix this scalar into a constant Yes, but I had already here fixed it fixed the wheelbine. I cannot fix fix the symmetry twice So i'm going to drop it here and turn this back into a six Yeah, then um, so i'm going to now add this list here to this list. I'm going to add Phi lambda alpha and f So how does it change then for me? It's easy. I just use the eraser here. You could just copy it or I will probably put my notes online as well or you can watch the video So here we make it a six and we only fix the general coordinate transformations um and so Oh, I'm sorry. I should put this here Phi lambda alpha and f So this is a complex scalar field if I have dilatations then if I have dilatations then um I can fix the real part of it or I can write this as a real part times a face And this complex scalar field is charged So it has a it has a weight under this one here And so i'm not going to eliminate a degree of freedom here I'm going to use the u1 symmetry from here and the dilatation from here to gauge fix this scalar If I do that I cannot go from four to three here So this remains a four And this becomes a zero What I have gauge fix here is that dilatations plus u1 similarly this eight here Was arising because I was using gauge fixing both q and s symmetry And so if you just use q supersymmetry, then I had 12 off-shell degrees of freedom And I'm going to use the s symmetry supersymmetry not to gauge fix the gravitino That would not be so smart because I want to have the gravitino as it is in point here I'm going to gauge fix this one away So then that this becomes a zero as well and I've used s supersymmetry to gauge fix the fermion in the chiral multiplet Then i'm done because this one stays there And that's two degrees of freedom And if all goes well, I should have now off-shell Equal bosons and fermions six plus four plus two is 12 and 12 from the gravitino So there's an off-shell counting of the While multiplet already including the compensator Well, if you go but that then then it's an on-shell. It's a little bit the same confusion as what I had here Of course if you write down actions for this this is an auxiliary field If you don't couple it to anything then this f is just zero But off-shell it is it has two components and this one you need for the closure of the super algebra Yes, well you have to go through all the details here And I that problem is going well It's not really a problem, but this this elimination of some of the auxiliary field is going to become a complicated exercise In fact also in the n equals two story Where I have lots of gauge fields that I can eliminate and they become functions of the other fields And then you have to figure out what that relation precisely is But for now I'm just giving you counting And so the action well, I'm going to write down the action You'll see what the equation of motion of f is it's still going to be zero And so the on-shell the on-shell action Sorry off-shell Supergravity This is called old minimal Is just s equals d 4 x square root of g 1 over 2 kappa squared r minus Psi bar mu gamma mu nu rho d nu psi rho Plus six a mu a mu Close to the bracket minus f times f bar You say a few words about it. This looks like I have a massive vector And that's precisely what it is But it doesn't couple to anything, but there is no gauge degree of freedom. How could it? How did it arise? Because this scalar field was charged under u1 And if it's charged under this u1, but I gauge fix it remember that The real part of it I gave an expectation value Yeah, our gauge fix it to a constant and of course in its kinetic term There is a d mu phi d mu phi covariant d mu So that's going to be an a mu a mu phi phi bar just like in the Higgs mechanism if you want So you gauge fix that and this term this term remains And this is just the ff bar term that you have for any of a chiral multiple of shell and in this particular case The eliminating it through this equation of motion is trivial Very good Are there any questions? Yes, yes Yes, yeah, that's right Are there any other questions? well, but that In this view you cannot do that because that's Against the equation of motion But yeah, you can I will do gauge supergravity later on that's one layer of complication after the other So you see here that this was the action that the on-shell action I wrote down before the equation of a mu is a mu equal zero So this whole exercise was pretty in some sense Well, you didn't gain much except that you have an off-shell formulation here. Okay So now uh, we repeat the same exercise now in a n equals two setting. So Um Let me first. Yes Oh the new minimal you start here with a tensor multiplet. So a tensor multiplet has what has a tensor multiplet has a tensor a real scalar and a fermion And now what you do is now you have a real scalar so you can only impose a gauge fixing condition on um On the on on on the real parts or the tensor remains And so then you're still left with the u1 symmetry. I cannot use it to gauge fix here So this mu really becomes a gauge field there and that's new minimal So all these old formalism the old minimal and new minimal they have a very natural interpretation Well, if you call this whole program natural, um a natural interpretation in terms of the super conformal method So the idea is make a bigger theory with more symmetry at compensators gauge fix and you get primary supergravity off-shell Only yes, uh, only if you go through this, uh Super conformal method. Well equivalents in the sense of um on shell you start to see that they become equivalent off-shell They use different compensators and then they become well classically they become equivalent I don't know what the status is about about the quantum theory. What are what are these these these? Differences show up. Maybe addition or something about it or somebody else So when you used to do rigid a super gravity to couple to some fields then the question is This is not completely quantum, but it's sort of something in between right you use a background an off-shell background to couple to For theory and the question is then again whether Um, I don't know what I understand the question. Um I don't know if the answer is known and if it is known. I don't know it Um, so I don't know I am absolutely sure the loop corrections have been studied in in in Both on shell or old minimal and new minimal, but I don't know what is the status of the art I I I think as far as people could judge they still lead to equivalent theories. Otherwise we this would have really been Emphasized more in the literature. I think yes, but maybe maybe some people in the audience know better Yes Well, there's not gauge invariant anymore. It's it's it's it has four degrees of freedom Um, so it's not it's not gauge invariant Well, it was gauge invariant in the theory where it was not gauge fixed But I gauge fixed the u1 and the dilatations such that Um, it is no longer gauge invariant. That's not a problem here because it doesn't couple to anything It doesn't give charge to anything and so it is just an auxiliary is just auxiliary here Or algebraic if you want. Well, I think it is the right way of saying it stuckelberg is uh before you do the gauge fixing Yeah, but not here anymore. Yeah, or maybe that's still called stuckelberg. Maybe it is. Yeah Okay, um, I have two minutes something um I just want for fun to um, say in two minutes the n equals two because now um Then we see what what will come to you tomorrow or no this afternoon So an n equals two super gravity It's just a simple remark that I will make here Well, we have of course two Gravitinos i equals one to two. That's what n equals two means And there's still only one metric Let's not worry about off shell off shell will come this afternoon So this had two degrees of freedom So on shell This had each gravitino on shell had two degrees of freedom It was this exercise that we didn't make here. There's 12 off shell But two on shell but now I have two of them. So that's four This one still has two so on shell. There's already a mismatch here. And so you have to compensate that or add another field That compensate for this and of course What immediately comes to your mind is an honest to god gauge fields Which in four dimensions has two on shell degrees of freedom And so now we have four plus two plus two That suggests already that n equals two supergravity at least without any matter couplings. This is called the gravity photon Because it's part of the gravity Multiplet that the minimal supergravity not coupled to any other matter Is just Einstein Maxwell Coupled to and then added to it. It's been three half particle And in Einstein Maxwell, of course, we have the reichner-nerston black hole and And we can start thinking about black holes and extremal and bps and so on But I think I should I should stop here and continue this afternoon. Thank you