 Σό, ε panels of this paper here, there is that prove that indeed both modes are normalizable, So when the questions of whether both boundary conditions, I mean, we can import both boundary conditions in principle and the corresponding boundary terms so for keeping the charge fixed, which is the boundary condition that Samir was talking about, just corresponds to adding these les постоянно transform, but then you want to impose the other boundary condition then you need to add some different boundary terms και αυτό είναι συμβουλήθηκε σε αυτό το πέραμα, μετά με άλλες ασπέτρες της H2-χολογραφίας. Είδαμε να μιλήσουμε εδώ, ότι στις κόντεξες της χώρσης, στις κόντεξες της χώρσης της χώρσης, ουσιακώς, για αυτοί της H2-χολογιακής, έχετε ένα factor H2-χολογιακής, αλλά σε κάποιες φορές, η H2-χολογιακή έρχεται από ένα H3-χολογιακή. Προστασandaag, όταν έχετε ένα H2-χολογιακό και ένα H3-χολογιακό, η H2-χολογιακή έρχεται από ένα H3-χολογιακό και της στις κόντεξες της χώρσης. Στην μητέρα της χωρις, αν έτσι έχετε ένα H2-χολογιακό και δύσκολα, η H2-χολογιακό κόντεξας δίδει από την χώρσης της χώρσης. Ερθόνι, όλος στην ίδια υπαγέντι απολαμβάνει, υπαγείας της χώρσης εστέψεται τέτοις. ορίτε θα δοκιμήσεις η αδύναση της αδύνασης. Άρα είναι η λόγια κοινωνή ότι πρέπει να πιθαριστούν τα κοντά που η αδύναση είναι καλύτερη. Λοιπόν, αν έχεις ένα αδύναση της αδύνασης, δηλαδή στην ΒΤΖ, στην ΑΤΣ, και οι δυέντρες πιθανές πραγματάς που έχουν ένα αδύναση της αδύνασης, δηλαδή η αδύναση της αδύνασης δεν αντιμετώ σε τα κοντά, άτω σε τόση τη δηλαδή μπορεί να πιθάρεις και διεθνούνται τα πράγματα. Βέβαια, αυτή ήταν κάποιες πράγματα, και τώρα μπορούμε να συνεχίσουμε από όπου είμαστε τώρα. Ναι, δεν θα μιλήσω. Βέβαια. Βέβαια. Βέβαια. Βέβαια. Βέβαια. Βέβαια. Βέβαια. Βέβαια. Βέβαια. Βέβαια. Βέβαια. Βγέβαια. Βέβαια. Βысπη. Β Şί irritatingLOT. ΒΓΙΣ this layer. Βγέβαια. Βγέβαια. Βγέβαια. by this, so this is the canonical momentum, the radial canonical momentum, and this is the conjugate induced metric on the radial slice, and there was this important conclusion that like the radial derivative of the symplectic form, if the radial slice is compact, is independent, is zero. So then what we did was, so this was like the generic Hamiltonian formulation, so next what we did was like we constructed the space of asymptotically locally ADS solutions of Einstein-Hilbert gravity in this case, and then we found that there are two independent functions, so one was like this G0, which was completely arbitrary, and then there was like this Tij I defined that was proportional to Gd, that was like another sub-leading coefficient in the asymptotic expansion, plus some other terms here, and we saw that if you plug this asymptotic expansion into the symplectic form, you get a symplectic form of the space of solutions, so then this omega reduced to this G0 ij wedge delta pi d ij, and this pi d was just like this Tij, just it had like the volume element, right, and from there we constructed the, so from this symplectic form we constructed the Poisson bracket like for functions on this reduced phase space, which is given by this, so now if you remember, so that is the structure now of this reduced phase space, and this is characterized by these variables, and it's a symplectic manifold that has like this symplectic structure, this Poisson bracket, so now let us look like at symmetries, so remember like when we discussed the Hamiltonian formulation of gravity before going to the space of asymptotic solutions, we also discussed like the local symmetries, and we saw that we had like two constraints, the Hamiltonian and the momentum constraint, and through the Poisson bracket this generated like bulk diffeomorphisms, so here now there are also, there is another set of local symmetries, so remember when we wrote the solutions, the asymptotic solutions, we were working in the particular gates that was of this form, right, and then we wrote down the expansion for gamma ij, so it was like e to the 2r plus subleading terms, right, so these are very specific gates, we could have chosen any other gates that we want, this is kind of the standard canonical gates, and now there is a question, are there residual diffeomorphisms that preserve these particular gates, but of course they will act on the coefficients here of the asymptotic solutions, so these are like, if you impose this condition, so now we do infinitesimal diffeomorphism, so let's look at generic diffeomorphism, so that would be like the g parameterized by psi, so we want the r component to be zero, and also delta psi of gri to be equal to zero, right, I mean so that means like to preserve these gates, like these coefficients here, and the mixed one, so these two give us conditions like for the parameter psi, and the two conditions are psi r equals zero, and psi dot i by j equals zero, so these are like the two equations psi dot, the dot here again is the radial derivative, and the solution takes this form, I will call it PbH, so it stands like for Penrose-Brown-Heno, so these are standard result in gravity, so you get subleading terms, but the important thing is that we have like two, so this class of diffeomorphisms is parameterized by two, well one function and one transverse vector, which are completely arbitrary, so we have like this sigma of x, which is an arbitrary function of the boundary, and then this psi zero i which is an arbitrary vector filled on the boundary, so this is the general solution, and all the subleading terms are determined in terms of, so here there is nothing subleading, and here all the subleading terms are determined in terms of there is some sigma dependence here, so here it's like the sigma. Right, so now that we know the residual symmetries, so we can see how they act on the coefficients of the expansion, right, and then basically the only coefficients that we're interested in is like is the g0, which is independent, it's arbitrary, and then the other like arbitrary coefficient which is like gd, so now you can act, so let me write it here, so you can easily do the calculation explicitly, so what you find here is that this part here is just like the standard, it's a boundary diffeomorphism, while this part here is an infinitesimal vial transformation of the boundary metric, so from this we can deduce that psi0 just like corresponds to boundary diffeomorphisms, whereas this sigma induces a vial transformation of the boundary, so this is vial, and this is... Now the important question now is like how does the PBH act on the symplectic conjugate of this g0, and that's a little bit more complicated calculation, so let me write the answer, ix, i0k, so this is kind of a toy model, like I mean for gravity, and then like I mean, so in a moment I'm going to apply the same thing, but I'm going to derive like the supersymmetric, supersymmetric transformations on the boundary, like on a carried boundary, so just bear with me like I mean a few minutes, so here there is another important term, sorry, which one? The right answer, if you point your finger I'll tell you where, to the right, This is psi0, sorry. Right, and then here, right, so remember there was like this, so πd was again this proportion to Tij, and remember that Tij, the trace of Tij, it was given by this function g0 that I didn't specify because it depends on the particular dimension, but it's a local function of g0, and then it appears in this transformation here, so now if you compute the transformation under these PBA transformations of this coefficient here, you get these homogeneous terms, so homogeneous because all of them are proportional to the original quantity of this tensor, including this one, so this part plus this part means that like this πd transforms as a regular tensor under the corresponding transformation, but this part here corresponds to some anomalous transformation. Sorry? Under the table, after the A... Ah, it's a sigma, so it's proportional to this... I multiply this... Yeah, it's proportional to this, the same sigma as here. Right, and now the point is that, so maybe I don't write it explicitly, so you can show that these transformations, they follow from the Poisson bracket, so namely if you compute... So if you define this function here, so if you now define... So remember that was analogous to defining the constraints, like on this phase page that we did before in the Hamiltonian formulation, so now you can define again an analogous constraint, and this was like the two constraints that we got on TIJ, if you remember one was like this trace condition and the other one was like the covariant derivative of TIJ was equal to zero. So now we multiply these two conditions like with the corresponding, like just some arbitrary functions, and now you compute the Poisson bracket of this with G0, and you get the same PBH of G0 and similar for pi DIJ. So now the point is that like these transformations here, one way to compute them is using like this Poisson bracket that we defined earlier, and taking the commutator, the Poisson bracket with the constraints, these constraints. So computing this is very easy, just directly from the asymptotic expansion actually, just doing the morphism. In principle you can do the same for this coefficient, but you have to go very high up in the expansion, so this Poisson bracket calculation is much simpler to do and you get the correct answer. So now an important, you can also compute the algebra, psi0, σ, and then π, σ' and then you find that this again it closes, the algebra closes, but now the point is now the structure constants, I'm not going to write them explicitly, independent of the fields. So remember this is different from what happened like in the phase space formulation of the theory, because in that case the algebra closed also, but the structure constants were dependent on the fields themselves. So once we go to the reduced phase space, this doesn't happen anymore. So here you get a normal Lie algebra. Okay, so no, these are not the ridgys symmetries, so that's where I'm going now. So now let's do an example. So now let's consider a generic two-dimensional conformal field theory and put it on a space with a metric g0, some arbitrary metric g0 and then we know that we have like these word densities. So the word densities are the two constraints that we have from before and then the trace of this stress tensor is given by the conformal anomaly, in this case it's c over 24 pi r of g0, so that's the rigid scalar, so that's like the standard word densities. So now you find that under local two-dimensional deformorphisms plus vile, first of all the metric transforms as we saw before and then you also get the transformation of the stress tensor which I'm not going to write the psi part plus here we get like a term, right. So here there's the standard term that, well you can see in the notes, so the notes are by the way already like on the website. So the stress tensor transforms as a proper tensor under deformorphisms, but under vile transformations like there is this anomalous term here, right, there is this inhomogeneous term. So now to define rigid symmetries like or killing symmetries in other way, so all we have to do is to set the transformation of the background metric, so we set the transformation of the background metric, this is g0, so this is the transformation of the psi0 and sigma, which is, we set it equal to zero. So that's the difference, like that's what defines global symmetry. So under these local symmetries there is no constraint on this transformation. So this is just the transformation under the local symmetries. Now imposing this condition here is the killing symmetry condition. So first of all this fix is that sigma is minus a half d0 psi0i. So the sigma is fixed in terms of psi0 for this case and then psi0 then is a conformal killing vector, right? I mean, so that becomes like the conformal killing vector condition for psi0. So in particular you can choose so choose like that ds, so the metric is just flat metric and choose holomorphic coordinates so this is dz, dz bar. So then you find the solutions is there are two arbitrary functions and then now you can use this transformation that we wrote down the general transformation here that we used under local symmetries. You can use it for this global symmetry transformation and then you find the standard expression for two dimensional CFTs where t is equal to t, dz and similar for the antiholomorphic coordinates. So from here you recognize this is the Schwarzian derivative and then from here by expanding it in Lorentz expansion you can you find that the modes satisfy the Virasoro algebra. So we see that now the structure of this reduced phase space for gravity like in the case of three dimensions precisely matches this phase space of two dimensional CFTs namely the space of coupling of the background metric and the corresponding stress tension. So that is a generic statement that you can make so that's like a reformulation of ADACFT if you want. So more generally the space of operators and couplings in the field theory is a symplectic manifold so you can see more like in the type lecture notes like on this point and so the symplectic form is we'll just tell the OIJI so a statement like of the ADACFT correspondence is that so this exists like for any local quantum field theory this symplectic manifold so a statement of the ADACFT correspondence is that the reduced phase space of asymptotic solutions in ADS is precisely identified with the symplectic manifold of renormalized operators and sources. So and these you can generalize to other contexts but it becomes more complicated like if you don't have a syndotically ADS. Sorry? Sure. Yeah so this I didn't talk now from here like I mean from the global symmetry now you can define the corresponding conserved charges and then on the space of the conserved charges then this for some bracket becomes the Dirac bracket and then you get the algebra of the charges. So that's one way to derive the Virasor algebra. I don't know if that was your question. One of the things suppose you don't put this condition delta G equals 0. Okay. Do those transformations still have interpretation in the boundary theory? Yeah so it's like toft anomalies right? I mean you put the C on field method. So when you put the field theory in the background like global would be global symmetries become gate symmetries. So these are precisely these symmetries. Okay. So the next half hour so I'm just going to make a few comments without writing explicitly much. So I'm just going to make some comments about boundary terms and then I suggest like I mean you can take a look in chapters 4 and 5 of the lecture notes if you want to read more about it. So here the idea is that like I discussed so far like this reduced phase space but then also in order to to renormalize the action we also need like to determine the corresponding boundary terms. And then the logic here is the following namely that like when you have like this asymptotically locally ADS spaces actually you don't have a hard boundary you have an asymptotic this is a conformal boundary and in practice mathematically this means that the bulk fields don't induce represent a particular metric like or fields like on the boundary but only a conformal class of metrics. So whenever you want to formulate the variational problem then you need to formulate in terms of this equivalence classes, these conformal classes. So in doing that you need in order to do that you need to add these boundary terms. So these boundary terms ensure that is a is a class is a is a variational problem on equivalence classes of under conformal transformations and you can show in general that this boundary term is minus a solution of the Hamilton Jacobi equation. So and there are algorithms to iteratively determine them. So again this algorithm is described in in subdetailing in the nodes. Okay, so in the next like half hour so I'm going to kind of repeat this story but for for supergravity. Equivalent class is like under so the class function is like two sigma, right? So it's basically like for any sigma. So what I keep fixed basically is like it's like this combination for arbitrary sigma. I don't demand that like g0 itself is kept fixed only the ray like I mean under my transmission is kept fixed. So if you demand that then you see that you have to add precisely the counter terms and this automatically ensures that actually the action is is finite. Yes. Yeah, all this analysis that I'm talking about here is asymptotic. So so basically you can apply everything I said here to the sitter. It's just like a double week rotation and then everything applies. So all that started is with ADS. I mean with the sitter is like global properties like I mean away from the boundary. So close to the boundary I mean everything I said applies. Okay, so it doesn't have to be conformally flat. You mean the boundary? The bulk? Well, no. I mean it depends like on the theory that you put on the boundary in the bulk, right? I mean so you need I mean it depends on the equations of motion that you solve. So you have to study like the asymptotic solutions of a given theory like in the manifold. Right. So let's see. So 4D N-quad 1 and 3D N-equal 2. Right. Okay. So now let's consider like the minimal gate supergravity in 5 or 4 dimensions. So I'm going to write it like with an arbitrary d here. So that's the bozonic part and then we have like fermionic parts. So here let me just write it schematically. So there's like an F nu there is gamma sigma and then here is nu. Right. So this is like minimal gate supergravity like in 5 dimensions. So here let me just put like delta d equal 4. So this term here the tensimost term only exists like in ADS 5. Right. So the spectrum is that we have like just the metric like a u1 gauge field and which is the gravity photon. So it's the gravity multiplet and then also a gravitino. So I should mention here that the way I wrote it is different from I think what appeared like in earlier lectures. So this time nu is Dirac and before we had like Majorana to Majorana simply Majorana fermions but in this formulation it's important to write in terms of Dirac fermions. Yes. You mean here? Yeah, that's lambda. Sorry. So that's lambda here. Thank you. Is that what? No, sorry. This is So this B is like in is the boundary dimension. Right. Okay. And then also there is like Gibbons hocking term which is a generalization of the pure gravity one. So now it gets contribution like also from the gravitino. So that's also necessary in order to formulate the theory. Sorry. The contraction here was a little bit sloppy. So if you want to make it right more precisely. So it's plus okay. So there is also like the super symmetry transformations. I'm not going to write them explicitly down. So now the first step here then is like the again the Hamiltonian and that's why it was important like to have this in the direct representation. So if you have it in the gravitino is in the direct representation then you can now formulate the Hamiltonian version of this theory but you need to project so to with this they're called radiality projectors. So this is the radial component. So this is the gamma matrix along the radial direction and then you have to decompose all the fermions in these two chiralities or radialities. So then we have like psi plus nu and psi minus nu and then this play the role of canonical momentum and coordinates. So because we have a first order action like for the fermions. So again the first right so like once we have the so from the Hamiltonian formulation what we can derive is the boundary terms which I'm not going to write down or explain how they are derived but we can also construct this reduced phase space reduced phase space which is the space of asymptotic solutions and then we have like for the field buying we have the analog of the this Pheferman-Greham expansions so this is in d equal 4 so that's like ads 5 and there are similar expressions like in for the ads 4 case so so this is the field buying so because we have fermions in the game now we need to trade the metric like for the field buying so we work like with the asymptotic expansions of the field buying so then there is the gates field and then let me just write like the expansions for the fermions and then so the point is that now this this term here so the story for the metric for the field buying is very similar to what I said before for gravity now for the for the gravitino here this this is an independent function here and it plays the role of the source of the dual operator and here this this term here corresponds to the dual dual momentum then this is like a source so like the different the intermediate coefficients here like they are all written like in the paper so the kind of lengthy expressions but the point is that again there are like this arbitrary coefficients here this and this and then also for the fermions and also for the gates field that I didn't write down so this now parameterize the space of asymptotic solutions this reduced phase space and again like the whole thing in here is like a symplectic structure from this generic Hamiltonian analysis of the theory so you can build the symplectic structure and the corresponding Poisson bracket so now we can define the operators so the operators are going to be related to this other mode so we identify this and this like as the sources and also for the gates field and then for the fermions like this and also this for the metric here now these modes have become proportional related to the one point functions but the one point functions have to be derived the counter terms so so we have a stress tensor here we have the R current which is given by the momentum of the gates field and then also we have the super current which is the operator conjugate to this source here ok and now these counter terms here I haven't I haven't shown you explicitly so they are in the paper they are pretty lengthy and they can be computed systematically but once you have these counter terms now and also the corresponding canonical momentum you can evaluate these limits and to leading order like this will go like ED and etc. all for the other modes so they will go like as the symplectic conjugates of the sources basically like this one point functions but they are the correct covariant one point functions so the counter terms you have to solve the Hamilton Jacob equation so this I didn't have time to go through in detail so there is a system multi algorithm also the finite counter terms yeah the finite counter terms also you can classify so they have to be like super conformal invariants and then there is a limited number of such local densities that you can construct but they are all constructed in the paper so if you are interested you can have a look but indeed like the whole all the possibilities like for the counter terms are there no the counter terms are unique right so there is nothing else that you can write so like if you solve the Jacob equation you will get like a unique answer so all the divergent terms they have a unique answer always which in this theory they are complicated so the only ambiguity is in the finite counter terms which is like in the coefficients of the super conformal invariants so anything else you write I mean it will not lead to the correct variation a problem so it's not that I mean you can write anything that I mean maybe it works like for some particular solution but I mean the counter terms have to work for all the possible solutions I mean to get the right structure are there any questions ok so so now once we have these operators so here when when I wrote the asymptotic expansions I didn't say about the constraints that actually this normalizable mode satisfy but as in the case of pure gravity they satisfy some constraints which will come out like automatically from this asymptotic analysis and also the Hamiltonian analysis so in other ways usually the constraints that you get like I mean now on this they come precisely from the constraints in the Hamiltonian formulation there is one to one correspondence so that's the most efficient way to determine these constraints so and as we saw in the example of two dimensional CFT these constraints now on these one point functions they become like the word dentist so and then here I'm not going to write so you have like you have bosonic and then you also have two fermionic word identities so let me write the terms plus j here and well there is some gammas times I'm writing a little bit schematically and then there is also the gamma trace identity which is of this form so here we have like the divergence of the super current and then there are other terms evolving the stress tensor and the r current and here also is the gamma trace of the super current and here the u1 current so on the right hand side however we have like these terms which are local so these terms are local in the sources so I'm not going to write them explicitly down so they are like similar to the to the conformal normal to the trace anomaly like in the two dimensional CFT they are expressed in terms of curvatures so they depend like on the Ricci curvature of the background metric and also the field strength of the r of the r current background good so so as we saw before like the fact that we have this this word dentist is a reflection of some local symmetries which can be generated again through the Poisson bracket so in this case so in quotes because it's a generalization of that so again we look for symmetries preserving the gates so in the gates in this case that I'm preserving is expressed in terms of the field bind psi r is equal 0 and then here we have two constraints here so it's psi plus and minus so these are the constraints that we sorry these are the gates that we preserve and then this leads to a set of local symmetries so and the local symmetries are again like the this Penrose-Brown-Henna diffeomorphisms but now we have more more transformations so we have like local Lorentz transformations so all of them have subleading terms here and then there is like u1 transformations and finally we have the supersymmetry transformations so these go like plus or minus r over 2l epsilon 0 plus or minus x plus all subleading terms that depend again on this parameter so the subleading terms do not introduce extra parameters so the important thing here now is that we have like two independent local spinors spinors here they correspond like to local symmetries and then epsilon plus is the q-suzi epsilon minus is the s-suzi and through the local so they are generated through the Poisson bracket basically they are generated so this word identity generates the epsilon plus transformations and this word identity generates the epsilon minus transformations so you can do explicitly so you can check explicitly the transformation of the sources under these residual transformations and what you will find is precisely the supersymmetry transformations well including like the theomorphisms and local Lorentz etc but like under these parameters here what you will find is precisely the local supersymmetry supergravity transformations on the background and the boundary so this would be like precisely like the off-cell n-qual one conformal supergravity like in living on the boundary so the supersymmetry the supergravity transformations like for the sources like under these epsilon plus and epsilon minus they are the standard ones however so I'll just write those in the stop so using the Poisson bracket we can compute the epsilon plus transformation of s-i so this is given by the let me write this schematically by this word identity Poisson bracket with s-i and what we find from this is this expression and then there is a lengthier expression for epsilon minus so now for epsilon minus is the same thing but here we take the the word identity corresponding to the gamma trace of s so with epsilon minus and then again s-i Poisson bracket and again there are a few terms so first we get this last year we get epsilon minus and then finally we get one more term which is i-k good ok so that was the final transformations and these transformations now actually they are the transformations like of the of the super current operator under supersymmetry like under q supersymmetry here and under s supersymmetry but they are local transformations however they are not the classical transformations that you would find just by doing the calculation like on background super gravity so here because we do the calculation holographically remember in the Gordon is a tie road like two anomaly terms on the right hand side that I didn't explicitly write down but like in the two-dimensional case where we saw there was like this conform anomaly this contributes this anomaly contributes to the transformation of the operator so in the you contribute to the transformation of the stress tensor in the two-dimensional CFT example that's why we got the spartzen derivative and then in this case we get again some contribution from these anomalies and these contributions we can easily figure out but this term here is proportional to the stress tensor so that's an operator and then this term here is also an operator but this term here so this term here is a local expression in terms of the background of the asymmetry background so that represents a kind of an anomaly and then similarly here so this term here so again this is a local term and also this is a local term and also this is a local term so here all the terms apart from the first one they are local anomalies so the point is that the supercurrent has an anomalous transformation under local supergravity transformations on this rigid supergravity background so now if you look at the if you try to find like killing symmetries so that would correspond to imposing the transformation of the source being zero so we would have like the transformation of this psi zero plus equal to zero which is like the epsilon plus minus gamma epsilon minus so now like in the case of the conformal killing spinors we would have to define like killing spinors in this case we have to set this to zero but it gives us like this is the conformal killing spinor equation so from here you can find the corresponding killing spinors but for generic curved background the spinors that satisfy this equation they have non zero both epsilon plus and epsilon minus so then you can plug it into the corresponding transformation of the supercurrent here and then you will see that the transformation under this rigid supersymmetry now that corresponds to the conformal killing vector would get anomalous contributions both from here and from all these terms here so this means that in principle supersymmetry like depending on the background supersymmetry might have like this anomalous transformation here when you put it on the curved background that admits like conformal killing spinors so even though like classically supersymmetry is preserved so I will stop here