 So I'm going to talk about some work that appeared in that paper like a year ago and Also some some work in progress with a number of people so Let me just Make like like motivate like I mean the the analysis that I'm going to discuss But just like Remind you that like so when we mean when we're talking about supersymmetric Bacchua in supersymmetric field theories, so Essentially like we can define the supersymmetric Bacchua by this property so that the supersymmetry like rigid supersymmetry variation of some Ferminic operator, let's say Which is given by this anti commutator here is is equal to zero. So this holds like in the quantum theory however The usual argument like for supersymmetric localization relies on a on a classical version of that statement Which is the following so if you just compute like the classical so here imagine that we have like some microscopic Lagrangian like With depending on like microscopic fields x over which we path integrate then we can Like the expectation value of this variation here is takes this form and then now the argument is that You can well Classical invariance and the supersymmetry means that you can pull this inside here. So we can always write like this. So that's fine However, now there is a non-trivial assumption in the next step, which is like putting this equal to zero because here we are assuming that Like we can pull this like like We can pull like the the path integral measure if you want like inside So so this means that like I mean we are assuming that supersymmetry has no anomaly Is preserved by the path integral measure which might not always be the case So another way to say this is that essentially we are assuming that the the Co-homology that the q-co-homology like at the classical q-co-homology is not is the same as the quantum q-co-homology So in this talk, I will provide some evidence That this might not always be the case and the two the classical and the quantum q-co-homology might not necessarily be the same So and this is not something that is entirely surprising because there are already well-known examples of that So and and here is like n-qual one super-virage or algebra in two dimensions and indeed there is such an anomaly here So like here. This is like the usual conform anomaly in the virage or algebra and and in this commutator here Which is the same as the like q-square like expression that I wrote earlier So here you see that I mean you get this expression, which is also the classical part of the algebra But you also get at the quantum level you get this anomalous contribution here So the the statement is that like at six dimensions like four dimensions or six dimensions There are anomalous terms of this type However, the difference is that in two dimensions this anomalous term survives in flat space But in in higher dimensions totally appears in in care space So Let me just summarize the result and then I will I will go through the explanation. So if you just Recall like the supersymmetry algebra like in in flat four-dimensional space It it can be written in this form like a little bit schematically maybe so this is like the stress tensor here So s is the super current This is the stress tensor and j is like is U1 current like so the r-symmetry current if it exists in the theory and Now what we're going to do is like generalize this to a curved background that admits rigid supersymmetry So in that case like several of these expressions here They become like co-varientized and and this zeta plus here and zeta minus are the two components of the killing spinoff so the killing spinoff satisfies this this Equation generically on on care manifolds in flat space you get zero here So zeta minus equal to zero so you don't get this term But in care space generically you only have like this kind of conformal killing spinners So so you get this other contribution as well But the important part is that you also get this Local term here what I call this a zeta Which depends like just on locally on the background fields So in particular in this case it depends like on the background metric and the background for the r-symmetry current so I will provide like some Like derivation of this of this anomaly and in some specific context and then discuss some of the consequence So the plan of the talk is first of all, I'm going to discuss a little bit like the super conformal word dainties and the relate of anomalies and and then like remind you like of the recent progress in Developing rigid supersymmetric backgrounds in Especially like on four dimensions in four dimensions and and then I will revisit a problem that has been starting in the context of Of supersymmetry on read on rigid supersymmetry on care backgrounds Which is like the dependence of the partition function on certain geometric data of this supersymmetric backgrounds and and then I will discuss Like the supersymmetry the rigid supersymmetry anomaly and the application like in in this context and finally I will mention some Quick comments about the customer charges and and the bps relation and how they affect it by this anomaly Okay, so if there are no questions, let me start with the with the super conformal word dainties. So As we we saw like in a couple of slides ago. So generically for any qual one Super symmetry on in four dimensions, which is what I'm going to focus on we have This the the super current multiplied contains like the stress tensor the super current and if we have like if it's a super conformal theory generically it also contains like the UNR current Plus also some auxiliary fields. So generically you can Package all these currents in a in a super field. So here So generically they appear in this in this form in terms of this vector super field But also these dots here indicate that depending on there are different ways of coupling it of packaging this like into a super multiplied and how you pack it Depends like on a number of auxiliary fields So besides the auxiliary fields, there is also some ambiguity that is called the The improvement terms. So for example, you can always shift the the currents like the stress tensor and the And the super current in by terms of this form without changing their defining properties Right, so now there are different multiplets that have been studying over the years So the most recent and most general one is the so-called s-multiplet that was found by Komar Gorski and Cyberg in 2010 and this multiplied always exists and It contains two auxiliary fields. So it's like a Like a chiral super field x and also an auxiliary spin or a super field chi alpha and the defining relations of these Super of these multiplets. So the soup is that they satisfy this this expressions here Then there are special cases of this s-multiplet that were historically found like before So the most well known is the so-called Ferrara zoom in a multi plate Which is obtained by setting the chi alpha auxiliary field to zero and there is also the r-multiplet, which is Relevant for super conformal theories that have like a u1 r symmetry and that corresponds to setting the x auxiliary field to zero so the way I would like to view this defining relations in the rest of the talk is some kind of Worded entities. So in other words, they are kind of classical conservation equations So if you expand this if you expand the super fields here in components What you will find is conservation equations for all the component currents So I'm saying classical because at this moment like we haven't turned on a background field So we cannot like see the the anomalies like the potential toph anomalies that can appear in these conservation equations Right. So now in order to see this potential anomalies like we want to couple this theory to Like at least in the beginning like linearized supergravity. So here I'm going to be a little bit schematic So most of these expressions that I'm going to write down they they apply like to strictly speak into the s-multiplet but the argument is not Like does not depend on the details of what of what I'm showing here so like to couple the theory linearly to To supergravity. We just add the coupling of this form where this s is again this Current super field and then this background h i contains like linearized supergravity fields and then we are in order to Like derive or encode if you want like this defining relations of the of of this Multiplet we assign like local gates transformations on this background fields. So so here we So these eights transforms like in this way and now if you expand This transformation in components, then you will find all the usual local symmetry So you will find like diffeomorphisms local frame rotations by transformations and also you want and Both local supersymmetry transformations the q and s supersymmetry transformations. So Precisely which transformations of this appear depend again like on the multiplying and how you gauge fix This s-multiplet, but this is not relevant or what we want to do here. So Right so then once you assign these gates transformations to to these background fields you can actually derive back the original Defining relations through an ether argument if you want. So namely you just demand that this coupling here is invariant under this transformation So you integrate by parts and then you you get this conservation equations oops, sorry, so this conservation equations here, right so So now we have the transformation under these local symmetries of the of the background fields like this these eights here and now First of all we have to see how we define the this super current This current super field which you we can obtain by if if the generating function is given by this expression So we can define it to be the functional derivative with respect to the background fields. So this is if you want like the standard Definition of the operators in the so-called locally normalization group of Osborne So here I just wrote this like for the for this current super field Now I should just mention this as a side comment this definition here is the so-called gives you the so-called consistent current which Is not the same as the so-called covariant current and the two differ by Being the zoom in no terms. So in particular the covariant current cannot be written as a gradient of the generating functional in this So now what I want to do is like I want to consider the transformation of the Of this operator of this current super field under these local gate symmetries And this is in principle straightforward to do you can just like take The definition here plus the transformation of of the background fields here And and then you just apply the chain rule here So you get the transformation of this derivative using the transformation of the source Plus here potentially you get this term here, which is non-zero provided. There are tough anomaly So if this generating functionally is not invariant under this This gate transformation So in other words if they are tough anomalies then you will get an anomalous Contribution here. So the the main point like I mean that I want to mean is precisely this fact that actually you get this anomalous transformation for for the For the operators for the current operators Okay, so now a So in principle you can just compute this expression directly by brute force But there is also like an elegant way of doing it which is using an underlying Simplactic structure that exists like on the space of couplings on the background fields and local operators and You can define on the symplectic space you can define a Poisson bracket that takes this form So here again is like this background fields and the dual operators. So you can Always define this symplectic with this Poisson bracket and then once you have this Poisson bracket then you can see that like this Local expression here. So this is just the defining property or this word entity that I mentioned for the super fields Now generates local Gate symmetries. So in other words This becomes like a first-class constraint on this space of couplings and local operators and if you take the Poisson bracket with a With a generalized coordinates. So this is like the background fields Then you will you obtain this and then you obtain back the original transformation that we we had like for the for the fields However, the interesting parties is to obtain the transformation of the super current Superfield which is again given by the Poisson bracket Of of the of the super field with the with the same constraint here. So and again, I mean you get this expression So notice now if there is an anomaly contribution here So because you're so the anomaly would depend on the background fields So that is 8 and and then you get a contribution from from this derivative here so this is a generic expression that you can always apply to find the transformation of the of the Of the operators or the super current Superfield in this particular case Okay, so now so this was like the transformation under the local Symmetries like on arbitrary care backgrounds, but now we we are interested in defining like global symmetries or like killing symmetries or rigid symmetries if you want and that simply corresponds to Defining like looking for solutions of the equation that the variation under these local symmetries of the background fields is equal to 0 So this is just like the local gates transformation of the background field But now we are setting this to 0 and and then we get like the generalized killing spin or equation or killing vector equation for for these parameters L 0 So this can be can be solved and then we can find the precise Like parameters L 0 that correspond to the killing symmetries and using the word that is then we can define the corresponding conserve charges and then the transformation of the of the operators under these Conserved charges now. So this is like the rigid symmetries the transformation of the operators under the rigid symmetries is again given by by this Commutator or anti-commutator depending on the spin of the component. So strictly speaking, this is a Dirac bracket now it's not a Poisson bracket anymore, but that's a technical complication, but you can use this expression here to compute the rigid variation of Of the operators under the global rigid symmetry So and this particularly includes like this supersymmetry variations of the super current So if you do this game in in flat space like in four dimensional flat space here I write again for the S-multiple you you will get precisely this Familiar looking expressions. So for the S-multiple you have like these two auxiliary fields, but then otherwise it looks precisely as I wrote In at the beginning of the talk. So it's just this is what you get like by just looking at rigid supersymmetry in flat space Okay, so so now we want to go one step further and couple the theory to non-linearly to background supergravity and one way to do that is is the so-called Festucia-Cyber argument where you generically couple the theory to some off-sale supergravity and and then freeze the gravitational degrees of freedom So it becomes a background and and then the corresponding super conformal toft anomalies can be determined For arbitrary anomaly coefficient. So in four dimensions, we have the A and C anomaly coefficients So for arbitrary values of those they can the anomalies can be determined by with zoom you know consistency condition argument and This has been done a long time ago for for the Ferrara zoom you know multiplied. So there is a a covariant care phase a covariant and Care of the super space formulation of Of the supergravity and then you can compute like using we assume in a condition conditions You can compute all the possible contributions to the super conform anomalies, but still it's kind of Tricky to extract the fermionic components of this anomaly that are relevant for the calculation that I'm interested in here, right? So however the there's an alternative way of computing This anomalies which relies on on holography So in particular if you start from minimal gay supergravity in five dimensions then this describes holographically n equal one Off-sale conformal supergravity on the four dimensional boundary The disadvantage is that in that case you only get Equal to see Anomaly coefficient so you cannot see the general form of the anomaly, but nevertheless, I mean this is already a General enough to to recover this rigid anomaly that I that we're going to see later on So in that context you have like a bulk of a bunch of bulk fields So in particular you have like the bulk field buying the gravitino and and the you engage field and Asymptotically they have arbitrary sources. So these arbitrary sources become the off-sale n equal one supergravity conformal supergravity background and Then you can define the the dual operator. So the The multi-plate that the super current multiplied in in the way that we saw earlier So you just take the variational the variation of the generating functional and and you define the operators in through this variation here Okay, so now This part is the technical part of this paper that I mentioned. So you you have to do holographic normalization for For this bulk supergravity theory like and be careful like to preserve super symmetry at all stages but after you do that then you can derive like the the super conformal word then it is for this particular supergravity multiplied and And the result so I'm not writing here the Bosonic ones You also get the Bosonic ones, but here I'm just writing the fermionic watch, which is like here the the derivative of the Supercurrent and the gamma trace of the super current and the important thing is that this is like how the ADS if it dictionary works So when you compute the classical classically Something in the bulk so that corresponds to the large and limit in the dual field theory But it's still a quantum calculation. So in particular you still get the Like the quantum anomalies like the top anomalies in the corresponding word dainties So we get this these two anomalies here and and the explicit form of this is Is this so notice that they are proportional to see so the central charts and Again a equal to see in this case. So you cannot distinguish between which parties Corresponds to a and which part corresponds to see So they are pretty complicated One point that is kind of interesting to notice is that like these terms here that I wrote in orange You can actually put them like to the left-hand side of this word dainties and combine them with this our current here and What it corresponds to is like sifting the the our current from the covariant expression, which is this J that enters here Sorry from the consistent, which is this J here to the covariant one that corresponds to this combination here Okay, but then there are also like these other terms here that depend like on the curvatures of the background metric and also of the of the background You one arc current Well, so I will show you like I mean the n equal one backgrounds that like Support like ridges per symmetry and like for generic backgrounds of that form They're actually non-zero this this expressions here, but this is what what you get like I mean from the calculation So I don't know if I answered like that's right Yeah, I mean so you can you can see I mean they have some nice properties this term you have nice I'm nice properties under like a bio transformations and thunder like I mean super bio transformation So they sold you can check that they sold the way Sumino consistency conditions for example, right? So so then I mean we can Like we can compute the supersymmetry transformations on the boundary So that is like the off-sale n equal one conformal supergravity transformations for the for the background fields and we can do this in the number of ways so one way is just like to Expand the bulk fields and the bulk supersymmetry transformations and and then deduce the transformations of these boundary sources another way is to use this Poisson structure like that that I mentioned earlier and This is actually much more powerful because it helps us like to evaluate the transformation of the of the Of the super current which is the one that we're interested in so Using this this Poisson bracket that I defined earlier you can compute now the transformation the supersymmetry transformation of the super current and And you find these expressions So so this is like the epsilon plus here. So there are two local supersymmetries So the epsilon plus is is the usual Q supersymmetry and the epsilon minus is the so-called S supersymmetry Or like superviolet transformations and then under those transformations the Again local transformations the super current transforms in this particular way. So you see now all these color terms They are local. They are they just depend on local Kermatchers of the background so they represent the top anomaly So and they precisely come from the top anomalies in the world entity So essentially this these terms here the orange in the red. They are the derivatives of of these terms and These terms here. So it's just like the derivative of those terms with respect to the source of the Of the super current which is like this Gravitilo background, right? I mean so this is linear in this psi and also all these terms are linearly psi So essentially if you take the derivative with respect to this psi you get precisely These these terms here that I wrote So this is the like the essential part like I mean of the calculation So it tells us that under local supersymmetric transformations this this super current transforms in an anomalous fashion okay, so now Let me review a little bit like of rigid supersymmetry on care backgrounds. So there are different versions of like rigid supersymmetry so the most straightforward and Like although very restrictive definition is like looking like for a covalently constant spinner then there are like Like there are conformal generalizations of that to the so-called Twister equation, which is similar to what I had and And then there are like also twist twist like by by line bundle. So you get like this this equation here and And and then there are also the The transformations that you get like I mean for from the coupling to some background supergravity So if you couple to conformal supergravity you get this killing spinor equation, which is the What you get from the setting the variation of the gravity no here So setting this variation to zero you get precisely This transformation here and then there is another supergravity which applies also to non-conformal theories the so-called new minimal supergravity and the killing spinner equation in that case that comes from the gravity no variation is given here so Now this killing spinner equations have been studied like extensively and the backgrounds that admit rigid symmetries or killing spinors they have been classified and Killing spinors of a new minimal and conformal supergravity they have been discussed firstly in these two papers here and So for n equal to n equal one theorists in four dimensions Like they can be coupled like two different supergravities using a given Festucia cyber argument and then you get like different killing spinors like on the corresponding backgrounds However, there are some differences like between the the backgrounds generically, but between conformal so between killing spinors of conformal supergravity and And new minimal supergravity the difference is only like in global properties of the background Which are not relevant from for what I want to talk talk about so I will Will not worry about this. I'm going to work like mostly with backgrounds of conformal supergravity, but they apply also to To new minimal supergravity so Finally an important point to keep in mind is that now this description of rigid supersymmetry is actually independent of the particular theory, so it only depends on the manifold and the killing spinner equations that that you put on it, okay, so So manifolds that admit two killing spinors z and z tilde so like this is like in euclidean signature Of opposite our charts are generically t2 vibrations over a Riemann surface And the metric can be parametrized in terms of these functions here So apologies for this here. It's not the same as the central charts. It's just like a conventional notation So Such manifolds they possess like a complex killing vector which can be built like is a sandwich of of this two killing spinors Which commutes like with its conjugate and in Lorentz and signature these two spinors become a complex conjugate So, however, I'm going to focus like for like the applications of what I'm going to talk about I'm going to focus on a special case of of these backgrounds, which are in which case one of the two Circles is trivially fibered. So in that case you can write the manifold the four-dimensional manifold like this over the three-dimensional base Which has metric Which has this metric so this is now you can dimensionally reduce here on the trivial cycle And then you get in three dimension you get cipher manifolds which I think like Cyril is going to talk about in on Friday and Also, if the second cycle is also trivially fibered that you get The A2 is like in two dimensions So and then there are like some specific examples here of manifolds that are of this type Okay, so so now let me review this analysis like on Partition of partition functions on on care backgrounds admitting two killing spins of opposite charts So this was a work done by a closer to mitrescu festucci and Komargov ski in in two papers from 2013 and 2014 so in the first phase paper, they started the linearized deformation problem around the flat space and In the second one, they generalize that like non-literally using the the holomorphic twist. So Now for four-dimensional manifolds that admit two killing spinors of opposite charts Then the the conclusion was that The partition function does not depend on the Hermitian metric on M4 and also it depends like holomorphically on a subset of the complex structure in light battle Moduli but in the case where One of the cycles is trivially fibered, which is the one that I'm I'm going to focus on then you can Boil down these conditions on the fact on the property that the partition function is independent of of these three Complex functions that parameterize the three-dimensional base this red one here Okay, so and then let me just like outline the proof they gave like in the first paper, which was like around flat space so if you have now The r-multiple in n equal one theories So it has like also these you and are and and has like this spectrum here So and the dual background fields are Given here So and then you can show that like in the flat space supersymmetry algebra Takes this form so this is like this super current and then this team you new here is is this combination of all the bosonic operators and It is conserved so So this again like it's in flat space But but you have like this conservation equation and and then J here that appears here is this complex structure Of the manifold m and and it can be in terms of spin or bilinear Right, so now the argument is that if you take the variation of the of the generating function of the partition function with respect to these parameters like the The background fields then you can always write the variation here. I'm just writing the variation of the Lagrangian, but You should think of it as integrate over So you can always write by the definition of these operators in in this form so Here I'm missing some deltas on the right-hand side, sorry, so Right, so now given now the the form of this background metric that like preserves like Admin's like reading supersymmetry, then you can you can express it So this background metric was like this this metric here again So you can express this metric In in this form, so this is a little bit of algebra But it's straightforward and then you you find that you can express this variation here this generic variation in In this form so basically the variation is proportional to these particular components of of this Bosonic operator But then the point is that this Bosonic operator was Q exact so it can be written as the Q variation of this Supercorrect so again revisiting the argument that we had in the beginning. So this means that The the partition function is invariant under these deformations so again, there is a caveat in in this In this argument namely that the algebra that we're using here namely this algebra is actually the Classical large plus so well strictly speaking this is flat space. So even the quantum algebra would not Will not change but if we apply the same argument like to non-linear deformations, then the anomaly here could potentially appear so Okay, so that is The the statement like I mean from from this analysis. So now let us look at there was a follow-up paper by this gentleman here in late 2016 where we decide they decide like to Holographically check this statement by computing the the partition function holographically on Like on spaces that admit like these boundary conditions that that are like these backgrounds that admit global supersymmetry So in particular the metric again here. So these are like asymptotically ADS 5 locally ADS 5 backgrounds with boundary data given by these expressions here So I just reparameterize these functions A and C slightly so here we have like this this mu and and then this this function w and this W and mu Sorry, and also this this constant gamma prime and gamma and then some total gates transformation Lambda so this locally this is like a pure gates. So It plays an important role like for the global analysis, but for the local nice and that I'm doing it doesn't play any role so then These are analytic continuations of of the T2 vibrations like in Euclidean signature. So these are like Lorentzian background So and then the killing speed or equations are the ones that I wrote earlier So it's just like the conformal super gravity killing speed or equations, but with Z minus non-zero in generically in these backgrounds So now you what you can do is like you can evaluate the renormalized on selection and then see if it depends on on this generic functions mu and w and What they find is that? So the W variation Gives this expression here and the new variation gives this expression So in particular It seems that they are not invariant. So this seems to be a this seems to be a contradiction to the previous claim like in field theory that The partition function should be independent of of these functions so But okay, so now let us see like the how we can Match like this resolve this tension. So let us go back to the quickly to the transformation here of Like that we computed earlier So this is like the transformation again of the super current under local super symmetry transformation Like Q and S super similar transformations, but now I want to focus on On solutions of the killing spin or equation of the conformal killing spin or equations So in that case this this epsilon zeros become like the different components of the killing spin or Z So now they are related and if I use this expression I get the following expression for the for the variation of Of the super current So so here we get right so now the killing spinner has like these two non-trivial components. So There is this terms here which depend again on On the arc current and also the stress tensor But the important thing is that like there are these color terms here the orange and the red which are local terms depending on the background so What is important here is that like this particular backgrounds I did mention is that actually they all the bosonic anomalies. They are numerically zero so although the central charge is non-zero so that there are generically Toft anomalies in the in the world entities when you evaluate numerically for example the the vial anomaly or the The u1 r anomaly. They are numerically zero on this particular Backgrounds that meet these killing spinors. However, if you evaluate This bosonic expression shears which enter in the variation in the rigid variation of the super current They they are actually non-zero generically only in very like only if you specialize to To particular backgrounds then you can actually set them to to zero But for a generic class of backgrounds that admits like to killing spinors of opposite archers. They are not zero So now let's Use this result and repeat the analysis will repeat the argument of Closet collaborators, so what we can do is like we can take again the variation with respect to w and new and Again, if you remember we can relate it to So this part here So the fact that you you can relate it You can write in this form so in terms of this Bosonic operator here, so this still applies So again you you can write in terms of all the Bosonic currents However, what does the apply now is is this expression here that this Bosonic current is the q of of the supercar So the difference is that now we have that the this Bosonic Current is is the q Variation of the super current plus the anomalous terms here So now then if I put this anomalous terms on the other hand side then by the same supersymmetry argument This term would give me zero, but I will get the extra terms like from the anomaly Right, so in particular I will get like the extra terms from from these expressions and if I evaluate them on the particular background so so I get these components here of of the anomalous transformation of the super current And then if I evaluate them on the background I get precisely this expression and the same from the new transformation, so I get some different components here of the transformation of the super current and Then if you evaluate them you get exactly these terms, so now if you compare these final expressions with the result Found by general linear collaborators, so they match precisely so however the transformation the the the calculation is slightly different because So here the calculation was done by evaluating explicitly the on selection and then varying with respect to the Bosonic transformations So there were no fermions in in this calculation however, if you keep the fermions in the analysis then you can you can actually repeat the argument of a closet and collaborators and then you find precisely that This this terms here the non invariance of the generating functional is indeed precisely because of the of the anomaly anomalous transformation of the super current so again the argument that we give is completely analogous to the one by Closet and collaborators and and then However, we do the variation around an arbitrary background, so it's not around flat space so it's closer like to the analysis like in the second paper that they have and Right, so so this seems to suggest that actually the generating functional does depend on on on these functions and then So I apply this Argument to this particular backgrounds that admit like I mean this much supersymmetry But the local transformation of the supercarry that I computed is more general So you can apply to any background like even if it means like one supercharge or like even more than than two So very quickly let me just Point out like some consequences of this anomaly so namely now for the word entity and the killing spinner Equations you can for the killer spinners you can construct the corresponding conserved charges so here is the electric charts and Here is the conserved charge associated with this killing vector K, which is the Bilinear the spinner bilinear so Now there is this arbitrary parameter omega here, which allows you to define in this case different type of Charges so for omega equal to minus two you get the so-called Maxwell charges and for omega equal to one you get the page charges So now if you use the fact that in a supersymmetric vacuum the variation of the super current is equal to zero and you contract this identity with the spinner then you You can relate This spinner bilinear is a tender here to the killing vectors and then you you obtain this bps condition For every value of omega so this holds like for any values for any choice of omega But then on the right-hand side you have an anomalous Contribution which depends precisely on this anomalous transformation of the super current So it's a length expression that is in the paper But it's precisely the the integral if you want of this anomalous transformation of the supercar So here this is just the mass and the and the angular momentum Okay, so let me conclude so So it seems that like in generic care backgrounds that admit like some notion of reading supersymmetry the this reading supersymmetry is is anomalous at the quantum level and In particular this implies that the supersymmetric partition function might not be Independent of the background data. So in particular this Hermitian metric and the complex structure and and also the this anomaly affects like the bps equation between like the the conserve charges like the Cosmere energy and the other cosmic charges So there are a number of future directions That Would be interesting to explore so one of them is like to determine the super current anomaly for arbitrary Coefficients a and c so for this and for different versions of super gravity directly from from the field theory point of view So I know there's like a number of people who are looking into that Already then it would also be interesting to see if there are concrete examples where localization computations are affected by by this anomaly and then finally there is some Discussion about whether whether you can actually like transform this supersymmetry anomaly into an anomaly of a different type such as a gravitational anomaly. So this was already hinted in the paper of Of David and collaborators. So in that case they give a counter term a local counter term that is actually not completely Defiomorphism invariant. So this would suggest that actually if you add this counter term You remove the anomaly from the supersymmetry, but then you you have some kind of gravitational anomaly. So and I also Like I think also there are like people looking into this as well. Thank you