 ...organizator, for inviting me here, and giving me this opportunity to speak, but also to come back to ICTP after so long meeting so many friends in this very stimulating environment, both scientifically and from the human point of view. So this work, this talk will be based on a series of work that started two years ago, but I'm mostly going to report about the work, which I started with a student of mine, and with Dario Rosa, which is not appeared yet. What I'm going to describe are two different topological structures, which sit inside supergravity, and the first one is very generic, and it exists in any dimension and for any supergravity, so in a certain sense being so generic is also less powerful. And the second structure exists for a certain class of supergravity theories, which I'm going to describe, and in this class there is n equal to and n equal 4 in d equal to supergravity, and n equal 2 in d equal 4 supergravity. And in the second part of this talk I will describe the application of these structures to localization, and describe also a relationship that emerges between them. So let me start to revisit in order to describe the structures, the BLST formulation of supergravity mostly for the purpose of setting the notation, since we already had a lot of talks about this. And as we all know, the BLST formulation of any theory, but in particular supergravity needs three ingredients. One, the anti-commuting ghost associated to the bosonic symmetries, the commuting ghost associated to the fermionic symmetries, and an ill potent operator, which I'm going to denote with S for traditional, respect certain traditions. Supergravity in particular I'm going to focus on Poincaré supergravity, also if this could be a standard to conformal supergravity, although we didn't really work out this yet, might be necessary in the future. The bosonic symmetries include the theomorphism, and I'm going to denote the ghosts of the theomorphism, as with Ximu, these are vectors, anti-commuting vectors. And there are going to be some Yambil symmetries, which certainly include local Lorentz and certain local R symmetries, and go to denote collectively all these Yambil symmetries, the ghosts associated with all these Yambil symmetries with C. So this C lives in the total Lie algebra of the bosonic symmetries. And then there are going to be the local ghosts of supersymmetry, which are going to be commuting, and if I talk about N supergravity, there are going to be N of them, which are going to be Majorana spinors. So as we know the transformations of supergravity are complicated, but there are certain transformations, and depends a lot on the model, on the dimension, and everything, but there are certain transformations which are universal. And in the BST formulation, in particular, what really counts, what defines the theories, are the transformations of the ghosts, in particular in this case of the commuting ghost. And the form of the supersymmetry of the BST transformation of the commuting ghost is the following. There are always defiomorphizing gauge transformations, which depends on the anti-commuting ghost associated with bosonic symmetries, but the really important piece is this one, and this one, the psi are the gravitinos here, so there are one forms, and one has this vector field, which is a commuting vector field, which is a bilinear in the supersymmetric ghost. It depends also on the fear-bind. So this term in terms is quadratic, of course, in the ghost of supersymmetry, but also depends on the gravitino, on the fear-bind. So it's quadratic in the fields, and that's what it means in the language of gravity, people that they algebra is soft. It simply means that the BST transformation doesn't, the definition of a ghost, it's not quadratic in the ghost, but it's quadratic, really. So they're quadratic in the ghost, but there are also other fields. From the point of view of BST, that's really not, especially if it comes from topological theories, it's not really a special thing. Simply there is no algebra associated with it. So this is a universal, this form, and the other transformation, which is universal, is this one, which is the transformation of the fear-bind. So given this transformation, one can use just the impotency of s, and arise to the conclusion that the BST algebra can be written in the following way. There is an operator that I'm going to call to distinguish from the small s, as big s, which essentially is defined on all the fields, but not on the bosonic ghost. So this operator s doesn't act on a reduced field space, the space, which does not contain the ghost of the bosonic symmetries. That's why it's called the equivariant BST, and so one subtract from it the action of the bosonic symmetries, that is, which are proportional to the bosonic ghost, anti-gomutin ghost. And so this s requiring the small s, so the BST operator is impotent, is equivalent to requiring that s square close to this algebra. So this algebra has two pieces, one is the lead derivative, and the parameter of this lead derivative is commuting, of course, is exactly the combination that I described before, the bilinear in the ghost of supersymmetry. And the other one is this general form, that is always a piece, which is the contraction of the young mill's fields, always with gamma. But there can be, and there is generally another piece, which also bilinear in the ghost, which I'm going to call phi. So this is the algebra, so if you want to... This is equivalent to the importance of s, just is a relation between the s square 2, it's a completely equivalent to the importance of the original operator. But this algebra is only valid on the reduced space, which does not contain the bosonic ghost. Yeah, this algebra is valid on all fields, but those. Yeah, thank you, that's important. So you see that supergravity is completely characterized by this algebra, and it's characterized by these two characters, which are these commuting combinations of fields. The first one, gamma mu, already talked about it. The fact that this gamma exist in any supergravity was actually pointed out in an old paper by Bolier and Belon in 86. This other guy, this scholar bilinear, which lives, which here he written more explicitly, in the case of supergravity, lives so in the total e-algebra, where ti, they noted the r-symmetries, Jammil's generator, and sigma ab are the Lorentz generators, so there are different components, this phi, is instead model dependent, and in a certain sense, so since gamma mu is always there, but this one, the form of this, the dependence of this phi on the ghost and the bosonic fields depends on the model. So in a certain sense, you give me this phi, I know the supergravity, that's what ultimately characterized supergravity. Just to be concrete, because this was very general, and those are the two cases that I'm going to discuss in detail in this talk. For n equal to supergravity in two dimension, phi, as since the bosonic li algebra has a Lorentz component, which is a bilion, and also u1 r-symmetry, which is a bilion, so the phi's are two components, which are a bilion, and they can be written as, just to be concrete, as these combinations, so this phi, fA, are the two possible, so the ghost, the super ghosts in this case, they are two super ghosts, which we can collect, Majorana, which we can collect in a Dirac super ghost, so we can make two combinations of them, f1 and f2, and these n are the scalars, which are duals of the gravity photon field strength, so there are auxiliary fields, scalar fields, there are two of them, and as you can notice this expression for phi, for both the Lorentz and the gauge component, is invariant under this global symmetry, which is a Lorentzian symmetry. The structure of n equal 4 is more complex, because in this case, there is a non-abilian r-symmetry group, the Lorentz, of course, we are still in two dimension, so the Lorentz component is still a bilion, and has a very similar structure than the n equal 2, but now we have four auxiliary fields, because we can make four combinations, bilion combinations of the super ghosts, we now have two Dirac super ghosts, in the fundamental of SU2R, and beyond these two combinations, we can make this other combination, which is complex, so this is both zz and zbar zbar, and there are also these scalar fields, which are duals of two forms, which we call in analogy with n equal 2 supergravity, gravity photon field strength, and again there is a global symmetry here, and this time is always a Lorentzian, but this time, of course, is a one, three symmetry. As I said, the gauge super ghosts, phi, the component in the non-abilian gauge algebra, is this one, and in this case, always these auxiliary fields appear, so this nA, but they are really also the combinations of non-gauge invariant combinations, combination ghost biliner, which live in the Lie algebra appears, so we can make these four combinations, these are one, two real combinations, and one complex combination, which are this time valued in the Lie algebra. However, eventually we will be interested in the, so this phi gauge, as I said, lives in the Lie algebra, so it's not gauge invariant, one can extract the gauge invariant trace of it, and this gauge invariant trace will depend also on the, like the Lorentz ghost biliner on the scalar gauge invariant ghost biliner. So these guys and nA and nFa are going to be important in what they follow, and that's the reason I put them here in evidence. So, okay, this, in a certain sense, I just restated the commutation relations of supersymmetries in BRST formalism, so it's just repackaging of this observation. Now essentially the simple but interesting observation is that this combinational ghost have universal and remarkable BRST transformation properties, which are these ones, and the first one, as I said, the first fact that this gamma mu is invariant under this S was already, as I said, remarked in the paper by Bolier Belon, even before, when they wrote this paper, the topological guy was not invented yet, but they remarked that this transformation are those, and here what I add here is the observation that those transformations are precisely the transformation, it's a piece of the transformation of topological gravity, and in topological gravity, you have gamma mu appears as the partner of the supersymmetric of the anti-commuting ghost of the filmorphism in this way, g mu nu transforms into topological gravitino, and S gamma, as I said before, is zero, and S mu nu is this one, and all these transformations are also valid for supergravity, once you define gamma mu, as I said, in the Bolier Belon form, and the topological gravitino by this relation, then this transformation coincide with the transformation of the BRST transformation of supergravity, so there is this sector, this topological gravity sector, which appear in the BRST transformation of supergravity. Now the new guy is this one, this phi, and one observes that the BRST transformation of phi are also of this form, this green is really not very visible, but I put it in green because if you consider this multiply phi, lambda and f, so f are the gauge field strength, which live both in the, as I said, in the Lorentz, in the arcimetry group, so phi, lambda and f belong to a multiplet, this is precisely the multiplet of topological yam mills, and the blue part would be the transformation of topological mills, that we know, which are very familiar, that is that phi is invariant, and lambda, the topological gravitino, goes into the derivative, the current derivative of the super ghost phi, and that variation of f is a total derivative, and if you couple topological yam mills to topological gravity, they can be coupled, and also those terms, those green terms appear, and those are precisely coincide with the transformation of BRST of supergravity. So again if you define, so there is a topological gaugeino, which is the BRST variation of the gauge field day, and then so this phi, the lambda and f form a multiplet under BRST transformation, which coincide precisely with topological yam mills, and topological, coupled to topological gravity. So this is the summary of what I said so far, so there is this universal subterzo composite, so this is an emergent structure, which appears, so that these are not elementary field, or rather there are multiplets, which the top component are the gauge fields, and the metric, but that other components of the multiplet are composites, made with the superghost, so the things that appear in the BRST formulation, because the important ingredient is the superghost, which is commuting. As I said, this structure is made of composites, so it's not clear what it means at, it's not clear yet, to me, what it means at quantum level, of course, I mean for the quantum supergravity, although I think it's an interesting challenge to see it as some relevance, but what I will discuss here will be later on, actually this application to the organization, which supergravity is used as a classical theory. Let me simply make a parenthesis here, topological admitts coupled to topological gravity can be defined, of course, as a microscopic theory, like topological theory, topological gravity can be defined as a microscopic theory, in which the gauginos and the gravitinos are elementary, unlike in supergravity, and this theory is interesting geometrically, because it computes the deram comology on the space of the, on the product space of the matrix, and the gauge connection on a given manifold, equivariant respect to the action of both the filmorphies and gauge transformation. Topological admitts only computes if you wanted the deram comology equivariant in respect to the comology of the space of connection equivariant in respect to gauge transformations, and topological gravity does the same for the matrix, and this theory would describe, this topological theory would describe this more complex deram comology. And as far as I know, there is not much study about this, but this theory by itself would be an interesting filthoretical model to study the magic dependence of Donaldson variance and various phenomena which we know occur in Donaldson theory. It means that, this space, on this space there is an action of the gauge symmetries. Yeah, okay. If this action would be free, you can make the quotient of this under this action, you obtain a nice manifold and that would be and then what you are doing here is the comology on that manifold. Okay. But since this action is not free, that will replace that is this. So what I am doing is, this is the comology of the modular space of this thing. So as I said, this structure is very generic, but there is more structure which emerges in certain theories whenever you can make gauging variance color bilinears, you are going to call F A because they already appeared in the previous slides, out of the commuting ghost which do not depend cell bilinears which are gauging variance and do not depend on other bosonic fields that are really bilinears, okay. Just because I don't have a better name, I will refer to this kind of supergravities as twistable supergravity for a little, just to give, since I already talked about it, to give you an example, the F1 that we found before are precisely bilinears of this sort, for D equal 2 in N equal 2 and for N equal 4 in D equal 2, we have this 4, but for example also for N equal 2 supergravity you have this ghost and you can make these combinations which are invariant. They are equally twistable. In other words, what you need is that the Lorentz indices and the SUR indices of the ghost has to combine in such a way that you can make invariance. So whenever you have these combinations, then something general occurs because as we said, the variation of the super ghost is universal and so if I have a combination of this form that means that the variation of this bilinear is high gamma or something, is high gamma trivial where the fermionic one form is this schematic structure where I indicated by xa, the matrix which you use to make the bilinear in the various cases. So since you have this algebra which holds generically then you have the relation that sfa is high gamma trivial since high gamma square is 0 you can apply the usual technique of descent equations. So you can construct a descent so there is a whole multiplet associated to fA which is the one form fermionic which I denoted by key, but there is going to be a sorry I think this is my thing I used it I wanted to use it not to forget it but I forgot it okay sorry So you have the whole multiplet I should say here that because this descent equation is a consequence of the algebra of a square or the fact that high gamma square is 0 since gamma is bosonic but it also relies on the fact that high gamma square, so it's a cobanded operator has trivial homology actually since gamma is a composite field you don't have a general proof that this is always true but in all the example that I worked out it happens to be so when high gamma you apply here a square so you get that high gamma of s key plus df is high gamma of something so you hear what I using is the triviality of the high gamma homology if high gamma homology were not trivial you could not descend so you stop at some point in the example that I worked out that never happens but I don't have a general proof in any case so typically you have a multiplet and you end up with something which has goes number 0 so you have to stop here so this is the same thing I just said so what you have is you have a super field associated with these bilinears who stop component is bosonic and these super fields satisfy this equation so in top of this s plus d minus high gamma invariant and you have as many of this scalar topological multiplet as there are where you have of this bilinear combinations which are gauging variants of the ghost now one important thing is that this bilinear combinations so that they are typically not independent they satisfy fields identities and these fields identities always involve the gamma that I talked about so the vector and the scalar combination are really related by nonlinear quadratic identities which for n equal to is equal to is equal to is equal to look like this and so they can be put again in a global duality invariant form in both cases and since gamma belongs to the topological gravity multiplet that as I said before that is the beginning of the relation between the scalar topological structure and the curvature topological structure between the two structure that I talked about so as I said I want to apply this so this is the second part of the talk I want to apply this to localization and since this is a workshop on localization there is no need to explain what it means as I said in the context of localization what counts is really classical supergravity so this structure that I talked about which is a classical structure can be used and as you know to find the localizable background one has to set to zero the supersymmetry variation of all the fermionic fields and the differential equation that one obtains by doing that are called generalized skilling spinor equations and they as we know they emit non-trivial solution only for special configuration of the bosonic fields which are the integrability condition for this equation to have solutions and of course this generalized skilling spinor equation are different in different dimensions and they are usually very complicated to solve again since these are the two cases that I am considering in this talk just to give a flash of how they look like for n equal 2 they look like this so they involve the spin connection the gauge fields and these auxiliary fields and for n equal 4 in n equal 2 they are more complicated all the all the 4 gravi photons appears in this equation as you know for this equation in two dimension have a very old solution which is due to Whitten which led to the topological twisted twisted model which is universal in the sense that it applies to all spacetime geometries but then in modern times other solution were explorers of this equation in for spheric topologies and so there has been a lot of work on this in d equal 2 but also in higher dimensions and since there is no general strategy to construct solution of the generalized skilling spinor equation one application of the topological structure that I described is to provide a systematic way to find and classify solution of this equation so also to find new solution of this equation and what I am going to describe in the rest of the stock is exactly the application of these two n equal 4 and this includes also as a specific case n equal 2 in d equal 2 so this is the most general solution in two dimension that I know of I am going to call the localization locals the set of bosonic backgrounds for which the variation of the bosonic fields is zero and on the localization logos not only the elementary the variation of the elementary fermions are zero but also the variation of the composite fermions so I get values equation exactly in parallel with what I do normally with analogies to the GKS equation one you obtain by put in to zero the variation of the topological gravity also I'm going to call this the topological gravity in equations and then I'm going to get the topological region and always to get that two equations and then in those pretty stable暗 days also I am going to get that next equation which are the topological skeleton equation so let me discussside turn this three equations the topological region topological gravity and topological color equation topological gravitinega, kaj ta formu je, a je taj, da je vzal, galima je to nisometrije, spesitivnja, vzal. To je vzal, ki je, kaj je, vsega, vzal, taj, taj, taj, taj, taj, taj, taj, Čosi je izvizelja na sefah in počinati ko seče. To je nepočbe izvizelja, in več neči je prejšt LA što, je izvizela na začelji gravid, tudi osobi gravid, ki se te dobra zelo ni jazrat, da so li pričas za telo toga teba. V odstanoj na super gravidin je to začelo, ki je zelo na začelji gravid, že je dvečje izvizela, The important thing is this equation, when either one of these algebras is non-ambilien, either the Lorents or the gauge, becomes non-linear. And to extract the gauging invariant content of this non-linear equation, one can define generalized field strength for the non-ambilien field strength, which is a polyform, if you want, which is, with a two-form component of this number, zero, And the phi, which plays the role of the zero of the equivalent extension of these two form component of those number two. And so one can construct polyforms, which are generalized chain classes. So they start with the chain class of the right given degree and end up with the trace of phi. So these are the equivalent extensions of the standard chain class. And the interesting thing is that even when you are in, for example, in two dimension, where you would not have the, let's say, trace of s square, because, you know, for dimension reason, you, the, so the, the, the chain classes are trivial, the second central trivial, you still have, in high, in all dimension, you will have all possible n. So these equivalent extensions are more refined invariants, because, of course, you are keeping the structure, the equivalent structure in. And the topological gaugin equation at gaugin variant level is equivalent to this. So it's an, it is really telling you that this chain, equivalent chain classes are d, d minus i gamma closed. And this operator, this cobando-reoperator, of course, square to l gamma. And that what defines that the rank homology of polyforms on space times, which are equivalent, respect to the killing vector action. So the idea is that the gravity equation sets you, defines you the killing structure. And this killing structure defines you a homology problem, equivalent respect to this gamma. And the chain classes are the equivalent extension of the standard chain classes, equivalent respect to this killing vector, in vector killing action. So we know that, in general, the chain classes are integral valued. And in all the examples that we computed, also this, so you get more invariants. So these invariants are also integral. So I don't have really, this could be my ignorance, but I don't know why this is so, but I found that this, that should be true for general reason, but that's what I verified. So this integral, this invariant, in other words, label different branches of the localization localization. So it's a classification of the localization, they give it topologically, but the equivalent, classification of the localization backgrounds. Of course, in each one of these branches, you can have different localizing backgrounds in the same topological class. So this topologic, in the end, in the summarizing, this first topological structure give a broad topological classification of the localizing backgrounds. Of course, as I said, they don't completely characterize the localization localization. They only give a topological classification. And to characterize the localization localization, you need extra equation. And when, the rest of the talk, so what we are going to see is that when you have these extra structures, then these are going to give you exactly the equation that you need to completely characterize the localizing backgrounds. So we come to the scalar topological equation associated to this, the fermions, which sit in the scalar topological multiplet. And again, the variation of the fermions to zero gives you this equation, which again gives you an element of the gamma-equivariant homologin. In other words, you have a polyform, which is made by the bilinear of the ghost and the scalar field, and this polyform is the gamma-closed. So this essentially lets you solve the localization equation in the sense that, given this FA, you can construct the scalar bosonic backgrounds by solving, if you want, the equation, the equivariant equation that I just show you. And on the other hand, as I already noticed, these FA are not independent, but satisfy these conditions. So you cannot choose, for example, for unequal four independent FA, but you have to choose four function FA, which satisfy this condition. So you have three independent functions, but there are no other conditions. So once you have chosen these three independent functions, which satisfy this equation, so at the point you have completely fixed the bilinear, so you have completely up-to-gaging bias, you have completely fixed the constant spinor, and then also the bosonic backgrounds. So in other words, just to give you a flavor, what you get for unequal four, you get a lot of solutions, which are parametrized. Here, I could not write this in a slide, because they are too big, and then there are six components of this, and so just to give you an idea, you get the rational function of the F and its derivatives, the ends are essentially derivatives. This is done for the round metric on the sphere. So in principle, so you have a space of solution or a space of localizing background, which is parametrized by three functions in the unequal four cases. Well, there is an observation here. Of course, this FA, satisfy this equation, as they are closed under the gamma. So once you find, given a solution of the equivalent closed solution of the equivalent closed under the gamma, an FA and an A, a bosonic background and A, you have, of course, a concept of equivalence in this solution. So this equation, with the terms in the backgrounds, FA and A, has a gauge invariance, which is, of course, the homological invariance. So there is a concept of equivalent solutions under, in other words, you have classes here, because this is a homological problem. So there are two solutions, which can be equivalent under a homological transformation. This homological transformation are labeled by invariant one-forms. And since this homological symmetry is really inherited by the original supergravity BSD symmetry, one is led to conjecture that localizing background would correspond to homological equivalence solution, give right to the same partition function. I cannot say this is sure, because, as I said before, this BSD symmetry, this invariance in supergravity is composite. So it's inherited, the parameter of the, which parameterize this invariance are not elementary fields. So it's not completely clear that this, at the quantum level, for the super, of course, if this were the topological matter field, a couple topological gravity would be automatic, but since this is a theory of composite, so all this topological matter and this topological gravity is composite, it's not obvious that changing the, there is an issue of Jacobian going from one to the other, and so it's not clear that this, I don't think this invariance is automatic, but let's say it's very tempting to think this is so, but if this is true, then there is a modular space of supersymmetrical localizing solutions, which would be finite dimension, because then I could choose in the words representatives for these classes, which I can put in this way, and these representatives would be labeled by constant, not by function. So instead of the background being parameterized by three functions, they would be parameterized by six constants. So this is something that should be checked, but let me finish with describing the relation between the two structure I talked about. Implicitly, I already talked about this, so I said that once I know the scalar solution of the topological scalar equation, I can reconstruct also the gauge background, and the reason for this is that, as I mentioned before, there are the first relations which connect the two supermultiplets, and this is a quadratic relation between superfields, so the scalar superfield, the quadratic combination, gives rise to a superfield which is completely made out of the topological gravity fields, which is this one, and so this allows to express the curvature polyform in terms of the scalar ones, and there are analogous relations which allow to express also the gauge polyforms, the curvature polyform of all the gauge fields in terms of the scalar ones, and in order to describe this relation in a contact and super symmetric, explicitly super symmetric form, one has to introduce a derivation which maps equivalently close polyforms to equivalently enclosed polyforms, which is this one, this is sort of dual operation on polyforms, so given a polyform which is F, A, and A, you can construct another polyform in which the role of the zero form and two forms are interchanged, and where there is this equivarian Laplacian which appears, so this is something which exists in the equivarian context, and once you have done that, introduced this operation, then there is this relation that you can show which relates exactly to the scalar, so this summarizes the relation between the scalar topological structure and the curvature topological structure, so the on-shell, this on the super symmetric background, the quadratic, these quadratic combinations of the scalar polyforms give rise to the curvature and the gauge field of the, both in n-equal 2 and n-equal to 4 theory. In n-equal 4 theory, what you get is an expression in terms of super fields of the chain classes, because as I said, in this case, the f are non-abilian, so they are not gauge invariant, so there is not a gauge invariant relation of course between the two, but the gauge invariant relation between these two structures is this one. So these, in other words, are the equivalent at the topological level of the integrability condition of the generalized covariant-killing spinor equations. In other words, it tells you how to derive the gauge fields once you know the scalar fields, and they are manifestly as 1, 3 invariant. So, to conclude, there is a host of new localizing background that one could explore for n-equal 2 and it would be interesting to compute the matter partition function for these new backgrounds and determine another thing if they are a homological invariant and they are going to be certainly duality invariant, because the formula is a soldier invariant. So my conclusion is that these ones, this is just a summary of what I said before, there are these two structures and there is a relation between the two and what is left, as I said, beyond exploring this new host of localizing backgrounds that we found in d-equal 2, of course these two apply this is to find the same structures in n-equal 2-d-equal 4, as I said, this is a twist about supergravity in the sense that I introduced before and if one could find the analogous relation between the curvature and the scalar multiplets for n-equal 2-d-equal 4 that would essentially be in solving the integrability condition of the GKS equation in d-equal 4 and of course most to me the most interesting issue open is to understand if this topological emergent structure of supergravity survive a quantum level and so let me conclude me with something which seems to me a little bit natural so beyond finding this localizing backgrounds what we really emerged here is that the classical supersymmetric of n-equal 4-d-equal 2 are parameterized by this topological multiplet and by topological gravity background on shell that is, you know, the H has to be the gamma closed and they had to satisfy this relation g mu nu and gamma mu has to be some on shell topological gravity background and there is this relation which connects the two backgrounds so when it's time to conjecture that there could be a non-linear topological sigma model whose coordinates are precisely the super-field HA that coupled to topological gravity which is described in an effective way the quantum fluctuation of supergravity in other words the equation of motion of this topological sigma model should reproduce this equation and so if one the fluctuation of this topological sigma model might describe effectively the quantum fluctuation of the supergravity field because if you want this degrees of freedom completely describe the space of vacua of the the classical space of vacua of supergravity so thank you