 First of all, for organizing this workshop and inviting me to speak here. It is always a pleasure to come back to ICTP. So in the last couple of years, I've been working on certain topics with my collaborators. All of them are here. OK, I don't see AD, but OK. So this is related to the work where we want to understand how the supersymmetrical localization, the technique which has been explored so much on the compact spaces, how does this apply in the case of quantum field theory on a non-compact spaces, in particular ADS spaces. So this is the title of my talk, Boundary Conditions and Localization on ADS. So these are the references for this talk. So most of my talk will be based on this second paper, which appeared a few months back. In first paper, we just looked at Churn-Saman's theory on ADS 2 times S1 background. This, the motivation here, was something different. Actually, it was related to some entanglement topic computation in Churn-Saman's theory. And it took us more than one and a half year to write the second paper. So in this paper, we developed some technique to compute the partition functions on ADS spaces, and in particular, incorporating the boundary conditions, especially normalizer boundary conditions. So most of my talk will be focused on this paper, specifically focusing on a simplest example, which is Karel multiplied in three-dimension. And near the end, I will mention some of the results, which we have obtained in this, which is still not a paper, but hopefully it will come soon. And also some results which we obtained in this paper. So in the last 10 years, there has been tremendous development in supersymmetric quantum field theory, in particular, gaze theories. And this is due to, thanks to the idea of supersymmetric localization, supersymmetric localization has provided us a powerful tool to evaluate or compute some of the observables exactly, and thus probe strongly coupled dynamics of a quantum field theory. This has led to various non-trivial checks of several conjecture dualities in several supersymmetric quantum field theory and defined in various dimensions. And also now we can compute various observables in quantum field theories using the technique of localizations. This has also provided highly non-trivial checks. Some of the work has been also applied. Some of these techniques have also been applied in the context of ADCFT. And it has also provided very non-trivial checks for this proposal. But most of this work has been focused mostly on supersymmetric quantum field theory, which is defined on the curved but compact spaces without boundary. Now rigid supersymmetric quantum field theory can also be defined on curved but non-compact spaces. And therefore, one can do the similar kind of computation, as people have done in the compact spaces and explore various kind of relations among supersymmetric quantum field theory, but defined on non-compact spaces. There are certain spaces which we are very much interested in. And these are spaces of this kind, ADS and SM. And these spaces are interesting because these appear in various physical examples, such as computing entropy of extramar black holes or entanglement entropy computation, conform field theory, or even as some surfaces describing Wilson loops in ADS. So therefore, these examples provide us motivation to study localization computation on such spaces. Now localization computation on non-compact space is a bit harder problem compared to compact spaces. And this is because of following two reasons. First, a simpler reason. And that is that now when we consider non-compact spaces, one is to worry about the boundary terms. In particular, the boundary terms which we get while considering the variation, supersymmetric variation of the action. And second, the most important one that now when we perform the path integral, we need to include boundary conditions or specifically, super suitable boundary conditions while doing the path integral computations. Now these boundary conditions, so these boundary conditions are integral part of any computation which we define on a non-compact spaces. In particular, non-compact spaces, boundary conditions defines the problem. If you choose different boundary condition while performing the same kind of computation, it computes different observable. And maybe in different quantum field theory. Now the boundary condition, when you impose these tell us in path integral which modes to integrate over. Typically, the natural boundary condition, one impose when you're considering a path integral or non-compact spaces is that a square integral mode, that is a normative mode with respect to standard L2 norm. And therefore, one would like to see whether one can do supersymmetric localization computation with such a square integral node. Now there are some motivation, the fact that we have been using this normalizable boundary condition with respect to this L2 norm. We have used this in the various other problems. And I mentioned some of them here, especially just a couple of them. And this will be for us as a motivation to use this normalizable boundary condition in the localization computation. So first is the entropy of a black hole, especially a black hole. So there's a proposal which is called a quantum integral. So that may define some alternate quantization kind of thing. But here I'm focusing only on normalized boundary condition. Well, this depends on what you're interested in. Because if you impose some other boundary conditions, which makes action finite but which is not a square integral, this may define some different theory, et cetera. So that's what I say. So the boundary condition is important. You need to specify beforehand. So that will define my theory and observable. So that's what I here. So this is the, I define the normalized boundary condition. So this is the standard normalizer I'm using. That's what I'm saying. So you define your own, but I'm using the standard L2 norm, which we use in the, I'm giving some example now, where we use this standard L2 norm to compute some observable. And yeah, so that's what Atis was asking. So that will define some other theory and maybe you compute some different observable. Yeah, that's fine. So that's why I say the boundary condition on non-compact space defines your problem. So that's important to say beforehand. So we are going to use this and see how far we can use it. So this describes a non-compact open space. This will describe me the fall of behavior. And that's enough for me. So one of the problems which we are interested in. Yeah, one can impose a normal boundary condition. But possibly, yeah. Well, I don't know. You have to change the norm. So if I use this standard norm, so I'm not so sure whether that would be normalized or not. OK, so one of the problems which we are interested in, for example, a black hole entropy. So for external black hole, there is the proposal to compute the entropy, which is given a following path integral. So we have a couple of talks describing this function. So I'm not going to describe it here. But this in the last few years, maybe last four, five years, there are several attempts to evaluate this path integral and compute some correction to the black hole entropy. For example, in one of this work, there was a non-trivial computation, which is a logarithmic correction to black hole entropy. And this computation was performed with the normalized boundary conditions, the norm which I defined just in the last slide, using the normalized boundary conditions on all the fluctuations. And this led to this coefficient which matched with the microscopic answers whenever this available. There was another interesting work done for the black holes in ADS-4. All these people are here. And these are the black holes, hyperbolic black holes. And for this, the entropy was computed from the twisted partition function, which are defined on ADS-2 times S1. This is a twisted partition function of ABGM theory. And they extract from this the entropy of this hyperbolic black hole. And in this work, they also impose a normalized boundary conditions. And they match the entropy of the black hole. Now, before going into localization with this L2 norm, normality condition, let me give you some brief, very briefly, what is localization. Since, again, this has been said several times in this workshop. So I'll not describe it here in detail. So just for the completion, so the basic idea of the supersymmetry localization is based on the following fact. That if you have a supersymmetry generator q, which is squared to some bosonic symmetry. And it is a symmetry of the theory in the sense that action is invariant. And the path integral measure is also invariant. Then the expectation will be of any q exact operator is 0. So Yanis, a few days back, talked about the subtleties in the path integral measure. I'm assuming here that the subtleties are not here, not there. If the subtleties present, then one needs to be careful. So using this fact, one immediately see that if you deform the action by any q exact deformation, such that square of q on such a deformation is 0, then the partition function is independent of the coupling constant of that q exact deformation. And therefore, one can evaluate this partition function for any value of t. In particular, for t goes to infinity. And if you have chosen v carefully, such that q v is positive definite, then this relates to the path integral over subspace of configuration. Where this configuration is defined by this equation. And this is the basic idea of localization. Okay, so let me, before going ahead, so we have applied this localization technique on two cases, two non-compact cases. One is ADS2 times S1, another ADS2 times S2. So for most of the analysis, I'll present the ADS2 times S1, but later on near the end, I would say something about ADS2 times S2. Now, so what do you want to do? So we want to have a background, ADS2 times S1, which is, I'll choose this metric here. L is some constant. And we want to put some n equal to supersymmetric theory, quantum field theory, with UNR symmetry on such a background. Now, so now this ADS2, so in order for a background, in order for a supersymmetric field theory, in order to have some super field theory on a background, it is necessary that the background must preserve one or more rigid supercharges. Now, typically we can achieve this by turning on some of the supergravity fields, present in my gravity multiplied. And such that they solve certain killing spinor equations. So this is, I'm using some formulation developed by these people. Now, so I'm not describing, these are some very super fields in the supergravity multiplied. So I'm not describing this very much here. But these are obtained, these are killing spinor equations, which we obtain by setting gravity norm and variation of gravity norm equal to zero. Now, ADS2 times S1 alone doesn't preserve supersymmetry. However, so what we want to do, we want to find out this, such some background value of these fields, such that ADS2 times S1 together with this background, satisfy this killing spinor equation. Also on the killing spinner, give us following killing vector. This will give me some equivalent algebra, which we can use for the localization computation. So these are the solutions. So here, these are very simple solutions. So these are the fields, value of the fields. So actually these are coupled to, this is later arithmetic case field, and this is some, I guess, relative to string currents and central charge current. Anyway, so these are the value of the fields and other components of the gauge field and HR set to zero. And this is my killing spinner. So this killing spinner is independent of time direction or the Euclidean time direction, the S1 direction. But this is anti-periodic Elingon theta. And using this super skilling spinner, one can realize supersymmetric transformation on the vector multiplet as well as current multiplet or charged current multiplet. And the supersymmetric algebra on this multiplet is something like this. So here, K is the same killing vector, which I mentioned here. And this lambda is some gauge transformation parameter, which is constructed out of the vector multiplet fields. And here we have some r-symmetric transformation with parameter L. Okay. Okay, so now we have a background. And we want to consider the partition function of chance-sample theory coupled to a matter in some representation. So for a moment, we'll focus only on fundamental matter, but one can generalize this to any arbitrary representation. So this theory is described by the following action. So here we have a chance-sample theory. And this is describing the matter part. Now in this paper, we looked at the only chance-sample theory. And one of the motivation was that this chance-sample theory is equivalent to, although it is supersymmetric, but it is equivalent to a bosonic chance-sample theory. And the reason is that the other fields like fermions and scalars are just auxiliary fields. Okay. So therefore one can compute the partition function of non-evalient chance-sample theory on ADS2 times S1, just by considering this one and considering and performing supersymmetric localization. But now we want to incorporate these matter fields. So here phi is some complex scalar field and there are different terms here. Like q here denotes the coupling of phi with the gauge field. And here delta is some r charge for the common multiplied. Here r is the Ritchie scalar and et cetera. Rest of the here dot-dot denotes the familiar terms which I'm not mentioning. Okay. So now as I mentioned earlier, so before going ahead to do localization computation, one important thing one has to do is to check that whether the action is supersymmetric or not. That means and where q of S equal to zero or not, okay? So it turns out that when you consider this chance-sample theory compared to matter, the action is supersymmetric up to a boundary terms, which is usually typically the case. In particular, the most important thing here is that the matter action itself is q exact up to boundary terms. So here I have just written down the boundary terms, which appear in the matter action. There is similar boundary term, similar kind of boundary terms for the vector multiplied. Okay. Now, so in order for localization computation, oh, I want to see matter action. Yeah. So this is the action, fermionic theory. What do you mean fermionic theory? Which is just this auxiliary field here. That is very fine. They're not dynamical, but they're very fine. Yeah, here f is auxiliary field in the current multiplier and g is auxiliary field in the vector multiplier. Yeah. Okay, so saying that the boundary terms, which we obtained, now we need to set them equal to zero by some condition. So one way to do it is that by adding especially a boundary term such that the supersymmetric variation of those boundary terms gives this term that is supersymmetric variation of those boundary terms gives exactly this term up to some negative sign. Or the other way to do it is that these boundary terms would go to zero with the boundary conditions. In particular, if you see these terms here, we find that all these terms go to zero with normalizable and smoothness condition on the fields. So with the normalizer boundary condition, I don't need to add a boundary term, but if you use some other boundary condition, then we may need to add a boundary terms. Okay, so let me make some more comment about the boundary conditions, especially the normalizable conditions and it's compatibility with the supersymmetry. So if you look the L2 norm, which I gave you earlier, so that fixes how the fields should fall off near the infinity as r goes to infinity. So r here is ads to coordinate and r goes to infinity as a boundary. So the normalizable condition requires that my fields like both scalar and the fermion should fall faster than exponential minus r over two or at most it should go like e to minus r over two. However, this condition is not consistent supersymmetry. And one way to say is by just looking at supersymmetry transformation. So what we find from supersymmetry transformation is that, okay, so, and this is due to the fact that the killing spinner which we had obtained earlier, which is given here, this killing spinner grows exponentially as r goes to infinity. In particular, it goes like e to the r over two. So therefore, this, since killing spinner grows like e to the r over two, we see that any normalizable fermionic fluctuation would give a bosonic fluctuation, which will not be a normalizable, okay? So thus we see that the supersymmetry, so we see also for the fermionic case also, in the sense bosonic, normalizable bosonic fluctuation will give a non-normalizable fermionic fluctuation. And this seems to be a generic fact for any ad spaces. In fact, if you look higher ad spaces, we have a similar feature of killing spinner, okay? So thus, naively, once it looks like the normalizable, if you want to do localization computation on ad spaces on a supersymmetric quantum filter, then we cannot use a normalizer boundary condition because it is not consistent supersymmetry. And the localization is just based on supersymmetry, okay? However, as I'll show you, the situation is not that bad. Okay, so as we usually do in the localization computation, we start with adding some q-exide deformation. So one such q-exide deformation is like this. So here psi me and psi are some fermionic bilinear constructed out of the fermions in the vector multiplied, okay? And these are the fermions of the carrier multiplied. This is chosen such that the qv, the bosonic part of qv is a manifestly positive definite. Now as a, so if you remember, I told you earlier that the matter action itself is a q-exide, okay? However, as localization argument suggests that the partition amongst the student depend on the choice of qv, okay? So whether we choose qv or whether we compute the partition function from the original as matter action, the partition function student depend on these choices, okay? Now, so the first step is to, in the localization computation is to find localization background and that is very simple. Just look at the, since it is positive definite, look at the minima of that. And this qv is minimized by the following configuration. So all the carrier multiplied fields are set to zero and the scalar and the auxiliary fields in the vector multiplied are given like this. So here alpha is some constant parameter. Let's say in a bilinear theory it is constant parameter but in non-ibilinear it will be some constant Lie algebra valued matrices, okay? So now once you have this localization background, so the next simple thing is to just evaluate the partition function on this background. So what we get here? So we are considering churn-simons coupled to matter. So first part here, exponential part comes from the churn-simons theory. So since carrier multiplied, all of the fields are set to zero, there's no contribution come from the action evaluated on the localization background. All the contribution come from just auxiliary field g and sigma. So that is this contribution. And the next thing we have to do is to evaluate this one of the determinants of the vector and the curl on this localization background. So yeah, okay, sorry. The action is what? This action, finite. Yeah, this is finite. This is finite. There's no word. On ad space here, this is finite. I don't know which other, but yeah, it's finite. Okay, so next we want to evaluate this one look determined for vector and the curl on this localization background. There are several ways to do it, but the way we, so the method which we are going to follow, which I am going to present here is something called Green's function method. And as you will see that in this method we're going to see some interesting features which probably we'll miss when we evaluate probably in some Eigen function method, et cetera. So in this method what we do, instead of evaluating the one look determined explicitly, we compute the variation of the logarithm of one look determinant, okay. So when you consider the variation of the logarithmic, we get the following thing. So here, the variation we do with some parameter. So here the parameter we choose was the localization, same as the localization parameter. So for the present discussion, we can think of this as a billion theory. So alpha is just one parameter, but if it is non-abillion, then one can choose any Cartian Eigen values. So this is a Green's function for the fermionic and bosonic part. And these are the kinetic operator for fermionic and bosonic fields, okay. So this involves, so this involves computation of Green's function for the fermionic fields and the bosonic fields. And then we need to, also we need to compute the variations of this differential operator with respect to this parameter alpha. Now, so just for the presentation, let me just give some basics of this, what is a Green's function and how do we use this to compute the partition function? So the Green's function very basically, so let's suppose we start with the differential operator D whose one loop determinant we are interested into compute, okay. Then the Green's function is actually nothing but the solution of this equation, which is if you solve this differential equation, we obtain the Green's function. So this is a differential operator which doesn't have a zero mode and we are computing the, I mean we are interested in computing this determinant of this D, okay. Now one way to compute this Green's function is as follows, very simply, there's a simple prescription to compute this Green's function. And that is that you start with the D and find the solution of this equation. This is nothing but solution of equation of motion, okay. Now since D doesn't have any zero mode, so there are no global solution. This global solution in the sense that the solution which is smooth as well as perfectly well behaved near the boundary, okay. So there will be no global solution to this equation that is just a statement that there are no zero modes. So let's suppose we have in the problem some second order differential operator, then we'll have a two linear independent solutions, okay. Let's suppose S1 of this, one of the linear combination, I'll call it S1, is smooth near x equal to zero. And the other one, let's say other, some other linear combination because since there are two linear independent solution, so there will be another linear combination. Let's suppose S2 and which is a valid solution that is in the sense satisfy the boundary condition near x goes to infinity, okay. Then the Green's function for this differential operator is simply constructed like this, okay. In our problem, in all these computations, whatever we are doing, all the normalizable and smoothness condition enters through this, okay. So by selecting the solutions which are admissible, which are not admissible, we are incorporating our boundary condition and smoothness condition in our problem. Now this Green's function which we constructed is very, these are some textbook material. So this very simply one can see that these are solutions of this equation, okay. And also it satisfy this user properties that first the Green's function is continuous at x equal to y. And also since we constructed four second or a differential operator, the first derivative of the Green's function is also discontinuous and this discontinuity will fix the constancy. But in the case when we are constructing the Green's function for many case, then usually they are the first differential operator. So in that case the Green's function itself will be discontinuous, okay. So now as I showed here, so the basic strategy to computing this variation of one loop determinant is just to compute this Green's function, follow this prescription, select which solution you want, which solution you want to throw away and just plug in here to compute the variation of one loop determinant, logarithm of one loop determinant. And once you have this result then we integrate with respect to alpha to get the one loop determinant, okay. Okay, so if you have zero mode, then you have to subtract out that zero mode. You have to consider something called generalized Green's function. What, sorry? Here I have not used any kind of regularization. But actually you encounter such a zero mode, let's suppose when you can computing the one loop determinant of the vector multiplied because the scalar fields now have a localization diagram and they will serve as zero mode. So you have to subtract out their contributions. Yeah, so but one can construct this is the Green's function for that problem also. Okay, okay, so first let me tell you the solutions which we use to construct the Green's function. So now let me make some statement. And the statement is that if the boundary conditions are consistent with supersymmetry then the Green's function for Bosonic field and for many fields are related. So let me explain what does it mean. So this slide is a bit generic here but in next few slides I'll be more specific. So let me explain what does it mean. So let's suppose I have X naught, which is my Bosonic field. For example, for chiral it is just simple phi and it satisfies this equation, okay. Then it's for many partner Q of X naught will satisfy the similar kind of equation, okay. This is not a remarkable statement. In fact, this is just a statement of supersymmetry of the action, okay. This just follows from the fact that the action supersymmetry. And the other my for many field because typically we have two sets of for many field one which is super part of X naught and another just X one. So the other for many field X one will be related with this X naught by some operator M. I'll mention just in next slide what is this operator M in this chiral multiplicative case. But typically this M will be constructed out of this DIJ operator. This is in the language of piston but this is nothing but so you have V and you express your V in this form, okay. So my full set of Bosonic fields are like X naught and Q X one and these are full set of for many fields, okay. So this M will be constructed out of this. So what one find here is a very simple fact that is that a solution for X naught which is solving this equation is also actually solution for Q of X naught because it is solving the same differential equation. However, this solution will administer provided it is consistent with their respective boundary condition and smoothness condition, okay. In other words, if I pick up a solution it is all the solution of this equation but may not be admissible because it may not satisfy their boundary condition. However, if the boundary condition and smoothness condition is consistent with supersymmetry then a solution for X naught is also solution for for many system. So let me for example give us X some explicit form. So this D which I have mentioned here which satisfy by this X naught is just in the Kara multiplied case is this one. So here I have a D mu some covariant derivative gauge covariant derivative which is involved which involves all the connection field and it will archimetry gauge field, et cetera. So it will, yeah exactly. And these are my other terms for phi this give like some kind of mass terms. In particular if we interested on localization background so all this G and sigma we have to evaluate we have to put it on the localization background. So this will give some sort of mass term but this will be also function of R because G and sigma are functions of R. R is the radial coordinate of ADS2 and et cetera. And the other thing which I was mentioning here this M which is I said that considered out of this operator is nothing but for Kara multiplied is this one. In particular this operator which we see here is nothing but D10 operator. The D10 operator which typically appears in the index computation. So actually as we will see there is a very interesting link between D10 and the Green's function computation which we still don't understand but yeah there is some, I feel that there is some interesting link. But then so, okay good point. So, okay good. So no, so this I'm computing for the part whose one loop determinant I'm interested in. So let's suppose I'm computing QV the one loop determinant from QV. So the Green's function which I need to compute is for the differential operator which appears in QV. Okay. Yeah, yeah. So, okay. No, actually, okay. So what I have to say that for Kara matter the action itself is also QX act. Okay. So there are two QX act here. So one which we added or deformation you can say QV and also action itself which is also QX act. So we'll compute the partition function for both the cases for action also and QV also. And localization computing idea say that it should be same. Okay. So as I said, they don't have a zero mode here. No, they don't have a zero mode. So that's what I was pointing here that so this equation shouldn't have a zero mode because this is what we are completely different this operator we are completely different we are completely different determinant of this operator. So typically they don't have a zero mode. But however if let's suppose consider vector multiplied scalar which has a localization background they will have a zero mode. So you have to subtract out that zero mode computing. So I'm computing right now for current multiplied. So for the whole professor I will just show the current multiplied and the vector material part I'll just mention the result near the end actually. Okay. Yeah. If there will be normalizable solution there will be also non-normalizable solution and you have to select which one of them you want to keep. Actually I'll show. I'll come to this point actually here in a moment. Okay. So this is the explicit form. Okay. So now coming to your question which you're asking. So now what we are saying that we have to solve the equations. Okay. And find out what are the solutions and based on the smoothness and normality we have to keep some and throw some. Okay. So explicitly solving the equation motion we find following. So since we are dealing on ADS2 times S1 so it has two circle directions. So I'm representing tau and theta. Tau is S1 direction and theta here is a circle direction of ADS2. Okay. So my solution, so my fields in the Fourier mode will be labeled with two quantum number N and P. Okay. So since this N which is the measure which measures the mass of the KK mode coming from this. So this will dictates the follow behavior of the fields near the r goes to infinity because this N determines the mass of the KK mode. On the other hand P associated with the theta which shrinks to zero as r goes to zero. Therefore P will dictates the smoothness behavior of the field near r goes to zero. Okay. So we find the following. So we find that so for a scalar field we have two sets, two solution because we have a second differential operator. So let me call it F plus and F minus. We find that F plus is normalizable for N greater than delta minus 1 over 12 and F minus is normalizable for N less than delta minus 1 over 12. Now as I said earlier in this slide that given a solution one can construct one can construct solution for many field. Okay. So we find that we can construct a normalizable solution for for many field from the F plus only if N greater than delta over 12. On the other hand we can construct the normalizable solution for for many field from F minus that is so B plus and C plus are constructed from F plus and B minus C minus are constructed from F minus. So we can construct the normalizable solution from F minus only for N less than delta over 12. In others what it implies that normalizability is compatible with supersymmetry for N greater than delta over 12 and L less than delta minus 1 over 12. And in particular in this interval if there is any integer the supersymmetry is not compatible or normalizability boundary condition is not compatible with supersymmetry. And this is and here we see especially the breaking of supersymmetry. Okay. So for example, so here I've shown here what normalizability boundary condition imply for various fields here. On the other hand let's suppose just for the presentation if you had to use supersymmetry boundary condition then it will allow the like other fields to blow up as R goes to infinity then the situation would have been different. In this case we find that the one which is normalized for Bosnian field also give the normalizable solution for Fominic field. And the one which is normalized for in this range also give the normalizable solution in this range. Okay. Now there are two points which I'm missing here N equal to delta over 12 and N equal to delta minus 1 over 12. So this is normalized boundary condition. And this is supersymmetry boundary condition. So here, okay. So there can be different supersymmetry boundary condition. So here I'm using the one where the Bosnian field is normalizable and then using supersymmetry transformation to detect what is the boundary condition for fermions. So we get this is the boundary condition. So this is that the C field falls much faster than E to minus 1 over 2 but B field blows up as R goes to infinity. But one could choose other way around also that you require the fermions to be normalizable and then from supersymmetry find out what is the boundary condition of a scalar. That will allow the scalar to blow up as R goes to infinity. Okay, so how much time do we have? Since we have 10 minutes gap anyway, so, okay. So as I said earlier, so what you have to do, you need to compute this quantity. We have a Green's function available because we know solutions about the admissible solution. We know Green's function. We know Green's function and we know we need to compute this one. Just for the explicit purpose. So for color multiplied, we got this one. This is two cross two matrix because fermions are two component and this for a scalar which we get something like this. So now the most important result of our paper. The most important result is that it's falling. If when normalizable boundary condition are consistent with supersymmetry, that is in the range n greater than delta over 12 and less than delta minus 1 over 12, then the variation of one loop determinant is actually total derivative up to an equation of motion. In fact, what we found that this fermionic term can be expressed using the equation of motion equal to a bosonic term plus a total derivative term and hence it becomes a total boundary term, okay. And this is one of the most important, I mean, this is one of the important result of our paper. On the other hand, if the boundary condition is not consistent supersymmetry, then there's no cancellation, okay. In particular, we find that that now the variation include boundary term up to equation of motion plus a bulk term also. The bulk term, which we cannot write as some total derivative term, okay. And then the complete one loop partition function given by the two terms, we have to add these two term depending on the intervals. And this is a complete one loop, okay. Let me state the result. So we compute the partition function. So this is a computation from the classical action, which is Qx act, okay. So we find the following. The first two terms come from the supersymmetry reason. The reason where the variation becomes a total derivative. So we have already integrated out. We have already done the integration with respect to alpha. So we have just mentioning the result. And then these two terms come from the reason where supersymmetry is not compatible with normalizability, okay. So this parameter, which here, X hat and Y hat are given here, which is a combination of these quantum numbers and R charge, et cetera, okay. On the other hand, if we had done this similar computation from the Qx act deformation, instead of a classical action, but from Qx act deformation, what we find is following. We find that first three terms is exactly similar as this one. But this term, which is a bulk term, this is a bulk term. This term is now become something like this. This A twos, B twos are very complicated quantities. They are functions of this quantum number N and P and delta, et cetera. And so what we find that the part, and this quantity is not equal to, this bulk term is not equal to the bulk term presented here, okay. So what we find here, the result is that the one loop determinant computed from with the normalizer boundary condition is not independent of Qx act deformation. And therefore, if there is an integer N lying between this interval, then we cannot perform localization computation. However, if there is no integer lying in this interval, N greater than delta minus over two L and less than delta over two L, then there's no problem. One can perform localization computation with normalizer boundary condition, okay. So some comments. So as I said, I said that if the boundary condition is consistent with supersymmetry, then the variation is given as a boundary term. And this is one of the important result of our paper. And these boundary terms are independent of Qx act deformation. However, bulk terms are dependent of Qx act deformation. And we have done also some other consistency check, especially in the free column multiplied by evaluating the using some different method. And we found that our result is consistent with other methods, okay. So however, one can ask that, okay, fine. But one can use some other boundary condition in particular supersymmetry boundary condition, okay. So one can find that in the supersymmetry boundary condition. So again, not going to describe here, but in last few days there talks how to compute this. So basically in this case, the computation is reduced to computing some index of some D10 operator. And evaluating this with supersymmetry boundary condition, we find this one, okay. In fact, this result is consistent with our result as long as there is no integer lying in that interval, delta minus 1 over 2L and delta over 2L. So if there are no integers, so our result consistent with this. So this is for the column multiplied part. What about vector multiplied? So we have also done the vector multiplied calculation using the index best calculation, which is given by this. This result is same as S3 remarkably. So the partition function of vector multiple on ADS times S1 is same as S3. However, recently we have gone back again to do this computation, but now with normalizable boundary condition. And what we find that this index result agrees with normalizable boundary condition, only if L square is greater than three over four. So if L square is not greater than three over four, then we have again bulk term. And normalizable boundary condition not consistent supersymmetric. We have been doing this. Okay, no, no, no, okay, see, okay. So this index computation, I'm not using any fixed point formula here. So I'm explicitly computing the Hilbert space of kernel and co-kernel. And this result based on that computation. So this is of course a question that whether fixed point formula, which people have been using, whether applied to this, I'm not using that formula here. I'm explicitly finding the Hilbert space of kernel, co-kernel, and then calculating this. So this, no, here it comes out to a finite dimension. Computation come out to a finite dimension. Okay, so I just, okay, okay. So just this, I have two more slides to finish first. So we have also done this hypermultiple calculation on ADS2 times S2. This has not come out yet, but hopefully it come out in soon. So we have done this computation with equal radius. This is relevant for black hole background. So ADS term S2 itself preserves supersymmetry. It has a killing spinner, but also for black hole background, where you have some gravi-photone background, then we have again ADS2 times S2 with that gravi-photone background is supersymmetric. So we have a two different background, and we can put theory supersymmetric quantum filter on this background and compute the partition function. So in this case, we have done the computation for hypermultiple using normalizer boundary condition. Again, from supersymmetric transformation, one finds that this normalizer boundary condition is not consistent with supersymmetry. However, in this case, we find something different. In this case, we find that, that when the case gravi-photone background equal to zero, that means just pure ADS2 times S2, the partition of the hypermultiple is given by this. There are no bulk term and the boundary conditions, normalizer boundary conditions consistent with supersymmetry. In fact, this answer is same as S4. On the other hand, if you turn on gravi-photone background, which is for the black hole background, then we find this result. And there are no bulk term again, normalizer boundary condition consistent supersymmetry and this result is exactly the same as the result obtained in this paper. So ADS2 times S2 calculation seems to work well, even with normalizer boundary condition. Okay, so just to summarize, so what we have found that normalizer boundary condition is consistent with supersymmetry, as long as for carrier-multiple case, let's say as long as there's no integer in this interval. Of course, one can, and we can see for, and we can have some delta and L and where there will be no such integer and there's no problem. We have done also computation for vector-multiple and we find that this is consistent with index calculation as long as this, that condition on L is satisfied. And for hypermultiple, we have done this computation, where it's normalizer boundary conditions consistent with supersymmetry, okay? And so in last slide, so we have developed this Green's function method to compute the one loop determinant and incorporate the boundary condition ADS spaces. These boundary conditions, when these boundary conditions are consistent supersymmetry, then we find that the one loop determinant is total derivative and that's the one important result. However, we still don't have a general proof that in general theory, whether that is always true or not. In general, it is very hard, Green's function is very hard to compute for higher spin field and also this difficulty increases space and dimension. But one of the advantages of Green's function method compared to index computation where expressly finding kernel or co-kernel is that when supersymmetry is consistent in normalizability, one just need to evaluate boundary terms and the boundary terms just depend on the boundary behavior of the fields. Okay, so accept that, okay. Thank you.