 우리는 이 좋은 작업실을 Tara Jaen-gemenขong-성으로 열일하고 제 기회에 대해서 concert 시작code가 이렇게 되었습니다. 이 기회에 대해 이 기회에 대해 기회의 로라 재생을 한 소식으로 Audra가 dèsauberrty으로 decided to remit research results which is done in collaboration with Samiye Mathi. previous session, Bernard talked about general formulation of localization in sprogravity. In this talk, I'm going to continue to talk about this localization in sprogravity. Here, from the title, you may be able to see the two keywords. One is equivariate cohomology, which is from the charge that we use for localization. And the other is twisting of fields. This is useful for computation of one of the terminates. Let me start with general motivation. Needless to say, sposmetic localization is a very powerful tool for exact computation. And there have been many, many applications for various field series in various backgrounds in many dimensions. And we have been looking at many such examples in many talks in this workshop. Then, natural question is, how about application to sprogravity? It could also possibly apply to the sprogravity because the argument of sposmetic localization principle is so general. One critical requirement is that we need an official formulation of sprogravity. And unfortunately, there is a restriction for such official formulation. For higher sposometric case, we don't have such official sprogravity. However, it is still fortunate that up to n equal 2 sposometric case, we have an official formulation of sprogravity in terms of four-dimensional language. Especially n equal 2 is constructed so-called super conformal construction in the 1980s. And for Euclidean sprogravity, Dewitt and Valentin presented paper, which turns out to be very useful for our application. Once we perform this localization computation, then this will provide an example of exact computation of sprogravity. Also, we will be able to see quantum or exact holography relations. And there can be many interesting examples that you may want to explore. But here, our physical interest is in the system of BPS black hole entropy for each ADS-2 CFT-1 correspondence. Let me briefly review about the history of black hole entropy formula. So black hole entropy formula was firstly suggested by Beckenstein and Hawking as the area of the near-horizon geometry of the black hole. And later, it receives the statistical interpretation. The Strominger and Waffa showed that the counting the microstate of string theory reproduces for the BPS black hole case reproduces the area low of Beckenstein and Hawking. And yesterday, there was a nice talk by Professor Waffa. But there is one caveat in this equality. This equality was only for large charge limit. But for the finite charge case, entropy formula receives the quantum correction. And this demands us to have the generalization of entropy function. For the extremal black hole case, such generalization, I mean entropy formula has been generalized by Ashok Sen, so-called quantum entropy function. And the definition is given by this, which is essentially a partition function in supergravity with this loop insertion. And its boundary condition is given by the near-horizon geometry of the black hole. And in general for extremal black hole has ADS2 factor as its near-horizon geometry, so ADS2 times some general geometry. But for the case of supersymmetric black hole, which is our current interest, this geometry is ADS2 times S2. And ADS2 is topological disk. And this Wilson loop wraps around the boundary of this Beckenkalli disk. And ADS2 geometry has infinite volume, so it has infrared divergences. So we introduce code of. And if we extract a finite value from this quantity, then this will define the quantum entropy function. And since each suggestion, there have been many successful tests for portability of quantum corrections. Those people computed one loop computation on the one-shot set point, and that result completely agrees with the microstate counting. And not only for the perturbative test, we also want to apply the supersymmetric localization for the exact test of this quantum entropy function. To apply the supersymmetric localization, we modified action by adding this q-exact term with real parameter lambda. Here we used canonical choice of this ferrimianic variable v as the summation of all the physical ferraments in the theory. And this algebra of our ferrimian symmetry closes into the compact bosonic symmetry, we call H. And in our case, this will be L' minus J'. Here L' is a rotation of the ADS2 disk. And J' is a rotation of ADS2. And since this partition function is independent of this real parameter lambda, and we can freely take lambda to be infinity, then the set point approximation becomes exact. And the new set point appears. That set point satisfies these localization equations, which called localization set point were equivalently, we say, BPS configuration. So we need to solve this equation for all the physical ferraments. And this localization set point was obtained. And it was quite remarkably simple. So for the vile multiplet, it turns out that the vile multiplet is localized to the ADS2 times S2 configuration. This is ADS2 times S2 metric. Here L is the radius of this metric, which is the scaleful parameter, which can be fixed to arbitrary constant by wire scaling metric. So this can be set to 1. And then the official contribution of the gravity comes in the physical metric, let's say, capital G menu, from the scalar in the vector multiplet through its scalar potential, because the relation between this physical metric and vile multiplet is related by this scalar potential vector. And this scalar is in vector multiplet. So in the vector multiplet sector, the solution, localization solution is also obtained, which is labeled by one parameter, let's say C, for each multiplet. Our pioneering work in this direction was done by Debalka, Gomez, and Murthy. And this is the general localization result of the quantum entropy function. And here phi is the redefinition of classical saddle point parameter. And this part is a classical action on the saddle point after renormalization. And this part is a one-loop determinant part. And then these people considered some n equal 2 truncation of n equal 8s for gravity with some assumption of one-loop partition function. And considered microstate counting of 1 a b p s black hole in type 2 theory in T6. And they showed that the integration over saddle point would give very precise remit, which will give better functions. But there was something to be more understood. Like, obviously, apparently, this one-loop partition function should be understood. Because the classical, yeah, this measure in the classical integration will be given through all the computation of one-loop determinant. So later, myself and my collaborator, and at the same time, these people showed that this one-loop determinant has the following universal form. This is given by the scalar potential on the saddle point with some coefficient. And for the vector multiply and hypermultiple case, this number is obtained as this one, 1 over 12, which agrees with the on-shell perturbative computation obtained by Ashok Sen. So now, yeah, we want to look at the one-loop computation for gravity multiply, which is a genuinely gravity computation in the context of localization. And yeah, in some sense, we already know this answer because this should be consistent with the perturbative computation by Ashok Sen, this one. So we want to know how and whether it really reproduces this consistent result, yeah. So to reach this result, we have addressed two main questions. Before that, is there any question? Okay, yeah. The first question is, what is the meaning of global charge q-equivalent in supergravity? Actually, this question was addressed in the previous talk by Bernhardt. But yeah, let me repeat this question again because this is very essential in our computation. And I showed the application to n equal to supergravity. And the main confusion may come from the usual slogan that there is no global supersymmetry in the theory of gravity. That is, in supergravity, supersymmetry is also gauged, which is not a real symmetry of functional integral. But yeah, this slogan is actually not always true. This is when it comes to the spacetime symmetry. So the global symmetry comes from the sum. If we fix the background, and in the computational partition function, we always have to fix the boundary condition. So we set the background as this boundary condition. Then the global spot charge q-equivalent is inherited from the symmetry of this background. Then it's a kind of problem of background field method. So we split the gravitational field into background part and quantum part. And we need to define the action of this global spot charge on the quantum fluctuation. So there is another interpretation of this global charge, which was usually obtained in the case of supersymmetric gauge theory. For brst quantization in the supersymmetric gauge theory, we have in many examples, we have used the equivalent charge as the combination of rigid spot charge and brst charge. Such that q-equivalent square becomes compact isometric transformation. This combination wasn't necessary because only this square of this q-charge doesn't give this only isometric transformation, but also involves the gauge transformation with field-dependent parameter. So to cancel this and obtain this, we have to define or we have to find the q-transformation of Gauss field. In this way, then we can finally find out that the square of this combined operator will give the isometric transformation. So technically, this is a kind of problem in the supergravity because this finding this q-transformation for all the quantum field including all the Gauss field in supergravity can be kind of demanding problem unless we already know the general formulation of background field method. And the main difficulty may come from the fact that the algebra in supergravity is not real algebra but the soft algebra. That means the structure constant is not a constant but field-dependent. Our second question is what are the twisted variables in supergravity and q-cohomology? This question is for computational one-loop determinant because we will use the so-called index theory. So we want to classify all the variables in supergravity in this homological representation. Here, phi is some bozonic variable and psi is some ferrimunic variable and q phi and q psi is corresponding ferrimunic and bozonic variable. So we will call this phi as an elementary bozon and psi is as an elementary for ferrimun. And there is a natural mapping between this bozon and ferrimun. We will call D1zero operator. So once we organize all the degree of freedom, all the field variables into this representation, then the one-loop determinant reduces to by the simple algebra to the determinant of this H operator which comes from q-equivalent scale with respect to the elementary ferrimun and elementary bozon. And this result can be reproduced by computing the equilibrium index. That index is defined by this. So this is some kernel of D1zero operator with respect to this u1 generator H minus core kernel of D1zero or equivalently trace over phi bozonic operator minus trace over ferrimunic variables. So once we obtain this index, then the result will be the form of some summation over e to the some eigenvalue of H with degeneracy an. Then we can easily reproduce the one-loop partition function by multiplying the eigenvalues with degeneracy. Therefore, this information of a cohomological variable, particularly the variable of phi and psi, elementary bozon and ferrimun, is essential in this computation. So let me recall our main question again. So what is the q-equivalent for sporgravity? And second question is what are the elementary variables, this phi and psi? And for the first question, we will start from the background field method of BRST quantization and its modification. And I briefly review the basic idea presented in the previous talk and show the application to n equal to sporgravity with some command. For the second question, to organize this variable, it's very useful to find the way of twisting of spinor variable. Here twisting means we can convert the spinor variable to scalar and vector by projecting some killing spinors. So let me start with the background field method of BRST quantization. We split all the fields, all the physical fields, as well as all the ghost fields into the background part and quantum part. Here background field is fixed by the boundary condition of the partition function, which is usually obtained from the solution of equation of motion. And usually this solution has redundant symmetry. For example, differential map one solution to another solution. So for this redundant gauge symmetry, we need also a ghost field for this redundant symmetry, which is for background symmetry. Then the usual BRST transformation for four fields can be written in this form. Simply replaced by all ghost fields into the total ghost field and total physical field. And this is, of course, very important. And now question is how to rid of the transformation of the fluctuating quantum field. It's very natural if you start from this full transformation. So just note that the background part, transformation of background part only involves the background field and background parameter. Then this transformation of fluctuating field, quantum field, just subtraction from the total transformation and by the background transformation. Note that finding the transformation of ghost field is very simply obtained. And if this algebra is real algebra, that means if structure function is constant, then this rule will reproduce the one of usual background field method in field theory. Now we want to modify this BRST symmetry. So consider we completely fixed the background by using the background symmetry, by completely gauge fixing the background symmetry. Then there is no background value for ghost field except isometric transformation. So to choose this background part of ghost to be an isometric parameter, in this case this will be the killing spinner. Then this background part symmetry transformation will go away. And one comment is that for non-compact space this isometric parameter is not normalizable. So this is no longer gauge symmetry. If we consider the compact space then this isometric parameter maybe this is constant is still normalizable. So this is still redundant gauge symmetry. So we need to introduce additional ghost for ghost. But in our case for non-compact space case we don't need to introduce additional ghost for ghost field. Now this background part of ghost field which is just isometric parameter. So we can impose this variation of this parameter to be zero. And this is really a deformation. Then this algebra equivalently closes to bosonics metric with rigid parameter given like this. Now we also have to look at the transformation of the anti-ghost field and auxiliary field. We can find out the transformation rule in such a way that the algebra closed in the same way as the other field. But there is one exception. The algebra for B field will involve some additional term which contains the derivative of this structure function. And if this structure function was constant then the algebra is completely closed. But it is possible by the observation that this index beta comes from the bosonics metric. And this structure function only comes from the commutator between bosonics metric and boson or ferrimianic metric. Fortunately in the case of supergravity that kind of thing happens. Generally for supergravity softness of this structure function appears only from the anti-commutator of super symmetries. So if one of the index is from the bosonics metric then this is constant. This is not a kind of general proof but kind of its observation. But in the case of our example that is n equal to super conformal gravity we have shown that this is really constant. And the modified VLST transformation gives the equivalent symmetry. In this form the isometric transformation with killing vector with other bosonics metric where the parameter is given by this. Now we can also consider metacoupling to the supergravity. And this general formulation can be also applied in the same manner when metacoupled to supergravity. And this formulation systemized the construction of the equivalent charge that was constructed in the various supersymmetric cases. That is if we take rigid limit of the supergravity coupled to for example Young's theory then this recovers the transformation of the field theory obtained in various places. Here note that the one supplementing was the Q transformation of ghost field particularly the rigid supersymmetric transformation of ghost field is reproduced by this part. Now we have the definition of Q equivalent charge. So now we want to find the organization of all fields into the representation of this homological complex. This organization can be thought of as a change of variable. So to make sense of change of variable this change should be local and invertible. And to find this change of variable we need to find an appropriate choice of twisting of spinners. Because the way of twisting of spinners is not unique so we will be able to find this choice of twisting. Such that we can find the homological variable and this change of variable is non-singular. So this is the procedure how we find the twisted variable and then homological classification. Let's start from choosing one way of twisting of spinners and make sure that this twisting. This twisting is also kind of a field redefinition so that make sure that this redefinition is invertible. And start with a given component of boson in some representation of representation R of gauge group or Lorentz symmetry or isymetry etc. Or start from ferrimion and consider variation of this bosonic field which may be very complicated. Maybe composite combination of other bosons and ferrimions with some coefficient made of killing spinners and background value of field. And in this complicated combination try to find the single term, single ferrimonic term which has same representation with this bosonic field R and which linearly appears. This single term, ferrimonic term should not involve any derivative. Also any coefficient involving this single term should be regular. Otherwise the invertibility will not be guaranteed. So if we can find such a ferrimonic term then we can classify this bosonic variable as an elementary boson and may exclude this ferrimion from the elementary variable. And keep this process until the end. If we fail then we may choose a wrong way of twisting. So we try to reconsider twisting and keep the procedure again and again. And this is kind of a change of variable which contains linear term and nonlinear term. And this procedure is to ensure that this linear term is regular transformation. So this procedure guaranteed invertibility if we consider small fluctuation. So for the case of large fluctuation we just assume it holds the invertibility holds even for large fluctuation. For example let's take an even gauge multiplet. So back term multiplet has this gauge field and complex color and gaugino and three auxiliary field. This has 9 plus 8 degree of freedom. And for the brst quantization we add u1 goes to multiplet which has 1 plus 2 degree of freedom. So we total have 10 plus 10 degree of freedom. So we choose a way of twist the variable using the some production by this ferrimunic basis. And we have we got some scalar and vector and some tensor. And this inverse relation is given by this. So inverse is guaranteed. So this is invertible. Actually we could have used another way of twisting using another set of spinner basis. But once we find one homological variable using one way of twisting then this may be enough to get the result. And the other way of twisting will give same result same result of partition function. So now investigate variation and following the previous procedure that I presented. So now that the 10 plus 10 degree of freedom first into representation of equivalent algebra. So we found the elementary boson to be gauge field and one of the scalar field. And ferrimunic elementary ferrimunic to be some ferrimunic field and gauss field. This is by looking this symmetry transformation. So just looking at the transformation of elementary boson we get the linear term of this ferrimun without any singular coefficient. Also starting from elementary ferrimun we get this linear term with regular coefficient. And this starting from this ferrimun we get the auxiliary field with regular coefficient. Now we want to classify this variable for viral multiply case. So viral multiply involves this graviton and gravitino. And all the asymmetry gauge field for asymmetry also auxiliary tens field and ferrimun and scalar field. So this was 24 plus 24 degree of freedom if we remove the gauge redundancies. But for the realistic continuation we want to look at all the degree of freedom which is 43 plus 40 degree of freedom if we keep all degree of freedom. So now we add 51 plus 54 gauss degree of freedom. This is quite many. And total we have 94 plus 94 degree of freedom. And similar classification can be done as the representation of equilibrium algebra. And this is the table of viral multiply field. For the general coordinate transformation we get a field vine I mean that graviton as a gauge field and we have various gauge fields and these are composite. We have auxiliary field with this degree of freedom. And we add all the gauss to multiply for all the gauss matrices. And we have total this much degree of freedom. Now we want to find a twisted variable. So by projecting of killing spinners this set of bases or this set of bases we kind of using the trial and error. We found one choice of twisting for gravitino and auxiliary ferrimun and gauss field for supercompensimetry and perincal supersimetry. And these are all invertible. And then we look at the variation of all fields. It looks quite complicated. We can write, we wrote down all the transformation rule etc etc. And write down this in terms of twisted variable and try to try the commercial classification following the previous rule. And the result is following. We got elementary version in this case and we got elementary ferrimun including this all this field by looking at this transformation rule. For example, starting from graviton we got some twisted variable of gravitino plus some nonlinear term. And starting from twisted variable of gravitino we got some gauss to field and gauss to field which is regular. One interesting thing is that the transformation of this tensor field will give some weird way of transformation involving this twisted variable of auxiliary ferrimun. Because in terms of SU2 cross SU2 cross SU2 r symmetry representation. This is from Lorentz symmetry and this is SU2 r symmetry. That representation is 131 and 311. But this chi field is 1 by 3. But yeah the twisting procedure given by this killing spinner maps this auxiliary field to this tensor originally tensor field depending on the point of the manifold. So the point where some of killing spinner is vanishing then the other combination is mapped to this. So yeah let me explain this twisting procedure later again. Now we want compute index and from there we want... Yeah I'm sorry this is kind of... We will use AtiaBot fixed point formula. AtiaBot fixed point formula tells us that the index computation reduced to the summation on the computation on the only fixed point of this bosonic generator. Here x tilde i mean fixed point means x tilde x coordinate doesn't move under this age transformation. In our case there are two fixed points. One is the center of age s2 together with north pole of s2. The other is center of age s2 together with south pole of s2. So at this fixed point the twisting procedure happened. So twisting between SU2 r symmetry and one of the SU2 in the Lorentz group the twisting of them happened. At the fixed point yeah let's look at the kaira part and at the kaira part of killing spinner. That is reduced to reduced to following way. So at north pole here at taikojero and saikojero then one of the killing spinner becomes zero. And at south pole the opposite part of the killing spinner becomes zero. So this killing spinner reduced I mean this SU2 r symmetry and SU2 one of the one precisely identified inverse of SU2 minus n north pole and inverse of SU2 plus at south pole. So that at north pole and south pole we classified all the representation of all the co-mological variables elementary co-mological variables. So yeah this is the all the representation in terms of this twisted representation twisted group for boson and ferrimian. And once we know this representation then it's very straight forward to obtain the I mean this trace the result yeah using the athia both fixed point formula. And the result will yeah we will get this kind of result and by expanding this and rid of the eigenvalue and degeneracy then we can rid off the contribution to the one loop determinant and then obtain this coefficient. Actually we got 11 of 12 and there's more. Recall that yeah there is overall general mode contribution of the bi-multipline. Yeah vector multiply also has also and bi-multipline also has general mode contribution. This was studied by Ashok Sen and this general mode contribution is kind of pure pure gauge pure gauge field but this parameter is non-normalizer field so that this is not gauge symmetry and we cannot neglect this we should take into account this contribution to the partition function. So by Ashok's study he obtained this general mode contribution to be one. So by adding this we obtain this very happy number which is consistent with original computation. Now let me summarize my talk and give brief I'll give brief outlook. So we have constructed equivalent to spur charge for n equal to spur gravity case and classified a homological variable with appropriate definition of twisting of variables. And the index computation gives one loop for bi-multipline which agrees with on-shell perturbative computation and we hope that this work brings some clarity to idea of twisting and localization in spur gravity and it may be useful in other direction also like other system like ADS CFT correspondent in other dimensions also it would be interesting to find a relation I typed some relation between twisting of our spur gravity and some topological gravity. Thank you for your attention.