 Thank you very much. First of all, I'd like to thank the organizers for giving me a chance to talk in this wonderful workshop. In this talk, I'd like to talk about our works on so-called buy-on southern points, which are expected to play an important role in understanding nonpartisan physics and research instruction in a CPM signal model. So this talk is based on the series of papers in collaboration with these people. So one of our motivation to study resurgence is to understand how to evaluate the path integral in physical system like field theory. Quantities like partition function is usually given in the form of path integral like this and formally it is said that this kind of path integral is a kind of integral over all field configurations. But of course, it is almost impossible to expressly evaluate this kind of integral over all field configuration. So we have to use some approximation method. One of the approximation methods, which we can use almost all cases, is the part of the expansion with respect to some coupling constant in the model. Such part of the expansion gives good approximation if the coupling constant is small. But it has to be known that such part of the partition series is a kind of factorially divergent asymptotic series and in many cases such part of the partition series is a model. So we have a kind of imaginary ambiguity, which is corresponding to the singularity on the plane. So we expect that we have to take into account nonproductive subtle points, the contribution of nonproductive subtle points, which usually takes this form. Here S-sigma is the value of action at the subtle point sigma and this power series is the kind of perturbation series around subtle point sigma. And then we construct trans-series by appropriately summing these subtle point configurations. So then we expect that this kind of trans-series gives a correct physical quantity and we expect that there is no ambiguity here. So what you'd like to do is to check if this really gives the correct physical quantity or not. So the first step to construct this kind of trans-series is to find the subtle point or critical points of the action. In terminology of physics, these subtle points correspond to the solutions of the equation of motion of the action S. But as in the case of ordinary finite dimensional integral, when we construct trans-series, we have to take into account not only real solutions, but also complex subtle point solutions. So we have to consider another analytic continuation of action by regarding the action as a function of the homomorphic variable, phi, by complexifying that model, and then we can find the subtle point by solving the Euler-Lanche equation, which can be obtained by taking the variation of action with respect to this homomorphic variable. Then we can find the subtle point configurations corresponding to, for example, part of the vacuum and instant configuration and so on. Then next step we have to take is to find, to determine how to sum this trivial subtle point contribution. So we have to determine which subtle points are and which integration contours are relevant in this trans-series. So this can be done by using the left-shift-temple method. So in this left-shift-temple method, we first determine the symbol j sigma and dr-temple k sigma associated to each subtle point sigma. Then we can use this symbol j sigma to calculate the subtle point contribution to the partition function, z sigma, and we can determine the coefficient in this trans-series by looking up the intersection number between this dr-temple and original integration contour. So this is the procedure of this left-shift-temple method, but in general it's very difficult to perform this kind of procedure in infinite dimensional configuration space. So this is a kind of challenging problem. In this, in our work, we kind of indirectly solve this kind of problem by using a kind of reduction from infinite dimensional configuration space to finite dimensional subspace of this infinite dimensional configuration space. And then by applying the left-shift-temple method in the finite dimensional subspace, we calculate, somehow calculate this kind of partition function and the physical quantities. So this is one of the main points I'd like to talk in this talk. I'd like to tell you in this talk. Okay, so the model we are interested in is the cpn-1 sigma model, which is described by fields, which can be interpreted as a map from to the base space M to target cpn-1 and action of this model is given like this. In this talk, we identify this field as the inhomogeneous coordinate of cpn-1 target space and this GAB is the Houdini study metric of the target manifold, target cpn. So because of this SN isometry, this model has SN symmetry. And this G is a constant, a bare coupling constant, and we are, we would So we consider expansion and nonpart of the effects with respect to this bare coupling constant or renormalized coupling constant which will be which will appear after renormalization procedure. Okay, so this is the basic model we are interested in. The reason why we are interested in this kind of cpn-1 model is that this is, the model can be understood, interpreted as a kind of 2D analog of 4D gauge theory which, like QCD, which describes our world. So it has been known that 2D, two-dimensional cpn-1 model can be interpreted as a kind of toy model of 4D gauge theory in the sense that they have many common properties like a sympathetic freedom and instanton and large end and mass gap and so on. So our, we expect that this 2D cpn-1 model has a similar research instruction as the 4D gauge theories. So we expect that we can we can accept that we can obtain some hint for so called IR renormal problem from relatively simple 2D models. So our motivation is to use nonlinear 2D cpn-2 model to get the hint for this IR renormal problem. So let me explain what IR renormal problem is. So a IR renormal problem, okay, so in generic physical theory, physical models, it is said that factorial growth of the perturbation theories is related to the factorial growth of the number of Feynman diagram contributing to the N's order in the perturbation theory. But in the theories like 4D gauge theories, it has been known that there is another source of factorial growth of the perturbation theories. For example, this kind of diagram which has N fermion groups in the larger loop gives a kind of factorial large contribution like in factorial. So by, so this is, this gives the factorial growth of the perturbation theories, and if we take the, what a resumination of this kind of diagrams, then we get singularity in the model plane at 1 over g square N. So, so there must be some non-trivial subtle point corresponding to this singularity, but this can cannot be the Yang-Mills instanton because the action, the value of action for this Yang-Mills instanton is like 1 over g squared. So this is this is much larger than the value of the this thing, the position of this singularity. So, and particularly this action of this instanton vanishes in the two-fifth limit, in the large end limit, and on the other hand, this singularity does not this, it does not go away in the large end limit. So this Yang-Mills instanton can never be the singularity, subtle point corresponding to this singularity. Do you understand, in the U.S. U.P.N. although you don't go, you can say, please stand or go to a large end limit, yeah? No, no. You cannot perform fields, it depends on the metric on the surface. Yeah. So, okay, anyway, so from this argument, we expect that there is another other subtle point that is called renormalon, and this has been a long-standing problem, because this we could not, the subtle point corresponding to this renormalon has never been found for a long time. So this is the IR renormalon program. But several years ago, there was Algiers and Instalt gave a proposal for this candidate for this IR renormalon. That is called Bayon. This Bayon is a field configuration, which consists of a pair of fractional instanton and anti-fractional instanton. So these two objects are so prior and important around in this talk. So let me explain what... Is it a zero-typical church? Okay, yes, right. They don't have that. Bayon doesn't have a typological church. Okay, let me explain what they are. So Oh, we heard that. Let me... So let us look at some figure just for fun. Okay, this is a figure of an instanton on cylinder. Instanton in CP1 signal model on cylinder. So here this red region correspond to the location of instanton. And if the size of instanton is much smaller than the radius of this compact direction x, this instanton looks like an ordinary one single instanton. But if we shrink the size of S1, then this instanton splits into two objects. And this is in the case of CP1 signal model. If you consider CPn-C1 signal model, single instanton splits into n-fractional instanton. So, okay, each of them are identified with a fractional instanton. So, okay, so this plays an important role. So let me explain this object more in detail. So, okay, so the model we consider is the CPn-1 signal model on cylinder. And in this talk, I will always call the non-compact direction as Euclidean time. And I will denote it by symbol tau. And spatial direction is denoted by x, and it is compactified with period 2 by r. Then action we consider is like this. So this is actually a little bit different from the previous one, because the derivative is now replaced with the covariant derivative. So this means that there is a, so we introduce the background gauge field for SCN global symmetry, MA, which takes value in the Coulton subarrangement of SCN global symmetry of this model. So we introduce this kind of background gauge field. So basically these parameters can be arbitrary, but if you consider this specific value of the background gauge field, and this action has enhanced SCN discrete symmetry. So this value of background gauge field is called a GN symmetric background. And the important point is that if we choose this background, then it is expected that this model has so-called adiabatic continuity. By using this, we can continuously connect the, we're coupling the description of this model to the strong coupling description on R2. So anyway, we are mainly interested in this, in this GN symmetric case. Okay, anyways, let us solve the equation of motion of this model to find the subtle point of action. We can easily find a solution of the equation of motion by rewriting the action into the sum of positive semi-definite term and topological charges. In this model, there are two relevant topological charges. One is ordinary instanton number, which is given by the integral of the k-reform, a proof of the k-reform. And another topological charge k is given by this quantity. So this is, so sigma is kind of obtained by the fairing of the moment map of SCN and the background gauge field. But anyway, this quantity gives a difference of values of this sigma in the future infinity and the infinite past. So this means that this quantity depends only on the boundary conditions. In this, in this sense, this is topological. So these two topological charges are invariant under continuous deformation of the field phi. So we can easily get the solution of the equation of motion by minimizing the positive semi-definite part, which takes this one. So this is positive semi-definite. So we can easily minimize this quantity just by a requiring that this equality holds. Then we get these PPS equations. I'm sorry, this m adjusts constants? M, M. Yeah, right. Yes. So you just deformed the standard of the ACPR model and zero set of M-rays. Okay. Just the question, because in series, there's a counterterm here, because there's an organization. What is the counterterm of this? Actually, at this moment, we don't consider, we just consider only bare action. We are, yeah, we'll introduce the counterterm later. Should we jump with this correction direction by something? Yes. Okay. So yeah, now we are considering only bare action. So this is a VPS equation. This can be easily solved by, this is just a very simple equation. So we can get the general solution like this. Here, F, A are arbitrary low-down polynomial of the coordinate on the cylinder defined in this way. Okay. So this is a general solution. So we, by choosing various F, A, then we can get various VPS solution, like solution of equation of motion. The most basic solution is the single instanton solution, which can be obtained by setting one of F, A to be degree one polynomial and others to be zero. Then this configuration is a kind of single instanton solution because the instanton number is one. And then we can calculate the action, the value of action for this configuration. It's given by 2 pi over g square. But this value is much larger than the value of action expected to be a sub-point for the iar phenomenon. So this... What is the definition of instanton number? It's all the characteristics of the bundle, vector bundle over r-class, but it's trivial bundle, so it's zero. No, no, no. But it's kind of normalized because it's not complex cases, integral curvature. Okay. So instanton cannot be the iar phenomenon. So we have to find another solution that can be responsible for this iar phenomenon. That is a fractional instanton, which can be obtained by setting one of F, A to be constant and others to be zero. So this is the solution of a fractional instanton. And this solution has two degrees of freedom corresponding to position-modulated parameter and internal phase-modulated parameter, phi. Anyway, if we plot the function, sigma, as a function of tau, then we can see this kind of kink profile. So basically this solution, fractional instanton solution, is a kink solution corresponding to a kind of tunneling process between different minimum of action. So, and we can calculate the value of action for this kink solution. Then we get this value. And if you set the value of this background gauge field to the gn symmetric point, then we get this value of action to pi over g square n. So this is one over n times single instanton solution action. So this can be, this is... Sorry, this is sort of a functional evaluation, right? Sorry. This is sort of a functional evaluation. Okay, so that you get solution plug-in back to the... Yes, yeah, yeah. So this is one over n times single instanton. And this is the same order of the expected value of the action for the IR renormalon. So this can be IR renormalon. But actually the fractional instanton does not contribute to the partition function because partition function contribution function receives contribution only from periodic configuration on this Euclidean time direction. It's compactified with period beta. So fractional instanton is a kind of kink tunneling process. So this cannot be periodic. But if we consider bion configuration, which consists of kink and anti-kink, then this kind of configuration can be periodic. So this kind of bion can have a non-trivial contribution to partition function. So the question is if bion, this kind of configuration, can be identified with the renormalon or not. This is the question we want to consider. Okay, so in our work, we explicitly calculate this bion contribution by using the lifted-temper method. So I would like to talk about this explicit variation of this bion contribution. Can I ask a question? How do we know that it exists in the m-acool zero limit? In the m-acool zero limit, this solution doesn't exist. But... So the point is that... So this is defined on the cylinder with this periodic boundary condition. And we expect that if we choose this value, then there is an adiabatic continuity. Then if we take the r-infinity limit, this vanishes. But because of this continuity, the theory on r2 without this m is related to this theory. How does one prove this assumption or just define it? Yeah, actually, in the large-end limit, there is a proof that there is no... Actually, in the large-end limit, there is a kind of, let's say, volume independence. So we can show about... For general end, it's a kind of conjecture, I think. So what do you mean? Can I... I will never obtain bion solutions? Or is it unstable? Sorry. In large-end limit, you don't have bion solutions? No, no, actually. I think we expect that there are bion solutions, also in the large-end limit. But then these will be different ones now. Different? Yeah. Why do you think so? I mean, because it will be equivalent to the equivalent of this formula, because this deformation parameter from the equation is gone. Actually... Oh, yeah, yeah. If we take the symbol of large-end limit, this will appear, but if we fix this r, then we can get some configuration. Okay. So this is the intercombination. Yeah, yes. So you're reading that if you don't time, it's just even, so it's... Well, there is no temperature, right? Okay. And then you introduced the temperature and reading the solution, which depends on your flow, but not on x, right? Right. Something lost. Okay. At this moment, I just considered zero-temperature limit. But when we calculate the partition function, we introduce the temperature. So we consider... The solutions which we will reach and they don't depend on x, right? Don't depend on... The solution, okay. The solution, right? There was only tau dependence, but not x dependence. No, no, next. And they still satisfy the equations of motion, right? Yes. So, okay. So let me explain our model. So the model I want to consider is the 2Dn equals 2 to supersymmetric cpn-miron-sponsible model. Because, actually, this supersymmetry is not necessary to calculate this biome contribution, but it's very convenient to consider supersymmetric model because we can use exactly that to check our biome solution is really correct or not. So this is the model. We added some fermion fermionic terms. And, okay. As I said, to calculate the partition function, we need to compactify the tau direction. So, okay. We introduced a compactified period of this compactified time direction beta. And x is also compactified. So we consider theory on torus. And as I said, we considered a background gauge field along this compactified space direction. And we introduced the same background gauge field for both terms and fermions because in this way, we can keep the supersymmetry in this model. So from now on, you'll be having toric partition functions, right? Actually, okay. Yeah. Actually, we want to consider a kind of generating function of this observable by adding a source term to this Lagrangian. So we want to consider this observable by evaluating the path integral for this action. And we consider endpoint correlation function of this operator by differentiating the generating function. What is M there? Sorry, M. M is the moment map of this. M is the background gauge field. This takes value in the SEM sub-algebra, the algebra of the SEM. So we consider this point correlation function. Actually, we can calculate, in our paper, we calculated the bionic contribution in general CPN model. But just for simplicity, let us focus on the case of Cp1 signal model. Then the explicit form of action looks like this. And we consider generating function for this height of this Cp1. And we consider the kind of correlation function of this operator. So the reason why we consider this kind of correlation function is that by taking the small radius limit of the space direction, then we get quantum mechanics in the 1D limit. In this 1D limit, we can exactly calculate this correlation function by using the Schwerding equation. So the zero-point function is just a grand state energy. And it's, of course, there, because of supersymmetry. And one-point function is given like this. So this contains infinitely many non-part-of-d terms. So this is completely fully non-part-of-d. And more interesting quantity is the two-point function, which takes this form. And actually this has a non-trivial research structure. So let us closely look at this quantity. So by expanding with respect to the coupling constant G, we get this part of the variation series for this two-point function. So the coefficient of this series is gamma n plus 1. So this is factorial divergent. But if we look at the non-part-of-d part, we have here, there's a, so imaginary part of this non-part-of-d contribution jumps at when coupling constant pathodility real axis. So this quantity has this kind of non-trivial research structure. So another question we want to check is that if this kind of structure can be reproduced from the analysis of biome. So that's another problem we want to discuss. Okay, so the first... What is E2? How do you compute this? Okay, E2 is the kind of this kind of two-point function integrated over this Euclidean time. And we can use... So in the 1D limit, this model deduces supersymmetric quantum mechanics. And we can solve the Schrodinger equation in the supersymmetric quantum mechanics. By using the solution of the Schrodinger equation, we can calculate this two-point function. Exactly. This is something that's trickling at zero radius or something. Zero radius remains, yes. So we can get this kind of non-trivial research structure and we want to check if this non-trivial structure can be reproduced from the biome analysis. Okay, so let us solve the... Let's first determine the biome solution by solving the equation of motion. So we consider the complex equation of the target space by the complex space. Then we use the coordinates of this space, phi and phi tilde, which... So this... The original CP1 target space corresponds to phi tilde equal to the complex conjugate of phi. And then we consider the analytic continuation of action by regarding the action as the holomorphic function of phi and phi tilde. And then we solve this equation of motion for this by taking the variation with phi and phi tilde. Because now we are considering the two-dimensional field theory. So the equation of motions are two partial differential equations. So it's very difficult to solve in general. So we consider these x-independent standards like this. So this x-dependent... If we use these x-dependent standards, we cannot consider several points which contains ordinary instanton, because instanton depends on x-direction. But if we are interested in the leading order non-tripartial contribution, which is expected to be given by Bion, then we can ignore x-direction because Bion's configurations are independent of x. So we use these x-independent answers. Then the equations of motion reduce to two ordinary differential equations. And actually this model has two symmetries. One is time shift symmetry and phase rotation symmetry. So we have two conservative charges in this model. And the degrees of freedom of this model is two. So we can use the two conservation laws to determine and to completely fix the form of the solution in this way. So the solution is essentially given by the Jacobi-Eleptik function. And there are several parameters A omega k. They are just complex constants depending on the parameter in the model and a pair of integers pq. So pq is our integers of the solutions, which takes in this range. So this means that this solution we have infinitely many subtle point solutions. And the important point is that these are complex subtle points because for generic p and q, phi tilde is not equal to phi bar because these are complex constants. So they are complex subtle points. So let us look at the simplest example of a solution. So this is a real biome solution corresponding to p1. This is a single biome solution which looks like this takes a form of sine hyperbolic in a very infinity limit. So this is the orbit in the target Cp1. So this corresponds to the tunneling from north pole to south pole and if we plot this height function as a function of tau, then we can see that there are kink and anti-kink. So this is a biome configuration. So this is called real biome solution because in this case phi tilde is equal to the complex conjugate of phi. So this is a solution in the original Cp1. But if we consider only this kind of solution, we can show that suji grand state energy becomes non-zero because of this non-part of the contribution. So we can index zero in this case. What is written index? Sorry, this is... From zero, then we can index past the zero, right? This is... Yeah, okay, so this is... Okay, written index is n. n. But then energy cannot be different from zero. If written index is non-zero, energy of the vacuum is zero. Vacuum is zero. Vacuum is... So this contradicts with the suji grand state. So there must be another contribution to cancel this contribution. So that is a complex biome contribution. This is also a p equal one solution with p equal to one which looks like a cosine hyperbolic in a better infinity limit. So this is called a complex biome because in this case phi tilde is not equal to phi bar. So if you brought the height function as a function tab, then we can see that this is also like a tunneling solution. But in the tunneling process this value of height becomes complex. So this is very peculiar solution. But we need this solution to guarantee that the suji grand state energy is zero. So this kind of complex solution is important to obtain the result consistent with suji. OK. So these are single biome solutions. And for general p and q, for example, this is a figure for p, q equal three one. So there are multi-bion in the configuration. So essentially p, integer p is a number of biomes. And q is a kind of a label of subtle points in p-bion sector, which basically controls the combination of complex and real biomes in the configuration. So OK. So this is biome solutions. So now we have subtle point solutions. Now let's consider the semi-classical biome contribution by calculating the d-ding order term in biome contribution this part. So naively, we can calculate this kind of quantity just by evaluating the determinant of this infinite dimensional version of matrix of the second derivative. But actually we cannot do this because in a weak coupling limit we are interested in, there are some nearly flat direction of action. More precisely we can show that this matrix in the weak coupling limit has some zero eigenvalues. So this means that in the weak coupling limit there are quasi-zero modes for which we cannot perform the Gaussian integration. So we have to be more careful carefully treat the integration along this direction. So to do so let us define a so-called body of action so roughly speaking this body of action is a kind of finite dimensional subspace on which the real part of action is nearly flat. To define this precisely let us define this set of configuration called value solution by this parameterized by a quasi-mosaic parameter corresponding to the direction of this zero eigenvalue of this operator. Then by this value solution can be obtained by solving this value equation which is obtained by projecting the equation of motion until the normal direction to this direction. Anyway so we can define this kind of set of configuration. So let me briefly before showing the solution of value equation let me briefly mention about the relation between value and the symbols. So symbols are defined by solving these flow equations in the value, this finite dimensional subspace, the flow equation is given like this. Here S effective is the effective action on the value and this GAV is indistimetric on this finite space finite dimensional space and we can show that if we have the solution of the flow equation in this body we can embed that solution into the solution of this flow equation of original space. So this means that symbols in value is contained in the symbols in the original configuration space. So this factor can be used, this is very useful when we consider intersection numbers and so on. Anyway so we can actually explicitly write down the solution of value solution in the case of single biome and weak coupling limit and better infinity limit. So this is the solution, this is actually a kind of kink anti-kink configuration because this is some of kink anti-kink and this value solution is parameterized by one quasi-modular parameter eta whose real part corresponds to the relative position of kink and imaginary part corresponds to the relative phase degrees of freedom. This is the solution and then actually this set of solution contains real biome start point and complex biome start point corresponding to these specific values of quasi-modular parameter. Okay, by using this value solution we can compose the degrees of freedom into the direction along the value and normal direction in this way. Then we can reduce the action in the weak coupling limit as in this way. So here this is effective action biome effective action which is defined in this way and this is basically the function on the value, action of the value of action on the value and start all points of this function correspond to the biome solution real biome and complex biome solution. And for normal direction and fermion direction we get Gaussian form. So by evaluating the integrating out the normal direction and fermion direction we basically get schematically we can write the single biome contribution in this way. So here this is the one loop determinant corresponding to the functional determinant which can be obtained by integrating out the normal direction and fermion direction. And we can evaluate this one loop determinant by using the counter-climbing mode expansion which is basically the Fourier expansion and then for each counter-climbing mode we get this operator, differential operator delta B and delta F becomes a kind of 1D shredding operator. So we can use a kind of technique to calculate the 1D calculator functional determinant for this kind of 1D shredding operator. Are you still considering 2,2 supersymmetric model? Sorry. Are you still considering 2,2 supersymmetric model? Yes. Yes. But the background is non-BPS so there is no such structure. That seems with the fermions. Yeah, right. So by using this kind of technique to calculate the 1D functional determinant of the 1D determinant then we get this x is given like this so this summation over the counter-climbing mode is divergent. So this is a kind of ordinary UV diverges in field theory so we can use some regularization in standard regularization to regularize this divergent sum. Then we get this expression for the 1D determinant. Here now there is a cut-off scale introduced to regularize this quantity. Then by using this 1D determinant we can combine this bare effective action to get the renormalized effective action which takes this one. This renormalized effective action takes depends on the renormalized parameter, epsilon tilde. This is shifted by the quantum effect and constant Y which depends on the renormalized coupling constant which is defined in this way. So we get renormalized effective action. R is here. R is equation. R is radius. R is compactification radius. Compactification radius, yes. Then finally we get this form for the single-bion contribution. So this is finite-dimensional integration over these quasi-modular parameters. But we are now considering the complexized theory. So quasi-modular parameters are also complex files, so we have to determine which subtle and which contours are relevant for this quantity. So that can be done by using the restive number method. Actually for this renormalized action we can explicitly determine, solve the flow equation to determine the symbol and dr-thimble. So this is a picture of the dr-thimble. Actually this is a 3D projection from a four-dimensional space. But anyway we can explicitly determine symbol and dr-thimble. Then we can find intersection number and integration path explicitly. Then we can leverage the quasi-modular integral. Then we get this result for the generating function. Single-bion contribution to the generating function. So this is very complicated. But the important point is that this contains a jump at this imaginary g equal to zero. So this is so yeah if the sergeant works in this model then this should be related to a variety of singularity at the value of this exponential. So furthermore we can see from this jump of this nonpartavity part we can how to say So this implies that the partavity part of this quantity should have this large order behavior. So this is kind of how to say prediction from the bion analysis. But it's very difficult to check this because high order, large order behavior is very difficult to calculate in the field theory. But if we take the 1D limit we can use the shredding equation and we can calculate this kind of high order behavior by using the partavity analysis by using the tools like Vendor package and then we can compare the Vendor analysis and bion analysis by taking the ratio of this coefficient then taking the large looking at the large behavior we can see that this ratio approaches one for any value of the source parameter epsilon. So this means that bion analysis is resurgent. So bion result is completely consistent with resurgence argument. Okay So this is the result in the CP1 model. We can easily generalize the result to the CPn-1 model and see that there are n-1 types of bion but for each bion we can repeat the same type of calculation then we can calculate the single bion contributions. So this is a result for the single bion contribution. This is a jump of the single bion contribution. So this is there is a jump very part of the generating function but this is a little bit so difficult complicated. So let us focus on the the geometric point by setting MA to this value. Then we get this expression for the generating function. So important point is that this is proportional to quantity. So this is mean that there is we have a jump in the imaginary part of this generating function. So this means that for the part of the perturbation series perturbation series should have variable singularity corresponding to this value of action. This point 4pi over gr2n so this is nothing but the expected value of the action for the IR normal. So this means that this result is completely consistent with the proposal that bion is responsible for IR normal. Okay, so this is okay. So yeah another thing I want to mention is that because in the 1d limit our bion solution is actually the general solution because there is no x dependence. So you can try to sum all other multi-bion contribution and p-bion contribution can be calculated in this way. So this is very complicated but also in this case there is a jump here and by summing over a bion number we can get the trans series for this. This is the result. We can see that the grand state energy is zero and this is completely consistent with the exact result and we get this expression for the two-point function and here we have a jump in the imaginary part of the two-point function. Actually this is completely consistent with the resultant structure and from the exact result. So this means that our bion calculation is completely consistent at all order in the bion number. Okay, so let me summarize the talk. We explicitly calculated bion contribution in cpn-1c model and we got completely consistent result. We expected the structure and we got consistent result with the proposal that bion is related to IR renormalon but to show that if this bion is really identified with IR renormalon we have to check that the higher order behavior of the perturbation series is really agree with our bion result. So we have to check explicitly the higher order perturbation series in cpn-1c model. This is an important future work. So as of here, thank you very much. Thank you very much. So any questions? Well I actually have two questions. One is did you try 0,2, and 0,1 supersymmetric signal models which kind of interpolate these two extremes where you have all control and no control? Because in four dimensions we expect that confinement is governed by QCD strings. So in what sense this is analog of QCD string solution? String solution. Okay. That's the right mechanism for confinement in this two-dimensional model. Yeah, actually that's okay, for the first question, we tried n-course 1,0 but in that case yeah, actually we couldn't calculate the So what was wrong? In a quantum mechanics limit we can calculate anything and the result is actually almost the same. But in the case of field theory because renormalization is much more complicated. So yeah we cannot do this kind of thing. So yeah we cannot we couldn't calculate anything in the case of field two-dimensional n-course 1,0 and so for the second question okay, so so it's about confinement Yes, actually yeah but I have to say that 2D CPU and second model is very similar to Gaetz theory but not exactly the same. There are several differences. For example 2D there's no, for example symmetry breaking phase transition in the 2D model but in 4D theory there can be such phase transition. So yeah, they are similar but there are differences. So I think we cannot use this completely this cannot be the complete analog of the 4D theory. Yeah, perfectly. This renormal also kind of improvisation on this. Yes, yes, but the renormalizations would provide that direction. And in general it kind of got confusing when we do this. But in a more complicated situation that several running conflict constants means that actual actions some of the very big numbers are actually seen since how one can justify this people's speech because you have sort of equations more than running conflict constants as well. In your situations just to scale a conflict constant is the same equation Yes, okay. So yeah, I think that the point so because we are now considering weak coupling regime which means that coupling constant is small but to access strong coupling regime we have to use something like this kind of adiabatic continuity. So we cannot perform this kind of analysis with small coupling regime but we can use some this kind of property by using to access that regime. Any other questions? Maybe I have just a small question which you talked about the Foubini-Strudy metric on CPN, last one. But did you ever use the singularities? That has some polls because there's a denominator. Do you use the singularities of that somewhere? Similarity. Foubini-Strudy metric. Yes. Do those appear in the calculations? The singularities of the Foubini-Strudy metric. Singularity. Complex file. Complex file. Okay, okay. Actually, yes. Actually, yes. It's chart similarity. It's not a physical similarity, right? Yeah, actually this complex It's singular in the complex Yeah, right. In a complex case. In this complex file solution hit that singularity if coupling constant is the real. But if we introduce complex file coupling constant, we can avoid that point. Actually, this is an important point. Thank you.