 Okay, so I'll start I'll start with the punchline the the summary of the the summary of the talk There will be some new words that I will introduce but the the the result is actually very you can formulate it and Very quickly. So there is there are three things one is the well given tiles are matrix for Whitton's R-spin class Whitton's R-spin class and I'm counting the words that I will have to explain The second one is also given tall given tiles are matrix again. I'll rewrite it for the Gromov Whitton theory of project of projective space of Cpn and the third one is the asymptotic expansion a Symptotic asymptotic expansion at Well as x goes to infinity of the solutions of this of this differential equation of phi of x satisfying Well and plus first derivative of phi is equal to x to the power s times phi of x and the okay, so the first two are Actually the first non-trivial examples of co-homological field theories, so let's say examples three and four of Co-homological field theories if you introduce co-homological field theories first to give two trivial examples And these are the next two ones which are the first non-trivial examples and the statement is that These three things to our matrices and this asymptotic expansion are all the same So this was partly known for So for Whitton's class and s equals one I will tell you in a well, I will tell you later what was known So now I have to introduce Co-homological field theories Whitton's class right and the and given tiles given tiles are matrix S is a parameter You will see the role of the parameter It will appear so s is just as s is a Is a real number Okay So the thing is I at some point I gave a six-hours lecture course on Come how co-homological field theories and our matrices so now it will be half an hour So it will be extremely efficient Maybe a little bit sketchy, but I'll try to I'll try to explain as much as as possible And then people who know it already can can sleep during half an hour in the nail wake you up in the end What is our x is a x is a variable it's the the pyramid so it's the This is the Dn plus one Fee with respect to Dx and plus one just just one variable. Yes, okay So first of all, what is a co-homological field theory? So I will I will start by example using these two examples since we have these two examples so a co-homological field theory is a bunch of Bunch of co-homology classes co-homology classes on Mg and bore so the modular space of Stable curves of genus g with n marked points like this is genus one for instance. I have an example in my t-shirt So for for all for all g and n So it's a collection of co-homology classes on different moduli spaces of Stable curves of genus g with n marked points So how does it work? For the Grom of Witten theory, so let's let's say let's call w The co-homological field theory Witten czar spin class and Omega will be the co-homological field theory that corresponds to the Grom of Witten theory of a projective space So first of all We put them. Well at every marked point here. We have n marked points. We put so on every marked point marked point maybe I'll maybe I'll Start with omega it will be a little bit better so on every marked point we put an A that goes from 0 to n and This means that the corresponding marked point on the curve will have to go through a hyperplane to the power a in Cpn and then we take the cycle so curves curves In mgn bar that can be mapped to Cpn in Such a way that so we we fix we fix the eighth power of the hyperplane for every marking in such a way that f of The i-th marking let me See will be the curve and x1 x2 xn will be the markings And then it maps to Cpn So the image of the the image of the i-th marking lies inside Well the eighth power of a hyperplane so actually a Projective subspace of co-dimension a and usually at this point you also fix the degree of the map, but I Will not fix the degree so usually there is also some parameter q to the power d where d is the degree of the map Degree of f but we will not do that so I cross it out. I plug q equals 1 and This will be a cycle of mixed degree so for different degrees of the map f This cycle will have different dimensions and they just take the sum of them all So for every genus and for every collection of Numbers a a1 a n Well, it's it's a cycle so it's a It's a if well if for every curve you have many maps it means the cycle is zero So the the proper way is to say that you look at the space of maps and then you projected to mg and bar So for all g for for any g for any n for any collection of a 1 a n from 0 to n You get a cycle in mg n bar Okay, so witness class works In a similar way except that the the cycle you get is completely different But the framework is very similar, which is why they are both co-homological field theories. They satisfy the same property so On every marking Here we put a remainder again. We put an a From 0 to r minus 2 so from this point on in order to treat both theories in parallel I will Set n equals r minus 2 or r equals capital n plus 2 and Then I will be able to talk about about both theories in parallel and then So then you look at the space of 1 over r differentials Differentials, let me call call it gamma for instance Such that gamma to the power r So if you take a 1 over r differential gamma to the power r will be a differential And I will I wanted to have zeros of orders ai at the marked points so this should be a section of the Co-tangent line bundle twisted by sum of a i x i So it's well, it's a 1 over r differential It is actually a section of a line bundle that is a an arth tensor root of this of this line bundle Then when you raise it to the power r you get a section of this line bundle with zeros of fixed orders at the at the marked points and Here I add this magic word virtual virtual cycle. So actually if you look at If you study 1 over r differentials like that you will see that most of the time The arth tensor root of this line bundle has no sections So the only 1 over r differential is the 0 1 over r differential But even this is too much because there is a 0 1 over r differential on every stable curve and Actually, if you compute the virtual dimension, it is smaller. So this thing cuts out a cycle inside. So this is a cycle Co-homology class in MGN bar So again for any genus for any n for any collection of ai's you have a you have a cycle Well, you have a co-homology class in MGN bar And now I will write the main the main axiom. So co-homological field theories have several axioms I will write one main axiom The main property of co-homological field theories, that's the factorization property. So there's a map q From MG minus 1 and plus 2 bar to MGN bar That is obtained by taking a Stable curve of genus minus 1 with n plus 2 marked points and gluing the last two marked points together so So here you have a Curve of genus 3 with 5 markings and then you took two markings and glued them together And you obtain a curve of genus 4 with 3 markings and you may ask what is the pullback? Under the map q of the co-homological field theory. So let's take omega, but it's the same formula for Witton's class Omega GN of a1 a n and The answer is that it is again given by the same co-homological field theory applied to this smaller moduli space So this is equal to actually sum over a prime plus a double prime equals capital N Omega G minus 1 n plus 2 a 1 a n a Prime a double prime So let me say it again You have this moduli space and GN bar on which you have a co-homology class omega GN of a1 a n Then you restrict this co-homology class on the boundary divisor composed of nodal curves with one node and You see that this boundary divisor is actually almost isomorphic to us to another moduli space and on this moduli space you also have Co-homology classes given by the same co-homological field theory because the co-homological field theory gives Co-homology classes on all moduli spaces So the axioms tells you that The pullback of the co-homological field theory on the bigger moduli space is equal to the co-homological field theory on the smaller space except that Here you have two extra marked points and the co-homological field theory only gives you a co-homology class when you assign These numbers a I to marked points So you need two more AIs and then you take the sum so you take the sum over a prime and a double prime These are the two extras That you plug here okay So this was the extra short introduction to co-homological field theories and now it is followed by another extra short introduction to our matrices No, so a co-homological field theory is a family of co-homology classes of mg n bar on mg n bar So you do not have to remember how they were obtained So this is a definition of the co-homological field theory. It's a co-homology class on mg n bar Once you have it you forget about one over our differentials The question about the definition right the prop so the proper definition of witness class is a little bit complicated because if you so as I said As I said on every curve you have the zero The zero one over our different you have the zero section of this Arthur so So if you just do that you will obtain the whole space But actually you have to you have to cut a piece of it using some virtual virtual class Right, so the our matrix. What is the our matrix? Okay, let me start here So first of all given to all our matrixes is not really a matrix It's a actually a matrix valued power series. So Z is a formal variable formal variable and this is our zero plus our one Z Plus our two Z squared plus and so on So it's a power series in Z and our zero or one or two or M is an N plus one times N plus one matrix That contains some information about the co-homological field theory So now I'm going to tell you how to find the our matrix if you have the co-homological field theory and then there is Telemann's classification theorem that tells you that you can actually go back in many cases So let me first tell you what information is contained in the our matrix so let's So let's take omega g1 Then you just have one marked point. So you just need one number a from zero to n plus one and What you're going to find is a co-homology class on mg1 bar. So I restrict it from mg1 bar to mg1 I Think it was a bad choice. I'll start I'll start again and on a bigger bigger spots so again omega g1 of A is a co-homology class on mg1 bar and I restrict it to mg1 So mg1 is the modulate space of smooth Riemann surfaces of genus G with one marked point the co-homology group of mg1 is much smaller and well in particular there is one co-homology class that is called the Psy the Psy class that's Rahul actually Rahul Penderipan to introduce this morning. So there is the class Psy 1 Which is the so it's the first-turn class of the co-tangent lines to this one marked point Remember there is one marked point on the on the curve. You take the co-tangent line That gives you a line bundle over mg1 and you take the first-turn class and Then there are some other classes that are called couple classes And you don't know you don't need to know what they are Because in a moment they will they will disappear, but just to be well, I list them in in parenthesis But they will disappear in a moment Okay, so here we have This co-homology class is equal to sum over m Psy to the power m This is this class Psy 1. Okay, let me call it Psy 1 and then I will write plus O of the couple classes which means that So just Set all kappa i is equal to 0 so whatever is in what whatever is well the The part of the expression that involves kappa classes. You just forget just leave the leave only the part that has that has only the only the class Psy 1 and Then here you have a coefficient That depends on a so a goes from 0 to n and It also depends on the genus. So here it's g minus 1 and G minus 1 Does not go from 0 to n. So I take its modulo n plus 1 rewrite it a little bit and this gives you the matrices are m So once again you take your co-homological field theory and you look at a small as well at a small part of it First of all you you restrict to the open part of the moduli space, which is already much simpler than the The compactification then you forget about the kappa classes if you don't know what they are It's even better. You don't even have to forget and So you just look at the well, basically one one part of one one component of this co-homological field theory along this this Particular co-homology class the Psy class Psy class to the power m and then for every m You get a bunch of numbers and they are and they form this n plus 1 by a n plus 1 matrix Yes Yes, this is of co-homological degree to m Yes, yes, so So remember the Omega so the my definition of omega for gromm of witton theory of of cpn Remember I said you look at at curves that map to cpn and that go through Some projective subspaces so it depends on the degree and you take the sum over all degrees So it's a mixed-degree co-homology class now Witton's class is actually pure degree, but Well, there is some there is some upper well, maybe I will mention there is some a Shift that you that you that you do to make it of of mixed-degree and semi-simple and I will I will mention that in a second So they're all of mixed-degree Okay, and then there is tell a month's classification theorem classification theorem That tells you that a semi-simple I Have to write it down, but I will not define it today a semi-simple co-homological field theory can be uniquely reconstructed uniquely reconstructed From its arm matrix so this Sorry Yes, these are so okay, so now is the good is now is the good the good moment to make this remark so omega is semi-simple automatically and Witton's class is actually not semi-simple, but there is a translate well an operation called a translation so you Okay, let's say W tilde sum over K greater than or equal to 0 push forward of Witton's class GN plus K So here is your Witton's class and then you add K more marked points and you put insertions one everywhere You could actually choose any any Any other remainder, but one is the one I'm using today So instead of taking just one Witton's class you take with with with n marked points You use Witton's class with n plus K mark points With a bunch of ones and you take their push forward so you forget these K marked points You take the push forward and you put them all on the same moduli space MGM bar and that gives you a modified Witton's class that has So the degree starts it starts with Witton's class and then it has degrees lower and lower so it's a triangular change of Change of well a triangular change where you start with Witton's class and then you add some smaller smaller degree terms that are push forwards of Witton's class on her spaces with more marked points and This thing becomes semi-simple and Then you can use the R matrix to reconstruct everything and if you are interested in Witton's class itself You just keep the highest or highest degree part So it's a again It's a technical thing if you if you know what I'm talking about then this is the precise definition If not just forget it forget it and think about Witton's class right And I managed to do it in half an hour. I think that was the fastest introduction I have ever done in so Co-homological field theories and our matrices so that Just one stupid question As in this sense has nothing to do with the our matrices from From a young Boxter equation. No, no, no, no, I don't think so. It would be it would be really surprising But it's the same letter. Okay, so the summary of all this is that we have Two important theories the Gromov Witton theory of projective spaces and Witton's R spin classes and The R matrix is the most complex way to encode all the information that they contain So you see Gromov Witton theory contains a lot of numbers you should you take the you should fix the degree of the map and all these numbers a 1 a n and the genus and the number of marked points and for each choice you have a number so there is a a lot of numbers and In these matrices so these matrices, of course, there's still a significant amount of numbers There's a well a square table n plus 1 by n plus 1 for For each RM, but so it's a much it's a much smaller amount of numbers And this is the most compact way you can you can encode a co-homological field theory So when you want to describe a co-homological field theory concretely you You try to write the R matrix Okay So now I will make the connection with the with the Differential equation So maybe I will start I will still continue about the arm arm matrices. I will start with a concrete recipe Recipe does it take a double e or not? I don't remember for computing The arm matrix So the reason I'm showing this is that this recipe actually Well, this is what makes the connection with the with the differential equation Okay, so Well, it's a little abstract, but okay, let's let's so I'll write it down the way the way it is So first of all there is there are two matrices two matrices related to the grading of a from 0 to n so you see H to the a is a co-homology class of Degree to a so there is a natural grading in the co-homology of Cpn and so there is one matrix that I will denote by a bold face mu Which is just the co-homological degree of a So this is called centered well centered Centered grading operator Grading operator So it starts with minus n then there is 2 minus n and then minus 2 and n in the end and I will multiply it by a parameter mu so for Omega for the Grom of Whitton theory of projective spaces Mu is equal to one half so it gives you the Co-homological degree of a just shifted in such a way to make this anti-symmetric This is where it is called centered and for Whitton's class. There is no co-homology Well, there is no co-homology. It's not a it's not the gram of Whitton theory of a variety But there is still a grading and in this case mu is equal to 1 over 2r or 1 over 2n plus 2 Okay, and there is another matrix bold face psi and that's the multiplication by the multiplication by the Euler field So I'm sorry. I am skipping a little bit too much, but this is the last thing I'm skipping So the Euler field is again related to the same grading when you have a grading you also have a you also have an Euler field and there is a matrix of Multiplication by this Euler field and this is given. Well, this is this matrix is just a shift by one modulo n plus one and again, I will multiply it by a By a by xi that is not bold face. So this is a this is a parameter. This is a matrix This is just a number and here in both cases The parameter xi is equal to n plus one But it would actually be possible to do to absorb all this parameter is easily absorbed into the differential equation So again the moral of all this is that for Witton's class and for the Gromov Witten theory of the projective space The actual computation of the arm matrix is almost the same You use the same matrix psi for multiplying by the Euler field and The the centered grading operator is the same up to a constant the constants are different But that's the only difference there is so you would think there are matrices for for For these two theories would be very similar actually, they are not so extremely similar, but They are Related to the same differential equation and so the recipe itself is the following Let me write it. Yeah RM plus one so the commutator of RM plus one with this bold face XI Is equal to M plus bold face mu? Times RM This is what I use every time. I want to compute an arm matrix completely. I have it on maple so you start you start with oh, I actually didn't tell you but but There's one more condition is that Our zero is actually the identity matrix So the arm matrix is complicated starting from R1, but R0 is just the identity matrix and plus one band plus one So you start with R0 R0, you know is the identity matrix Then you know the commutator of R1 with XI Right, so the commutator for one with XI gives you almost all coefficients of R1 except for some of them, so there are still Capital N plus one in determinants in determinants and then you write the next equation and you see that for it to be Compatible these unknown coefficients have to have some precise values So this is actually This is actually related to the condition of semi simplicity So these equations do determine there are matrix completely in the well in the case where the Co-homological field theory is semi-simple and It is not completely straightforward. I mean it's not it's not like you compute them one by well You can you you actually need this equation to find the remaining unknown coefficients of RM and Some coefficients of RM plus one and then if you want to find the remaining coefficients of RM plus one you go to the next equation Okay, so this is the way you compute the arm matrix and once you have computed the arm matrix You get you have all the information about the Gromov-Witton theory So now let's write Let's look at this differential equation So maybe I have 20 minutes. I think I still can digress a little bit so It is actually easy to solve this differential equation. It's a okay It's an ordinary differential equation of order n plus one so it has a vector space of dimension n plus one of linearly independent solutions and It is actually not so hard to construct a basis of these solutions So I'll write one solution That starts with one So I start with one and then I want the n plus first derivative of the next term Divided by x to the power s to be one So I write x to the power n plus s plus one divided by n plus s plus one and I stop at s plus one I guess something like that right, so if you differentiate this n plus one times You get x to the s and this is x to the s times one Then I write the next term So that its derivative is equal to this term multiplied by x to the power s So this will be x to the power two times n plus s plus one and Then I rewrite all these and I add some more So 2n plus 2s plus 2 right, I guess what What is it going to be n plus 2s plus 2 something like that? I hope I'm not doing it wrong, but I think it's something like I think it's I think this is correct and so on So this is a convergent series that is almost hyper geometric with your table with some Small change of variables you can make it into a hyper geometric series and this gives you a perfectly well-defined solution of this differential equation and Then you can make another one that starts with an accent another one that starts with x squared and The last one that starts with x to the n and If you start with x Times n plus one it doesn't work because then the n plus first derivative of the first term will already be non-zero So this is a basis of solutions however, it does not really It does not really help To have this basis of solutions if you want to if you want to find the asymptotic at infinity Sorry S is a real number no not necessarily n is an integer, but s is a s is a is a real number By the way, there is a remark. I'm not sure if it's useful. I haven't found any use for it so far But this differential equation so it actually n plus one and s are exchanged if you By the by the Fourier transform at the Fourier transform Transforms multiplication by x into the derivative into derivation and vice versa. So it's a maybe there is some Some meaning about that It's it's defined for positive x positive real x Nothing so for yeah, so for If s is an integer you can you can extend it analytically to the whole complex plane with some some singularities, but In this case, it's just on the real axis real positive axis okay, but so Now I claim that all of these series that I wrote so I wrote the first series and the first terms of the other ones I claim that all of them have the same asymptotic expansion at infinity So not just the same asymptotic, but the whole asymptotic expansion That means that they are all The same except well up to exponentially decreasing terms So what is known about the asymptotic expansion of these solutions? Let me write it down First of all asymptotic at infinity 15 minutes, okay, so let's try let's try to Let's try to write the asymptotic in this form Times x to the power beta. Let's say so what will happen when you Differentiate this so when you differentiate the exponential you will find Omega times the derivative of this so x to the power alpha minus one Well times the same thing and then there will be a smaller term actually when you differentiate when you differentiate x to The beta there will be a that will be a smaller term so Basically when you differentiate this you multiply everything by omega times x to the power alpha minus one so if you want This to be a solution of the of the differential equation You need s to be equal to alpha minus one Times n plus one n plus one is the number of times you differentiate and alpha minus one is the power of x by which You multiply every time And omega to the power n plus one must be equal to one Right, so there's this parameter alpha so up so From now on I will actually switch to alpha instead of s there is a relationship between s and alpha You can use either one as a parameter and from now on I will use alpha So there is one parameter alpha that that's that can work So any asymptotic so any asymptotic of any solution of this equation Has this form with the same alpha the only possible alpha is the alpha that satisfies this and Then omegas are different so omegas n plus one through to unity So if you take omega equals one this gives you the Asymptotic that is well the fastest increasing asymptotic and this is the asymptotic of all those solutions that I wrote and Then any other omega will give you well an oscillating or a decreasing decreasing solution Well, which is exponentially smaller So that means that if you look at the vector space of all solutions of this differential equation Almost all of them grow like this without with omega equals one and then there is a subspace of dimension one that goes small that goes Well less fast and then another subspace of the co-dimension to with this even slower growth and so on so in the end You find one unique solution that increases that sorry decreases What the fastest fastest? fastest rate of decreases Okay, but now but now I will just look at the generic solution and take omega equals one So I take omega equals one. This is enough for us. So this is the asymptotic expansion of this is the asymptotic expansion of sorry, this is the asymptotic so far for almost any solution of this of this equation and Then there is There is also a way to determine beta. So it doesn't it does not say beta cannot be determined by this crude estimations, but better can actually be all you can also determine the value of beta and beta is equal to Minus and over two times alpha minus one Okay, so this is the asymptotic for all solutions are almost all solutions of this differential equation And if I want the whole asymptotic expansion, it will have the form sum for m greater than or equal to zero B m X to the power minus alpha m and then you can take derivatives of this So if you take derivative number a it will be The asymptotic expansion will have again the same exponential term Then there will be x to the power a minus n over two Times alpha minus one times some For m greater than or equal to zero B a m X to the power minus alpha m so we actually found yeah, and If you take a equals n plus one That's the same as a equals zero because of the differential equation if you differentiate n plus one times You get back to Back to fight So we have found n power series or n series of coefficients B zero m B 1 m B 2 m up to B capital n m. So these are n power series and I claim that so if you write this out actually if you differentiate. So if you compare comparing maybe Okay, just one one thing. I didn't tell you about the armatrices. So maybe I'll I'll do it now Then yes So you can Nothing So the asymptotic okay the asymptotic expansion is so this is the leading term and the asymptotic expansion is Everything that is comparable to powers of X. So this thing multiplied by a power of X This is what I call an asymptotic expansion, but the solution itself has exponentially smaller terms Exponentially smaller terms and actually they are given by similar things with other roots of unity here instead of one So it is also known so it is also known what they are Okay, so one thing I Wanted to say about the armatrix and for God. So let me say it now so when you look at this this relation This commutator It is easy to deduce that So the armatrix or zero as I said is the unit matrix is the identity matrix and then our one Will have a zero diagonal then something non-zero just above and here and Zeroes everywhere else and then our two will have two zero diagonals and Something non-zero on the third diagonal Two non-zero coefficients here and zeros everywhere else. So actually Because of the particular form of psi as you see psi just adds one modulo n plus one Because of this particular form of the matrix psi the armatrices will have a lot of zeros and So there will be only one non-zero coefficient on each line or in each column. So if you add them up If you add all these columns, for instance, you will find a power series that you can call f1 of z and Then here you will have f2 of z And so on So the armatrix is determined by Sorry f0 f1 because I numbered everything from 0 to n So the armatrix is actually described by n power series f0 of z fn of z and This is what we found here. Here. We also found n n plus one power series With these coefficients so actually F a of z is equal to some V a m if I'm not mistaken something like z z z over alpha and there is maybe to the power m So basically the power series that you see here are precisely the power series that appear in the armatrix and Thus we have found them all so I have five minutes left for some examples. I think it will be quite useful because this is This is a little bit abstract. So Before before giving it the examples, so let me just sum up. So here you have this asymptotic expansion and its derivatives when you differentiate The eighth derivative and find the derivative a plus one that automatically gives you a relation between these numbers b a m and the next sequence of numbers b a plus one m and this relation simply is exactly the same as the This relation between the armatrices so between the coefficients when you when you rewrite this coefficient on the armatrices in terms of These power series of zero F n you find exactly the same relations as when you differentiate the asymptotic expansion So the proof of the main result is actually a simple computation. There is there is nothing nothing complicated That what was No, no, no, no, no these equations determine everything just not just not I don't know how to say so Let so let let me say it again. You start with those are zero then, you know, then you have this equation where I have r1 and r0 This does not determine r1 completely just up to some okay But then you look at the next equation with r1 and r2 and This equation can be solved only for some particular values of the remaining coefficients of r1 So in the end it does determine r1 completely It's just not that's not that you know our am and you determine r and plus one for from this equation you need two two equations to determine each matrix So these FAs give you our matrix for this W code for everything. So, okay, so let me okay, so Of course now now we want to know what are the particular So you see here you have this parameter mu different values of the parameter mu and and In the differential equation you have the parameter alpha which replace the parameter s So now we need to express alpha in terms of mu and the expression is alpha equals 1 minus 1 over 2 mu and s Maybe I can recall is alpha minus 1 times n plus 1 So let's look what it gives for Witton's class and what it gives for CPN so for Witton's class Mu is equal to 1 over 2 r So this gives us alpha equals 1 divided by 1 Minus 1 over r which is r over r minus 1 and n plus 1 is equal to r minus 1 So alpha minus 1 times n plus 1 is equal to 1 So in this case we are just using the equation n plus first derivative of psi is equal to x times psi and This was actually known from singularity theory. So this is not a this is not a new result Now let's try for CPN and For CPN we run into a small problem because if you plug mu equals one half You find alpha equals infinity Maybe this is the reason why people don't didn't realize the relation with the differential equation before It's because it does not work precisely for this value of the parameter But for us it is actually not a problem at all because we know the dependence on alpha So plugging alpha equals infinity is not is not such a big problem. So let me give you one example And you will see immediately how it works So let's take the example of n equals 1 So we're studying the gram of Witten theory of CP1 or the three-spin Witten class okay, and Then we have the formula for following formula for BM. So BM is equal to alpha minus 1 times alpha plus 1 3 alpha minus 1 times 3 alpha plus 1 and So it goes until 2m minus 1 alpha minus 1 2m minus 1 alpha plus 1 divided by 8 to the power m m factorial alpha to the m so plugging Alpha equals r to the r minus 1 Which in this case is three halves Well, this is pretty straightforward. You just plug it in this formula and what you find is the well-known power series sum for m greater than equal to 0 6m factorial over 2m factorial over 3m factorial z to the m and this is Let's call this series usually called a and this is one of the two power series that appears in the r matrix for Witten's three-spin class The other one is obtained by differentiating this one in the asymptotic expansion It's almost the same, but you write one let me check what is Yeah, one plus 6m over 1 minus 6m and Then the rest is the same So these are the coefficients you actually see in the r matrix of Witten's three-spin class Now if you want to plug Alpha equals infinity So if you plug alpha equals infinity just gives you the leading coefficient of this of this polynomial You see it's a polynomial in alpha divided by alpha to the power m Actually if you look Over there you will see that it is divided by alpha to the power 2m in the r matrix So in the r matrix you have a polynomial of power of degree 2m upstairs divided by alpha to the power 2m So when you take the limit as alpha goes to infinity it just gives you the leading term 1 times 1 times 3 times 3 times and so on And so for cp1 Plugging alpha equals infinity You find again two power series. So one is sum for m greater than equal to zero 2m factorial squared over m factorial to the power 3 z to the m And differentiating this in the asymptotic expansion You find the second one 1 plus 2m 1 minus 2m times the same thing 2m factorial squared m factorial to the power 3 z to the m Very nice simple expressions Unfortunately, they're only so nice for n equals 1 so don't expect them to be as nice for for For larger values of n, but still you're okay I think I I think I'll have to have a couple more remarks, but I think I have to stop I have to stop here So the maybe okay one one one last remark maybe here So bm Is actually a polynomial well up to this up to this alpha to the power m It is actually a polynomial in both alpha and n So right now I kind of plugged n equals 1 But actually there is one sequence of polynomials in alpha and n that contains all the information about gram of witton theory of all projective spaces for any n and all witton's classes for every r And the asymptotic expansions for all our differential equations. Thank you very much Any question What is the geometric meaning of the differential equation in case of gram of witton theory of ctf for example? um so it is Yeah, it is what is called the the Frobenius potential of the of the Frobenius manifold so so So witton's class is the co-homological field theory that is associated with the Frobenius manifold That is actually a well known Frobenius manifold in singularity theory So it's good corresponds to the singularity a r minus one and um So the singularity a a r minus one is actually obtained from this well this so This thing here Right, maybe x squared This thing here has the singularity Just yeah just Has the singularity x to the a r minus one singularity at the origin And uh, well there is a way to construct our matrices for for For these Frobenius manifolds using oscillating integrals And the oscillating integral solves the differential equation. So it's a well it's a it's a bit Long to explain maybe it's something for a discussion after after the after the talk But it's a it's a let's say it's a different part of the theory For any for any Frobenius manifold that comes from a From a singularity you can automatically construct its r matrix using an oscillating integral Well for this for for the potential of this that well this this function that gives you the singularity But So why do you need to take asymptotics not like the whole solution? The the r matrix is the asymptotic of the solution That's the Yeah, it's a philosophical question. I don't know they don't so the Yeah, it's all right. Yeah, well, it's it's uh, yeah, I don't know. So the this is let's say This is the asymptotic expansion right if you take the asymptotic expansion of e to the power minus x at infinity The asymptotic expansion is just zero So there is a non-zero function, but the the asymptotic expansion is just zero and the asymptotic expansion is what appears in the Appears in the r matrix So I yeah, I don't know how to I don't know how to answer why you why you don't take take into account the remaining parts But the this is the how it works Yes, is there other varieties where you know Are matrices explicitly as in this case? No it's actually so I think So it is it is not frequent for an r matrix to be explicit So far, I think the only explicit example is actually cp1 And this is what this was the starting point of the this was cb computed this the the r matrix for cp1 It is rather surprising that no one realized that before but the the thing is there is another way to compute the r matrix for cp1 using localization and then you get A little bigger results the so it's the r matrix for the equivalent gram of wooden theory of cp1 So it has the equivalent parameter and and then it is not a not as explicit And if you have ever used the localization theorem, you know that's when you Uh, so when you you when the localization parameter tends to zero You get then the well the non-equivariant limits, but it's not so easy to find usually it it it is the result of some complicated simplifications So it is not at all easy to look at the formula for the equivalent r matrix and see that it reduces to such a simple form Uh, the non-equivariant limits Yeah, and another case. Yeah, I don't know no it's uh It's not at all not at all common for it to be explicit Yes Yeah, thanks demo for the next step I wanted to ask you whether uh, this method allows it to compute large genus as in topics uh for the written Yeah, this is uh, this is a question that we asked ours. I didn't I don't know it's a it's work in progress So maybe maybe but For instance Yeah, yeah So I'm not I'm not ready for that. That's a very good question. That's yeah, yeah, but I'm not I'm not ready to answer So I'm sure there will be some applications But so far I'm not sure The second good question I want to ask is whether you have an understanding on this with respect to resurgence Resurgence, what is resurgence? That's a Maybe In the interpretation of this as in typically small terms as instant instant terms Ah, okay. No, yeah Okay, good. Okay. Good question too, but Yeah, no, okay. Thank you very much