 I'm thankful to organize this for possibility to give a talk. The subject of my talk is the computations of, I will talk about some technique, new technique for computations of flat coordinates on the Frobenius manifolds, which are connected with isolated singularities. And this talk is based on joint works with last one with Spadenaika and Vladimir Belyvin and also, in fact, with other colleagues, such as Molochikov, Boris Dubrovin, and Grisha Tarnopolskiy, Mohamed Zhanoff, and so probably I forget some one. Why we are interested in Frobenius' manifold structure? The reason is that the structure arises at least in three kinds of models of quantum field theory and string theory. First one, and it was known many years ago, this structure appears in two-dimensional topological conformal field theories. The models of topological conformal field theories appears after a written twist in an restriction by sectoral Kyle fields in N equal to super conformal field theory. N equal to super conformal field theories are interesting because of two, if you want, two-dimensional super string, six of 10 dimensions super strings and to have space-time supersymmetry, we need to simplify on N equal to super conformal field theory or on the Kala-Biga, what is equivalent. And it was found by Dishgraphia Lindeverlinda that this structure, in fact, but then non-trivial and concrete case of Frobenius' manifold structure appears in the case which they exactly solved, is connected with SO2, SO2 algebra. But then this structure was mathematically synthesized by Boris Dubrovnik and there is a huge class of N equal to super conformal field theories which was classified by Kazama Suzuki and it would be interesting to solve some models which connected with these different cases of this Kazama Suzuki series and the sub-series of this Kazama Suzuki class is so-called Gepner-Carol Rinds, which I will consider. And another class of models, it was found recently that special cases of Polikov non-critical string series especially so-called minimal models of minimal models of alluvial gravity which were considered many years ago by Kniznik Balakov Zomoluchikov. In these models also we have Frobenius' manifold structure and it's first and also like it was in the case considered by Tishgraf-Hirlind and Verlinde to solve exactly these models we need in fact to found some special... We need also to use flat coordinates of Frobenius' manifold namely we need to know this to find a special solution of so-called Douglas string equation which has a simple form namely in flat coordinates. It's the second class of the models and the third one is models which is the third class of problems is the... arises in consideration of computations on Kalabiyao manifolds. In this case we to get so-called special Kepler geometry on model space of Kalabiyao manifolds we need to compute so-called periods of Kalabiyao manifolds and in fact it is the same as the problem of computations of flat coordinates. I will not talk about this last point but it is right. I want to start from the... I would like to remind how Frobenius' manifold structure appears in the one of this case namely in the topological conformal... models of topological conformal field series and namely the models which describe topological sector in N equal to super conformal field series connected with Landau-Gensburg model. In these models we have super potential which depend on a few fundamental Karel fields and these fields generated Karel ring R0. The basis of this ring I will denote here alpha. Alpha will equal M which is the dimension of our Karel ring. The first of this ring is the generator's fundamental Karel rings. We generate all the rings and this is the basis and this ring R0 is a morphic to the ring of polynomials which factorized by the ideal generated by derivatives of the super potential. And it was shown by Desgraf-Hirlinde-Wirlinde that to compute when we are interested in this model computation of the lightest of Karel fields and their super partners, which looks like this. It was shown by Desgraf-Hirlinde-Wirlinde that, yes, I forgot to define g1, 1, alpha. In this paper it was shown that to compute arbitrary Karel lattice of such kind you need to know two-point function together with so-called perturbed three-point function and it was shown that from word identities for this and other properties, associativity that this two-function satisfies the following properties. This three-point function is thought derivative of some scalar function which is called pre-potential in respect to these parameters which is called coupling constant. This two-point function connected with this one. In this ring I should mention we have one element with this unit of the ring. Two-point function is connected with three-point function like this and of course they satisfy this equation. This equation for this function for pre-potential is called Desgraf-Hirlinde-Wirlinde equation. And when s is equal to zero, I should explain what is this alphabet. The gamma is three-point function multiplied by a converse of this two-point function and this equation means that we have a ring and I will call this structure constant of the ring which I will call r. And this ring, this new ring is this new ring inside with this one when s alpha is equal to zero. And in fact we have two structure. In fact s alpha can be considered as a coordinate on this some manifold, m-dimensional manifold and we have two structures, the remaining structure and structure of the commutative associative algebra with unit in each tangents space and the crucial effect for exact solution of the model is to solve the model you need to find pre-potential in this coordinate. I will explain that in this function of these parameters. The crucial fact for exact solution of the model which was discovered also by Desgraf-Hirlinde-Wirlinde, that this ring which defined by this two objects structure constant and this pairing, this ring which has some additional of course properties, it's not only commutative associative algebra but it's provided by some special pairing which is compatible in the way which I described there. This ring coincide with another one which I will denote double way, where double way is deformation of our super potential, it is double way zero of x plus t alpha is parameters of deformation, e alpha is some basis of the ring R0, we can choose basis of this ring, I will denote alpha of x, this basis and the structure constant of this ring defined by this definition, e alpha modulo derivatives, this structure constant of the ring R0, these are structure constant which already depends of coupling constant and this c-alphabet gamma of S is structure constant of R without zero. At last we have another ring by definition, it's ring of polynomials, here it's already deformed super potential after deformation and then again we can define structure constant of this ring and I choose the basis for this deformed ring to be the basis of the initial ring and we can compute structure constant tilde gamma modulo derivative zero x. And the statement is the following, that we can define additionally to this structure constant, we can define metric on the manifold with coordinate now, t alpha coordinates on the space of deformation of our super potential, of our singularity defined by this super potential and we can define now Riemann tensor metric by formula, it is residue and the theorem of M cytos, not curge cytos but M cytos, I forgot the first name, says that there exist such n form omega of xt which is lambda some function of xt dx1 dxn, such form that this metric together with c tilde satisfy to all axiom of Frobenius algebra, it means that they satisfy to the following properties and first the alphabet of gamma of t is flat metric Riemann tensor for this metric is equal zero, then c tilde satisfy to associativity equation, I will not write indexes here, then c mu lambda is equal to c mu lambda nu where c mu lambda is symmetric tensor in all these variables and last covariant derivatives which correspond to this metric applied to c nu alpha beta is equal lambda nu c mu alpha beta and from this equation follows say that c mu lambda tilde is equal to third covariant derivative of some function and because this metric is flat there exist such coordinate frames that were in this new frame there exist flat coordinates such function of deformation parameters such that in this coordinate our metric is just constant and the equation is the form which we saw here and the important discovery by Dishgraph Verlinder Verlinder is that and it is important for exact solution of the model is that this ring coincide coincide with ring R which I denote R here which was defined by this in terms of this deformed 3 point function and this 2 point function they coincide and therefore because we know in fact we know if we know flat coordinates and this form it is called by K-Site primitive form if we know flat coordinates and primitive form we in fact has exact solution for this model sir louder please C hat C hat what C at the beginning yes of course yes so in the formula yes I it's right it's right but it's you need to compute using this definition you can compute in principle you can compute like serious in respect to deformation parameters this C tilde and this is the question is no no I no I explain I'm just the concept of Frobenius manifold I will tell about what I do in some I will explain I will talk about some technique of computation of flat coordinates and primitive form yet it can be applied for computation all of all of this of all this object prepotential C and so on for I will I will demonstrate it into into for two models of series but because it's depends the form of structure concepts C tilde depends on the model okay now maybe maybe now I will try to to use now let let us because much time consider let's consider some super potential and which is quasi quasi which is quasi homogeneous homogeneous homogeneous it means that it has this this this property and it's convenient it's convenient to use to use integer number row and D also not not rational sometimes people use because it's possible D equal one and then we have the fallen deform formal terminology it here this this this this potential or this singularity define some some ring like I explained here some Jacobi ring and we can introduce bases in this ring e alpha and we will denote the weight weights of this elements degree and if this degree of this element less than degree of singularity itself then it's called relevant equal marginal and more than D irrelevant should mention that the problem when when all elements has relevant relevant degrees for this cases for such kind singularity the flat coordinates are known the problem is how to to compute them when we have also marginal deformation and irrelevant and it's important also for it's important also for Colabia border spray geometry of Colabia the space of Colabia many of us okay the main conjecture which we use we we didn't haven't managed to to prove it but we check the conjecture the conjecture for some non-trivial cases the conjecture is the following we want to find flat coordinates flat coordinates I explained what what is it for given singularity and the conjecture is that this the expression for flat coordinates is given by this expression here we have some of integral some of such element such expressions here we have in brackets we have a slating integral which and here there are some circles it is circles of homology group Cn with this boundary where our not disturb super potential it's real part is equal minus and minus infinity and it it it's you meet us to integrate to integrate here it's this integral convergent here we have our elements of our base e alpha in this degrees M alpha M alpha is integer numbers and at last here we have some monomials which looks exactly like which are dual to these monomials of elements monomials of deformation deformation parameters T alpha and the condition for this integer is just that whites of this monomials because we can we can we have we define weights for for elements of basis but we can define the weight of degree of defined degree T alpha like my definition d minus degree e alpha so the weights should be equal this one and then we I will explain more about definition of this restriction for this number so we we need to if this conjecture says that we need to compute this oscillating integral for such polynomials we impose on flat coordinates there are some arbiters we can impose this normalization equations in the first order C mu coincide with deformation parameter T mu and also the weight of C mu is equal of course T the weight of this one and to compute this oscillating integral we can we will use we will use this property of oscillating integral integral of some form it is in fact it d x is n form it's denotation d x is denoted d x 1 d x n if integral of some p 1 this p 1 is polynomial multiplied by this form of this form and this form of the coincide if the difference is exact form in exact not not in not to relation to the RAM differential but such way deformed differential it's simple to choose to check that by integrating by path that they coincide in this case and then this differential define kagamologist in our space of n forms on our in our space and and the statement is that there exists only n forms and this forms the basis the dimension of basis of kagamology coincide with the dimension of our ring it's equal m like like ring defined here and we can choose basis of this kagamology elements of our ring multiplied by d x and therefore we need integral of such elements and alpha let now primitive form is just naive form d x we need to compute this integral with this exponent to do this we expand our monomial to the basis of this it means that we should solve this linear problem this is some additional term which is exact it disappears after this in our oscillating integral so we need to find only this constant and they can be completed recursively from this definition and it's very simple okay and then when we get this expansion we insert it into integral and what we need to know is just integral where we have the basis element and it gives some pairing between cycles and it is some constant and we can choose one of the simplest choice of the basis in kagamology and cycles in kagamology group is to take it as just dual to consider the army like this but it's interesting that it is not the choice is not correct for getting flat general solution for flat coordinates because this theorem predicts that there exists some modulus in primitive form and there are also models for flat coordinates therefore we see some, I have no time to explain this but there is some freedom here which is connected with so called resonances resonances is that we have sometimes we can have the elements of our basis of the ring whose weights differs in the d or integral multiplied by d where d is the weight of singularity itself. In this case we can sort this not have this has more general form and we get some parameters and I will show what we have also when we have marginal and relevant elements then we have non-trivial primitive form it's already not naive dx but some polynomials and this polynomials is written here in general form so to compute flat coordinates we need to compute also primitive form and we do it, we insert this form in our integral where we apply this properties which I mentioned before compute this and we stay with such kind expression for flat coordinates be here we know computation from solving of linear problem but we don't know but we have additional relation which follows from the unit in our ring is the same in flat coordinates and in another coordinate system and it fix from this equation we fix all these parameters and get some expression we apply it for case of this ring which I call ring they connected with I find group it depends of two parameters n and k and we can consider this super potential like our singularity homogenous we can choose the basis of our ring to be sure polynomials and we can see the two cases first is a su-3 of level 3 in this case potential has this form it's ten dimensional it's ten dimensional we have in this ring nine relevant elements in one marginal this one because only one parameter we have and it's marginal this form primitive form looks like some function of this unknown of this deformation parameter which is correspond to marginal this is a marginal deformation and it has this form we compute this by technique which I described I tried to describe and in result we get here this factor lambda for primitive form it is geometric function and we get say we get say flat coordinates as ten which correspond to marginal it looks like this it's ratio of two and then we want to get to check this form because it's found this computation was found on the conjecture and we use we do alternative computation of all these things using the idea from the work Clem, Tyson and Schmidt it's old work and we do this computation I will skip this case it's interesting because for next S3 of level 4 models we have one marginal and one irrelevant and we also compute this and then compare it with direct computation I will tell you two words about the direct computation we have this definition of formatic then we do the following formula for product like this and then we rewrite this like this and we denote this function it's h lambda of our deformation we know this function from definition from this structure constant but these are not known and then we impose on this metric and on this structure constant Dubrovin axioms the first we impose the property that this metric should be flat Riemann tensor should be equal 0 and it give us hundreds equation for a few function h function h this function and they are all this equation is compatible and they leave some solution in fact it can be for simplest case for level 3 it give us hypergeometric equation can be transformed to hypergeometric equation for level 4 when we have many variables we can solve this equation by mathematical and we solve this and we get some parameters here for k we get one parameter solution for metric and for flat coordinates which we get from this equation when we know metric and crystal then we have one parameter for k equal 1 for k equal 3 and then we impose this extra axioms and it give us nothing because it follows this property from this definition what about this in fact it is written in this graph it impose it is satisfied automatically for case of level 3 for case of level 4 it solving of this equation give two parameters but this m-cyte theorem predicts one parameter one model of primitive form and this is fixed really fixed by this equation and we get so it shows that our conjecture is right but we don't know how to check it and it is some formals and conclusion is that so we have some efficient way we don't need to get differential equation what is very difficult to do when we have many many marginal say deformation parameters the problem of computational flat coordinates in fact is linear problem so maybe even by hands and we do in this way but mathematics is not difficult and it has natural application to possible generational topological conformal field series to dimensional gravity so to say generation of gravity and the most interesting it seems to me that it can be applied to computation of the geometry on the model space and in fact this flat coordinates is nothing but at least for one case when we have the family of deformed quintics really so the flat coordinates give us periods of which we need for computational geometry thank you very much