 I'm thankful to organize this possibility to give a talk. The subject of my talk is the computations of, I will talk about some 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 Spadenaik and Vladimir Belavin, and also, in fact, with other colleagues, Sasha Zamolochikov, Boris Dubrovin, and Grisha Tarnopolskaya, Mohamed Zhanov, and probably I forget some one. Why we are interested in Frobenius' manifold structure? The reason is that the structure arises in at least three kinds of models of quantum field theory and string theory. The first one, and it was known many years ago, is this structure appears in two-dimensional topological conformal field series. The models of topological conformal field series appears after written twist in an restriction by sector of Kyle fields in n equal to super conformal field series. n equal to super conformal field series are interesting if you want to compactify ten-dimensional super string, six of ten dimensions super strings, and to have space-time supersymmetry, we need to compactify on n equal to super conformal field series or on the Calabria, what is equivalent. And it was found by Digrafio Lindeverlinda that this structure, in fact, but as a non-trivial and concrete case of Frobenius' manifold structure appears in the case which is exactly solved, it's connected with SO2, SO2 algebra. But then the structure was mathematically summarized by Boris Dubrovnik and there exists a huge class of n equal to super conformal field series which was classified by Kazamo Suzuki and it would be interesting to solve some models which connected with these different cases of this Kazamo Suzuki series and the sub-series of this Kazamo 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 Louisville gravity which were considered many years ago by Kniznik Balakov Zomolochikov. In these models also we have Frobenius' manifold structure and first and also like it was in the case considered by Tishgraf-Irland-Everländer to solve exactly these models we need in fact to found some special, we need also to use flat coordinates of Frobenius' manifold. And finally 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 rises in consideration of modifications on Kalabiyau manifolds. In this case we to get so-called special Kepler geometry on model space of Kalabiyau manifolds we need to compute so-called periods of Kalabiyau 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 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. 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 depends on a few fundamental carol fields and these fields generated carol ring R0. The basis of this ring I will denote here alpha. Alpha will equal m which is the dimension of our carol ring. The first of this ring is the generator's fundamental carol rings which generate all the ring and this is the basis. This ring R0 is a morphic to the ring of polynomials which factorized by the ideal generated by derivatives of the super potential. It was shown by Dishgraf Verlinde-Verlinde that to compute when we are interested in this model computation of the lightest of carol fields and their super partners which looks like this. It was shown by Dishgraf Verlinde-Verlinde that yes, I forgot to define here alpha. In this paper it was shown that to compute arbitrary correlates 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 as a activity 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 Dishgraf Verlinde-Virton equation. And when s is equal to zero I should explain what is this. The gamma is three-point function multiplied by a converse of this two-point function and this equation means that we have 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. 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 fact for exact solution of the model is to solve the model you need to find pre-potential in this coordinate in this coordinate I will explain that it really affects in these parameters as function of these parameters. The crucial fact for exact solution of the model which was discovered also by Dishgraf Verlinde-Virlinde that this ring which defined by this two objects structure constant in 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 coincides with another one which I will denote double way where double way is deformation of our of our super potential it is double way zero of x plus t alpha is a parameters of deformation e alpha is some basis of the ring of the ring r zero we can choose basis of this ring I will denote the alpha of x this basis and the structure constant of this ring defined by this definition e alpha modulo derivatives this is a structure constant of the ring r zero this is a 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 the ring of polynomials here it's already deformed 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 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 residuum and the theorem of m cytos not curge cytos but m cytos I forgot the first name says that there exists such n form omega of x t which is lambda some function of x t d x one d x n such form that this metric together with c tilde satisfy to all axi-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 c tilde satisfy to associativity equation I will not write indexes here then c mu nu lambda is equal c mu lambda nu where c mu nu lambda is symmetric tensor in all these variables and last derivatives which correspond to this metric applied to c mu c alpha nu alpha beta is equal number of nu c mu alpha beta and from this equation follows say that c mu nu lambda tilde is equal third covariant derivative of some function and because this metric is flat there exist such coordinate frame that 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 has the form which we saw here and the important discovery by Desgraf-Eurlind de Verlinder is that and it is important for exact solution of the model is that this ring 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 this form was called by K-Site primitive form if we know flat coordinates and primitive form in fact has exact solution for this model sir, louder please in c-hat c-hat, what? in c-hat yes, of course yes in the formula yes it's right it's right but you need to compute using this definition you can compute in principle you can compute like series in respect to deformation parameters this c tilde and this this is the question it is no no now I explain 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 it can be applied for computation all of this all this object pre-potential and so on I will demonstrate it for two models of series but because it depends the form of structure consistency it depends on the model ok, now maybe now I will try to to use now let us because much time consider let's consider some super-potential and which is quasi quasi which is quasi homogenous homogenous it means that it has this property and it's convenient to use integer number rho and d also not not rational sometimes people use of course it's possible d equal 1 and then we have the full and deformed formal terminology here this this potential for 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 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 de-relevant and I should mention that the problem when when all elements has 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 Calabria geometry of Calabria so in the case of Calabria many of us the main conjecture which we use we didn't haven't managed to to prove it but we check the conjecture the conjecture for non some non-trivial cases the conjecture is the following we want to find flat coordinates flat coordinates explain 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 elements such expressions here we have in brackets we have a slating integral integrals which and here there are some circles it is circles of homology group CN with this boundary so our non-disturbed super potential is part is equal minus infinity and it it permit 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 have we define weights for for elements of bases but we can define the weight or degree of defined degree T alpha like by definition like D minus degree 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 numbers so 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 model with deformation parameter T mu and also the weight of C mu is equal of course the weight of this one and to compute this oscillating integral we will use this property of oscillating integral integral of some form it is in fact dx is n form it is denotation dx is denoted dx1 dxn if integral of some p1 this p1 is polynomial multiplied by this form of this form and this form they coincide the difference is exact form exact not in not to relation to the RAM differential but such way deformed differential it is simple to choose to check that by integrating by paths that they coincide in this case then this differential define cogomologists in our space of n forms on our in our space and the statement is that there exist only n forms and this forms the basis the dimension of basis of cogomologists coincide with the dimension of our ring it is equal m like ring defined here and we can choose basis of this of this cogomology elements of our ring multiplied by dx and therefore we need to integrate integral of such elements of such element let now primitive form is just naive form dx we need to compute this integral with this exponent to do this we expand our monomial to the basis of this cogomologist it means that we should we should solve this linear problem just this is some additional term which is exact exact form it disappeared after when we insert this in our oscillating integral so we need to find only this constant these are constants and they can be computed 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 our integral where we have the basis element and it give some pairing between between cycles and cosicles it is some constant and we can choose one of the simplest simplest simplest choice of the basis in homolog homology and cycles in homology group is to take it's just dual to consider the army like like this like written here but it's interesting that that it that it is not that choice is not correct for getting flat general solution for flat coordinates because this theorem predicts that there exists some modulus in primitive format there are also models in for flat coordinates therefore and we we see some I have no time to explain it but we there is some freedom here which is connected with so called resonance resonance is means that we have some times we can have the elements of our basis of the ring whose weights differs in the D or integer multiplied by D where D is the weight of singularity itself in this case we can this not do not have this it has more general form we get some parameters and I will show what we have ok also when we have marginal and irrelevant irrelevant 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 and to compute so to compute flat coordinates we need to compute also primitive form and we do it we inserted this form in our selective integral where we apply this these properties which I mentioned before compute this and we stay with with such kind expression for flat coordinates be here we know from computation for 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 this from this equation all these parameters and and get some expression we apply it my time is finishing we apply it for case of this ring which I call Jepner-Karil ring is the ring they depend of they connected with SON of level K I find group it depends of two parameters N and K and we can consider this super potential like our singularity we can choose the basis of our ring to be sure polynomials and we consider two cases S3 of level 3 in this case the potential has this form it's 10-dimensional Milner number is 10 it's 10-dimensional Jacobi ring and we have in this ring 9 relevant elements in one marginal this one because only one parameter 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 we get here this factor lambda for primitive form it is it is a hyper geometric function and we get say we get say flat coordinates S10 which correspond to marginal it looks like this it looks it's ratio of 2 and then we want to come to get to check this this form because it's found this computation was found on the conjecture and we use we do alternative direct 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 miss this case skip this case because it's interesting because for next three of level four 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 a four metric then we do the following we apply a formal for product and then we rewrite this and we just denote this function H lambda it's H lambda of our deformation parameters we know this function from definition from this structure constant but these are not known and then we impose on this metric structure constant Dubrovian axioms the first we impose the property that this metric should be flat Riemann term at tensor should be equal zero this first one and it give us thousands of different equation for a few function H function H this function and they all this equation is compatible and they leave some solution and in fact it can be for simplest case for level three it give us a hyper geometric equation can be transformed to a hyper geometric equation for level four when we have many variables we can solve this equation by Mathematica and we solve this and we get some parameters it seems to me it's written here for K for K equal one parameters solution for metric and for flat coordinates which we get from this equation when we know metric and Christophage then we have one parameter for K equal one for K equal three and then we impose this extra axioms it give us nothing because it follows this property follows in fact from this definition what about this in fact it's written this graph here in the equation it imposes it it satisfies automatically for case of level three but in case of level four in the next it solving of this equation give two parameters but this M-cyte theorem predicts one parameter, one model of primitive form and all of permittance manifolds and this is fixed really fixed by this equation W-W-W equation and we get so it shows that our conjecture is right but we don't know how to to to check it and it is some some formals additional formals and conclusion is that so we have some efficient, really efficient way we don't need to get differential equation what is very difficult to do when we have many many deformation parameters the problem of computational flat coordinates in fact is linear problem it can be so maybe even by hands but and we do this way but mathematics is not difficult and it has natural application to possible generational topological conformal field series two-dimensional W-W gravity so to say definition of levial gravity and the most interesting it seems to me that it is can be applied to computation of geometry on model space and in fact this flat coordinates is nothing but I think at least for one case when we have the family of deformed quintics it's a series so that coordinates give us periods for which we need for computational geometry thank you very much