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ABSTRACT (original article): Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide polynomial Frobenius manifolds. Flat coordinates of the Frobenius manifold metric η are Saito polynomials which are distinguished basic invariants of the Coxeter group.
Algebraic Frobenius manifolds are typically related to quasi-Coxeter conjugacy classes in finite Coxeter groups. We find explicit relations between flat coordinates of the metric η and flat coordinates of the intersection form g for most known examples of algebraic Frobenius manifolds up to dimension 4. In all the cases, flat coordinates of the metric η appear to be algebraic functions on the orbit space of the Coxeter group. CITATION (original article): Misha Feigin, Daniele Valeri, Johan Wright (2024), Flat coordinates of algebraic Frobenius manifolds in small dimensions, Journal of Geometry and Physics, 200, 105151. https://doi.org/10.1016/j.geomphys.2024.105151

How this notebook is organised

In this notebook, you will find programs that compute the relations between the flat coordinates of the metric and the flat coordinates of the intersection form for 9 algebraic Frobenius manifolds of dimensions 3 and 4. These Frobenius manifolds are each related to a particular Coxeter group which plays a key role in finding the relations. Each section below covers a particular Coxeter group and the algebraic Frobenius manifolds related to it. The preamble on a Coxeter group should be evaluated before evaluation of each particular Frobenius manifold corresponding to this Coxeter group. A technical comment appears after all programs which is referred to throughout the file.

Note: a notebook with full outputs is attached to this post.

H
3

Coxeter Group
H
3

We input a set SimpleRoots of simple roots for a finite Coxeter group in the coordinates x[1], x[2], x[3]

φ1=(1+Sqrt[5])/2;φ2=(1-Sqrt[5])/2;SimpleRoots={x[1],-(1/2)*(φ1*x[1]+φ2*x[2]+x[3]),x[3]};

We calculate dim to be the rank of the Coxeter group

dim=Length[SimpleRoots]xCoords=Table[x[i],{i,1,dim}];

We define the Coxeter invariant InnerProduct and Reflection functions, and we use them to calculate the reflections of the basis vectors by the simple roots and record these in the table ReflectedCoords

InnerProduct[alpha_,beta_]:=Sum[D[alpha,x[k]]*D[beta,x[k]],{k,1,dim}]Reflection[alpha_,beta_]:=beta-2*alpha*InnerProduct[alpha,beta]/InnerProduct[alpha,alpha]ReflectedCoords=Table[Table[Simplify[Reflection[SimpleRoots[[alpha]],x[i]]],{i,1,dim}],{alpha,1,dim}];

Calculating the root system RootSystem for the finite Coxeter group with simple roots SimpleRoots

RootSystem=Simplify[Union[SimpleRoots,Table[-SimpleRoots[[i]],{i,1,dim}]]];OldLength=Length[RootSystem];NewLength=0;While[NewLength≠OldLength,OldLength=Length[RootSystem];NewRoots=RootSystem;For[i=1,i≤OldLength,i++,For[j=1,j≤OldLength,j++,AppendTo[NewRoots,Simplify[Reflection[RootSystem[[i]],RootSystem[[j]]]]]]];RootSystem=DeleteDuplicates[NewRoots];NewLength=Length[RootSystem]]Clear[i,j];

We input basic invariants yPolyX[i] for the Coxeter group letting yPolyX[dim] be the lowest degree invariant

ϵ1[Coords_]:=Sum[Coords[[i]]^2,{i,1,dim}]ϵ2[Coords_]:=Sum[Coords[[i]]^2*Coords[[j]]^2,{i,1,dim-1},{j,i+1,dim}]ϵ3[Coords_]:=Product[Coords[[i]]^2,{i,1,dim}]δ[Coords_]:=Product[Coords[[i]]^2-Coords[[j]]^2,{i,1,dim-1},{j,i+1,dim}]

yCoords=Table[y[i],{i,1,dim}];yPolyX[1][Coords_]:=95*ϵ2[Coords]*ϵ3[Coords]-32*ϵ1[Coords]^2*ϵ3[Coords]-5*ϵ1[Coords]*ϵ2[Coords]^2+2*ϵ1[Coords]^3*ϵ2[Coords]+3*Sqrt[5]*δ[Coords]*ϵ2[Coords];yPolyX[2][Coords_]:=Sqrt[5]*δ[Coords]+ϵ1[Coords]*ϵ2[Coords]-11*ϵ3[Coords];yPolyX[dim][Coords_]:=Sum[Coords[[i]]^2,{i,1,dim}];YXPolys=Table[yPolyX[i],{i,1,dim}];Yx=Table[y[i]YXPolys[[i]][xCoords],{i,1,dim}];

We calculate the degrees Wdegrees of the basic invariants

WdegreesEquations=Table[Sum[x[i]*D[YXPolys[[j]][xCoords],x[i]],{i,1,dim}]Wdeg[j]*YXPolys[[j]][xCoords],{j,1,dim}];WdegreesSols=SolveAlways[WdegreesEquations,xCoords];Wdegrees=Table[Wdeg[i]/.WdegreesSols[[1]],{i,1,dim}]

{10,6,2}

We check that yPolyX[i] are invariants

Counter=0;For[u=1,u≤dim,u++,For[v=1,v≤dim,v++,If[Expand[YXPolys[[u]][ReflectedCoords[[v]]]-YXPolys[[u]][xCoords]]0,Counter++,Print[u,v],Print[u,v]]]]Counter-dim^20Clear[u,v];

True

Below we define the intersection form in the x coordinates gInvX to be the identity matrix

gInvX=Table[KroneckerDelta[i,j],{i,1,dim},{j,1,dim}];gInvX//MatrixForm

1	0	0
0	1	0
0	0	1

We calculate the Jacobian matrix JacobianYX of the y coordinates with respect to the x coordinates

JacobianYX=Table[Expand[D[YXPolys[[j]][xCoords],x[k]]],{j,1,dim},{k,1,dim}];JacobianYX//MatrixForm

We find the intersection form gInvYx in the y coordinates expressed using the x coordinates

gInvYx=Expand[JacobianYX.gInvX.Transpose[JacobianYX]];gInvYx//MatrixForm

We calculate the ratio of the determinant of gInvYx in the x coordinates to the product of all roots in the root system RootSystem

xRandom=Table[x[i]RandomInteger[{1,100}],{i,1,dim}];Det[gInvYx]/Product[RootSystem[[i]],{i,1,Length[RootSystem]}]/.xRandom//Simplify

-85899345920

We find all monomials MonomialsYY in the y coordinates with degrees in x coordinates equal to the degrees of the entries of the intersection form gInvYx . See technical comment for details

MonomialsYY=Table[{},{i,1,dim},{j,1,dim}];For[u1=1,u1≤dim,u1++,For[u2=1,u2≤dim,u2++,BaseSequence=Flatten[Append[{1},Table[Product[Floor[(Wdegrees[[u1]]+Wdegrees[[u2]]-2)/Wdegrees[[dim-i]]]+1,{i,1,j}],{j,1,dim-1}]]];For[v=0,v<BaseSequence[[dim]],v++,PossibleMonomial={Floor[v/BaseSequence[[dim-1]]]};For[w=2,w<dim,w++,NextDigit=Floor[(v-Sum[PossibleMonomial[[i]]*BaseSequence[[dim-i]],{i,1,w-1}])/BaseSequence[[dim-w]]];AppendTo[PossibleMonomial,NextDigit]];LastDigit=(Wdegrees[[u1]]+Wdegrees[[u2]]-2-Sum[PossibleMonomial[[i]]*Wdegrees[[i]],{i,1,dim-1}])/Wdegrees[[dim]];AppendTo[PossibleMonomial,LastDigit];If[IntegerQ[PossibleMonomial[[dim]]]True,If[PossibleMonomial[[dim]]≥0,AppendTo[MonomialsYY[[u1,u2]],PossibleMonomial]]]]]];Clear[u1,u2,v,w];

We calculate the intersection form gInvY in the y coordinates as expressed in the y coordinates

HomogeneousPolysYY=Table[Sum[c[i,j,k]*Product[y[l]^MonomialsYY[[i,j]][[k]][[l]],{l,1,dim}],{k,1,Length[MonomialsYY[[i,j]]]}],{i,1,dim},{j,1,dim}];HomogeneousPolysYYx=Expand[HomogeneousPolysYY/.Yx];EquationsYY=Flatten[Table[HomogeneousPolysYYx[[i,j]]gInvYx[[i,j]],{i,1,dim},{j,1,dim}]];gInvSolutionsYY=SolveAlways[EquationsYY,xCoords];gInvY=HomogeneousPolysYY/.gInvSolutionsYY[[1]];gInvY//MatrixForm

30 3 y[2] +36 2 y[2] 3 y[3] +8y[1] 4 y[3]	28 2 y[2] y[3]+8y[2] 4 y[3]	20y[1]
28 2 y[2] y[3]+8y[2] 4 y[3]	8y[1]+8y[2] 2 y[3]	12y[2]
20y[1]	12y[2]	4y[3]

We define a function that calculates the Laplacian of any function that is expressed in the x coordinates

LaplaceX[f_]:=Sum[D[f,{x[i],2}],{i,1,dim}]

Below we find all monomials MonomialsY of the y coordinates with degrees in x coordinates equal to the degrees of the basic invariants minus 2. See technical comment for details

MonomialsY=Table[{},{i,1,dim-1}];For[u=1,u<dim,u++,BaseSequence=Flatten[Append[{1},Table[Product[Floor[(Wdegrees[[u]]-2)/Wdegrees[[dim-i]]]+1,{i,1,j}],{j,1,dim-1}]]];For[v=0,v<BaseSequence[[dim]],v++,PossibleMonomial={Floor[v/BaseSequence[[dim-1]]]};For[w=2,w<dim,w++,NextDigit=Floor[(v-Sum[PossibleMonomial[[i]]*BaseSequence[[dim-i]],{i,1,w-1}])/BaseSequence[[dim-w]]];AppendTo[PossibleMonomial,NextDigit]];LastDigit=(Wdegrees[[u]]-2-Sum[PossibleMonomial[[i]]*Wdegrees[[i]],{i,1,dim-1}])/Wdegrees[[dim]];AppendTo[PossibleMonomial,LastDigit];If[IntegerQ[PossibleMonomial[[dim]]]True,If[PossibleMonomial[[dim]]≥0,AppendTo[MonomialsY[[u]],PossibleMonomial]]]]];Clear[u,v,w];

We find a set of harmonic basic invariants HarmonicY expressed as polynomials in y coordinates

We input the prepotential F and the algebraic equation ZPolyT for Z in the coordinates t[1], t[2], t[3], such that the unity vector field is the partial derivative over t[1]

Here we calculate the derivatives DerivZT of Z with respect to the t coordinates and we define the command DerivT that calculates the derivatives with respect to the t coordinates of functions that depend on the t coordinates and Z=Z(t)

Here we calculate the charge d for the prepotential, the degrees deg[i] of the t coordinates and the Euler vector field Euler

We calculate the algebraic degree AlgDegZt of Z and the degree DegZ of Z

We calculate the third order derivatives Cijk of the prepotential F and present the metric eta in t coordinates

We check that the prepotential F satisfies the WDVV equations

We calculate the cotangent structure constants CotangentStrCons in t coordinates

We calculate the intersection form gInvT in the t coordinates

Here we define the Laplacian LaplaceT for a function expressed in the t coordinates and Z

Below we find all monomials MonomialsT of the t coordinates and Z with degrees equal to the degrees of the basic invariants. See technical comment for details

We find a set HarmonicT of harmonic polynomials in the t coordinates and Z with the same degrees as the basic invariants

We equate the harmonic basic invariants HarmonicY with the harmonic polynomials HarmonicT in the t coordinates and Z, and we rearrange to find the y coordinates as polynomials YTPolys in the t coordinates and Z with some coefficients to be found

We calculate the Jacobian matrix JacobianYT of the y coordinates with respect to the t coordinates

We find the intersection form gInvYt in the y coordinates expressed in the t coordinates via a tensorial transformation of the intersection form gInvT in the t coordinates

We equate the two above expressions for the intersection form gInvY and gInvYt in the y coordinates expressed in the t coordinates, and we equate them in order to find the unknown coefficients

We present the y coordinates expressed as polynomials ytPolys of the t coordinates and Z

We calculate the Jacobian matrix ytJacobian of the y coordinates with respect to the t coordinates, we observe that the entries are polynomial in the t coordinates and Z

The determinant of the above Jacobian matrix ytJacobian

We find the intersection form gInvYT in the y coordinates expressed in the t coordinates

We check that gInvYT gives the correct intersection form

The unity vector field in y coordinates

We calculate the expressions for t[2] and t[3] in terms of the y coordinates and Z

We substitute these relations into the expression for y[2] to find t[1] in terms of the y coordinates and Z

We substitute these relations into the expression for y[1] to find the algebraic equation that Z satisfies in the y coordinates

We input a set SimpleRoots of simple roots for a finite Coxeter group in the coordinates x[1], x[2], x[3], x[4]