Flat coordinates of algebraic Frobenius manifolds in small dimensions

Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide polynomial Frobenius manifolds. Flat coordinates of the Frobenius metric $\eta$ 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 Frobenius metric $\eta$ 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 $\eta$ appear to be algebraic functions on the orbit space of the Coxeter group.


Introduction
A Frobenius manifold with a polynomial prepotential is known as a polynomial Frobenius manifold. Coxeter orbit spaces, constructed from finite irreducible Coxeter groups, can be given the structure of a semisimple polynomial Frobenius manifold [10]. Dubrovin conjectured that these classify all irreducible semisimple polynomial Frobenius manifolds, which Hertling proved with the added assumption that the Euler vector field has positive degrees [16]. A Frobenius manifold with an algebraic prepotential is known as an algebraic Frobenius manifold. This is a natural case to consider after classifying the polynomial Frobenius manifolds.
The first non-rational algebraic Frobenius manifolds were found by Dubrovin and Mazzocco in 2000, which they derived from the Coxeter group H 3 in relation to Painlevé VI equation [14]. Explicit prepotentials of these Frobenius manifolds were given more recently by Kato, Mano and Sekiguchi [19] (see also Remark 6.1 in that paper).
The local monodromy group of a semisimple Frobenius manifold is generated by finitely many reflections [12]. It comes together with a particular set of generating reflections R 1 , . . . , R n , where n is the dimension of the Frobenius manifold. In the case of an algebraic Frobenius manifold there is a finite orbit of the braid group B n acting on n-tuples of reflections in the monodromy group, this action is known as the Hurwitz action [14]. The local monodromy group is then necessarily a finite group [20], and the product of reflections R i gives a quasi-Coxeter element w in this group. An equivalent property of w is that it does not belong to any proper reflection subgroup of the Coxeter group (see [9]).
It is expected that irreducible semisimple algebraic Frobenius manifolds are closely related to the quasi-Coxeter conjugacy classes of finite irreducible Coxeter groups, where polynomial Frobenius manifolds correspond to the conjugacy class of a Coxeter element [4]. We recall some findings of algebraic Frobenius manifolds below together with their links to quasi-Coxeter elements. It seems not clear though whether these constructions give the same quasi-Coxeter conjugacy class as described above following [9].
Pavlyk constructed bi-Hamiltonian structures of hydrodynamic type by considering dispersionless limit of generalised Drinfeld-Sokolov hierarchies associated to a regular element of a Heisenberg subalgebra H w of an affine Lie algebra g [22]. To make dispersionless limit finite 1 INTRODUCTION 2 one has to restrict analysis to a suitable submanifold of the phase space. In this construction the Heisenberg subalgebra H w is associated with a regular quasi-Coxeter element w of the Weyl group of the finite-dimensional Lie algebra g (in general, non-equivalent Heisenberg subalgebras are in one-to-one correspondence with conjugacy classes of the Weyl group [18]). Dubrovin had previously shown that bi-Hamiltonian structures of hydrodynamic type have a correspondence with Frobenius manifolds [11]. Pavlyk claimed that his construction produces algebraic Frobenius manifolds and he gave an explicit expression for the prepotential in the case of the conjugacy class D 4 (a 1 ) [22] (in the notation for conjugacy classes of Weyl groups from Carter [1]).
Dinar also gave a construction of algebraic Frobenius manifolds [7]. Starting with a regular quasi-Coxeter element w in a Weyl group there is a distinguished nilpotent element e in the associated simple Lie algebra g [3], [25]. Dinar constructed a bi-Hamiltonian structure of hydrodynamic type on a subvariety of the Slodowy slice S e ⊆ g * using Dirac reduction and gave an explicit expression for the prepotential in the case of nilpotent orbit F 4 (a 2 ) [4] (in the notation for nilpotent orbits from [2]). He also derived prepotentials for D 4 (a 1 ) [6] and E 8 (a 1 ) [5], the latter of which was simplified in a joint work with Sekiguchi [8]. The eigenvalues of the quasi-Coxeter element w have the form e 2πi |w| η j , where |w| denotes the order of w and 0 ≤ η j ≤ |w| − 1. The degrees d j of the corresponding Frobenius manifold are d j = η j +1 |w| . Two algebraic prepotentials related to Weyl groups E 6 and E 7 , and six algebraic prepotentials related to the Coxeter group H 4 were found by Sekiguchi [24] who used degrees of the latter Frobenius manifolds conjectured by Douvropoulos (see further details in [9]). These prepotentials are denoted by H 4 (k), where k = 1, 2, 3, 4, 7, 9. The degrees of these Frobenius manifolds are determined as follows [9]. For a regular quasi-Coxeter element w with eigenvalues e 2πi |w| η j , there exists a regular element w 0 ∈ W such that w is conjugate to w l 0 for some l ∈ N, |w| = |w 0 | and the eigenvalues of w 0 have the form e where the remainder (η j + l) (mod |w|) is between 1 and |w|. The algebraic degree of these Frobenius manifolds also have a combinatorial interpretation [9].
Any Frobenius manifold has two compatible flat metrics η and g, where metric g is usually referred to as the intersection form. It is a complicated problem in general to express one flat coordinate system in terms of the other. For polynomial Frobenius manifolds expressing flat coordinates of η via that of g gives a distinguished set of basic invariants of a Coxeter group, known as Saito polynomials [23]. These polynomials play an important role in the representation theory of Cherednik algebras [15].
Let us explain the relation between the two sets of flat coordinates for two-dimensional algebraic Frobenius manifolds. Prepotentials for two-dimensional (semisimple) algebraic Frobenius manifolds have the following form [10]: where k ∈ Q \ {−1, 0, 1}. The degrees of the Frobenius manifold are d 1 = 1 and d 2 = 2 k , and the charge is d = k−2 k . Let x 1 , x 2 be flat coordinates of the intersection form. Then we have the relations which can be checked similarly to the polynomial case k ∈ N considered in [10]. Now, let w be a quasi-Coxeter element in the dihedral group I 2 (m). It must be the product of two reflections that generate I 2 (m). Hence w = c l , where c is a Coxeter element of I 2 (m) and (m, l) = 1. The eigenvalues of w are e ± 2πi m l . We can assume l ≤ m 2 as the corresponding elements w = c l give representatives for all the quasi-Coxeter conjugacy classes. Then the smallest positive integer r such that w is conjugate to c r is r = l. Thus, the degrees of the Frobenius manifold, using prescription (1.1) and following [9], are since η 1 = m − l, η 2 = l and |w| = m. From the general form (1.2) of a prepotential of an algebraic two-dimensional Frobenius manifold, we see that k = m l and thus and this has charge d = m−2l m . Note that when l = 1 we get the polynomial two-dimensional Frobenius manifolds.
The Coxeter group I 2 (m) has basic invariants We can express these basic invariants in terms of the flat coordinates of the metric in the following form: Formulas (1.5) may be thought of as inverse relations to formulas (1.3), where we replace flat coordinates x 1 , x 2 with basic invariants given by (1.4). Note that in the above analysis we relate an algebraic Frobenius manifold (1.2) with a conjugacy class of a quasi-Coxeter element in a dihedral group provided that k ≥ 2. Twodimensional Frobenius manifolds with 0 < k < 2 have positive degrees but a relation to quasi-Coxeter elements seems unclear in this range. Note that the charge d < 0 in this case, whereas d ≥ 0 when k ≥ 2. The general conjecture on the relation of algebraic Frobenius manifolds with quasi-Coxeter elements in [4] assumes that the degrees are positive. A possible way to exclude the examples (1.2) with 0 < k < 2 is to impose an additional assumption to the conjecture that the charge d ≥ 0. For k < 0 the Frobenius manifolds with prepotential (1.2) have d 2 < 0.
In this work we establish relations between the two sets of flat coordinates for all but one of the known non-rational algebraic Frobenius manifolds of dimensions 3 and 4. Thus, we deal with the two Dubrovin-Mazzocco examples (H 3 ) ′ and (H 3 ) ′′ , Pavlyk's example for D 4 (a 1 ) and Dinar's example for F 4 (a 2 ). We also cover most of the examples from [24] related to H 4 . The only known algebraic Frobenius manifold in dimensions 3 or 4 which we do not deal with is the example H 4 (9) from [24]. In this case the prepotential is a rational function, rather than a polynomial function, of the flat coordinates and an additional variable Z, which is algebraic in the flat coordinates. In all the cases we consider the flat coordinates of the metric η appear to be functions on a finite cover of the orbit space of the corresponding Coxeter group, while coordinates on the orbit space are basic invariants in flat coordinates of the intersection form g. Formulas (1.4) and (1.5) demonstrate this in dimension 2.
We note that the inversion symmetry [10] of a polynomial Frobenius manifold gives a Frobenius manifold with a rational prepotential, and, more generally, the inversion of an algebraic Frobenius manifold gives a Frobenius manifold with an algebraic prepotential. For the polynomial cases and the algebraic cases listed above their inversions have a negative degree. Relations between the two sets of flat coordinates of the resulting Frobenius manifolds can be deduced directly from the relations between the original two sets of the flat coordinates by considering how the inversion changes the flat coordinates of the intersection form following [21] and how the inversion changes the flat coordinates of the metric following [10].
The structure of the paper is as follows. In Section 2 we present some preliminary results on Frobenius manifolds, Coxeter invariant polynomials, the Laplace operator and we explain the method we use in order to relate the two flat coordinate systems. We assume that flat coordinates of η are algebraic in the basic invariants of the corresponding Coxeter group which turns out to be the case as we manage to find flat coordinates in such a form. The key tool to use is the Laplace operator in the flat coordinates of the intersection form g. This operator can also be rewritten in the flat coordinates of the Frobenius manifold metric η, see Proposition 2.1. This allows us to investigate harmonic functions in the flat coordinates. On the other hand we find harmonic basic invariants (see Proposition 2.3) which we equate to harmonic functions of suitable degree in the flat coordinates. We still have some coefficients to be found in these relations. In order to find them we compute the intersection form g in two different ways. After the general method is explained in subsection 2.4 we deal with all the examples in the subsequent Sections.
In all the examples we express explicitly the basic invariants of the flat coordinates of the intersection form g in terms of the flat coordinates of the Frobenius manifold metric η, generalising formulas (1.4) and (1.5) from the two-dimensional case. The explicit formulas for the flat coordinates of the metric η in terms of the basic invariants of the flat coordinates of the intersection form g are given for the examples related to H 3 , D 4 , H 4 (1) and H 4 (2). These formulas are also obtained but are too long to include for the example related to F 4 and for H 4 (3). Calculations are done by Mathematica.
In the final Section 7, we present some observations on the dual prepotentials of algebraic Frobenius manifolds. In subsection 7.1 we find the dual prepotentials for Frobenius manifolds with prepotentials of the form (1.2) with k = 1 l . For l ≥ 2 we express the dual prepotential using a hypergeometric function and we apply the inversion symmetry to find the dual prepotentials for k = − 1 l . In subsection 7.2, for (H 3 ) ′′ and D 4 (a 1 ) we analyse singularities of the third order derivatives of their dual prepotentials on the Coxeter mirrors. While determining dual prepotentials for algebraic Frobenius manifolds was a motivation for the present work results in Section 7 demonstrate that these prepotentials are considerably more involved comparing to the polynomial Frobenius manifolds.
2 General approach

Notations
For any two vectors a = (a 1 , . . . , a n ), b = (b 1 , . . . , b n ) ∈ C n , we define Let M be a smooth manifold with coordinate system z 1 , . . . , z n . If f ∈ C ∞ (M ) is homogeneous in the z coordinates, and has degree k, then we may write For an (r, s)-tensor field T on M expressed in the z coordinates we denote For a vector field X = X λ (z)∂ z λ on M, the Lie derivative L X T of an (r, s)-tensor field T along X is an (r, s)-tensor field which is defined in the z coordinates by the following formula: Note that, throughout, we are assuming summation over repeated upper and lower indices.

Laplace operator on Frobenius manifolds
Let M be an n-dimensional Frobenius manifold M with prepotential F (t 1 , . . . , t n ) [10]. The third order derivatives of the prepotential are used to define the symmetric (0, 3)-tensor c as We define the metric η, which is constant in the t coordinates, as Let us denote η ij = η −1 ij to be the inverse of the metric. We assume that The prepotential F satisfies the WDVV equations We assume that F is quasihomogeneous: where the Euler vector field E has the form with d 1 = 1 and the charge d = 1. Let us use the shorthand notations Then c i jk are structure constants of a commutative Frobenius algebra defined on the tangent space T t M and e = ∂ t 1 is its unity. It follows that The intersection form g is defined in the t coordinates by the formula which is known to be a flat metric on a dense open subset of M. Let x 1 , . . . , x n be flat coordinates for g such that g ij (x) = δ ij . (2.7) In these coordinates, x α ∂ xα .
Let us denote g ij = g −1 ij to be inverse of the intersection form. We know that where E −1 is the multiplicative inverse of the Euler vector field E. We define the tensor field * c i jk := g jλ c iλ k , (2.8) and we see that * Thus Define ∆ to be the Laplace operator in the x coordinates and ∇ to be the gradient operator in the x coordinates, so for a function f ∈ C ∞ (M ) we have Let z 1 , . . . , z n be any coordinate system on M. Then (2.10) We also have g iλ (z)g λj;k (z) = (g iλ g λj ) ;k (z) − g iλ ;k (z)g λj (z) = δ i j;k − g iλ ;k (z)g λj (z) = −g iλ ;k (z)g λj (z). (2.11) Let g Γ i jk (z) be the Christoffel symbols for the metric g in the z coordinates. Then, in the coordinate system x 1 , . . . , x n , the Christoffel symbols satisfy the following coordinate transformation law: The following proposition can be extracted from [10] (see formula (G.6) and Lemma 3.4). We include a complete proof below. Furthermore, where we sum over the index λ and i = 1, . . . , n is fixed.
Proof. We have which gives the equality (2.13) by formula (2.10). Now we find ∆(t j ). By relation (2.12) we have that Using equations (2.10) and (2.11) we get that (2.14) By relation (2.6) we can rearrange equation (2.14) as From relations (2.4) we see that c ij k;l (t) = c ij l;k (t), since η is constant in the t coordinates, hence By relation (2.1) the Lie derivative of the tensor field c ij k has the form Therefore By relations (2.5) and (2.9) we have that The statement follows by formula (2.3).

Coxeter-invariant coordinates
Let W be a finite irreducible Coxeter group of rank n acting on its complexified reflection representation V ∼ = C n by orthogonal transformations with respect to ( · , · ). Consider an orthonormal basis e 1 , . . . , e n , and the coordinates x 1 , . . . , x n defined as for all v ∈ V and all i = 1, . . . , n. Let y 1 , . . . , y n be a set of homogeneous generators of the algebra C[x 1 , . . . , x n ] W . It is well-known that such a set always exists [17] and we have an algebra isomorphism C[y 1 , . . . , y n ] ∼ = C[x 1 , . . . , x n ] W .
These generators are called basic invariants. The degrees d W i of basic invariants y i do not depend on the choice of basic invariants [17]. We assume that d W Proof. The first claim follows from the invariance of ∆ under orthogonal transformations. We have which implies the second statement.
We will use the following statement.
Proof. Let y 1 , . . . , y n ∈ C[x 1 , . . . , x n ] W be a set of basic invariants. Define Y n as where H = Ker(∆) is the vector space of harmonic polynomials. Consider the vector spaces V k of homogeneous W -invariant polynomials of degree k and the linear maps for j = 1, . . . , n − 1. Since the dimension of the domain is larger than the dimension of the range, there must be a nontrivial kernel that is not contained in Y n V deg y j (x)−2 by the direct sum decomposition (2.15). Let Y j be a nonzero element of this kernel. The polynomials Y j , 1 ≤ j ≤ n, are homogeneous and each basic invariant y i can be expressed as a polynomial in Y j , thus Y j generate C[x 1 , . . . , x n ] W and we have that ∆(Y j ) = 0 for all j ≤ n − 1.
Suppose we can write the prepotential F of a Frobenius manifold M as a polynomial F (t, Z) in the t coordinates and Z, where Z satisfies an equation of the form where a k ∈ C[t 2 , . . . , t n ], such Frobenius manifolds are called algebraic. We say that an algebraic Frobenius manifold M is associated to the Coxeter group W if there exist basic invariants y 1 , . . . , y n in the flat coordinates x 1 , . . . , x n of the intersection form g which are simultaneously polynomial in t 1 , . . . , t n and Z. All of the examples of Frobenius manifolds which we consider below are associated to Coxeter groups. We will sometimes need to consider the t coordinates and Z as independent variables (see e.g. Proposition 2.4 below). In such cases, for a rational function f of n + 1 variables, we will write f F (t, Z) instead of f (t, Z).

Relating flat coordinates with basic invariants
In this subsection we will explain how to relate flat coordinates t i with flat coordinates x j of the intersection form g, or rather with basic invariants y j of a Coxeter group. It is known [10] that for d = 1 we have The general method for finding basic invariants as polynomials y i (t, Z) below will go through the following steps: Choose y 1 , . . . , y n−1 so that y 1 , . . . , y n form a set of basic invariants for a finite irreducible Coxeter group W.
2) Let Y 1 , . . . , Y n be a set of basic invariants such that ∆(Y n ) = 1 and ∆(Y j ) = 0 for j = 1, . . . , n − 1, which exist by Proposition 2.3. Each Y i can be expressed as a polynomial in y 1 , . . . , y n . In particular, Y n = 1 2n y n . 3) Let V j be the vector space of polynomials in t 1 , . . . , t n and Z which are homogenous in the x coordinates of degree d W j . Find the harmonic elements of V j using Proposition 2.1, for j = 1, . . . , n − 1.

4)
Equate Y j = Y j (y) with a general harmonic element of V j . Rearrange these equations to find each y j as a polynomial in t 1 , . . . , t n and Z up to some coefficients to be found. This can be done successively for j = n − 1, n − 2, . . . , 1.

5)
Find the intersection form g ij in the y coordinates by the formula g ij (y) = (∇(y i ), ∇(y j )) , and express the entries as polynomials in the y coordinates, which can be done by Lemma 2.2. Substitute the expressions for y i (t, Z) into these entries, so we have g ij (y(t)).

6)
Calculate the components g ij (y(t)) of the intersection form g in the y coordinates by performing a change of coordinates y = y(t) on the intersection form g λµ (t) given by formula (2.6): Here, the derivatives ∂y i ∂t λ are found via their expressions in the t coordinates and Z which still contain some coefficients to be found.

7)
We equate the two expressions for g ij (y(t)) from steps 5) and 6), and find the values for the remaining coefficients, which is possible in all the examples we consider. Thus we get basic invariants y j expressed as polynomials in t i and Z. Note that the polynomials we find may not be unique if the Coxeter graph of W has non-trivial symmetries.
One may alternatively try to skip steps 2) and 4), but this increases the difficulty of the calculations needed to equate the two expressions for g ij (y(t)).
Proposition 2.4. Let e = e i (y)∂ y i be the unity vector field of an algebraic Frobenius manifold associated to W with prepotential F (t, Z). Then e i (y) ∈ C[t, Z] for each i = 1, . . . , n.
Proof. We know that e = ∂ t 1 . Hence since ∂Z ∂t 1 = 0 by relation (2.16). Proposition 2.5. Let g be the intersection form of an algebraic Frobenius manifold associated to W with root system R W . Then is the Jacobi matrix and c ∈ C.
Proof. It follows from [17] that for some c ∈ C. From relation (2.10), we see that 3 Algebraic Frobenius manifolds related to H 3 There are two non-polynomial algebraic Frobenius manifolds which we can be associated to H 3 , both found by Dubrovin and Mazzocco [14]. Prepotentials of these three dimensional Frobenius manifolds were given explicitly by Kato, Mano and Sekiguchi [19] (see also Remark 6.1 in [19]). Let R H 3 be the following root system for H 3 : and A 3 is the alternating group on 3 elements. Let us introduce the following basic invariants for H 3 (cf. [23]):

2)
where The basic invariants y 1 , y 2 , y 3 have degrees 10, 6, 2, respectively. [23]) The intersection form g ij (y) takes the form Consider another set of basic invariants for H 3 given by The following statement can be checked directly.
The Euler vector field is the unity vector field is e(t) = ∂ t 1 , and the charge is d = 2 5 . The intersection form (2.6) is then given by , and the harmonic elements of V 2 are proportional to A general element of V 1 is of the form where a i ∈ C. By calculating the Laplacian of this general element (3.16) using Proposition 2.1 and formulas (3.13)-(3.15) we find that the only harmonic elements of V 1 are as claimed. A general element of V 2 has the form By calculating the Laplacian of this general element (3.17) using Propostion 2.1 and formulas (3.13)-(3.15) we find that the only harmonic elements of V 2 are as claimed.
Theorem 3.4. We have the following relations Proof. Note that Y 3 = 1 6 y 3 = 10 9 t 3 . We now equate Y 1 and Y 2 given by relations (3.8)-(3.10) with general harmonic elements of V 1 and V 2 , respectively, given by Proposition 3.3. We then rearrange these equations to find y i in terms of t j and Z. We find where a, b ∈ C. In order to find a and b we perform steps 5-7 from Section 2.4. That is, we transform the intersection form (3.12) into y coordinates by applying formulas (3.21)-(3.23) and compare it with the expression given by Lemma 3.1. We find that a = 133120000 6561 and b = − 16000 729 , which implies the statement.
Proof. We have P (t, Z) = 0 by relation (3.11). Hence We thus have that The first term is polynomial in t 1 , t 2 , t 3 and Z. The polynomial P F is irreducible over is the field of rational functions in t 1 , t 2 and t 3 . Hence the second term in equality (3.24) can be represented as an element of the ring C(t 1 , t 2 , t 3 )[Z], when we reduce it modulo P F as a polynomial in Z. It can be checked that it is a polynomial in t 1 , t 2 and t 3 .
Proposition 3.6. We have that By Proposition 2.5, we need only find det ∂y i ∂t j . It can be calculated by Theorem 3.4, which leads to Proposition 3.6.
In the next statement we express flat coordinates t i via basic invariants y j and Z, which is an inversion of the formulas from Theorem 3.4.

(H 3 ) ′′ example
The prepotential for (H 3 ) ′′ is The Euler vector field is the unity vector field is e(t) = ∂ t 1 , and the charge is d = 2 3 . The intersection form (2.6) is then given by and the harmonic elements of V 2 are proportional to Proof. Using Proposition 2.1 we can directly calculate A general element of V 1 is of the form where a i ∈ C. By calculating the Laplacian of this general element (3.34) using Proposition 2.1 and formulas (3.31)-(3.33) we find that the only harmonic elements of V 1 are as claimed. A general element of V 2 has the form where b i ∈ C. By calculating the Laplacian of this general element (3.35) using Propostion 2.1 and formulas (3.31)-(3.33) we find that the only harmonic elements of V 2 are as claimed.
Theorem 3.10. We have the following relations Proof. Note that Y 3 = 1 6 y 3 = 2t 3 . We now equate Y 1 and Y 2 given by relations (3.8)-(3.10) with general harmonic elements of V 1 and V 2 , respectively, given by Proposition 3.9. We then rearrange these equations to find y i in terms of t j and Z. We find where a, b ∈ C. In order to find a and b we perform steps 5-7 from Section 2.4. That is, we transform the intersection form (3.30) into y coordinates by applying formulas (3.39)-(3.41) and compare it with the expression given by Lemma 3.1. We find that a = 62208 25 and b = 640 7 , which implies the statement.
Proof is similar to the one for Proposition 3.5.
Proposition 3.12. We have that where c = −2 14 ·5 5 and By Proposition 2.5, we need only find det ∂y i ∂t j . It can be calculated Theorem 3.10, which leads to Proposition 3.12.
In the next statement we express flat coordinates t i via basic invariants y j and Z, which is an inversion of the formulas from Theorem 3.10.
Theorem 3.13. We have the following relations:
Proof. We have that which gives the statement by applying the relations from Theorem 3.10.

Algebraic Frobenius manifold related to D 4
The D 4 (a 1 ) Frobenius manifold has been described by Pavlyk [22] and Dinar [4], with a prepotential given explicitly by Pavlyk. It is a four dimensional Frobenius manifold which can be associated to the Coxeter group D 4 , it is denoted with the conjugacy class a 1 in the Coxeter group D 4 [1]. The prepotential for D 4 (a 1 ) is The Euler vector field is the unity vector field is e(t) = ∂ t 1 , and the charge is d = 1 2 . We note that slightly different prepotentials and coordinates are used in Pavlyk [22] and Dinar [4]. The intersection form (2.6) is then given by Let R D 4 be the following root system for D 4 : Let us introduce the following basic invariants for D 4 (cf. [23]): 10) The basic invariants y 1 , y 2 , y 3 , y 4 have degrees 6, 4, 4, 2, respectively.
The harmonic elements of V 1 are proportional to and the harmonic elements of V 2 are of the form where a, b ∈ C are constants.
Proof. Using Proposition 2.1 we can directly calculate A general element of V 1 is of the form where a i ∈ C. By calculating the Laplacian of this general element (4.20) using Proposition 2.1 and formulas (4.16)-(4.19) we find that the only harmonic elements of V 1 are as claimed. A general element of V 2 has the form where b i ∈ C. By calculating the Laplacian of this general element (4.21) using Propostion 2.1 and formulas (4.16)-(4.19) we find that the only harmonic elements of V 2 are as claimed.

Under the corresponding tensorial transformation the intersection form given by formulas (4.2)-(4.7) takes the form given in Lemma 4.1.
Proof. Note that Y 4 = 1 8 y 4 = t 4 . We now equate Y 1 with a general harmonic element of V 1 , and we equate Y 2 and Y 3 with general harmonic elements of V 2 , where Y 1 , Y 2 and Y 3 are given by formulas (4.12)-(4.14) and the harmonic elements of V 1 and V 2 are given by Proposition 4.3. We then rearrange these equations to find y i in terms of t j and Z. We find where a i , b j ∈ C. In order to find a i and b j we perform steps 5-7 from Section 2.4. That is, we transform the intersection form (4.2)-(4.7) into y coordinates by applying formulas (4.26)-(4.29) and compare it with the expression given by Lemma 4.1. A particular solution is given by which implies the statement.
By Proposition 2.5, we need only find det ∂y i ∂t j . It can be calculated by Theorem 4.4, which leads to Proposition 4.7.
In the next statement we express flat coordinates t i via basic invariants y j and Z, which is an inversion of formulas from Theorem 4.4.

Algebraic Frobenius manifold related to F 4
The F 4 (a 2 ) Frobenius manifold was described by Dinar with a prepotential given explicitly [4] (there seem to be some typos for the prepotential in [4], we include a corrected version below which was communicated to us by Dinar). It is a four dimensional Frobenius manifold which can be associated to the Coxeter group F 4 , and is denoted by the conjugacy class a 2 in the Coxeter group F 4 [1]. The prepotential for F 4 (a 2 ) is F (t) = 2 5 3 7 67·521749 5 9 7 t 7 4 + 2 4 3 10 13·693097 5 9 7 t 6 4 t 3 + 2 2 3 8 13 2 23 2 7·97 5 9 t 5 4 t 2 3 + 3 7 13 3 18224639 2 6 The Euler vector field is the unity vector field is e(t) = ∂ t 1 , and the charge is d = 2 3 . The intersection form (2.6) is then given by Let R F 4 be the following root system for F 4 : Let us introduce the following basic invariants for F 4 (cf. [23]): The basic invariants y 1 , y 2 , y 3 , y 4 have degrees 12, 8, 6, 2, respectively.
Under the corresponding tensorial transformation the intersection form given by formulas (5.1)-(5.7) takes the form given in Lemma 5.1.

Remark 5.4.
There is in fact one other way to choose y i in Theorem 5.3 as polynomials of t j and Z. This non-uniqueness is due to the Z 2 symmetry of the Coxeter graph of F 4 .

Proposition 5.5. The derivatives ∂y
Proof is similar to the one for Proposition 3.5.
Note that we do not include relations for t i , Z and e as functions of the basic invariants y j for this example as they are too long to present here.
Note that for examples H 4 (3), H 4 (4) and H 4 (7) we will omit the relations for t i and Z as functions of the basic invariants y j , as they become too long. Likewise, we omit analogues of Propositions 4.3, 4.7 and 4.9.
the harmonic elements of V 2 are proportional to 10 4 , and the harmonic elements of V 3 are proportional to Proof. Using Proposition 2.1 we can directly calculate A general element of V 1 is of the form a 1 t 1 t 3 + a 2 t 1 t 5 4 + a 3 t 1 t 2 4 Z + a 4 t 2 2 t 3 4 + a 5 t 2 2 Z + a 6 t 2 t 3 t 4 4 + a 7 t 2 t 3 t 4 Z + a 8 t 2 t 9 4 + a 9 t 2 t 6 4 Z + a 10 t 3 3 + a 11 t 2 3 t 5 4 + a 12 t 2 3 t 2 4 Z + a 13 t 3 t 10 4 + a 14 t 3 t 7 4 Z + a 15 t 15 4 + a 16 t 12 4 Z, (6.28) where a i ∈ C. By calculating the Laplacian of this general element (6.28) using Proposition 2.1 and formulas (6.24)-(6.27) we find that the only harmonic elements of V 1 are as claimed. A general element of V 2 has the form where b i ∈ C. By calculating the Laplacian of this general element (6.29) using Propostion 2.1 and formulas (6.24)-(6.27) we find that the only harmonic elements of V 2 are as claimed. A general element of V 3 has the form where c i ∈ C. By calculating the Laplacian of this general element (6.30) using Propostion 2.1 and formulas (6.24)-(6.27) we find that the only harmonic elements of V 3 are as claimed.
Proposition 6.5. The derivatives ∂y i ∂t j ∈ C[t 1 , t 2 , t 3 , t 4 , Z]. Proof is similar to the one for Proposition 3.5. Proposition 6.6. We have that where c = 5 and By Proposition 2.5, we need only find det ∂y i ∂t j . It can be calculated by Theorem 6.4, which leads to Proposition 6.6.
In the next statement we express flat coordinates t i via basic invariants y j and Z, which is an inversion of the formulas from Theorem 6.4. Proof. Formula (6.42) follows immediately from Theorem 6.4. Using relations (6.16) and (6.33) we see that We can solve this system of equations to find t 2 and t 3 which gives us formulas (6.40) and (6.41). Substituting formulas (6.40)-(6.42) into relation (6.32) and solving for t 1 we get formula (6.39). Finally, substituting relations (6.39)-(6.42) into formula (6.31) we get the formula (6.43).
Proof. We have that which gives the statement by applying the relations from Theorem 6.4.
Proof is similar to the one for Proposition 3.5.
By Proposition 2.5, we need only find det ∂y i ∂t j . It can be calculated by Theorem 6.10, which leads to Proposition 6.12.
Theorem 7.2. Let M be a two-dimensional Frobenius manifold with prepotential (7.3) with k = −l −1 , where l ∈ Z ≥2 . Then the dual prepotential of M has the form where F * (x) is the function given by formula (7.9).
Proof. Given a two-dimensional Frobenius manifold M, with charge d = 1 and η 11 = 0 one can construct a two-dimensional Frobenius manifold M with charge d = 2 − d using a symmetry of the WDVV equations known as an inversion [10]. The flat coordinates x of the intersection form of M may be expressed in terms of the flat coordinates x of the intersection form of M via the following relation: for i = 1, 2. Moreover, the dual prepotential F * of M may be expressed as where F * is the dual prepotential of M [21]. In two dimensions, semisimple Frobenius manifolds with d = 1 and η 11 = 0 are uniquely parametrized, up to isomorphism, by their charge [10]. A Frobenius manifold with prepotential (7.3) has charge d = k−2 k . Let M be the Frobenius manifold with prepotential (7.3) with k = l −1 , thus M has charge d = 1 − 2l. We know from Theorem 7.1 that this Frobenius manifold has a dual prepotential of the form (7.9). The inversion M must have charge d = 2l + 1 and therefore its prepotential must be of the form (7.3) with k = −l −1 . The dual prepotential of M is given by equation (7.22) from which the statement follows.

(H 3 ) ′′ and D 4 (a 1 )
Recall that for a polynomial Frobenius manifold associated to a Coxeter group W with root system R = R W , the dual prepotential has the form (7.2). Let α ∈ R and define α i = (α, e i ), then we have the following relations (for generic points on (α, x) = 0): for all i, j, k = 1, . . . , n. Below we give related results for the algebraic Frobenius manifolds (H 3 ) ′′ and D 4 (a 1 ). To check that the polynomial P M (x, Z) factorises on the hyperplanes (α, x) = 0 we first substitute the expressions for y i (x) from relations (3.1)-(3.7), or (4.8)-(4.11), into the left-hand side of equation (3.45), or equation (4.34), respectively. We then restrict to the hyperplane (α, x) = 0 and see that the polynomial factorises as claimed.