V-systems, holonomy Lie algebras and logarithmic vector fields

It is shown that the description of certain class of representations of the holonomy Lie algebra associated to hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated to $\Delta.$ The flat sections of the corresponding $\vee$-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$-system is free in Saito's sense and show this for all known $\vee$-systems and for a special class of $\vee$-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic $\vee$-systems.


Introduction
The ∨-systems are special finite covector configurations introduced in [41,42] in relation with certain class of solutions of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations, playing a fundamental role in 2D topological field theory, N = 2 SUSY Yang-Mills theory and the theory of Frobenius manifolds [7,8,24].
Let V be a complex vector space and A ⊂ V * be a finite set of non-collinear vectors in the dual space V * (covectors) spanning V * . To such a set one can associate the following canonical form G A on V : where x, y ∈ V . Let us assume that this form is non-degenerate and thus establishes the isomorphism ϕ A : V → V * . Let α ∨ = ϕ −1 A (α) be the corresponding inverse image of α ∈ A. The system A is called ∨-system if the following ∨-conditions (2) β∈Π∩A β(α ∨ )β ∨ = να ∨ are satisfied for any α ∈ A and any two-dimensional plane Π ⊂ V * containing α and some ν, which may depend on Π and α. If Π contains more than 2 covectors then (2) imply that ν does not depend on α ∈ Π and (3) β∈Π∩A β ∨ ⊗ β| Π = ν(Π)Id.
If Π contains only two covectors from A, say α and β, then (2) imply that (4) G A (α ∨ , β ∨ ) = 0. 1 The examples of ∨-systems include all two-dimensional systems, Coxeter systems and the so-called deformed root systems [25,38,41], but the full classification is still an open problem (see the latest results in [12,13,23,36]). The combinatorial (or matroidal) structure of all known ∨-systems is quite special, but there are no general results known so far. In this paper we would like to make some steps in this direction, using the framework of the theory of the hyperplane arrangements [29].
For any finite set of non-collinear covectors A ⊂ V * one can consider the associated arrangement of complex hyperplanes ∆ = ∆ A := ∪ α∈A H α in V given by α(x) = 0, α ∈ A and the corresponding holonomy Lie algebra g ∆ with generators {t α } α∈A and the relations (5) [t α , where Π is any two dimensional subspace of V * (see Kohno [19,20]). This Lie algebra coincides with the Lie algebra of the unipotent completion of the fundamental group of the corresponding complement Σ = V \ ∆ [19]. Its enveloping algebra is the quadratic dual of the cohomology algebra H * (Σ, C) in the cases when the latter is quadratic [44]. The relations (5) are equivalent to the flatness of the universal logarithmic connection [20] (6) In particular, for the standard arrangement of hyperplanes H ij in C n given by z i − z j = 0, 1 ≤ i < j ≤ n we have the Kohno-Drinfeld Lie algebra t n with generators t ij = t ji , 1 ≤ i < j ≤ n and relations [19] (7) [t ij , t kl ] = 0, [t ij , t ik + t jk ] = 0 for all distinct i, j, k, l.
The first result of this paper is a one-to-one correspondence between the certain linear representations of holonomy Lie algebras and ∨-systems (see Theorem 1 below). It is essentially a reformulation of the known equivalence of the ∨-conditions and the flatness of the corresponding ∨-connection [42] (8) where ξ ∈ V, x ∈ Σ and κ ∈ C is a parameter. Similar result was also pointed out recently by Arsie and Lorenzoni in [4]. By identifying T x Σ with V we can view the flat sections of the ∨-connection as the vector fields on Σ, which are parallel with respect to ∇ ξ (∨-parallel vector fields). The monodromy of the system (9) gives a linear representation of the corresponding fundamental group π 1 (Σ) in V. Important examples of ∨-systems are the following classical series found in [6]: respectively with non-zero parameters c 0 , . . . , c n with non-zero sum.
In the A n (c) case the corresponding system (9) is equivalent to the classical Jordan-Pochhammer system with the solutions, which can be given by the Pochhammer type integrals (see [2,30] and Section 3 below). The monodromy of this system is closely related to the classical Gassner representation of the pure braid group [5] (see the precise statement and the relation with bending of polygons in [18]). For a review of the higher rank representations of the braid group in relation with KZ equation we refer to Kohno [21,22].
In the main part of the paper we study the polynomial solutions of the systems (9), which are polynomial ∨-parallel vector fields. Such solutions may exist only for special values of parameter κ, which can be shown to be equal to the degree of the corresponding solution.
We call ∨-system A harmonic if there are n = rank A linearly independent polynomial ∨-parallel vector fields of degrees κ 1 , . . . , κ n such that κ 1 + · · · + κ n = |A| is the number of covectors in A. We show that for any harmonic ∨-system the corresponding vector fields are gradient and freely generate all logarithmic vector fields Der(log ∆) as a module over polynomial algebra, which means that the corresponding arrangements are free in Saito's sense [29]. As a corollary by Terao's factorisation theorem [29] the Poincare polynomial of Σ in that case has the form We conjecture that all the arrangements of ∨-systems are free, so the corresponding Poincare polynomials are always factorizable in such a form. We show also that all known ∨-systems [12,13] are combinatorially equivalent to Coxeter restrictions, which are known to be free [31].
In Section 4 we prove that the classical series of ∨-systems (10), (11) are harmonic and present the residue formulae for the potentials of the corresponding gradient vector fields (see Theorems 4 and 5). This fact seems to be remarkable since as we show even the restrictions of Coxeter systems in general are not harmonic.
In the last section we discuss the Coxeter case and the relation of harmonic ∨systems with Saito flat coordinates [34]. We prove that all Coxeter ∨-systems are harmonic and find the corresponding potentials. In the case when all the roots are normalised to have the same length these potentials are known to be precisely the Saito flat coordinates [15], so in the non-simply laced cases we have one-parameter deformations of these coordinates, which we describe explicitly.

∨-systems and representations of holonomy Lie algebras
Let ∆ be a hyperplane arrangement in V and A ⊂ V * be a set of equations of the hyperplanes from ∆, which we will call an equipment of ∆. We will assume that the set A generates V * . Arrangement ∆ is called irreducible if one cannot decompose Assume now that V is a complex Euclidean space with symmetric non-degenerate bilinear form G. Denote by α = G −1 α the vector corresponding to α ∈ V * and look for representations ρ : g ∆ → End(V ) of holonomy Lie algebra g ∆ of the form (12) ρ(t α ) = α ⊗ α, α ∈ A for some equipment A of ∆. In general, there are no such equipments, so these representations exist only for special hyperplane arrangements.
To state the theorem we will need the following notion of complex Euclidean ∨-system introduced in [13].
Let A be a finite set of non-collinear vectors in a complex Euclidean vector space V ∼ = V * . We say that the set A is well-distributed in V if the canonical form (1) is proportional to the Euclidean form G. The set A is called complex Euclidean ∨-system if it is well-distributed in V and any its two-dimensional subsystem is either reducible (consists of two orthogonal vectors) or well-distributed in the corresponding plane.
Note that we allow here the canonical form to be degenerate. If the canonical form (1) is non-degenerate, then we can use it to define the Euclidean structure on V and we have the definition of the usual ∨-system.
defines a representation of the associated holonomy Lie algebra g ∆ .
Conversely, if (12) is a representation of the holonomy Lie algebra g ∆ for an irreducible arrangement ∆ with equipment A then A is a complex Euclidean ∨system.
Thus for given hyperplane arrangement ∆ the description of all representations of holonomy Lie algebra g ∆ of the form (12) of g ∆ is essentially equivalent to the classification of all the ∨-systems A associated to ∆. Note that ρ depends not only on the arrangement, but also on the choice of the equations of the hyperplanes.
To prove this we first use the result by Kohno [20], who showed that the flatness conditions of the logarithmic connection (6) [∇ ξ , ∇ η ] = 0 are equivalent to the relations (5).
A similar interpretation of the ∨-conditions as flatness of the corresponding ∨connection on the tangent bundle was pointed out in [42]. Indeed, it is easy to see that the relation which in its turn is equivalent to the commutation relations for all α ∈ A and all 2-dimensional subspaces Π ⊂ V * containing α. Now if Π contains only two covectors α and β then we have which is zero for non-proportional α and β only if which is ∨-condition (4). If Π contains more than two covectors then the commutation relations (14) are equivalent to the property that the restriction of the operator β∈A∩Π β ∨ ⊗ β on Π is proportional to the identity, which coincides with ∨-condition (3). Now to prove the theorem we note that substitution of (12) into the holonomy Lie algebra relations (5) gives (15) [ α ⊗ α, for all α ∈ A and all 2-dimensional subspaces Π ⊂ V * containing α (cf. [43]). Comparing this with a version of ∨-conditions (14) we have the first claim of the theorem. Now fixing α and summing these relations over all 2-dimensional Π containing α we have (16) [ Since this is true for all α ∈ A, the set A generates V * and the arrangement is irreducible this implies that the operator β∈A β ⊗ β is proportional to the identity, or equivalently, that If µ = 0 then G A is non-degenerate and α ∨ = µ −1 α satisfy ∨-conditions (14). If µ = 0 then we have complex Euclidean ∨-system. This completes the proof.

∨-systems and gradient logarithmic vector fields
One of the main problems in the theory of ∨-systems is the characterisation of the corresponding hyperplane arrangements, see e.g. [36]. Since in dimension 2 any covector system is a ∨-system, the problem starts from dimension 3.
Here we discuss some connection of ∨-systems with the theory of logarithmic vector fields initiated by K. Saito [33]. We start with a brief review of this theory, mainly following Orlik and Terao [29].
Consider a hyperplane arrangement ∆ ⊂ C n . A vector field X = ξ i (z) ∂ ∂zi on C n is called logarithmic if it is tangent to every hyperplane Π ∈ ∆. The hyperplane arrangement ∆ is free if the space of all logarithmic vector fields Der(log ∆) is free as the module over polynomial algebra P n = C[z 1 , . . . , z n ] (see [29,33]). The degrees b 1 , . . . , b n of the corresponding homogeneous generators X 1 , . . . , X n are called the exponents of the arrangement: Here the degree of a homogeneous polynomial vector field X = ξ i (z) ∂ ∂zi is defined as the degree of any of its components: deg X = deg ξ i .
Saito's criterion [29] implies that ∆ is free if there are n homogeneous linearly independent over P n logarithmic vector fields X 1 , . . . , X n such that the sum of the degrees equals the number of hyperplanes |∆|: Such fields can be chosen as the free generators of the module Der(log ∆).
Conjecturally the property that the arrangement ∆ is free depends only on combinatorics of ∆ (see [29], page 154). This is in agreement with the following remarkable Factorization Theorem proved by Terao [40]: Poincare polynomial of the complement Σ = C n \ ∆ for a free arrangement ∆ has the form with some positive integers b 1 , . . . , b n . This is a far-going generalisation of Arnold's formula P Σn+1 (t) = (1 + t)(1 + 2t) . . . (1 + nt) for the Poincare polynomial of the configuration space of n + 1 distinct points on the plane, corresponding to A n -type arrangement, see [3].
It is known (Arnold, Saito) that all Coxeter arrangements are free with the exponents b i being the exponents of the corresponding Coxeter group W . The corresponding generators X i = grad f i , i = 1, . . . , n, where f 1 , . . . , f n are basic W -invariants, which by Chevalley theorem freely generate the corresponding algebra of polynomial W -invariants C[z 1 , . . . , z n ] W . Indeed, it is easy to see that the corresponding fields are logarithmic and, by Saito's criterion, generate Der(log ∆) because the sum of the exponents of a Coxeter group is known to be the number of reflection hyperplanes, see e.g. [16].
It is known also that any linear arrangement in C 2 is free and that a generic arrangement in C n with n > 2 is not free [29].
The arrangement ∆ is called hereditarily free if it is free and all restriction arrangements to the hyperplanes of ∆ and their intersections are also free [29]. The property of ∆ being free is not hereditary [29], but it is known that all Coxeter arrangements are hereditarily free [31]. Conjecture 1. For any ∨-system A the associated arrangement ∆ A is hereditarily free.
We have shown that the class of ∨-systems is closed under the restriction [13], so it is enough to prove that ∆ A is free. In particular, this would imply by Terao's theorem that the corresponding Poincare polynomial P ΣA (t) is factorizable in the form (18).
According to Terao's conjecture [29] the freeness is combinatorial property, so Conjecture 1 would follow from a stronger conjecture that for any ∨-system A the associated arrangement ∆ A has combinatorial structure of a restriction of a Coxeter system, which is true for all known ∨-systems.
Indeed, for the classical series we have to check only the cases when some of the coefficients vanish. In the A n case this does not happen, while in the B n case this happens when c 0 + c i = 0 for some i, which leads to Zaslavsky configurations D k n , combinatorially equivalent to some restrictions of the systems of type D N , see [28].
The arrangements of the exceptional generalised systems and their deformations [13] are also combinatorially equivalent to the Coxeter restrictions. In the AB(1, 3) case we have the arrangement equivalent to the restriction (E 7 , A 3 ) (see [12]) with Poincare polynomial in type G(1, 2) the arrangement has the combinatorial structure of the restriction (E 7 , A 2 2 ) (or, equivalently, (E 8 , A 5 )) with Poincare polynomial in type D(2, 1, λ) the arrangement is equivalent to Zaslavsky configuration D 1 3 = (D 6 , A 3 ) with Poincare polynomial Note that for the complex Euclidean ∨-systems the conjectures are not true. The counterexample is given by the ∨-system of type F 3 (t) with t 2 = − 1 2 , consisting of the following 10 vectors in C 3 A = {e 1 ± e 2 , e 1 ± e 3 , e 2 ± e 3 , i(e 1 ± e 2 ± e 3 )}.
The corresponding Poincare polynomial is not factorizable, so the arrangement is not free and is not combinatorially equivalent to any Coxeter restriction. Note that the corresponding canonical form G A = 0 in this case.
We are going to show now that at least for a subclass of ∨-systems we can find the corresponding generating logarithmic vector fields X 1 , . . . , X n as polynomial ∨-parallel vector fields (9) for special values of κ being the exponents b 1 , . . . , b n . Theorem 2. The polynomial solutions ψ of (9) are gradient logarithmic vector fields for the corresponding arrangement ∆ A with the degrees (19) deg ψ = κ.
Proof. To prove this it is convenient to choose an orthonormal basis in V , so that the canonical form G A becomes standard. Then we can identify V and V * with C n , so that α ∨ = α and α∈A α i α j = δ ij , i, j = 1, . . . , n, where α i is the i-th coordinate of α. The system (9) takes the form for some polynomial potential F (z). The fact that ψ is logarithmic follows from the regularity of the left hand side on the the hyperplane (α, z) = 0, which implies that (α, ψ) = 0 on this hyperplane, so that ψ is tangent. To find the degree of ψ multiply the relations (20) by z i β j and add over all i, j to have is the Euler vector field.
The potential F of a ∨-parallel vector field ψ can be defined in coordinate-free way by the relation for any α ∈ V * . The parallel transport condition implies that the potential F satisfies compatible system of the Euler-Poisson-Darboux type equations So the question is for which integer values of parameter κ do the polynomial solutions of (9) exist, and whether we can find enough such solutions to generate all logarithmic vector fields over polynomial algebra. Note that for κ = 1 we always have the solution ψ i = z i , i = 1, . . . , n corresponding to the Euler vector field ψ = E.
To understand the situation better let us consider the case of rank 2 systems A. In this case the gradient generators of logarithmic vector fields may not exist. It is well-known [29] that any such arrangement is free and Der(log ∆ A ) is generated by Euler vector field E and where Q = α∈A (α, z). The last vector field is gradient if and only if Q is harmonic: which in general is not the case. Indeed, consider a particular case of 4 lines with Then ∆Q = 2(1 + a)(−x 2 1 + 3x 1 x 2 − x 2 2 ), which vanishes only when a = −1, so the lines form a harmonic bundle (projectively equivalent to B 2 case). Adding to X a multiple of Euler field E also would not make it gradient.
Since vector field ξ has closed circular orbits, the necessary condition for the existence of polynomial f is γ ∆Qdt = 0, where γ is the circle x 1 = cos t, x 2 = sin t. In our case γ ∆Qdt = 2(1 + a) This means that we are dealing with a special subclass of both free arrangements and ∨-systems. This motivates the following definition.
We say that ∨-system A of rank n is harmonic if the corresponding system (9) has n linearly independent (at generic point) polynomial solutions for κ = κ 1 , . . . , κ n such that (25) κ 1 + · · · + κ n = |A|, where |A| is the number of covectors in A.
Theorem 3. The arrangement ∆ of any harmonic ∨-system is free with exponents b i = κ i , i = 1, . . . , n and the Poincare polynomial of Σ = V \ ∆ has the form The proof follows immediately from Theorem 2 and the Saito criterion. As one can see from the Euler-Poisson-Darboux type equations (24) the corresponding potentials F 1 , . . . , F n belong to the algebra of quasi-invariants of A It would be interesting to understand their role for these algebras (cf. [11]). As we will see now for the classical series the corresponding potentials turn out to be certain deformations of Saito's generators of the algebra of invariants.

Analysis of the classical series
Consider first ∨-systems of type A n from [6]: One can check that the corresponding canonical form is non-degenerate if σ = c 0 + c 1 + · · · + c n = 0, and the vector α ∨ for α = √ c i c j (e i − e j ) has the form [6]). The corresponding KZ equations ∇ ξ ψ = 0 with ∇ ξ given by (13) and ψ = (ψ 0 , . . . , ψ n ) ∈ V * have the form with ∂ i ψ i determined from the relation ψ 0 + · · · + ψ n = 0 : These equations are nothing but the Jordan-Pochhammer linear system for the integrals of the hypergeometric type (see e.g. Aomoto [2], Orlik and Terao [30], formula (1) on page 71). More precisely, we have the following Note that for the chosen contour γ the integral (30) is well-defined if and only if λ 0 + λ 1 + · · · + λ n is an integer, which we will assume to be the case. Then we have Consider the derivative which coincides with the equations (28), (29) with λ j = κ cj σ . Note that since λ 0 + · · · + λ n = κ, so we need κ to be integer. We claim that if we choose simply the smallest κ = 1, 2, . . . , n then we will have the basic gradient logarithmic vector fields X with components . . , n (note that the canonical form is not standard in this case). Indeed, Φ λ is meromorphic in x at infinity with the expansion The contour integral (30) is simply the coefficient at x −1 (times 2πi) in this expansion, so it is clearly polynomial in x 0 , . . . , x n . Since the degrees of these polynomials are 2, 3, . . . , n+ 1 are the same as in non-deformed case, by Saito's criterion we have the claim.
Consider now the ∨-systems of B n -type [6] B n (c) = √ c i c j (e i ± e j ), 1 ≤ i < j ≤ n; Let us assume for the beginning that c i + c 0 = 0 for all i = 1, . . . , n, so the corresponding arrangement is of type B n .
The canonical form has the matrix G = 2σC, C = diag (c 1 , . . . , c n ) with σ = c 0 + c 1 + · · · + c n , so for α = √ c i c j (e i ± e j ) we have and for α = 2c i (c i + c 0 )e i we have The corresponding equations (13) for ψ = (ψ 1 , . . . , ψ n ) ∈ V * have the form Consider the product and the corresponding integral where γ as before is a large circle. The integral is well-defined if the sum 2(λ 0 + λ 1 + · · · + λ n ) ∈ Z.
Theorem 5. ∨-systems B n (c) with c j + c 0 = 0 for all j = 1, . . . , n are harmonic with the corresponding potentials F k = J λ given by contour integrals (34) with λ i = (2k − 1) ci 2σ and k = 1, . . . , n. The corresponding value of κ is 2k − 1. Proof. We have (35) ψ One can easily check that Since the difference of the right hand sides of the last two formulas is the integral of the total derivative γ d xΦ λ , we see that the integrals (35) satisfy equations (32), (33) with λ j = κ cj 2σ . Note that 2(λ 0 + λ 1 + · · ·+ λ n ) = κ, so κ must be an integer. It is easy to see that the integral (34) vanishes for even κ, so the minimal values of κ are 1, 3, 5, . . . , 2n−1. Since they coincide with the exponents of the Weyl group the corresponding vector fields X with ξ j = c j ψ j are the generators of Der(log ∆ A ).
For special c this is however possible. Let then σ = 2n − k − 1 and the integral (34) becomes Taking now small contour γ surrounding x = 0 we have up to a non-essential multiple which is the remaining potential for the arrangement D k n (cf. [29]). Note that this case corresponds to the restriction of the Coxeter arrangement of type D k+2n to the subspace x k+1 = x k+n+1 , x k+2 = x k+n+2 , x k+n = x k+2n . So one might expect that the restrictions of Coxeter systems are always harmonic. This however is not true as the following example shows.
Direct check shows that there are no quasi-invariants of degree 3 and the spaces of quasi-invariants of degree 4 is two-dimensional. We have κ 1 ≥ 1, κ 2 ≥ 3, so κ 3 ≤ 3 and the only possible choice is κ 1 = 1, κ 2 = κ 3 = 3. As the space of quasi-invariants of degree 4 contains the square of the quasi-invariant of degree 2 one cannot have three independent solutions of the system (20) at the specified κ i .
Note that the corresponding arrangement can be given by It is free with a basis of logarithmic vector fields [29], page 251). Note that the restriction of the D 5 invariant x 1 . . . x 5 gives the polynomial x 1 x 2 x 3 3 of degree 5.

Coxeter arrangements and Saito flat coordinates
Let G be an irreducible finite Coxeter group generated by reflections in a real Euclidean space V of dimension n and ∆ be the set of all corresponding reflection hyperplanes. Define the corresponding Coxeter root system R as a set of normals chosen in a G-invariant way. Note that we have either 1 or 2 different orbits of G on R, so such a system in general depends on the additional parameter q = |α|/|β|, which is a ratio of the lengths of the roots from two different orbits.
The positive part A = R + of Coxeter root system is known to be a ∨-system ( [41], see also [25]), which we call Coxeter ∨-system. We are going to show that it is harmonic and that the corresponding potentials of the gradient logarithmic vector fields are given by Saito flat coordinates [34].
Recall briefly the definition of these remarkable coordinates, which can be considered as a canonical choice of generators in the algebra of G-invariant polynomials S G (V ). Let y 1 , . . . , y n be any set of homogeneous generators in S G (V ) of degrees d 1 > d 2 ≥ d 3 ≥ · · · > d n = 2. The image of the Euclidean contravariant metric on V is degenerate on the orbit space V /G, but its Lie derivative along well-defined vector field ∂ ∂y1 gives flat metric η (called Saito metric), which is non-degenerate everywhere [33,9].
The corresponding flat coordinates t 1 , . . . , t n ∈ S G (V ) are called Saito flat coordinates. They were found explicitly by K. Saito et al in [35] for all the cases except E 7 , E 8 (for the latter cases see [27], [1], [39]). These coordinates play an important role in 2D topological field theory [7] and related theory of Frobenius manifolds developed by Dubrovin [8,9,10]. In the A n case they appear in the theory of the dispersionless KP hierarchy [26].
For the classical Coxeter groups of types A n and B n the Saito coordinates can be written as the residues at infinity [17,7]: in type B n . Comparing this with the formulas (30), (34) we see that they coincide with the potentials of the A n (c)-type ∨-systems with c 0 = c 1 = · · · = c n and of B n (c)-type ∨-systems with c 0 = 0, c 1 = · · · = c n respectively. It turns out that this link with harmonic ∨-systems is not accidental.
Theorem 7. The Coxeter ∨-system R + is harmonic. In the case when all the normals have the same length the potentials of the corresponding gradient logarithmic vector fields are the Saito flat coordinates t 1 , . . . , t n .
Proof. In the case when all the vectors are normalised to have the same length this follows from the results of [15], where it was shown that the Saito polynomials satisfy the corresponding system (24) with κ = deg t i − 1 being the corresponding exponent of the Coxeter group.
This covers completely one-orbit cases: simply laced ADE as well as H 3 , H 4 and odd dihedral groups I 2 (2k + 1). The B n case follows from Theorem 11: for a general choice of normals B n = {e i ± e j , 2(1 + c 0 )e i , 1 ≤ i < j ≤ n} the potentials are given by 2k−1 2n+2c 0 , k = 1, . . . , n (the case of equal lengths corresponds to c 0 = 0). Thus it remains to consider only the case F 4 and even dihedral groups I 2 (2p). The Coxeter ∨-system of type F 4 consists of the following covectors: (e 1 ± e 2 ± e 3 ± e 4 ), 1 ≤ i < j ≤ 4.
In the case t = 1 all the roots have equal length, the case t = 1/ √ 2 corresponds to the root system F 4 . In the complex case we have to add that t 2 = −1 for the non-degeneracy of the corresponding bilinear form.
One can show that in the complex coordinate z = x 1 + ix 2 the potentials of the corresponding ∨-system are Note that when a = b we have the basic invariants zz, z 2p +z 2p , known to be Saito flat coordinates in this case [35].

Concluding remarks
Dubrovin discovered a remarkable almost duality between polynomial Frobenius structure on the orbit spaces of Coxeter groups and corresponding logarithmic Frobenius structures with the prepotential where R is the corresponding Coxeter root system with all the roots of the same length [10]. A natural question is what is the dual structure in the case when R is a general Coxeter ∨-system with roots of different length, or more generally, if R is any harmonic ∨-system.
A related question is about differential-geometric interpretation of the corresponding potentials. As we have seen above these potentials are certain deformations of Saito flat coordinates. We hope to address these questions elsewhere soon.