The quadratic WDVV solution $E_8(a_1)$

A Frobenius algebra is a commutative associative algebra with unity e and an invariant nondegenerate bilinear form (., .). A Frobenius manifold is a manifold M with a smooth structure of Frobenius algebra on the tangent space T t M at any point t ∈ M with certain compatibility conditions [7]. Globally, we require the metric (., .) to be flat and the unity vector field e is constant with respect to it. In the flat coordinates (t 1 , ..., t r ) where e = ∂ ∂t r-1 the compatibility conditions implies that there exist a function F(t 1 , ..., t r ) such that

and the structure constants of the Frobenius algebra is given by

where η ij denote the inverse of the matrix η ij . In this work, we consider Frobenius manifolds where the quasihomogeneity condition takes the form (0.1)

This condition defines the degrees d i and the charge d of the Frobenius structure. If F(t) is an algebraic function we call M an algebraic Frobenius manifold. The associativity of Frobenius algebra implies the potential F(t) satisfies a system of partial differential equations known as the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations:

In topological field theory a solution to WDVV equations describes the a module space of two dimensional topological field theory [5].

Dubrovin conjecture on classification of algebraic Frobenius manifolds and hence algebraic WDVV solutions, is stated as follows: semisimple irreducible algebraic Frobenius manifolds with positive degrees d i correspond to quasi-Coxeter (primitive) conjugacy classes in irreducible Coxeter groups. A quasi-Coxeter conjugacy class in an irreducible Coxeter group is a Conjugacy class which has no representative in a proper Coxeter subgroup.

There are two major results support the conjecture. First, the conjecture arises from studying the algebraic solutions to associated equations of isomonodromic deformation of algebraic Frobenius manifolds [7], [8]. It leads to quasi-Coxeter conjugacy classes in Coxeter groups by considering the classification of finite orbits of the braid group action on tuple of reflections obtained in [12]. Therefore, it remains the problem of constructing all these algebraic Frobenius manifolds. Second, Dubrovin constructed polynomial Frobenius structures on the orbit spaces of Coxeter groups [6]. Then Hertling [9] proved that these are all possible polynomial Frobenius manifolds. The isomonodromic deformation of Polynomial Frobenius manifolds lead to Coxeter conjugacy classes [7].

The classification of polynomial Frobenius manifolds reveals a relation between the order and eigenvalues of the conjugacy class, and the charge and degrees of the corresponding Frobenius manifold. More precisely, If the order of the conjugacy class is κ + 1 and the eigenvalues are exp 2η i πi κ+1 then the charge of the Frobenius manifold is κ-1 κ+1 and the degrees are η i +1 κ+1 . We depend on this weak relation in considering a new examples of algebraic Frobenius manifolds. In [2] we continue the work of [10] and we began to develop a construction of algebraic Frobenius manifolds using classical W -algebras. This means we restrict ourself to conjugacy classes in Weyl groups. The examples obtained correspond, in the notations of [1], to the conjugacy classes D 4 (a 1 ) and F 4 (a 2 ).

In [3] we uniform the construction of polynomial Frobenius manifolds from classical W -algebras associated to regular nilpotent orbits. In [4] we extend this result and we uniform the construction of algebraic Frobenius manifolds form the classical W -algebras associated to subregular nilpotent orbits in the Lie algebra of type D r where r is even and E r . These subregular nilpotent orbits correspond to the regular quasi-Coxeter conjugacy classes D r (a 1 ) where r is even and E r (a 1 ), respectively.

In this work we write explicitly the potential of the algebraic Frobenius manifold E 8 (a 1 ). One of the main obstacles was that the minimal faithful matrix representation of the Lie algebra E 8 is the adjoint representation. This means that the elements of the Lie algebra are represented by square matrices of dimension 248. We overcome this problem by constructing instead the Weyl-Chevalley normal form. Another problem arises was the following. The Frobenius structure lives in a hypersurface which obtained by calculating the restriction of the invariant polynomials of the adjoint action to Slodowy slice. We avoid this calculation by returning to the method we used in [2] to obtain the algebraic Frobenius manifold F 4 (a 2 ). Which depends on the existence of a local Poisson structure compatible with the classical W -algebras (bihamiltonian structure).

In this paper we will not review the rich and deep theory behind the construction of the WDVV solution from classical W -algebras associated to nilpotent orbits. The interested reader may consult the paper [4] and [2] for details.

The resulting Frobenius manifold has the following potential F(t In the end we would like to add that we ar

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