From the Darboux–Egorov system to bi-flat F-manifolds

Abstract

Motivated by the theory of integrable PDEs of hydrodynamic type and by the generalization of Dubrovin's duality in the framework of F -manifolds due to Manin [22], we consider a special class of F -manifolds, called bi-flat F -manifolds.A bi-flat F -manifold is given by the following data (M, ∇ 1 , ∇ 2 , •, * , e, E), where (M, •) is an F -manifold, e is the identity of the product •, ∇ 1 is a flat connection compatible with • and satisfying ∇ 1 e = 0, while E is an eventual identity giving rise to the dual product * , and ∇ 2 is a flat connection compatible with * and satisfying ∇ 2 E = 0. Moreover, the two connections ∇ 1 and ∇ 2 are required to be hydrodynamically almost equivalent in the sense specified in [2].First we show that, similarly to the way in which Frobenius manifolds are constructed starting from Darboux-Egorov systems, also bi-flat F -manifolds can be built from solutions of suitably augmented Darboux-Egorov systems, essentially dropping the requirement that the rotation coefficients are symmetric.Although any Frobenius manifold possesses automatically the structure of a bi-flat F -manifold, we show that the latter is a strictly larger class.In particular we study in some detail bi-flat F -manifolds in dimensions n = 2, 3. For instance, we show that in dimension three bi-flat F -manifolds are parametrized by solutions of a two parameters Painlevé VI equation, admitting among its solutions hypergeometric functions. Finally we comment on some open problems of wide scope related to bi-flat F -manifolds.

Complex reflection groups, logarithmic connections and bi-flat F-manifolds

“…As far as we know, the presence of these parameters has never been observed before. The results of the present Section also prove that the relation between generalized WDVV equations and the full family of Painlevé VI equation (in standard form) obtained in [20] is the counterpart in flat coordinates of the relation between three dimensional semisimple bi-flat F -manifolds and the full family of Painlevé VI equation (in sigma form) obtained in canonical coordinates in [28,4] (see also [2]).…”

“…Proof. From the above definition it follows that, in canonical coordinates for • (the product compatible with ∇ (1) ), we have (see [1,2]):…”

“…is a manifold M equipped with a pair of flat connections ∇ (1) and ∇ (2) , a pair of products • and * on sections of the tangent bundle T M and a pair of vector fields e and E satisfying the following axioms:…”

“…Proof: By definition of almost hydrodynamical equivalence, we have (d ∇ (1) − d ∇ (2) )(X * ) = 0 for every vector fields X (and equivalently with X•). By straightforward computation we get…”

Regular Flat Structure and Generalized Okubo System

“…where T, B ∞ are constant square matrices, is an Okubo system if T is diagonalizable. If T is not necessarily diagonalizable, (1) is said to be a generalized Okubo system, which was studied by H. Kawakami [20,19] in order to generalize the middle convolution to linear differential equations with irregular singularities.…”