Flat Structure on the Space of Isomonodromic Deformations

Abstract

Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of systems of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of Frobenius manifold. As its consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlevé equation.

The Saito determinant for Coxeter discriminant strata

“…It would also be interesting to investigate generalisations to the discriminant strata of the extended affine Weyl groups orbit spaces, the corresponding Frobenius manifolds were considered in [11][12][13]. Furthermore, Frobenius structures for the orbit spaces of complex reflection groups were discovered in [18] (see also [19], [10]). In this case one does not have the full structure of Frobenius manifold in general but rather the weaker Saito structure without metric [26] (see also [20], [4] for further studies of the complex reflection groups case).…”

Meromorphic connections over F-manifolds

“…In the case m even, a 2-parameter family exists. It was discovered recently independently in [KMS15] and [AL17]. Finally, Theorem 8.6 will calculate for these normal forms the regular singular exponents (see the Remarks 7.1).…”

“…Remarks 3.4. (i) Kato, Mano and Sekiguchi [KMS15], Arsie and Lorenzoni [AL17] and Konishi, Minabe and Shiraishi [KMS18] all establish and study the structures of flat F-manifolds with Euler field on the orbit spaces of most finite complex reflection groups.…”

“…Beautiful generalizations were found in [Lo14] [KMS15] [AL15] [KM19], relations between all generic 3-and 4-dimensional regular F-manifolds and all generic solutions of the Painlevé equations of types VI, V, IV, III and II (see the Remarks 3.5). Also, the constructions of Frobenius manifolds with Euler fields on the orbit spaces of the Coxeter groups [SaK79] [Du96] were generalized to constructions of flat F-manifolds with Euler fields on the orbit spaces of most finite complex reflection groups [KMS15] [KMS18] [AL17] (see the Remarks 3.4).…”

3-dimensional F-manifolds

“…Recently, flat F -manifolds with Euler fields were subject to work by Arsie and Lorenzoni [AL13] [Lo14] [AL17] [AL19], Kato, Mano and Sekiguchi [KMS15], Kawakami and Mano [KM19], Konishi, Minabe and Shiraishi [KMS18] [KM20]. They established such structures on orbit spaces of complex reflection groups.…”