Talks## Flat structure on the space of isomonodromic deformations (II)

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Toshiyuki Mano

2015-05-12

10:30:00 - 12:00:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

The WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equation was found by physicists in 2D topological field theory. B. Dubrobin proved by introducing the notion of Frobenius manifolds that there exists a correspondence between solutions to the WDVV equation and isomonodromic deformations of linear differential equations with special conditions. (In n=3 case, the (n=3) WDVV equation is equivalent to a one-parameter family of the sixth Painleve equations.) The main purpose of this talk is to generalize the WDVV equation so that the generalized equation will be equivalent to isomonodromic deformations of generic linear differential equations of Okubo type. I will also show the existence of a "flat generator system" of invariant polynomials for the standard action of a finite complex reflection group. This is a generalization of K. Saito's result for finite Coxeter groups.

In the first lecture, I will introduce a completely integrable system of differential equations of Okubo type in several variables and explain some basic facts about logarithmic vector fields along a divisor. In the second lecture, I will introduce a geometric structure called "Saito structure without metric" which is presented in C. Sabbah's textbook. Then I will show that it is possible to construct a Saito structure without metric on the space of the independent variables of a generic Okubo type system in several variables. The existence of flat generator systems for finite complex reflection groups is a consequence of this construction.

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