Harmonic bundles and Toda lattices with opposite sign

TL;DR

This paper explores the classification of real solutions to the Toda equation using wild harmonic bundles and twistor structures, providing explicit computations of Stokes factors and criteria for integral structures.

Contribution

It introduces a novel classification method for Toda solutions via harmonic bundles and details explicit computations of associated flat bundles and Stokes factors.

Findings

Classification of real Toda solutions via parabolic weights

Explicit computation of Stokes factors for meromorphic flat bundles

Criteria for the existence of integral structures in twistor theory

Abstract

We study a certain type of wild harmonic bundles in relation with a Toda equation. We explain how to obtain a classification of the real valued solutions of the Toda equation in terms of their parabolic weights, from the viewpoint of the Kobayashi-Hitchin correspondence. Then, we study the associated integrable variation of twistor structure. In particular, we give a criterion for the existence of an integral structure. It follows from two results. One is the explicit computation of the Stokes factors of a certain meromorphic flat bundle. The other is an explicit description of the associated meromorphic flat bundle. We use the opposite filtration of the limit mixed twistor structure with an induced torus action.