Classification of Toda-type tt*-structures and $\mathbb{Z}_{n+1}$-fixed points

TL;DR

This paper classifies Toda-type tt*-structures using anti-symmetry conditions, linking them to fixed points of certain multiplications, and explores their relation to representation theory.

Contribution

It provides a new classification of Toda-type tt*-structures via fixed point analysis and simplifies the anti-symmetry conditions to two cases.

Findings

Classification of Toda-type tt*-structures based on anti-symmetry.

Description of structures as fixed points of $e^{rac{2 ext{pi}i}{n+1}}$-multiplication.

Application to the connection between tt*-Toda equations and representation theory.

Abstract

We classify Toda-type tt*-structures in terms of the anti-symmetry condition. A Toda-type tt*-structure is a flat bundle whose flatness condition is the tt*-Toda equation (Guest-Its-Lin). We show that the Toda-type tt*-structure can be described as a fixed point of $e^{\sqrt{-1}\frac{2\pi}{n+1}}$-multiplication and this ``intrinsic'' description reduces the possibilities of the anti-symmetry condition to only two cases. We give an application to the relation between tt*-Toda equations and representation theory.