Horizontality with infinite complexity in the twistor spaces on tori
Naoya Ando, Anri Yonezaki
TL;DR
The paper investigates horizontality complexity in twistor spaces $\hat{E}$ over the torus, distinguishing finite versus infinite holonomy-driven behavior. Building on prior work that finite complexity corresponds to finite subgroups of $SO(3)$, it analyzes eigenvalues of products of holonomy generators to show that infinite complexity arising from those finite cases yields dense orbits in the twistor fiber and that certain eigenvalues take the form $\zeta= e^{i\psi\pi}$ with an irrational rotation parameter $\psi$. It provides a main theorem establishing density of $X(\omega)$ for specific triplets and characterizes the algebraicity of $\zeta$ with degrees $1$, $2$, or $4$; cyclotomic-polynomial arguments are used to rule out rational rotation parameters in the infinite case. Together with a classification of finite holonomy groups ($\mathbb{Z}/n\mathbb{Z}$, $D_{2n}$, $A_4$, $S_4$, $A_5$) and explicit angle relations, the work clarifies when horizontality on $\hat{E}$ becomes dense and highlights open problems in holonomy-group density and classification.
Abstract
We study the complexity of horizontality in the twistor space $\hat{E}$ associated with an oriented vector bundle $E$ of rank $4$ with a positive-definite metric over a torus. If the horizontality has finite complexity of degree $d>2$ for an element of a fiber of $\hat{E}$, then the complexity is expressed in terms of a finite subgroup of $SO(3)$ ([3]). In the present paper, we observe that if the horizontality has infinite complexity derived from one of the cases studied in [3], then the complexity is expressed by a dense subset of $S^2$.