The topological holonomy group and the complexity of horizontality
Naoya Ando, Anri Yonezaki
TL;DR
This work analyzes horizontality in twistor spaces $\hat{E}_{\varepsilon}$ associated with oriented rank-$4$ vector bundles over the 2-torus $T^2$ and classifies the resulting topological holonomy groups in $SO(3)$. It develops a framework in which horizontality is encoded by the holonomy group $G_{\hat{\nabla},\varepsilon}$, and shows that finite holonomy groups are precisely dihedral groups $\mathcal{D}_{2n}$ and the groups $\mathcal{A}_4$, $\mathcal{S}_4$, $\mathcal{A}_5$, with explicit angle data; infinite holonomy arises with noncommuting, order-two generators or higher-order generators. The paper proves density phenomena: under certain noncommutative, infinite-order conditions, $G_{\hat{\nabla},\varepsilon}$ is dense in $SO(3)$, and via the double cover $SO(4)\to SO(3)\times SO(3)$ this yields density in $SO(4)$ and, in the Hermitian case, density in $SU(2)$ and $U(2)$. The results rely on constructing $h$-connections realizing prescribed holonomies and apply eigenvalue/trigonometric arguments (including cyclotomic-polynomial considerations) to distinguish finite, infinite, and dense behavior, thereby illuminating horizontality on torus bundles and its global holonomy consequences.
Abstract
Based on [1], we study the complexity of horizontality in each twistor space $\hat{E}_{\varepsilon}$ associated with an oriented vector bundle $E$ of rank $4$ with a positive-definite metric over the $2$-torus $T^2$, and obtain classification of the topological holonomy groups in $SO(3)$. We observe that there exist many topological holonomy groups in $SO(3)$ generated by two finite order elements and equipped with noncommutative pairs which consist of infinite order elements. We find topological holonomy groups which are dense in $SO(4)$.