Triality rotation of connections and Dirac operators on SU(3)
Abstract
We show how triality, the symmetry permuting the three 8-dimensional representations of Spin(8), can also appear as a symmetry of geometric objects such as connections and Dirac operators. Take any Riemannian metric g on the manifold M = SU(3) and let ∇ be a metric connection on TM. Using an explicit model of Clifford multiplication by deformed Okubo products, we combine spinorial lifts with isometries TM ≃ S+ ≃ S- to show that ∇ can be rotated into two new metric connections on TM. So there is a canonical ℤ3-orbit of metric connections determined by (g, ∇), with no other choices. The three connections coincide when g is bi-invariant and ∇ belongs to the Cartan-Schouten line, but in general they are different. The rotations of ∇ preserve the norm of the curvature, so they define a new ℤ3-symmetry of the moduli space of Yang-Mills connections on TM. They also preserve the Chern-Simons 3-form and adjoint Yang-Mills-Higgs functionals. After choosing an orientation on M, the rotated connections define three self-adjoint, Dirac-type operators on the spinor bundle. We prove that these operators are natural, in the sense that they commute with the Kosmann derivatives of spinors along all invariant Killing vector fields that preserve ∇. We also show that they are not isospectral, in general. This is verified by computing the simplest eigenvalue for the Levi-Civita connection and observing that it splits among the three operators as the bi-invariant metric on M is deformed to one with isometry group (SU(3) × SU(2) × U(1)) / ℤ6. Finally, we outline how the constructions in the paper may extend to general 8-manifolds with a PSU(3)-structure.
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BibTeX
@misc{baptista2026triality,
title = {{Triality rotation of connections and Dirac operators on SU(3)}},
author = {Joao Baptista},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12548},
eprint = {2610.12548},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12548},
}
RIS
TY - UNPB
TI - Triality rotation of connections and Dirac operators on SU(3)
AU - Joao Baptista
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12548
UR - https://arxiv.org/abs/2610.12548
AB - We show how triality, the symmetry permuting the three 8-dimensional representations of Spin(8), can also appear as a symmetry of geometric objects such as connections and Dirac operators. Take any Riemannian metric g on the manifold M = SU(3) and let ∇ be a metric connection on TM. Using an explicit model of Clifford multiplication by deformed Okubo products, we combine spinorial lifts with isometries TM ≃ S^+ ≃ S^- to show that ∇ can be rotated into two new metric connections on TM. So there is a canonical ℤ_3-orbit of metric connections determined by (g, ∇), with no other choices. The three connections coincide when g is bi-invariant and ∇ belongs to the Cartan-Schouten line, but in general they are different. The rotations of ∇ preserve the norm of the curvature, so they define a new ℤ_3-symmetry of the moduli space of Yang-Mills connections on TM. They also preserve the Chern-Simons 3-form and adjoint Yang-Mills-Higgs functionals. After choosing an orientation on M, the rotated connections define three self-adjoint, Dirac-type operators on the spinor bundle. We prove that these operators are natural, in the sense that they commute with the Kosmann derivatives of spinors along all invariant Killing vector fields that preserve ∇. We also show that they are not isospectral, in general. This is verified by computing the simplest eigenvalue for the Levi-Civita connection and observing that it splits among the three operators as the bi-invariant metric on M is deformed to one with isometry group (SU(3) × SU(2) × U(1)) / ℤ_6. Finally, we outline how the constructions in the paper may extend to general 8-manifolds with a PSU(3)-structure.
ER -