How does triality rotate connections and Dirac operators on SU(3)?

Triality becomes a rotation of geometric objects

The paper asks whether triality, usually a statement about representations of Spin(8), can act on geometric data rather than only on abstract vector spaces. Its starting point is the observation that on M = SU(3) the tangent bundle TM and both chiral spinor bundles S+ and S− are isomorphic, and share the same structure group PSU(3). Because the adjoint action on su(3) cuts out PSU(3) inside SO(8), and this lifts to Spin(8), the two half-spin representations restricted to this subgroup coincide with the adjoint one.

That coincidence is what makes the construction possible. A connection on TM compatible with the Ad-invariant inner product can be carried across the isomorphisms to connections on S+ and S−, and back again. The paper's contribution is to make this transport explicit for general metrics on SU(3), not only the bi-invariant one, and to check whether the transported connection agrees with the standard spinorial lift of a connection with torsion. It does not always agree, and the discrepancy is what produces three distinct connections rather than one.

A deformed Okubo product supplies Clifford multiplication

To define spinors and Dirac operators one needs Clifford multiplication, the rule that lets a vector act on a spinor. On su(3) with its Ad-invariant metric there is an orthogonal product, the Okubo product, that satisfies Clifford-like relations. The paper lifts this product to bundle maps between TM, S+ and S−, identifying all three with M × su(3), and then deforms those maps so that they model Clifford multiplication for an arbitrary metric g on TM rather than only the bi-invariant metric induced by the inner product.

The existence of this Clifford module structure on M × (su(3) ⊕ su(3)) is what justifies identifying that bundle with the spinor bundle S = S+ ⊕ S−. In other words, the algebraic product is not a bookkeeping device: it is the mechanism that lets the author write down spin geometry on SU(3) for metrics away from the highly symmetric case, which is exactly the regime where the three rotated connections can differ.

One connection yields a canonical Z3 orbit of three

Given a metric g and a metric connection ∇ on TM, the paper transports the spinorial lifts of ∇ on S+ and S− back to TM using the natural isometries TM ≃ S+ ≃ S−. This produces connections ∇1, ∇2 and ∇3 on TM and on each of S±. The result is a canonical Z3 orbit determined by the pair (g, ∇), with no further choices: the rotation is a cyclic transformation T of order 3 on the space of metric connections.

The three connections collapse to one when g is bi-invariant and ∇ belongs to the Cartan-Schouten line. Away from that point they are generally distinct, and the paper states the precise conditions for distinctness in theorem 5.1. The rotated connections generally carry torsion, and the paper gives explicit formulas for their curvature tensors, Ricci scalars and torsion traces in terms of the curvature of ∇ and the Okubo products. A condensed formula writes the new connections as ∇rX Y = ∂X Y + ρ(r−1)AX(Y), where AX records the entries g(∇X Vj, Vk) in a g-orthonormal trivialization and the ρ(r−1) are the three inequivalent so(8) representations related by triality.

The rotation preserves curvature norms and Yang-Mills functionals

Because the rotation is canonical and of order three, it descends to a map of moduli spaces Ag/G → Ag/G, where Ag is the space of metric connections and G the gauge group. Proposition 5.2 shows that T preserves the Chern-Simons 3-form and the norm of the curvature, |R(T∇)|²g = |R(∇)|²g. Preserving the curvature norm means the rotation maps Yang-Mills connections to Yang-Mills connections, so it defines a new three-fold symmetry of the moduli space of Yang-Mills connections on (TM, g).

The paper extends this to Higgs fields. It studies Hg, the space of g-skew endomorphisms of TM, interpreted as adjoint Higgs fields, and defines a natural triality action on it that preserves the natural inner product and satisfies the compatibility relation T(∇X φ) = (T∇)X(Tφ). Theorem 5.4 shows that the simultaneous rotation of connections and Higgs fields descends to a map of moduli spaces (Ag × Hg)/G → (Ag × Hg)/G, and the natural Yang-Mills-Higgs functionals are preserved. The rotation is therefore a symmetry of the Euler-Lagrange solution spaces of those functionals, not merely of the underlying bundles.

Self-adjointness requires starting from the spinor side

The rotated connections on TM generally have non-zero trace torsion, and the classical Dirac operators of such connections are not formally self-adjoint. To obtain self-adjoint operators the paper chooses one chiral bundle, say S+, which amounts to choosing an orientation on M, and starts from the connections ∇r on S+ instead. It defines elliptic operators Dr : Γ(S+) → Γ(S−) by Drψ = Σk Vk · ∇rVk ψ, computes their L² adjoints D†r, and builds the self-adjoint Dirac operator as the anti-diagonal operator on S = S+ ⊕ S− from Dr and D†r.

Expanding in gamma matrices, the operator takes the form Drψ = Σj γj(∂Vj + ½ divg Vj)ψ + ¼ Σj≠k≠l Γrjkl γjγkγlψ − ½ τr · ΓS ψ, where ΓS is the chiral operator, τr are the trace torsion vector fields of ∇r, and Γrjkl = g(∇rVj Vk, Vl). The terms with a single gamma matrix coincide across the three connections; the axial terms with seven gamma matrices do not. The paper also shows in section 8 that the τr can be non-zero even when ∇ is the Levi-Civita connection, so the torsion is not an artifact of an exotic starting choice.

The operators commute with Kosmann derivatives of invariant Killing fields

The construction depends on a pair (g, ∇) and an orientation on SU(3), so one expects every symmetry preserving those inputs to preserve the three connections and Dirac operators. Theorems 7.1 and 7.5 confirm this: if X is a left- or right-invariant Killing vector field with LX∇ = 0, then LX∇r = 0 as well, and the Kosmann-Lichnerowicz derivative of spinors satisfies LX(∇rY ψ+) = ∇r[X,Y]ψ+ + ∇rY(LXψ+) for r = 1, 2, 3. The theorems also describe what happens when LX∇ does not vanish.

A consequence, proposition 7.6, is that the Kosmann derivatives LX commute with the operators Dr. This is the paper's precise sense in which the three Dirac operators are natural: they are tied to the isometries of the metric and the symmetries of the connection rather than being arbitrary elliptic operators on the spinor bundle. That property is what later lets the author compute adjoint actions on a restricted subspace in the explicit example.

Spectra split when the bi-invariant metric is deformed

The paper's sharpest concrete result is that the three operators are not isospectral in general. It considers a family of left-invariant metrics on M obtained by rescaling the circle generated by a vector v8 in su(3), with scalar curvature Rg = (3/(4a²))(9 − µ²), where a and µ are positive parameters of the initial bi-invariant metric and of the circle's rescaling. The connected component of the isometry group becomes (SU(3) × SU(2) × U(1))/Z6.

Choosing ∇ to be the Levi-Civita connection, the paper exhibits a spinor ψ0 with D1ψ0 = −(3/(2a))ψ0, D2ψ0 = −(3/(4a))(1 + µ)ψ0, and D3ψ0 = −(3µ/(2a))ψ0. As µ → 0+ with a fixed, the eigenvalue of D3 becomes arbitrarily small while the scalar curvature tends to a positive constant, and the eigenvalue of D2 tends to a finite constant below Friedrich's bound. Hence D2 and D3 cannot be isospectral to the classical Dirac operator D1. The same example gives trace torsion fields τ2 = −τ3 = (3/(4a))(1 − µ)v8, which are non-zero whenever µ ≠ 1 and g is not bi-invariant.

PSU(3)-structures carry the same construction

The paper outlines how the construction may extend beyond M = SU(3) to general 8-manifolds carrying a topological PSU(3)-structure, a reduction of the frame bundle to PSU(3) ⊂ SO(8) in the sense of Hitchin and Witt. Such a structure determines a metric, an orientation, and an identification TM ≃ P ×Ad su(3); since PSU(3) lifts canonically to Spin(8), it also determines a spin structure and chiral spinor bundles S± ≃ P ×PSU(3) Δ±. The two half-spin representations restricted to PSU(3) are again equivalent to the adjoint one, so the Ad-equivariant isometries Δ± ≃ su(3) give global bundle isometries I± : TM → S±, unique up to sign, and the three connections can be defined as before.

The author also flags two further directions without carrying them out: 8-manifolds whose spin bundle reduces to G2 ⊂ Spin(8), where the vector and two spin representations again coincide and a para-octonionic product may replace the Okubo one, and higher-dimensional Riemannian submersions with 8-dimensional fibre M, where one could rotate the vertical projection of a higher-dimensional connection while leaving its horizontal and mixed components unchanged. These are stated as possibilities, not as proved results, and the paper's verified claims remain on SU(3).

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