Spin(9) in nLab

Contents

Contents

Idea

The spin group in dimension 9.

Properties

Relation to octonionic Hopf fibration

The octonionic Hopf fibration is equivariant with respect to the Spin(9)-action, the one on S 8=S(ℝ 9)S^8 = S(\mathbb{R}^9) induced from the canonical action of Spin(9)Spin(9) on ℝ 9\mathbb{R}^9, and on S 15=S(ℝ 16)S^{15} = S(\mathbb{R}^{16}) induced from the canonical inclusion Spin(9)↪Spin(16)Spin(9) \hookrightarrow Spin(16).

This equivariance is made fully manifest by realizing the octonionic Hopf fibration as a map of coset spaces as follows (Ornea-Parton-Piccinni-Vuletescu 12, p. 7):

S 7 ⟶fib(h 𝕆) S 15 ⟶h 𝕆 S 8 = = = Spin(8)Spin(7) ⟶ Spin(9)Spin(7) ⟶ Spin(9)Spin(8) \array{ S^7 &\overset{fib(h_{\mathbb{O}})}{\longrightarrow}& S^{15} &\overset{h_{\mathbb{O}}}{\longrightarrow}& S^8 \\ = && = && = \\ \frac{Spin(8)}{Spin(7)} &\longrightarrow& \frac{Spin(9)}{Spin(7)} &\longrightarrow& \frac{Spin(9)}{Spin(8)} }

Relation to standard model gauge group

The exact gauge group of the standard model of particle physics (see there) is isomorphic to the subgroup of the Jordan algebra automorphism group of the octonionic Albert algebra that “stabilizes a 4d sub-Minkowski spacetime” (see there for details).

More concretely, it is identified with the subgroup of Spin(9) which respects a splitting ℍ⊕ℍ≃ ℝℂ⊕ℂ 3\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3 (Krasnov 19, see also at SO(10)-GUT)

References

• Thomas Friedrich, Weak Spin(9)-Structures on 16-dimensional Riemannian Manifolds, (arXiv:math/9912112)

• Maurizio Parton, Paolo Piccinni, Spin(9) and almost complex structures on 16-dimensional manifolds, (arXiv:1105.5318)

• Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Spin(9) geometry of the octonionic Hopf fibration, (arXiv:1208.0899, doi:10.1007/s00031-013-9233-x)

• Maurizio Parton, Paolo Piccinni, The Role of Spin(9) in Octonionic Geometry, (arXiv:1810.06288)

• Kirill Krasnov, SO(9)SO(9) characterisation of the Standard Model gauge group (arXiv:1912.11282)