Contents

Idea

The spin group in dimension 5.

Properties

Exceptional isomorphism

This is an indirect consequence of triality, see e.g. Čadek-Vanžura 97. Alternatively, it can be shown as follows.

Let $V$ be a 4-dimensional complex vector space with an inner product and a compatible complex volume form. As explained here, this structure can be used to define a conjugate-linear Hodge star operator on $\Lambda^2 V$ whose $+1$ and $-1$ eigenspaces, say $\Lambda_{\pm}^2 V$, are each 6-dimensional real inner product spaces. Thus, the group $\mathrm{SU}(V)$ acts as linear transformations of $\Lambda_{\pm}^2 V$ that preserve the inner product, giving a homomorphism $\rho: \mathrm{SU}(V) \to \mathrm{O}(\Lambda_{\pm}^2 V)$. In fact $\rho$ maps $\mathrm{SU}(V)$ in a 2-1 and onto way to $\mathrm{SO}(\Lambda_{\pm}^2 V)$. Taking $V = \mathbb{C}^4$ this shows $\mathrm{SU}(4) \cong \mathrm{Spin}(6)$.

Now suppose $V$ is additionally equipped with an complex symplectic structure, i.e. a nondegenerate skew-symmetric complex-bilinear form $J \in \Lambda^2 V$. The subgroup $\mathrm{Sp}(V)$ of $\mathrm{SU}(V)$ preserving this extra structure is isomorphic to the compact symplectic group $\mathrm{Sp}(2)$, which is also the quaternionic unitary group. This subgroup $\mathrm{Sp}(V)$ acts on $\Lambda_+^2 V$ and $\Lambda_-^2 V$ preserving $J \in \Lambda^2 V = \Lambda_+^2 V \oplus \Lambda_-^2 V$. Since $\mathrm{Sp}(V)$ is compact, every invariant subspace has an invariant complement, so one or both of the 6-dimensional subspaces $\Lambda_+^2 V$ and $\Lambda_-^2 V$ must have a 5-dimensional subspace invariant under the action of $\mathrm{Sp}(V)$. This shows that the double cover $\rho: \mathrm{SU}(4) \to \mathrm{SO}(6)$ restricts to a 2-1 homomorphism $\sigma : \mathrm{Sp}(2) \to \mathrm{SO}(5)$. Since

the differential $d\sigma$, being injective, must also be surjective. Thus $\sigma : \mathrm{Sp}(2) \to \mathrm{SO}(5)$ is actually a double cover. Since $\mathrm{Sp}(2)$ is connected this implies $\mathrm{Sp}(2) \cong \mathrm{Spin}(5)$.

Action on quaternionic Hopf fibration

This is almost explicit in Porteous 95, p. 263

Cohomology

This is a special case of the general statement in Pittie 91, see e.g. Kalkkinen 06, Section 3).

(Bott-Cattaneo 98, Lemma 2.1)

Coset spaces

$G$-Structure and exceptional geometry

Related concepts

• Spin(5,1)

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References

• Ian Porteous, Clifford Algebras and the Classical Groups, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1995)

• Martin Čadek, Jiří Vanžura, On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)

• Raoul Bott, Alberto Cattaneo, Integral Invariants of 3-Manifolds, J. Diff. Geom. 48 (1998) 91-133 [arXiv:dg-ga/9710001, doi:10.4310/jdg/1214460608]

• Harsh Pittie, The integral homology and cohomology rings of SO(n) and Spin(n), Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (doi:10.1016/0022-4049(91)90108-E)

• Jussi Kalkkinen, Global Spinors and Orientable Five-Branes, JHEP 0609: 028, 2006 (arXiv:hep-th/0604081)