nLab Spin(5)

Idea

The spin group in dimension 5.

Properties

Exceptional isomorphism

Proposition

There is an exceptional isomorphism

$$Spin(5) \;\simeq\; Sp(2)$$

between Spin(5) and the quaternionic unitary group $Sp(2) = U(2,\mathbb{H})$.

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

$$\dim(\mathrm{Sp}(2)) = 10 = \dim(\mathrm{SO}(5))$$

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

Proposition

(Spin(5)-equivariance of quaternionic Hopf fibration)

Consider

1. the Spin(5)-action on the 4-sphere $S^4$ which is induced by the defining action on $\mathbb{R}^5$ under the identification $S^4 \simeq S(\mathbb{R}^5)$;

2. the Spin(5)-action on the 7-sphere $S^7$ which is induced under the exceptional isomorphism $Spin(5) \simeq Sp(2) = U(2,\mathbb{H})$ (from Prop. ) by the canonical left action of $U(2,\mathbb{H})$ on $\mathbb{H}^2$ via $S^7 \simeq S(\mathbb{H}^2)$.

Then the quaternionic Hopf fibration $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$ is equivariant with respect to these actions.

This is almost explicit in Porteous 95, p. 263

Cohomology

Proposition

The integral cohomology ring of the classifying space $B Spin(5)$ is spanned by two generators

1. the first fractional Pontryagin class $\tfrac{1}{2}p_1$

2. the linear combination $\tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2$ of the half the second Pontryagin class with half the cup product-square of the first Pontryagin class:

$$H^\bullet \big( B Spin(5), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \left[ \tfrac{1}{2}p_1, \; \tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2 \right]$$

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

Proposition

Let

$$\array{ S^4 &\longrightarrow& B Spin(4) \\ && \big\downarrow^{\mathrlap{\pi}} \\ && B Spin(5) }$$

be the spherical fibration of classifying spaces induced from the canonical inclusion of Spin(4) into Spin(5) and using that the 4-sphere is equivalently the coset space $S^4 \simeq Spin(5)/Spin(4)$ (this Prop.).

Then the fiber integration of the odd cup powers $\chi^{2k+1}$ of the Euler class $\chi \in H^4\big( B Spin(4), \mathbb{Z}\big)$ (see this Prop) are proportional to cup powers of the second Pontryagin class

$$\pi_\ast \left( \chi^{2k+1} \right) \;=\; 2 \big( p_2 \big)^k \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,,$$

for instance

$$\begin{aligned} \pi_\ast \big( \chi \big) & = 2 \\ \pi_\ast \left( \chi^3 \right) & = 2 p_2 \\ \pi_\ast \left( \chi^5 \right) & = 2 (p_2)^2 \end{aligned} \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,;$$

while the fiber integration of the even cup powers $\chi^{2k}$ vanishes

$$\pi_\ast \left( \chi^{2k} \right) \;=\; 0 \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,.$$

(Bott-Cattaneo 98, Lemma 2.1)

Coset spaces

coset space-structures on n-spheres:

standard:
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$	this Prop.
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$	this Prop.
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$	this Prop.
exceptional:
$S^7 \simeq_{diff} Spin(7)/G_2$	Spin(7)/G₂ is the 7-sphere
$S^7 \simeq_{diff} Spin(6)/SU(3)$	since Spin(6) $\simeq$ SU(4)
$S^7 \simeq_{diff} Spin(5)/SU(2)$	since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere
$S^6 \simeq_{diff} G_2/SU(3)$	G₂/SU(3) is the 6-sphere
$S^15 \simeq_{diff} Spin(9)/Spin(7)$	Spin(9)/Spin(7) is the 15-sphere

see also Spin(8)-subgroups and reductions

homotopy fibers of homotopy pullbacks of classifying spaces:

(from FSS 19, 3.4)

$G$-Structure and exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reduction	from spin group	to maximal subgroup
Spin(7)-structure	Spin(8)	Spin(7)
G₂-structure	Spin(7)	G₂
CY3-structure	Spin(6)	SU(3)
SU(2)-structure	Spin(5)	SU(2)
generalized reduction	from Narain group	to direct product group
generalized Spin(7)-structure	$Spin(8,8)$	$Spin(7) \times Spin(7)$
generalized G₂-structure	$Spin(7,7)$	$G_2 \times G_2$
generalized CY3	$Spin(6,6)$	$SU(3) \times SU(3)$

see also: coset space structure on n-spheres

Related concepts

• Spin(5,1)

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

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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)