Contents

group theory

spin geometry

string geometry

# Contents

## 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)

### 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:

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

(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

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

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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)

• Jussi Kalkkinen, Global Spinors and Orientable Five-Branes, JHEP0609:028, 2006 (arXiv:hep-th/0604081)

Last revised on March 24, 2019 at 10:56:45. See the history of this page for a list of all contributions to it.