nLab Cayley plane

Idea

The Cayley plane is the projective plane over the octonions, i.e. the octonionic projective space $\mathbb{O} P^2$.

Definition

The Cayley plane can’t be constructed using homogeneous coordinates the way it works for projective planes over division rings, since multiplication of octonions is not associative. However, it can be constructed in several other ways that generalize the approach for division rings:

• One can start with $\mathbb{O}^2$ and give it the structure of an affine plane? in the obvious way, and then add “points at infinity” in the usual way to obtain a projective plane. This is the most straightforward approach, but as always it has the defect that it makes the line at infinity appear special.

• One can consider the space of $3\times 3$ matrices over $\mathbb{O}$ that are “Hermitian” and idempotent, hence can be imagined as “projections onto dimension-1 subspaces of $\mathbb{O}^3$”.

• Writing out the components of such a matrix explicitly, one obtains a Veronese vector $(x_1,x_2,x_3;\xi_1,\xi_2,\xi_3)$ where $x_i\in \mathbb{O}$ and $\xi_i\in \mathbb{R}$, such that $\xi_i \overline{x_i} = x_j x_k$ and $\Vert x_i\Vert^2 = \xi_j \xi_k$ for all cyclic permutations $(i,j,k)$ of $(1,2,3)$. The Cayley plane can be identified with the space of nonzero such vectors modulo the scalar action of $\mathbb{R}$.

Properties

General

The octonionic projective plane is a non-Desarguesian plane, that is, Desargues' theorem does not hold. See projective plane.

Isometries

• F4 is the isometry group of $\mathbb{O} P^2$, with the stabilizer of a point being Spin(9). Hence $\mathbb{O} P^2 \cong F_4/Spin(9)$.

Cell structure

Proposition

There is a homeomorphism

$$\mathbb{O}P^2 \,\simeq\, D^{16} \underset{h_{\mathbb{O}}}{\cup} \mathbb{O}P^1$$

between the octonionic projective plane and the attaching space obtained from the octonionic projective line along the octonionic Hopf fibration.

(Lackmann 19, Lemma 3.4.)

(Mimura 67, page 166)

See also at cell structure of projective spaces.

Homotopy groups

Proposition

The homotopy groups of octonionic projective plane are

(1)$$\pi_k \big( \mathbb{O}P^2 \big) \;=\; \left\{ \array{ 1 &\vert& k \leq 7 \\ \pi_k \big( S^8 \big) &\vert& 8 \leq k \leq 14 } \right.$$

Further homotopy groups are

$$\pi_{15}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_{120}$$

$$\pi_{16}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_2^3$$

$$\pi_{17}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_2^4$$

$$\pi_{18}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_{24}\times\mathbb{Z}_2$$

$$\pi_{19}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_{504}\times\mathbb{Z}_2$$

$$\pi_{20}\big(\mathbb{O}P^2\big) \cong 1$$

$$\pi_{21}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_6$$

$$\pi_{22}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}_4$$

$$\pi_{23}\big(\mathbb{O}P^2\big) \cong\mathbb{Z}\times\mathbb{Z}_{120}\times\mathbb{Z}_2^2\,.$$

(While $\pi_{15}\big(S^8\big) \cong\mathbb{Z}\times\mathbb{Z}_{120}$, which includes the homotopy class of the octonionic Hopf fibration.)

(Lackmann 19, page 7)

(Mimura 67, Theorem 7.2.)

Cohomology

Proposition

For $A \in$ Ab any abelian group, then the ordinary cohomology groups of octionionic projective plane $\mathbb{O}P^2$ with coefficients in $A$ are

$$H^k(\mathbb{O}P^2,A) \simeq \left\{ \array{ A & for \; k=0,8,16 \\ 0 & otherwise } \right. \,.$$

(Lackmann 19, Corollary 4.1.)

AKM-Theorem

Proposition

(octonionic AKM-theorem)

The 13-sphere is the quotient space of the (right-)octonionic projective plane by the left multiplication action by Sp(1):

$$\mathbb{O}P^2 / \mathrm{Sp}(1) \simeq S^{13}$$

(Atiyah-Berndt 02)

Related concepts

• Cayley 4-form

References

General

• Malte Lackmann, The octonionic projective plane, in MATRIX Book Series 4, Springer (2021) [doi:10.1007/978-3-030-62497-2_6, arXiv:1909.07047]

See also:

• Wikipedia, Cayley plane

• Salzmann et. al., Compact Projective Planes, with an introduction to Octonion Geometry

The AKM-theorem for $\mathbb{O}P^2$:

• Michael Atiyah, Jürgen Berndt, in: Surv. Differ. Geom. VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (International Press, Somerville, MA, 2003) 1-27 (arXiv:math/0206135, doi:10.4310/SDG.2003.v8.n1.a1)

In string theory

Discussion of the Witten genus of Cayley plane-fiber bundles is in

• Carl McTague, The Cayley Plane and the Witten Genus (arXiv:1006.0728)

Indications that M-theory in 10+1 dimensions may be understood as the KK-compactification on Cayley-plane fibers of some kind of bosonic M-theory in 26+1 dimensions:

• Hisham Sati, $\mathbb{O}P^2$ bundles in M-theory, Commun. Num. Theor. Phys 3:495-530,2009 (arXiv:0807.4899)

• Hisham Sati, On the geometry of the supermultiplet in M-theory, Int. J. Geom. Meth. Mod. Phys. 8 (2011) 1-33 (arXiv:0909.4737)

• Michael Rios, Alessio Marrani, David Chester, Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory, Phys. Lett. B, 808, (2020)

  (arXiv:1906.10709)

• Mamoru Mimura, The Homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ. 6 (2) 131 - 176, 1967. https://doi.org/10.1215/kjm/1250524375