nLab quaternionic projective plane

Idea

The quaternionic projective plane $\mathbb{H}P^2$ is the projective plane over the skew-field of quaternions, hence the quaternionic 2-dimensional quaternionic projective space.

Properties

Cell structure

See at cell structure of projective spaces.

As a quaternion-Kähler manifold / Wolf space

The quaternionic projective plane $\mathbb{H}P^2$ the first of the list of examples of spaces that are both quaternion-Kähler manifolds as well as symmetric spaces, called Wolf spaces.

AKM-Theorem

In higher dimensional analogy to the Arnold-Kuiper-Massey theorem identifying the quotient of the complex projective plane by its O(1)-action as the 4-sphere, we have:

Proposition

(quaternionic AKM-theorem)

The 7-sphere is the quotient space of the (right-)quaternionic projective plane by the left multiplication action by U(1) $\subset$ Sp(1):

$$\mathbb{H}P^2 / \mathrm{U}(1) \simeq S^7$$

(Arnold 99, Atiyah-Witten 01, Sec. 5.5)

Related concepts

• quaternionic projective space

  • quaternionic projective line

• real projective plane

• complex projective plane

• octonionic projective plane

References

• Vladimir Arnold, Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation, Tr. Mat. Inst. Steklova, 1999, Volume 224, Pages 56–67; English translation: Proceedings of the Steklov Institute of Mathematics, 1999, 224, 46–56 (mathnet:tm691)

M-theory on the 8-manifold $\mathbb{H}P2$, hence on a quaternion-Kähler manifold of dimension 8 with holonomy Sp(2).Sp(1), is considered in

• Michael Atiyah, Edward Witten, p. 75 onwards in $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)

and argued to be dual to M-theory on G₂-manifolds in three different ways, which in turn is argued to lead to a a possible proof of confinement in the resulting 4d effective field theory (see there for more).