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quaternionic projective plane

Contents

Contents

Idea

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

Properties

As a quaternion-Kähler manifold / Wolf space

The quaternionic projective plane P 2\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):

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

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

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

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).

Created on February 6, 2021 at 23:39:19. See the history of this page for a list of all contributions to it.