nLab metric cone over complex projective 3-space

Idea

The metric cone over complex projective 3-space carries the structure of a G₂-manifold whose Riemannian metric is invariant under the canonical Sp(2) action by left-matrix multiplication on homomogeneous coordinates in $\mathbb{H}^2 \simeq_{\mathbb{R}} \mathbb{C}^4 \to \mathbb{C}P^3$ (Bryant-Salamon 89, see also Acharya-Bryant-Salamon 20).

Related concepts

• twistor fibration

References

• Robert Bryant, Simon Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. Volume 58, Number 3 (1989), 829-850 (euclid:euclid.dmj/1077307681)

• Bobby Acharya, Robert Bryant, Simon Salamon, A circle quotient of a $G_2$ cone, Differential Geometry and its Applications Volume 73, December 2020, 101681 (arXiv:1910.09518, doi:10.1016/j.difgeo.2020.101681)

As a compactification-space for M-theory on G₂-manifolds:

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