Cayley 4-frames and a quaternion-K\"ahler reduction related to Spin(7)

TL;DR

The paper explores a specific group action on a quaternionic projective space to derive a quotient of the Cayley Grassmannian using quaternionic K"ahler reduction, linking geometric structures related to Spin(7).

Contribution

It introduces a new group action on HP^7 that facilitates a quaternionic K"ahler reduction of the Cayley Grassmannian, revealing novel geometric connections.

Findings

Constructed a U(1)×Sp(1) action on HP^7.

Derived a Z_2-quotient of the Cayley Grassmannian.

Connected Cayley geometry with Spin(7) structures.

Abstract

We describe an action of $U(1)\times Sp(1)$ on $HP^7$ that allows to obtain a $Z_2$-quotient of CAYLEY (the Grassmannian of oriented 4-planes in $R^8$ closed under the three-fold cross product) by quaternionic K\"ahler reduction.