Geometry and Physics of Sp(3)/Sp(1)^3

Abstract

The action of $Sp(3)$ on a vector space $V_3\in \mathbb H^3$ is analyzed. The transitive action of the group is conveyed by the flag manifold (coset space) $Sp(3)/Sp(1)^3\sim G/H$, a Wallach space. The curvature two-forms are shown to mediate pair-wise interactions between the components of the $\mathbb H^3$ vector space. The root space of the flag manifold is shown to be isomorphic to that of $SU(3)$, suggesting similarities between the representations of the flag manifold and those of $SU(3)$. The passage from $SU(3)$ to $Sp(3)$ and the interpretation given here encompasses the spin of the fermionic components of $V_3$. Composite fermions are representable as linear combinations of product states of the eigenvectors of $G/H$.