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

TL;DR

This paper explores the geometric and physical properties of the symmetric space Sp(3)/Sp(1)^3, revealing its curvature, root structure, and implications for fermionic spin and composite states.

Contribution

It provides a detailed analysis of the Sp(3)/Sp(1)^3 space, connecting its geometry to physical concepts like fermionic spin and composite particles, and compares it to the SU(3) structure.

Findings

Curvature forms mediate pairwise interactions.

Root space of the flag manifold is isomorphic to SU(3).

Interpretation of fermionic spin within the geometric framework.

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 $G/H$.