Hamiltonian dynamics and geometry on the two-plectic six-sphere

TL;DR

This paper explores the two-plectic geometry of the six-sphere derived from a G2-invariant three-form, revealing its non-flatness and automorphisms, and presents solutions to Hamilton-de Donder-Weyl equations.

Contribution

It explicitly proves the non-flatness of the two-plectic structure and characterizes its automorphisms by the G2 Lie algebra, providing new insights into higher symplectic geometry.

Findings

Proves the non-flatness of the two-plectic structure on the six-sphere.

Shows that infinitesimal automorphisms form the G2 Lie algebra.

Constructs solutions to Hamilton-de Donder-Weyl equations with various sources.

Abstract

We study the two-plectic geometry of the six-sphere induced by pulling back a canonical $G_2$-invariant three-form from $\mathbb{R}^7$ . Notably we explicitly prove non-flatness of this structure and show that its infinitesimal automorphisms are given by the exceptional Lie algebra $\mathfrak{g}_2$. Several interesting classes of solutions of the dynamical Hamilton-de Donder-Weyl equations with one- and two-dimensional sources are exhibited.