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

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.