A family of left-invariant SKT metrics on the exceptional Lie group $G_2$

TL;DR

This work constructs a left-invariant integrable almost complex structure on the exceptional Lie group $G_2$ via the Samelson method and builds a 7-parameter family of left-invariant $\mathcal{J}$-Hermitian metrics. By imposing the SKT condition $dc=0$, the authors extract a 3-parameter subfamily of SKT metrics, which necessarily includes all bi-invariant metrics and is invariant under the right action of a maximal torus $T$ of $G_2$; they also prove a converse: any left-invariant $\mathcal{J}$-Hermitian metric that is SKT and right-$T$-invariant must belong to this subfamily. The results provide explicit, highly symmetric SKT structures on $G_2$ and connect Samelson complex structures with SKT geometry, offering concrete examples for studying Bismut connections on exceptional groups. Overall, the paper delivers an explicit classification of a family of left-invariant SKT metrics on $G_2$ and shows how torus symmetries constrain allowable metrics.

Abstract

For a complex manifold $(M,J)$, an SKT (or pluriclosed) metric is a $J$-Hermitian metric $g$ whose fundamental form $ω:=g(J\cdot,\cdot)$ satisfies the condition $\partial\overline{\partial}ω=0$. As such, an SKT metric can be regarded as a natural generalization of a Kähler metric. In this paper, the exceptional Lie group $G_2$ is equipped with a left-invariant integrable almost complex structure $\mathcal{J}$ via the Samelson construction and a 7-parameter family of $\mathcal{J}$-Hermitian metrics is constructed. From this 7-parameter family, the members which are SKT are calculated. The result is a 3-parameter family of left-invariant SKT metrics on $G_2$. As a special case, the aforementioned family of SKT metrics contains all bi-invariant metrics on $G_2$. In addition, this 3-parameter family of left-invariant SKT metrics are also invariant under the right action of a certain maximal torus $T$ of $G_2$. Conversely, it is shown that if $g$ is a left-invariant $\mathcal{J}$-Hermitian metric on $G_2$ such that $g$ is invariant under the right action of $T$ and for which $(g,\mathcal{J})$ is SKT, then $g$ must belong to this 3-parameter family of left-invariant SKT metrics.

Theorems & Definitions (30)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Remark 3.1
• Definition 4.1
• Proposition 4.2
• proof