# On certain classes of $Sp(2,R)$ symmetric $G_2$ structures

## TL;DR

This paper constructs two distinct families of $G_2$ structures with $Sp(2,R)$ symmetry in seven dimensions, distinguished by their torsion properties and underlying homogeneous spaces, expanding understanding of special geometric structures related to split real forms.

## Contribution

It introduces two new families of $G_2$ structures with $Sp(2,R)$ symmetry, characterized by different torsion conditions and realized on different homogeneous spaces.

## Key findings

- First family has $ au_2
eq 0$, second has $ au_1= au_2=0$
- Each family is associated with a different homogeneous space of $Sp(2,R)$
- The structures are related to split real form of $G_2$ and root diagram distinctions

## Abstract

We find two different families of $Sp(2,R)$ symmetric $G_2$ structures in seven dimensions. These are $G_2$ structures with $G_2$ being the split real form of the simple exceptional complex Lie group $G_2$. The first family has $\tau_2\equiv 0$, while the second family has $\tau_1\equiv\tau_2\equiv 0$. The families are different in the sense that the first one lives on a homogoneous space $Sp(2,R)/SL(2,R)_l$, and the second one lives on a homogeneous space $Sp(2,R)/Sl(2,R)_s$. Here $SL(2,R)_l$ is an $SL(2,R)$ corresponding to the $\mathfrak{sl}(2,R)$ related to the long roots in the root diagram of $\mathfrak{sp}(2,R)$, and $SL(2,R)_s$ is an $SL(2,R)$ corresponding to the $\mathfrak{sl}(2,R)$ related to the short roots in the root diagram of $\mathfrak{sp}(2,R)$.