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

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 $\mathfrak{sp}(2,R)$.