On certain classes of mathbf{Sp}(4,mathbb{R}) symmetric G_2 structures

Abstract

We find two different families of mathbf{Sp}(4,mathbb{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 _2equiv 0, while the second family has tau _1equiv tau _2equiv 0, where tau _1, tau _2 are the celebrated G_2-invariant parts of the intrinsic torsion of the G_2 structure. The families are different in the sense that the first one lives on a homogeneous space mathbf{Sp}(4,mathbb{R})/mathbf{SL}(2,mathbb{R})_l, and the second one lives on a homogeneous space mathbf{Sp}(4,mathbb{R})/mathbf{SL}(2,mathbb{R})_s. Here mathbf{SL}(2,mathbb{R})_l is an mathbf{SL}(2,mathbb{R}) corresponding to the mathfrak{sl}(2,mathbb{R}) related to the long roots in the root diagram of mathfrak{sp}(4,mathbb{R}), and mathbf{SL}(2,mathbb{R})_s is an mathbf{SL}(2,mathbb{R}) corresponding to the mathfrak{sl}(2,mathbb{R}) related to the short roots in the root diagram of mathfrak{sp}(4,mathbb{R}).