Another infinite tri-Sasaki family and marginal deformations

Several Einstein–Sasaki seven-metrics appearing in the physical literature are fibred over four-dimensional Kähler–Einstein metrics. Instead we consider here the natural Kähler–Einstein metrics defined over the twistor space $Z$ of any quaternion Kähler four-space, together with the corresponding Einstein–Sasaki metrics. We work out an explicit expression for these metrics and we prove that they are indeed tri-Sasaki. Moreover, we present a squashed version of them which is of weak $G2$ holonomy. We focus in examples with three commuting Killing vectors and we extend them to supergravity backgrounds with T3 isometry, some of them with $AdS4 × X7$ near horizon limit and some others without this property. We would like to emphasize that there is an underlying linear structure describing these spaces. We also consider the effect of the $SL(2,R)$ solution-generating technique presented by Maldacena and Lunin to these backgrounds and we find some rotating membrane configurations reproducing the $E–S$ logarithmic behaviour.