Timelike U-dualities in Generalised Geometry

We study timelike U-dualities acting in three and four directions of 11-dimensional supergravity, which form the groups $SL(2)\times SL(3)$ and SL(5). Using generalised geometry, we find that timelike U-dualities, despite previous conjectures, do not change the signature of the spacetime. Furthermore, we prove that the spacetime signature must be $(-,+,...,+)$ when the U-duality modular group is either $\frac{SL(2)\times SL(3)}{SO(1,1)\times SO(2,1)}$ or $\frac{SL(5)}{SO(3,2)}$. We find that for some dual solutions it is necessary to include a trivector field which is related to the existence of non-geometric fluxes in lower dimensions. In the second part of the paper, we explicitly study the action of the dualities on supergravity solutions corresponding to M2-branes. For a finite range of the transformation, the action of $SL(2)\otimes SL(3)$ on the worldvolume of uncharged M2-branes charges them while it changes the charge of extreme M2-branes. It thus acts as a Harrison transformation. At the limits of the range, we obtain the "subtracted geometries" which correspond to an infinite Harrison boost. Outside this range the trivector field becomes non-zero and we obtain a dual solution that cannot be uniquely written in terms of a metric, 3-form and trivector. Instead it corresponds to a family of solutions linked by a local SO(1,1) rotation. The SL(5) duality is used to act on a smeared extreme M2-brane giving a brane-like solution carrying momentum in the transverse direction that the brane was delocalised along.