Timelike U-dualities in generalised geometry

Abstract

We study timelike U-dualities acting in three and four directions of 11-dimensional supergravity, which form the groups SL(2) × 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 $\frac{{\mathrm{SL}(2)\times \mathrm{SL}(3)}}{{\mathrm{SO}\left( {1,1} \right)\times \mathrm{SO}\left( {2,1} \right)}}$ or $\frac{{\mathrm{SL}(5)}}{{\mathrm{SO}\left( {3,2} \right)}}$. 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) × 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.