Membrane Duality Revisited

Just as string T-duality originates from transforming field equations into Bianchi identities on the string worldsheet, so it has been suggested that M-theory U-dualities originate from transforming field equations into Bianchi identities on the membrane worldvolume. However, this encounters a problem unless the target space has dimension $D=p+1$. We identify the problem to be the nonintegrability of the U-duality transformation assigned to the pull-back map. Just as a double geometry renders manifest the $O(D,D)$ string T-duality, here we show in the case of the M2-brane in $D=3$ that a generalised geometry renders manifest the $SL(3)\times SL(2)$ U-duality. In the case of M2-brane in $D=4$, with and without extra target space coordinates, we show that only the $\mathrm{G}\mathrm{L}(4,R)⋉R^4$ subgroup of the expected $SL(5,R)$ U-duality symmetry is realised.