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+1D=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)O(D,D) string T-duality, here we show in the case of the M2-brane in D=3D=3 that a generalized geometry renders manifest the SL(3)×SL(2)SL(3)×SL(2) U-duality. In the case of M2-brane in D=4D=4, with and without extra target space coordinates, we show that only the GL(4,R)⋉R4GL(4,R)⋉R4 subgroup of the expected SL(5,R)SL(5,R) U-duality symmetry is realized.