Membrane Duality Revisited

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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) × 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 GL(4,R) R⁴ subgroup of the expected SL(5,R) U-duality symmetry is realised.