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Membrane duality revisited. (English) Zbl 1332.81172

Summary: 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 \(\operatorname{O}(D, D)\) string T-duality, here we show in the case of the M2-brane in \(D = 3\) that a generalized geometry renders manifest the \(\operatorname{SL}(3) \times \operatorname{SL}(2)\) U-duality. In the case of M2-brane in \(D = 4\), with and without extra target space coordinates, we show that only the \(\operatorname{GL}(4, R) \ltimes R^4\) subgroup of the expected \(\operatorname{SL}(5, R)\) U-duality symmetry is realized.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T20 Quantum field theory on curved space or space-time backgrounds
81T60 Supersymmetric field theories in quantum mechanics
81V17 Gravitational interaction in quantum theory
83E30 String and superstring theories in gravitational theory
83E50 Supergravity
53Z05 Applications of differential geometry to physics
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