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Double field theory was developed by theoretical physicists as a way to encompass $T$-duality.
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O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, J. High Energy Phys. 1008, (2010) 008
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O. Hohm and S. K. Kwak, Frame-like geometry of double field theory, J. Phys. A, 44 (2011), 085404, 1-28
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C. Hull and B. Zwiebach, The gauge algebra of double field theory and Courant brackets. J. High Energy Phys. 0909 (2009) 099
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I. Vaisman, Dirac structures on generalized Riemannian manifolds, arXiv:1105.5908[math.DG]
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I. Jeon, K. Lee and J.-H. Park, Differential geometry with a projection: Application to double field theory, J. High Energy Phys. 1104 (2011) 014
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I. Jeon, K. Lee and J.-H. Park, Stringy differential geometry, beyond Riemann, Phys. Rev. D, 84 (2011) 044022
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