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After many years, the deep nature of spacetime in string theory remains an enigma.
C. Hull and B. Zwiebach, JHEP 0909
1907
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There are many examples of fundamental physics structures that stay unchanged under the interchange of conjugate variables, such as spatial and momentum coordinates: x a → p a x_{a}\to p_{a} and p a → − x a p_{a}\to-x_{a} . See M. Born, Nature 136
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For a recent review on bi-Lagrangian manifolds and references see, F. Etayo, R. Santamaría and U. R. Trías, Differential Geometry and its Applications 24
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Born reciprocity was mentioned in the context of string theory in G. Veneziano, Europhys. Lett. 2
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D.J. Gross and P.F. Mende, Phys.Lett. B197
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E. Witten, Phys. Rev. Lett. 61
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Note that the usual Hamiltonian and diffeomorphism constraints of string theory in flat target space already implement Born reciprocity!
This is reminiscent of an almost hyper-Kähler structure, which arises when ( I , J , K ) (I,J,K) are all Kähler structures. In the mathematical literature this is sometimes called an almost hyper-para-Kähler structure or a para-quaternionic manifold or a 3-web; see, S. Ivanov, S. Zamkovoy, Differ.Geom.Appl. 23
2005
Later among the works it cites.
See also, G. W. Gibbons, J. Geom. Phys. 8
2007
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L. F. Alday and J. M. Maldacena, JHEP 0706 (2007) 064; N. Berkovits and J. Maldacena, JHEP 0809 (2008) 062
2008
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See also, D. S. Berman, N. B. Copland and D. C. Thompson, Nucl. Phys. B 791
2010
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For discussions of curved momentum and phase space consult: Yu. A. Gol’fand, Sov. Phys. JETP, 10
2011
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Cited in the paper.
In our new formulation of string theory we do not have to restrict M M to be either compact or spacelike; in particular, we can allow compactifications on light-like directions
Cited in the paper.
L. Freidel, R.G. Leigh and D. Minic, Phase Space String Theory
Cited in the paper.
Our conventions are such that in the conformal frame the 2d metric is − d τ 2 + d σ 2 -d\tau^{2}+\mathrm{d}\sigma^{2} , ∗ d σ = d σ *\mathrm{d}\sigma=\mathrm{d}\sigma , ∗ d τ = d τ *\mathrm{d}\tau=\mathrm{d}\tau and d σ ∧ d τ = d 2 σ \mathrm{d}\sigma\wedge\mathrm{d}\tau=\mathrm{d}^{2}\sigma
Cited in the paper.
Here neutral means that η \eta is of signature ( d , d ) (d,d) , while H H is of signature ( 2 , 2 ( d − 2 ) ) (2,2(d-2))
Cited in the paper.