Abstract
We provide the canonical formulation of double field theory. It is shown that this dynamics is subject to primary and secondary constraints. The Poisson bracket algebra of secondary constraints is shown to close on-shell according to the C-bracket. A systematic way of writing boundary integrals in doubled geometry is given. By including appropriate boundary terms in the double field theory Hamiltonian, expressions for conserved energy and momentum of an asymptotically flat doubled space-time are obtained and applied to a number of solutions.
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References
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].
C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].
C. Hull and B. Zwiebach, The Gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].
A.A. Tseytlin, Duality symmetric formulation of string world sheet dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].
A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [INSPIRE].
M.J. Duff, Duality rotations in string theory, Nucl. Phys. B 335 (1990) 610 [INSPIRE].
M.J. Duff and J.X. Lu, Duality rotations in membrane theory, Nucl. Phys. B 347 (1990) 394 [INSPIRE].
O. Hohm and S.K. Kwak, Frame-like geometry of double field theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [INSPIRE].
S.K. Kwak, Invariances and equations of motion in double field theory, JHEP 10 (2010) 047 [arXiv:1008.2746] [INSPIRE].
O. Hohm, T-duality versus gauge symmetry, Prog. Theor. Phys. Suppl. 188 (2011) 116 [arXiv:1101.3484] [INSPIRE].
O. Hohm and S.K. Kwak, Double field theory formulation of heterotic strings, JHEP 06 (2011) 096 [arXiv:1103.2136] [INSPIRE].
O. Hohm, S.K. Kwak and B. Zwiebach, Unification of type II strings and T-duality, Phys. Rev. Lett. 107 (2011) 171603 [arXiv:1106.5452] [INSPIRE].
O. Hohm, S.K. Kwak and B. Zwiebach, Double field theory of type II strings, JHEP 09 (2011) 013 [arXiv:1107.0008] [INSPIRE].
O. Hohm and S.K. Kwak, Massive type II in double field theory, JHEP 11 (2011) 086 [arXiv:1108.4937] [INSPIRE].
O. Hohm and S.K. Kwak, N = 1 supersymmetric double field theory, JHEP 03 (2012) 080 [arXiv:1111.7293] [INSPIRE].
O. Hohm and B. Zwiebach, On the Riemann tensor in double field theory, JHEP 05 (2012) 126 [arXiv:1112.5296] [INSPIRE].
O. Hohm and H. Samtleben, Gauge theory of Kaluza-Klein and winding modes, Phys. Rev. D 88 (2013) 085005 [arXiv:1307.0039] [INSPIRE].
O. Hohm and B. Zwiebach, Large gauge transformations in double field theory, JHEP 02 (2013) 075 [arXiv:1207.4198] [INSPIRE].
C.M. Hull, Finite gauge transformations and geometry in double field theory, JHEP 04 (2015) 109 [arXiv:1406.7794] [INSPIRE].
U. Naseer, A note on large gauge transformations in double field theory, JHEP 06 (2015) 002 [arXiv:1504.05913] [INSPIRE].
D.S. Berman and M.J. Perry, Generalized geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].
D.S. Berman, E.T. Musaev and M.J. Perry, Boundary terms in generalized geometry and doubled field theory, Phys. Lett. B 706 (2011) 228 [arXiv:1110.3097] [INSPIRE].
D.S. Berman, H. Godazgar and M.J. Perry, SO(5, 5) duality in M-theory and generalized geometry, Phys. Lett. B 700 (2011) 65 [arXiv:1103.5733] [INSPIRE].
D.S. Berman, H. Godazgar, M. Godazgar and M.J. Perry, The local symmetries of M-theory and their formulation in generalised geometry, JHEP 01 (2012) 012 [arXiv:1110.3930] [INSPIRE].
D.S. Berman, M. Cederwall and M.J. Perry, Global aspects of double geometry, JHEP 09 (2014) 066 [arXiv:1401.1311] [INSPIRE].
P. West, E 11 , generalised space-time and IIA string theory, Phys. Lett. B 696 (2011) 403 [arXiv:1009.2624] [INSPIRE].
A. Rocen and P. West, E 11 , generalised space-time and IIA string theory: the R-R sector, arXiv:1012.2744.
I. Jeon, K. Lee and J.-H. Park, Differential geometry with a projection: Application to double field theory, JHEP 04 (2011) 014 [arXiv:1011.1324] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Stringy differential geometry, beyond Riemann, Phys. Rev. D 84 (2011) 044022 [arXiv:1105.6294] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Incorporation of fermions into double field theory, JHEP 11 (2011) 025 [arXiv:1109.2035] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Supersymmetric double field theory: stringy reformulation of supergravity, Phys. Rev. D 85 (2012) 081501 [Erratum ibid. D 86 (2012) 089903] [arXiv:1112.0069] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Ramond-Ramond cohomology and O(D, D) T-duality, JHEP 09 (2012) 079 [arXiv:1206.3478] [INSPIRE].
J.-H. Park, Comments on double field theory and diffeomorphisms, JHEP 06 (2013) 098 [arXiv:1304.5946] [INSPIRE].
M.B. Schulz, T-folds, doubled geometry and the SU(2) WZW model, JHEP 06 (2012) 158 [arXiv:1106.6291] [INSPIRE].
N.B. Copland, Connecting T-duality invariant theories, Nucl. Phys. B 854 (2012) 575 [arXiv:1106.1888] [INSPIRE].
N.B. Copland, A double σ-model for double field theory, JHEP 04 (2012) 044 [arXiv:1111.1828] [INSPIRE].
D.C. Thompson, Duality invariance: from M-theory to double field theory, JHEP 08 (2011) 125 [arXiv:1106.4036] [INSPIRE].
C. Albertsson, S.-H. Dai, P.-W. Kao and F.-L. Lin, Double field theory for double D-branes, JHEP 09 (2011) 025 [arXiv:1107.0876] [INSPIRE].
G. Aldazabal, W. Baron, D. Marques and C. Núñez, The effective action of double field theory, JHEP 11 (2011) 052 [Erratum ibid. 11 (2011) 109] [arXiv:1109.0290] [INSPIRE].
D. Geissbuhler, Double field theory and N = 4 gauged supergravity, JHEP 11 (2011) 116 [arXiv:1109.4280] [INSPIRE].
M. Graña and D. Marques, Gauged double field theory, JHEP 04 (2012) 020 [arXiv:1201.2924] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry I: type II theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, \( {E_d}_{(d)}\times {\mathbb{R}}^{+} \) generalised geometry, connections and M-theory, JHEP 02 (2014) 054 [arXiv:1112.3989] [INSPIRE].
I. Vaisman, On the geometry of double field theory, J. Math. Phys. 53 (2012) 033509 [arXiv:1203.0836] [INSPIRE].
B. Zwiebach, Double field theory, T-duality and Courant brackets, Lect. Notes Phys. 851 (2012) 265 [arXiv:1109.1782] [INSPIRE].
G. Aldazabal, D. Marques and C. Núñez, Double field theory: a pedagogical review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [INSPIRE].
O. Hohm, D. Lüst and B. Zwiebach, The spacetime of double field theory: review, remarks and outlook, Fortsch. Phys. 61 (2013) 926 [arXiv:1309.2977] [INSPIRE].
R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997 [gr-qc/0405109] [INSPIRE].
M. Blau, Lecture notes on general relativity, Albert Einstein Center for Fundamental Physics, Bern Germany (2011).
P.A.M. Dirac, Lectures on quantum mechanics, Belfer Graduate School of Science Monographs Series volume 2, Belfer Graduate School of Science, New York U.S.A. (1964).
T. Regge and C. Teitelboim, Role of surface integrals in the hamiltonian formulation of general relativity, Annals Phys. 88 (1974) 286 [INSPIRE].
J. Berkeley, D.S. Berman and F.J. Rudolph, Strings and branes are waves, JHEP 06 (2014) 006 [arXiv:1403.7198] [INSPIRE].
D.S. Berman and F.J. Rudolph, Branes are waves and monopoles, JHEP 05 (2015) 015 [arXiv:1409.6314] [INSPIRE].
J.-H. Park, S.-J. Rey, W. Rim and Y. Sakatani, O(D, D) covariant noether currents and global charges in double field theory, arXiv:1507.07545 [INSPIRE].
C.D.A. Blair, Conserved currents of double field theory, arXiv:1507.07541 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752.
J.W. York, Jr., Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972) 1082 [INSPIRE].
A.H. Taub, Empty space-times admitting a three parameter group of motions, Ann. Math. 53 (1951) 472
E. Newman, L. Tamubrino and T. Unti, Empty space generalization of the Schwarzschild metric, J. Math. Phys. 4 (1963) 915 [INSPIRE].
C.W. Misner, The flatter regions of Newman, Unti, and Tamburino’s generalized Schwarzschild space, J. Math. Phys. 4 (1963) 924.
D.J. Gross and M.J. Perry, Magnetic monopoles in Kaluza-Klein theories, Nucl. Phys. B 226 (1983) 29.
K. Sfetsos, Rotating NS five-brane solution and its exact string theoretical description, Fortsch. Phys. 48 (2000) 199 [hep-th/9903201] [INSPIRE].
M. Gualtieri, Generalized complex geometry, Ph.D. thesis, Oxford University, Oxford U.K. (2003), math/0401221.
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Naseer, U. Canonical formulation and conserved charges of double field theory. J. High Energ. Phys. 2015, 158 (2015). https://doi.org/10.1007/JHEP10(2015)158
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DOI: https://doi.org/10.1007/JHEP10(2015)158