Abstract
Equations of motion of low-energy string effective actions can be conveniently described in terms of generalized geometry and Levi-Civita connections on Courant algebroids. This approach is used to propose and prove a suitable version of the Kaluza-Klein-like reduction. Necessary geometrical tools are recalled.
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G. Aldazabal, D. Marques and C. Núñez, Double field theory: a pedagogical review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [INSPIRE].
A. Alekseev and P. Xu, Derived brackets and Courant algebroids, http://www.math.psu.edu/ping/anton-final.pdf.
D. Bailin and A. Love, Kaluza-Klein theories, Rept. Prog. Phys. 50 (1987) 1087 [INSPIRE].
D. Baraglia and P. Hekmati, Transitive Courant algebroids, string structures and T-duality, Adv. Theor. Math. Phys. 19 (2015) 613 [arXiv:1308.5159] [INSPIRE].
O.A. Bedoya, D. Marques and C. Núñez, Heterotic α ′ -corrections in double field theory, JHEP 12 (2014) 074 [arXiv:1407.0365] [INSPIRE].
E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Ten-dimensional Maxwell-Einstein supergravity, its currents and the issue of its auxiliary fields, Nucl. Phys. B 195 (1982) 97 [INSPIRE].
E.A. Bergshoeff and M. de Roo, The quartic effective action of the heterotic string and supersymmetry, Nucl. Phys. B 328 (1989) 439 [INSPIRE].
P. Bouwknegt, Courant algebroids and generalizations of geometry, lecture at StringMath, UPenn, Philadelphia U.S.A., (2011).
P. Bressler, The first Pontryagin class, math/0509563 [INSPIRE].
H. Bursztyn, G.R. Cavalcanti and M. Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007) 726 [math/0509640] [INSPIRE].
G.F. Chapline and N.S. Manton, Unification of Yang-Mills theory and supergravity in ten-dimensions, Phys. Lett. B 120 (1983) 105 [INSPIRE].
A. Coimbra, R. Minasian, H. Triendl and D. Waldram, Generalised geometry for string corrections, JHEP 11 (2014) 160 [arXiv:1407.7542] [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, Supergravity as generalised geometry II: E d(d) × ℝ + and M-theory, JHEP 03 (2014) 019 [arXiv:1212.1586] [INSPIRE].
M.J. Duff, Kaluza-Klein theory in perspective, in The Oskar Klein centenary. Proceedings, Symposium, Stockholm Sweden, 19-21 September 1994, pg. 22 [hep-th/9410046] [INSPIRE].
M. Garcia-Fernandez, Torsion-free generalized connections and heterotic supergravity, Commun. Math. Phys. 332 (2014) 89 [arXiv:1304.4294] [INSPIRE].
M. Garcia-Fernandez, Lectures on the Strominger system, arXiv:1609.02615 [INSPIRE].
M. Garcia-Fernandez, Ricci flow, Killing spinors and T-duality in generalized geometry, arXiv:1611.08926 [INSPIRE].
M. Gualtieri, Generalized complex geometry, math/0401221 [INSPIRE].
M. Gualtieri, Branes on Poisson varieties, arXiv:0710.2719 [INSPIRE].
N. Hitchin, Brackets, forms and invariant functionals, math/0508618 [INSPIRE].
N. Hitchin, Instantons, Poisson structures and generalized Kähler geometry, Commun. Math. Phys. 265 (2006) 131 [math/0503432] [INSPIRE].
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].
O. Hohm and S.K. Kwak, Double field theory formulation of heterotic strings, JHEP 06 (2011) 096 [arXiv:1103.2136] [INSPIRE].
O. Hohm and S.K. Kwak, Frame-like geometry of double field theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [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].
O. Hohm and B. Zwiebach, Towards an invariant geometry of double field theory, J. Math. Phys. 54 (2013) 032303 [arXiv:1212.1736] [INSPIRE].
O. Hohm and B. Zwiebach, Double field theory at order α ′, JHEP 11 (2014) 075 [arXiv:1407.3803] [INSPIRE].
O. Hohm and B. Zwiebach, Double metric, generalized metric and α ′ -deformed double field theory, Phys. Rev. D 93 (2016) 064035 [arXiv:1509.02930] [INSPIRE].
O. Hohm and B. Zwiebach, Green-Schwarz mechanism and α ′ -deformed Courant brackets, JHEP 01 (2015) 012 [arXiv:1407.0708] [INSPIRE].
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].
B. Jurčo and J. Vysoký, Leibniz algebroids, generalized Bismut connections and Einstein-Hilbert actions, J. Geom. Phys. 97 (2015) 25 [arXiv:1503.03069] [INSPIRE].
B. Jurčo and J. Vysoký, Courant algebroid connections and string effective actions, in Tohoku Forum for Creativity, (2016) [arXiv:1612.01540] [INSPIRE].
B. Jurčo and J. Vysoký, Heterotic reduction of Courant algebroid connections and Einstein-Hilbert actions, Nucl. Phys. B 909 (2016) 86 [arXiv:1512.08522] [INSPIRE].
T. Kaluza, Zum Unitätsproblem der Physik (in German), Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1921 (1921) 966.
O. Klein, Quantum theory and five-dimensional theory of relativity (in German and English), Z. Phys. 37 (1926) 895 [Surveys High Energ. Phys. 5 (1986) 241] [INSPIRE].
Z. Liu, Y. Sheng and X. Xu, Pre-Courant algebroids and associated Lie 2-algebras, arXiv:1205.5898 [INSPIRE].
D. Marques and C.A. Núñez, T-duality and α ′ -corrections, JHEP 10 (2015) 084 [arXiv:1507.00652] [INSPIRE].
D. Martelli and J. Sparks, Non-Kähler heterotic rotations, Adv. Theor. Math. Phys. 15 (2011) 131 [arXiv:1010.4031] [INSPIRE].
P. Ševera, Poisson-Lie T-duality and Courant algebroids, Lett. Math. Phys. 105 (2015) 1689 [arXiv:1502.04517] [INSPIRE].
W. Siegel, Manifest duality in low-energy superstrings, in Proceedings, Strings ′93, Berkeley U.S.A., (1993), pg. 353 [hep-th/9308133] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
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ArXiv ePrint: 1704.01123
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Vysoký, J. Kaluza-Klein reduction of low-energy effective actions: geometrical approach. J. High Energ. Phys. 2017, 143 (2017). https://doi.org/10.1007/JHEP08(2017)143
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DOI: https://doi.org/10.1007/JHEP08(2017)143