Abstract
We consider the symmetries of a closed bosonic string, starting with the general coordinate transformations. Their generator takes vector components ξμ as its parameter and its Poisson bracket algebra gives rise to the Lie bracket of its parameters. We are going to extend this generator in order for it to be invariant upon self T-duality, i.e. T-duality realized in the same phase space. The new generator is a function of a 2D double symmetry parameter Λ, that is a direct sum of vector components ξμ, and 1-form components λμ. The Poisson bracket algebra of a new generator produces the Courant bracket in a same way that the algebra of the general coordinate transformations produces Lie bracket. In that sense, the Courant bracket is T-dual invariant extension of the Lie bracket. When the Kalb-Ramond field is introduced to the model, the generator governing both general coordinate and local gauge symmetries is constructed. It is no longer self T-dual and its algebra gives rise to the B-twisted Courant bracket, while in its self T-dual description, the relevant bracket becomes the θ-twisted Courant bracket. Next, we consider the T-duality and the symmetry parameters that depend on both the initial coordinates xμ and T-dual coordinates yμ. The generator of these transformations is defined as an inner product in a double space and its algebra gives rise to the C-bracket.
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References
G.P.L. Courant, Dirac manifolds, Trans. Am. Math. Soc. 319 (1990) 631.
Z.-J. Liu, A. WEinstein and P. Xu, Manin Triples for Lie Bialgebroids, J. Diff. Geom. 45 (1997) 547 [dg-ga/9508013] [INSPIRE].
A. Alekseev and T. Strobl, Current algebras and differential geometry, JHEP 03 (2005) 035 [hep-th/0410183] [INSPIRE].
N. Halmagyi, Non-geometric String Backgrounds and Worldsheet Algebras, JHEP 07 (2008) 137 [arXiv:0805.4571] [INSPIRE].
N. Halmagyi, Non-geometric Backgrounds and the First Order String Sigma Model, arXiv:0906.2891 [INSPIRE].
I. Ivanišević, Lj. Davidović and B. Sazdović, Courant bracket found out to be T-dual to Roytenberg bracket, Eur. Phys. J. C 80 (2020) 571 [arXiv:1903.04832] [INSPIRE].
C. Hull and B. Zwiebach, The Gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].
W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
Lj. Davidović and B. Sazdović, The T-dual symmetries of a bosonic string, Eur. Phys. J. C 78 (2018) 600 [arXiv:1806.03138] [INSPIRE].
D. Roytenberg, A Note on quasi Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys. 61 (2002) 123 [math/0112152] [INSPIRE].
K. Becker, M. Becker and J. Schwarz, String Theory and M-Theory: A Modern Introduction, Cambridge University Press, Cambridge, U.K. (2007).
B. Zwiebach, A First Course in String Theory, Cambridge University Press, Cambridge, U.K. (2004).
Lj. Davidović and B. Sazdović, T-duality in a weakly curved background, Eur. Phys. J. C 74 (2014) 2683 [arXiv:1205.1991] [INSPIRE].
T.H. Buscher, A Symmetry of the String Background Field Equations, Phys. Lett. B 194 (1987) 59 [INSPIRE].
E. Alvarez, L. Álvarez-Gaumé and Y. Lozano, An Introduction to T duality in string theory, Nucl. Phys. B Proc. Suppl. 41 (1995) 1 [hep-th/9410237] [INSPIRE].
A. Giveon, M. Porrati and E. Rabinovici, Target space duality in string theory, Phys. Rept. 244 (1994) 77 [hep-th/9401139] [INSPIRE].
A. Giveon, E. Rabinovici and G. Veneziano, Duality in String Background Space, Nucl. Phys. B 322 (1989) 167 [INSPIRE].
C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].
M. Gualtieri, Generalized complex geometry, Ph.D. thesis, Oxford University, U.K. (2003) math/0401221 [INSPIRE].
P. Ševera and A. WEinstein, Poisson geometry with a 3 form background, Prog. Theor. Phys. Suppl. 144 (2001) 145 [math/0107133] [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
J. Shelton, W. Taylor and B. Wecht, Nongeometric flux compactifications, JHEP 10 (2005) 085 [hep-th/0508133] [INSPIRE].
J.A. de Azcarraga, A.M. Perelomov and J.C. Perez Bueno, The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures, J. Phys. A 29 (1996) 7993 [hep-th/9605067] [INSPIRE].
Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Annales Inst. Fourier 46 (1996) 1243.
Lj. Davidović, B. Nikolic and B. Sazdović, Canonical approach to the closed string non-commutativity, Eur. Phys. J. C 74 (2014) 2734 [arXiv:1307.6158] [INSPIRE].
Lj. Davidović, I. Ivanišević, B. Sazdović, Courant bracket twisted both by a 2-form B and by a bi-vector θ, in preparation.
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Davidović, L., Ivanišević, I. & Sazdović, B. Courant bracket as T-dual invariant extension of Lie bracket. J. High Energ. Phys. 2021, 109 (2021). https://doi.org/10.1007/JHEP03(2021)109
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DOI: https://doi.org/10.1007/JHEP03(2021)109