Abstract
Yang-Baxter deformations of superstring σ-models have recently inspired a supergravity solution generating technique. Using the open/closed string map and a Killing bi-vector as a deformation parameter, new solutions can be built, such that the (generalised) supergravity field equations were conjectured to always reduce to the classical Yang-Baxter equation (CYBE) [1]. In this work we provide a proof of this conjecture, using a systematic approach based on the so-called β-supergravity, which is a dynamical theory for the field βmn instead of the NSNS 2-form bmn.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Bakhmatov, Ö. Kelekci, E. Ó Colgáin and M.M. Sheikh-Jabbari, Classical Yang-Baxter Equation from Supergravity, Phys. Rev. D 98 (2018) 021901 [arXiv:1710.06784] [INSPIRE].
I. Bakhmatov, E. Ó Colgáin, M.M. Sheikh-Jabbari and H. Yavartanoo, Yang-Baxter Deformations Beyond Coset Spaces (a slick way to do TsT), JHEP 06 (2018) 161 [arXiv:1803.07498] [INSPIRE].
M. Jimbo, Introduction to the Yang-Baxter Equation, Int. J. Mod. Phys. A 4 (1989) 3759 [INSPIRE].
T. Matsumoto and K. Yoshida, Integrable deformations of the AdS 5 × S 5 superstring and the classical Yang-Baxter equation — Towards the gravity/CYBE correspondence, J. Phys. Conf. Ser. 563 (2014) 012020 [arXiv:1410.0575] [INSPIRE].
T. Matsumoto and K. Yoshida, Lunin-Maldacena backgrounds from the classical Yang-Baxter equation — towards the gravity/CYBE correspondence, JHEP 06 (2014) 135 [arXiv:1404.1838] [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
T. Araujo, I. Bakhmatov, E. Ó Colgáin, J.-i. Sakamoto, M.M. Sheikh-Jabbari and K. Yoshida, Yang-Baxter σ-models, conformal twists and noncommutative Yang-Mills theory, Phys. Rev. D 95 (2017) 105006 [arXiv:1702.02861] [INSPIRE].
T. Araujo, I. Bakhmatov, E. Ó Colgáin, J.-i. Sakamoto, M.M. Sheikh-Jabbari and K. Yoshida, Conformal twists, Yang-Baxter σ-models & holographic noncommutativity, J. Phys. A 51 (2018) 235401 [arXiv:1705.02063] [INSPIRE].
C. Klimčík, Yang-Baxter σ-models and dS/AdS T duality, JHEP 12 (2002) 051 [hep-th/0210095] [INSPIRE].
C. Klimčík, On integrability of the Yang-Baxter σ-model, J. Math. Phys. 50 (2009) 043508 [arXiv:0802.3518] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS 5 × S 5 superstring action, Phys. Rev. Lett. 112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, Jordanian deformations of the AdS 5 × S 5 superstring, JHEP 04 (2014) 153 [arXiv:1401.4855] [INSPIRE].
D. Osten and S.J. van Tongeren, Abelian Yang-Baxter deformations and TsT transformations, Nucl. Phys. B 915 (2017) 184 [arXiv:1608.08504] [INSPIRE].
B. Hoare and A.A. Tseytlin, Homogeneous Yang-Baxter deformations as non-abelian duals of the AdS 5 σ-model, J. Phys. A 49 (2016) 494001 [arXiv:1609.02550] [INSPIRE].
R. Borsato and L. Wulff, Integrable Deformations of T-Dual σ Models, Phys. Rev. Lett. 117 (2016) 251602 [arXiv:1609.09834] [INSPIRE].
R. Borsato and L. Wulff, On non-abelian T-duality and deformations of supercoset string σ-models, JHEP 10 (2017) 024 [arXiv:1706.10169] [INSPIRE].
D. Lüst and D. Osten, Generalised fluxes, Yang-Baxter deformations and the O(d, d) structure of non-abelian T-duality, JHEP 05 (2018) 165 [arXiv:1803.03971] [INSPIRE].
J.-i. Sakamoto, Y. Sakatani and K. Yoshida, Homogeneous Yang-Baxter deformations as generalized diffeomorphisms, J. Phys. A 50 (2017) 415401 [arXiv:1705.07116] [INSPIRE].
J.J. Fernandez-Melgarejo, J.-i. Sakamoto, Y. Sakatani and K. Yoshida, T-folds from Yang-Baxter deformations, JHEP 12 (2017) 108 [arXiv:1710.06849] [INSPIRE].
J.-i. Sakamoto and Y. Sakatani, Local β-deformations and Yang-Baxter σ-model, JHEP 06 (2018) 147 [arXiv:1803.05903] [INSPIRE].
T. Araujo, E. Ó Colgáin, Y. Sakatani, M.M. Sheikh-Jabbari and H. Yavartanoo, Holographic integration of \( T\overline{T}\kern0.5em \&\kern0.5em J\overline{T} \) via O(d, d), arXiv:1811.03050 [INSPIRE].
B. Hoare and A.A. Tseytlin, Type IIB supergravity solution for the T-dual of the η-deformed AdS 5 × S 5 superstring, JHEP 10 (2015) 060 [arXiv:1508.01150] [INSPIRE].
D. Orlando, S. Reffert, J.-i. Sakamoto and K. Yoshida, Generalized type IIB supergravity equations and non-Abelian classical r-matrices, J. Phys. A 49 (2016) 445403 [arXiv:1607.00795] [INSPIRE].
G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A.A. Tseytlin, Scale invariance of the η-deformed AdS 5 × S 5 superstring, T-duality and modified type-II equations, Nucl. Phys. B 903 (2016) 262 [arXiv:1511.05795] [INSPIRE].
L. Wulff and A.A. Tseytlin, κ-symmetry of superstring σ-model and generalized 10d supergravity equations, JHEP 06 (2016) 174 [arXiv:1605.04884] [INSPIRE].
R. Borsato and L. Wulff, Non-abelian T-duality and Yang-Baxter deformations of Green-Schwarz strings, JHEP 08 (2018) 027 [arXiv:1806.04083] [INSPIRE].
T. Araujo, E. Ó Colgáin and H. Yavartanoo, Embedding the modified CYBE in Supergravity, Eur. Phys. J. C 78 (2018) 854 [arXiv:1806.02602] [INSPIRE].
D. Andriot and A. Betz, β-supergravity: a ten-dimensional theory with non-geometric fluxes and its geometric framework, JHEP 12 (2013) 083 [arXiv:1306.4381] [INSPIRE].
D. Andriot, M. Larfors, D. Lüst and P. Patalong, A ten-dimensional action for non-geometric fluxes, JHEP 09 (2011) 134 [arXiv:1106.4015] [INSPIRE].
D. Andriot, O. Hohm, M. Larfors, D. Lüst and P. Patalong, A geometric action for non-geometric fluxes, Phys. Rev. Lett. 108 (2012) 261602 [arXiv:1202.3060] [INSPIRE].
D. Geissbuhler, D. Marques, C. Núñez and V. Penas, Exploring Double Field Theory, JHEP 06 (2013) 101 [arXiv:1304.1472] [INSPIRE].
D. Andriot and A. Betz, NS-branes, source corrected Bianchi identities and more on backgrounds with non-geometric fluxes, JHEP 07 (2014) 059 [arXiv:1402.5972] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].
D.S. Berman and D.C. Thompson, Duality Symmetric String and M-theory, Phys. Rept. 566 (2014) 1 [arXiv:1306.2643] [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].
J.-i. Sakamoto, Y. Sakatani and K. Yoshida, Weyl invariance for generalized supergravity backgrounds from the doubled formalism, Prog. Theor. Exp. Phys. 2017 (2017) 053B07 [arXiv:1703.09213] [INSPIRE].
Y. Sakatani, S. Uehara and K. Yoshida, Generalized gravity from modified DFT, JHEP 04 (2017) 123 [arXiv:1611.05856] [INSPIRE].
A. Baguet, M. Magro and H. Samtleben, Generalized IIB supergravity from exceptional field theory, JHEP 03 (2017) 100 [arXiv:1612.07210] [INSPIRE].
D.S. Berman, H. Godazgar, M.J. Perry and P. West, Duality Invariant Actions and Generalised Geometry, JHEP 02 (2012) 108 [arXiv:1111.0459] [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].
A. Ashmore, Marginal deformations of 3d \( \mathcal{N} \) = 2 CFTs from AdS 4 backgrounds in generalised geometry, JHEP 12 (2018) 060 [arXiv:1809.03503] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1811.09056
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bakhmatov, I., Musaev, E.T. Classical Yang-Baxter equation from β-supergravity. J. High Energ. Phys. 2019, 140 (2019). https://doi.org/10.1007/JHEP01(2019)140
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2019)140