Abstract
We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to D ≥ 5 dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted Y , that provides exactly the necessary properties to also describe compactifications to D = 4 dimensions. We classify “M-exact” E7-algebroids and show that this precisely matches the form of the generalised tangent space of E7(7) × ℝ+-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].
M. Gualtieri, Generalized Complex Geometry, Ph.D. thesis, Oxford University, Oxford OX1 3RH, U.K. (2004) [math/0401221].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as Generalised Geometry I: Type II Theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [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].
C.M. Hull, Generalised Geometry for M-Theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].
P. Pires Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Ed(d) × ℝ+ generalised geometry, connections and M theory, JHEP 02 (2014) 054 [arXiv:1112.3989] [INSPIRE].
Z.-J. Liu, A. Weinstein and P. Xu, Manin Triples for Lie Bialgebroids, J. Diff. Geom. 45 (1997) 547 [dg-ga/9508013] [INSPIRE].
C. Klimčík and P. Ševera, Dual non-Abelian duality and the Drinfeld double, Phys. Lett. B 351 (1995) 455 [hep-th/9502122] [INSPIRE].
P. Ševera, Letters to Alan Weinstein about Courant algebroids, arXiv:1707.00265 [INSPIRE].
P. Ševera, Poisson-Lie T-duality as a boundary phenomenon of Chern-Simons theory, JHEP 05 (2016) 044 [arXiv:1602.05126] [INSPIRE].
M. Bugden, O. Hulík, F. Valach and D. Waldram, G-Algebroids: A Unified Framework for Exceptional and Generalised Geometry, and Poisson-Lie Duality, Fortsch. Phys. 69 (2021) 2100028 [arXiv:2103.01139] [INSPIRE].
M. Bugden, O. Hulík, F. Valach and D. Waldram, Exceptional Algebroids and Type IIB Superstrings, Fortsch. Phys. 70 (2022) 2100104 [arXiv:2107.00091] [INSPIRE].
O. Hulík and F. Valach, Exceptional Algebroids and Type IIA Superstrings, Fortsch. Phys. 70 (2022) 2200027 [arXiv:2202.00355] [INSPIRE].
G. Inverso, Generalised Scherk-Schwarz reductions from gauged supergravity, JHEP 12 (2017) 124.
Y. Sakatani, U-duality extension of Drinfel’d double, PTEP 2020 (2020) 023B08 [arXiv:1911.06320] [INSPIRE].
E. Malek and D.C. Thompson, Poisson-Lie U-duality in Exceptional Field Theory, JHEP 04 (2020) 058 [arXiv:1911.07833] [INSPIRE].
E. Malek, Y. Sakatani and D.C. Thompson, E6(6) exceptional Drinfel’d algebras, JHEP 01 (2021) 020 [arXiv:2007.08510] [INSPIRE].
J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris, Sér. A–B 264 (1967) A245.
O. Hohm and H. Samtleben, Exceptional field theory. III. E8(8), Phys. Rev. D 90 (2014) 066002 [arXiv:1406.3348] [INSPIRE].
M. Cederwall and J.A. Rosabal, E8 geometry, JHEP 07 (2015) 007 [arXiv:1504.04843] [INSPIRE].
D.S. Berman, M. Cederwall, A. Kleinschmidt and D.C. Thompson, The gauge structure of generalised diffeomorphisms, JHEP 01 (2013) 064 [arXiv:1208.5884] [INSPIRE].
T. Dereli and K. Doğan, ‘Anti-commutable’ local pre-Leibniz algebroids and admissible connections, J. Geom. Phys. 186 (2023) 104752 [arXiv:2108.10199] [INSPIRE].
J. Grabowski, D. Khudaverdyan and N. Poncin, The Supergeometry of Loday Algebroids, arXiv:1103.5852.
B. Jurčo and J. Vysoký, Leibniz algebroids, generalized Bismut connections and Einstein-Hilbert actions, J. Geom. Phys. 97 (2015) 25 [arXiv:1503.03069] [INSPIRE].
Z. Chen, Z. Liu and Y. Sheng, E-Courant algebroids, Int. Math. Res. Not. 2010 (2010) 4334 [arXiv:0805.4093] [INSPIRE].
D. Baraglia, Conformal Courant Algebroids and Orientifold T-Duality, Int. J. Geom. Meth. Mod. Phys. 10 (2013) 1250084 [INSPIRE].
M. Grützmann and T. Strobl, General Yang-Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical framework or From Bianchi identities to twisted Courant algebroids, Int. J. Geom. Meth. Mod. Phys. 12 (2014) 1550009 [arXiv:1407.6759] [INSPIRE].
G. Bossard et al., Generalized diffeomorphisms for E9, Phys. Rev. D 96 (2017) 106022 [arXiv:1708.08936] [INSPIRE].
M. Cederwall and J. Palmkvist, Extended geometries, JHEP 02 (2018) 071 [arXiv:1711.07694] [INSPIRE].
J.-H. Park and Y. Suh, U-gravity: SL(N), JHEP 06 (2014) 102 [arXiv:1402.5027] [INSPIRE].
K. Lee, C. Strickland-Constable and D. Waldram, Spheres, generalised parallelisability and consistent truncations, Fortsch. Phys. 65 (2017) 1700048 [arXiv:1401.3360] [INSPIRE].
O. Hohm and H. Samtleben, Consistent Kaluza-Klein Truncations via Exceptional Field Theory, JHEP 01 (2015) 131 [arXiv:1410.8145] [INSPIRE].
C. Eloy, M. Galli and E. Malek, Adding fluxes to consistent truncations: IIB supergravity on AdS3 × S3× S3× S1, JHEP 11 (2023) 049 [arXiv:2306.12487] [INSPIRE].
C. Strickland-Constable, Subsectors, Dynkin Diagrams and New Generalised Geometries, JHEP 08 (2017) 144 [arXiv:1310.4196] [INSPIRE].
G. Bossard et al., Extended geometry of magical supergravities, JHEP 05 (2023) 162 [arXiv:2301.10974] [INSPIRE].
M. Gunaydin, G. Sierra and P.K. Townsend, Exceptional Supergravity Theories and the MAGIC Square, Phys. Lett. B 133 (1983) 72 [INSPIRE].
M. Gunaydin, G. Sierra and P.K. Townsend, The Geometry of N = 2 Maxwell-Einstein Supergravity and Jordan Algebras, Nucl. Phys. B 242 (1984) 244 [INSPIRE].
M. Gunaydin, H. Samtleben and E. Sezgin, On the Magical Supergravities in Six Dimensions, Nucl. Phys. B 848 (2011) 62 [arXiv:1012.1818] [INSPIRE].
F. Hassler and Y. Sakatani, All maximal gauged supergravities with uplift, arXiv:2212.14886 [https://doi.org/10.1093/ptep/ptad104] [INSPIRE].
B. de Wit and H. Nicolai, The Consistency of the S7 Truncation in D = 11 Supergravity, Nucl. Phys. B 281 (1987) 211 [INSPIRE].
A. Ashmore, C. Strickland-Constable, D. Tennyson and D. Waldram, Generalising G2 geometry: involutivity, moment maps and moduli, JHEP 01 (2021) 158 [arXiv:1910.04795] [INSPIRE].
G.R. Smith and D. Waldram, M-theory moduli from exceptional complex structures, JHEP 08 (2023) 022 [arXiv:2211.09517] [INSPIRE].
P. Ševera, Poisson-Lie T-Duality and Courant Algebroids, Lett. Math. Phys. 105 (2015) 1689 [arXiv:1502.04517] [INSPIRE].
C.D.A. Blair, D.C. Thompson and S. Zhidkova, Exploring Exceptional Drinfeld Geometries, JHEP 09 (2020) 151 [arXiv:2006.12452] [INSPIRE].
C.D.A. Blair and S. Zhidkova, Generalised U-dual solutions in supergravity, JHEP 05 (2022) 081 [arXiv:2203.01838] [INSPIRE].
C.D.A. Blair and S. Zhidkova, Generalised U-dual solutions via ISO(7) gauged supergravity, JHEP 12 (2022) 093 [arXiv:2210.07867] [INSPIRE].
Acknowledgments
The authors would like to thank Anthony Ashmore for a helpful discussion and the organisers of the “Supergravity, Generalized Geometry and Ricci Flow” programme at the Simons Center for Geometry and Physics for hospitality while this work was being completed. The authors also thank the anonymous referee for their valuable comments and suggestions which helped to improve the work. E. M. also thanks Imperial College London for hospitality. O. H. was supported by the FWO-Vlaanderen through the project G006119N and by the Vrije Universiteit Brussel through the Strategic Research Program “High-Energy Physics”. E. M. was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the Emmy Noether program “Exploring the landscape of string theory flux vacua using exceptional field theory” (project number 426510644). F. V. was supported by the Postdoc Mobility grant P500PT203123 of the Swiss National Science Foundation. D. W. was supported in part by the STFC Consolidated Grant ST/T000791/1 and the EPSRC New Horizons Grant EP/V049089/1.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2308.01130
Rights and permissions
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.
About this article
Cite this article
Hulík, O., Malek, E., Valach, F. et al. Y-algebroids and E7(7) × ℝ+-generalised geometry. J. High Energ. Phys. 2024, 34 (2024). https://doi.org/10.1007/JHEP03(2024)034
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2024)034