Abstract
We show that M-theory admits a class of supersymmetric eight-dimensional compactification background solutions, equipped with an internal complex pure spinor, more general than the Calabi-Yau one. Building-up on this result, we obtain a a particular class of supersymmetric M-theory eight-dimensional non-geometric compactification backgrounds with external three-dimensional Minkowski space-time, proving that the global space of the non-geometric compactification is again a differentiable manifold, although with very different geometric and topological properties respect to the corresponding standard M-theory compactification background: it is a compact complex manifold admitting a Kähler covering with deck transformations acting by holomorphic homotheties with respect to the Kähler metric. We show that this class of non-geometric compactifications evade the Maldacena-Nuñez no-go theorem by means of a mechanism originally developed by Mario García-Fernández and the author for Heterotic Supergravity, and thus do not require l P -corrections to allow for a nontrivial warp factor or four-form flux. We obtain an explicit compactification background on a complex Hopf four-fold that solves all the equations of motion of the theory, including the warp factor equation of motion. We also show that this class of non-geometric compactifications are equipped with a holomorphic principal torus fibration over a projective Kähler base as well as a codimension-one foliation with nearly-parallel G 2-leaves, making thus contact with the work of M. Babalic and C. Lazaroiu on the foliation structure of the most general M-theory supersymmetric compactifications.
Article PDF
Similar content being viewed by others
References
M. Zabzine, Lectures on Generalized Complex Geometry and Supersymmetry, Archivum Math. 42 (2006) 119 [hep-th/0605148] [INSPIRE].
F. Denef, Les Houches Lectures on Constructing String Vacua, arXiv:0803.1194 [INSPIRE].
H. Ooguri, Geometry As Seen By String Theory, arXiv:0901.1881 [INSPIRE].
P. Koerber, Lectures on Generalized Complex Geometry for Physicists, Fortsch. Phys. 59 (2011) 169 [arXiv:1006.1536] [INSPIRE].
G.W. Moore, Physical mathematics and the future, in Vision talk, Strings conference, Princeton U.S.A. (2014).
T. Ortin, Gravity and strings, Cambridge University Press, Cambridge U.K. (2004).
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).
S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge U.K. (1975).
M. Graña, Flux compactifications in string theory: A Comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [INSPIRE].
K. Becker and M. Becker, M theory on eight manifolds, Nucl. Phys. B 477 (1996) 155 [hep-th/9605053] [INSPIRE].
K. Becker, A Note on compactifications on spin(7) — holonomy manifolds, JHEP 05 (2001) 003 [hep-th/0011114] [INSPIRE].
D. Martelli and J. Sparks, G structures, fluxes and calibrations in M-theory, Phys. Rev. D 68 (2003) 085014 [hep-th/0306225] [INSPIRE].
D. Tsimpis, M-theory on eight-manifolds revisited: N = 1 supersymmetry and generalized Spin(7) structures, JHEP 04 (2006) 027 [hep-th/0511047] [INSPIRE].
C. Condeescu, A. Micu and E. Palti, M-theory Compactifications to Three Dimensions with M2-brane Potentials, JHEP 04 (2014) 026 [arXiv:1311.5901] [INSPIRE].
D. Prins and D. Tsimpis, Type IIA supergravity and M -theory on manifolds with SU(4) structure, Phys. Rev. D 89 (2014) 064030 [arXiv:1312.1692] [INSPIRE].
E.M. Babalic and C.I. Lazaroiu, Singular foliations for M-theory compactification, JHEP 03 (2015) 116 [arXiv:1411.3497] [INSPIRE].
E.M. Babalic and C.I. Lazaroiu, Foliated eight-manifolds for M-theory compactification, JHEP 01 (2015) 140 [arXiv:1411.3148] [INSPIRE].
M. Graña, C.S. Shahbazi and M. Zambon, Spin(7)-manifolds in compactifications to four dimensions, JHEP 11 (2014) 046 [arXiv:1405.3698] [INSPIRE].
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [INSPIRE].
P. Libermann, Sur le probleme d’equivalance de certaines structures infinitesimales regulieres, Annali Mat. Pura Appl. 36 (1954) 27.
P. Libermann, Sur les structures presque complexes et autres structures infinitesimales reguliers, Bull. Soc. Math. Plane 83 (1955) 195.
I. Vaisman, On locally conformal almost kähler manifolds, Israel J. Math. 24 (1976) 338.
F. Bonetti, T.W. Grimm and T.G. Pugh, Non-Supersymmetric F-theory Compactifications on Spin(7) Manifolds, JHEP 01 (2014) 112 [arXiv:1307.5858] [INSPIRE].
F. Bonetti, T.W. Grimm, E. Palti and T.G. Pugh, F-Theory on Spin(7) Manifolds: Weak-Coupling Limit, JHEP 02 (2014) 076 [arXiv:1309.2287] [INSPIRE].
J. Shelton, W. Taylor and B. Wecht, Nongeometric flux compactifications, JHEP 10 (2005) 085 [hep-th/0508133] [INSPIRE].
B. Wecht, Lectures on Nongeometric Flux Compactifications, Class. Quant. Grav. 24 (2007) S773 [arXiv:0708.3984] [INSPIRE].
M. Graña, R. Minasian, M. Petrini and D. Waldram, T-duality, Generalized Geometry and Non-Geometric Backgrounds, JHEP 04 (2009) 075 [arXiv:0807.4527] [INSPIRE].
D. Andriot, Non-geometric fluxes versus (non)-geometry, arXiv:1303.0251 [INSPIRE].
A. Malmendier and D.R. Morrison, K3 surfaces, modular forms and non-geometric heterotic compactifications, Lett. Math. Phys. 105 (2015) 1085 [arXiv:1406.4873] [INSPIRE].
J. Gu and H. Jockers, Nongeometric F-theory heterotic duality, Phys. Rev. D 91 (2015) 086007 [arXiv:1412.5739] [INSPIRE].
L. Martucci, J.F. Morales and D.R. Pacifici, Branes, U-folds and hyperelliptic fibrations, JHEP 01 (2013) 145 [arXiv:1207.6120] [INSPIRE].
A.P. Braun, F. Fucito and J.F. Morales, U-folds as K3 fibrations, JHEP 10 (2013) 154 [arXiv:1308.0553] [INSPIRE].
P. Candelas, A. Constantin, C. Damian, M. Larfors and J.F. Morales, Type IIB flux vacua from G-theory I, JHEP 02 (2015) 187 [arXiv:1411.4785] [INSPIRE].
P. Candelas, A. Constantin, C. Damian, M. Larfors and J.F. Morales, Type IIB flux vacua from G-theory II, JHEP 02 (2015) 188 [arXiv:1411.4786] [INSPIRE].
E.M. Babalic and C.I. Lazaroiu, The landscape of G-structures in eight-manifold compactifications of M-theory, arXiv:1505.02270 [INSPIRE].
E.M. Babalic and C.I. Lazaroiu, Internal circle uplifts, transversality and stratified G-structures, arXiv:1505.05238 [INSPIRE].
J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].
E. Cremmer, B. Julia and J. Scherk, Supergravity Theory in Eleven-Dimensions, Phys. Lett. B 76 (1978) 409 [INSPIRE].
J.P. Gauntlett and S. Pakis, The Geometry of D = 11 Killing spinors, JHEP 04 (2003) 039 [hep-th/0212008] [INSPIRE].
J.P. Gauntlett, J.B. Gutowski and S. Pakis, The Geometry of D = 11 null Killing spinors, JHEP 12 (2003) 049 [hep-th/0311112] [INSPIRE].
S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66 (2002) 106006 [hep-th/0105097] [INSPIRE].
M.J. Duff, J.T. Liu and R. Minasian, Eleven-dimensional origin of string-string duality: A One loop test, Nucl. Phys. B 452 (1995) 261 [hep-th/9506126] [INSPIRE].
T.W. Grimm, T.G. Pugh and M. Weissenbacher, On M-theory fourfold vacua with higher curvature terms, Phys. Lett. B 743 (2015) 284 [arXiv:1408.5136] [INSPIRE].
T.W. Grimm, T.G. Pugh and M. Weissenbacher, The effective action of warped M-theory reductions with higher derivative terms — Part I, arXiv:1412.5073 [INSPIRE].
T.W. Grimm, T.G. Pugh and M. Weissenbacher, The effective action of warped M-theory reductions with higher-derivative terms — Part II, arXiv:1507.00343 [INSPIRE].
I. Vaisman, Locally conformal symplectic manifolds, Int. J. Math. Math. Sci. 8 (1985) 521.
S. Dragomir and L. Ornea, Locally Conformal Kähler Geometry, Springer-Verlag, Berlin Germany (2012).
L. Ornea and M. Verbitsky, A report on locally conformally Kähler manifolds, arXiv:1002.3473.
L. Álvarez-Gaumé and E. Witten, Gravitational Anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
I. Vaisman, Generalized hopf manifolds, Geometriae Dedicata 13 (1982) 231.
B.-Y. Chen and P. Piccinni, The canonical foliations of a locally conformal kähler manifold, Ann. Mat. Pura Appl. 141 (1985) 249.
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: 1503.00733
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Shahbazi, C.S. M-theory on non-Kähler eight-manifolds. J. High Energ. Phys. 2015, 178 (2015). https://doi.org/10.1007/JHEP09(2015)178
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2015)178