Abstract
We show that the D = 11 supermembrane theory (M2-brane) compactified on a M9 × T2 target space, with constant fluxes C± naturally incorporates the geometrical structure of a twisted torus. We extend the M2-brane theory to a formulation on a twisted torus bundle. It is consistently fibered over the world volume of the M2-brane. It can also be interpreted as a torus bundle with a nontrivial U(1) connection associated to the fluxes. The structure group G is the area preserving diffeomorphisms. The torus bundle is defined in terms of the monodromy associated to the isotopy classes of symplectomorphisms with π0(G) = SL(2, Z), and classified by the coinvariants of the subgroups of SL(2, Z). The spectrum of the theory is purely discrete since the constant flux induces a central charge on the supersymmetric algebra and a modification on the Hamiltonian which renders the spectrum discrete with finite multiplicity. The theory is invariant under symplectomorphisms connected and non connected to the identity, a result relevant to guarantee the U-dual invariance of the theory. The Hamiltonian of the theory exhibits interesting new U(1) gauge and global symmetries on the worldvolume induced by the symplectomorphim transformations. We construct explicitly the supersymmetric algebra with nontrivial central charges. We show that the zero modes decouple from the nonzero ones. The nonzero mode algebra corresponds to a massive superalgebra that preserves either 1/2 or 1/4 of the original supersymmetry depending on the state considered.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.M. Hull, Gauged D = 9 supergravities and Scherk-Schwarz reduction, Class. Quant. Grav. 21 (2004) 509 [hep-th/0203146] [INSPIRE].
G. Dall’Agata, N. Prezas, H. Samtleben and M. Trigiante, Gauged supergravities from twisted doubled tori and non-geometric string backgrounds, Nucl. Phys. B 799 (2008) 80 [arXiv:0712.1026] [INSPIRE].
G. Dall’Agata and S. Ferrara, Gauged supergravity algebras from twisted tori compactifications with fluxes, Nucl. Phys. B 717 (2005) 223 [hep-th/0502066] [INSPIRE].
M. Trigiante, Gauged supergravities, Phys. Rept. 680 (2017) 1 [arXiv:1609.09745] [INSPIRE].
N. Kaloper and R.C. Myers, The odd story of massive supergravity, JHEP 05 (1999) 010 [hep-th/9901045] [INSPIRE].
A. Dabholkar and C. Hull, Duality twists, orbifolds, and fluxes, JHEP 09 (2003) 054 [hep-th/0210209] [INSPIRE].
R.A. Reid-Edwards, Flux compactifications, twisted tori and doubled geometry, JHEP 06 (2009) 085 [arXiv:0904.0380] [INSPIRE].
M. Graña, R. Minasian, M. Petrini and A. Tomasiello, A scan for new N = 1 vacua on twisted tori, JHEP 05 (2007) 031 [hep-th/0609124] [INSPIRE].
J. Shelton, W. Taylor and B. Wecht, Nongeometric flux compactifications, JHEP 10 (2005) 085 [hep-th/0508133] [INSPIRE].
R. D’Auria, S. Ferrara and M. Trigiante, E7(7) symmetry and dual gauge algebra of M-theory on a twisted seven-torus, Nucl. Phys. B 732 (2006) 389 [hep-th/0504108] [INSPIRE].
J. Scherk and J.H. Schwarz, How to get masses from extra dimensions, Nucl. Phys. B 153 (1979) 61 [INSPIRE].
E. Bergshoeff, T. de Wit, U. Gran, R. Linares and D. Roest, (Non)Abelian gauged supergravities in nine-dimensions, JHEP 10 (2002) 061 [hep-th/0209205] [INSPIRE].
D.A. Lowe, H. Nastase and S. Ramgoolam, Massive IIA string theory and matrix theory compactification, Nucl. Phys. B 667 (2003) 55 [hep-th/0303173] [INSPIRE].
R.A. Reid-Edwards, Geometric and non-geometric compactifications of IIB supergravity, JHEP 12 (2008) 043 [hep-th/0610263] [INSPIRE].
L. Andrianopoli, M.A. Lledó and M. Trigiante, The Scherk-Schwarz mechanism as a flux compactification with internal torsion, JHEP 05 (2005) 051 [hep-th/0502083] [INSPIRE].
H. Samtleben, Lectures on gauged supergravity and flux compactifications, Class. Quant. Grav. 25 (2008) 214002 [arXiv:0808.4076] [INSPIRE].
J.J. Fernandez-Melgarejo, T. Ortín and E. Torrente-Lujan, The general gaugings of maximal d = 9 supergravity, JHEP 10 (2011) 068 [arXiv:1106.1760] [INSPIRE].
A. Chatzistavrakidis and L. Jonke, Matrix theory compactifications on twisted tori, Phys. Rev. D 85 (2012) 106013 [arXiv:1202.4310] [INSPIRE].
C.M. Hull and R.A. Reid-Edwards, Flux compactifications of string theory on twisted tori, Fortsch. Phys. 57 (2009) 862 [hep-th/0503114] [INSPIRE].
C.M. Hull and R.A. Reid-Edwards, Flux compactifications of M-theory on twisted tori, JHEP 10 (2006) 086 [hep-th/0603094] [INSPIRE].
S. Thangavelu, Harmonic analysis on Heisenberg nilmanifolds, Rev. Unión Matem. Argentina 50 (2009) 75.
Y. Shi, Partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds and holonomy maps, Comptes Rendus Math. 352 (2014) 743.
S. Kachru, M.B. Schulz, P.K. Tripathy and S.P. Trivedi, New supersymmetric string compactifications, JHEP 03 (2003) 061 [hep-th/0211182] [INSPIRE].
I.V. Lavrinenko, H. Lü and C.N. Pope, Fibre bundles and generalized dimensional reductions, Class. Quant. Grav. 15 (1998) 2239 [hep-th/9710243] [INSPIRE].
J.R. Weeks, The shape of space, 2nd edition, Marcel Dekker, U.S.A. (2002).
M.P. Garcia Del Moral, C. Las Heras, P. Leon, J.M. Pena and A. Restuccia, M2-branes on a constant flux background, Phys. Lett. B 797 (2019) 134924 [arXiv:1811.11231] [INSPIRE].
C.M. Hull, Massive string theories from M-theory and F-theory, JHEP 11 (1998) 027 [hep-th/9811021] [INSPIRE].
M.P. Garcia del Moral, J.M. Pena and A. Restuccia, Supermembrane origin of type-II gauged supergravities in 9D, JHEP 09 (2012) 063 [arXiv:1203.2767] [INSPIRE].
M.P. Garcia del Moral, J.M. Pena and A. Restuccia, Classification of M2-brane 2-torus bundles, U-duality invariance and type-II gauged supergravities, Phys. Rev. D 100 (2019) 026005 [arXiv:1604.02579] [INSPIRE].
B. de Wit, J. Hoppe and H. Nicolai, On the quantum mechanics of supermembranes, Nucl. Phys. B 305 (1988) 545 [INSPIRE].
I. Martin and A. Restuccia, Magnetic monopoles over topologically nontrivial Riemann surfaces, Lett. Math. Phys. 39 (1997) 379 [hep-th/9603035] [INSPIRE].
I. Martín, A. Restuccia and R.S. Torrealba, On the stability of compactified D = 11 supermembranes, Nucl. Phys. B 521 (1998) 117 [hep-th/9706090] [INSPIRE].
L. Boulton, M.P. Garcia del Moral and A. Restuccia, Discreteness of the spectrum of the compactified D = 11 supermembrane with nontrivial winding, Nucl. Phys. B 671 (2003) 343 [hep-th/0211047] [INSPIRE].
E. Bergshoeff, E. Sezgin and P.K. Townsend, Supermembranes and eleven-dimensional supergravity, Phys. Lett. B 189 (1987) 75 [INSPIRE].
M.J. Duff and K.S. Stelle, Multimembrane solutions of D = 11 supergravity, Phys. Lett. B 253 (1991) 113 [INSPIRE].
K.S. Stelle, Lectures on supergravity p-branes, in ICTP summer school in high-energy physics and cosmology, Trieste, Italy (1996), pg. 287 [hep-th/9701088] [INSPIRE].
B. de Wit, K. Peeters and J. Plefka, Superspace geometry for supermembrane backgrounds, Nucl. Phys. B 532 (1998) 99 [hep-th/9803209] [INSPIRE].
I. Martin, J. Ovalle and A. Restuccia, Compactified D = 11 supermembranes and symplectic noncommutative gauge theories, Phys. Rev. D 64 (2001) 046001 [hep-th/0101236] [INSPIRE].
M.P. Garcia del Moral and A. Restuccia, Spectrum of a noncommutative formulation of the D = 11 supermembrane with winding, Phys. Rev. D 66 (2002) 045023 [hep-th/0103261] [INSPIRE].
M. Abou-Zeid, B. de Wit, D. Lüst and H. Nicolai, Space-time supersymmetry, IIA/B duality and M-theory, Phys. Lett. B 466 (1999) 144 [hep-th/9908169] [INSPIRE].
B. de Wit, U. Marquard and H. Nicolai, Area-preserving diffeomorphisms and supermembrane Lorentz invariance, Commun. Math. Phys. 128 (1990) 39.
P.J. Kahn, Symplectic torus bundles and group extensions, New York J. Math. 11 (2005) 3555 [math.SG/0405109].
M.P. Garcia del Moral et al., SL(2, Z ) symmetries, supermembranes and symplectic torus bundles, JHEP 09 (2011) 068 [arXiv:1105.3181] [INSPIRE].
M.P. Garcia del Moral, C. Las Heras and A. Restuccia, String bound states from the supermembrane with fluxes, work in progress.
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: 2005.06397v1
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
del Moral, M.P.G., Heras, C.L., Leon, P. et al. Fluxes, twisted tori, monodromy and U(1) supermembranes. J. High Energ. Phys. 2020, 97 (2020). https://doi.org/10.1007/JHEP09(2020)097
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2020)097