Abstract
We introduce a magnetic analogue of the seven-dimensional nonassociative octonionic R-flux algebra that describes the phase space of M2-branes in four-dimensional locally non-geometric M-theory backgrounds. We show that these two algebras are related by a Spin(7) automorphism of the 3-algebra that provides a covariant description of the eight-dimensional M-theory phase space. We argue that this algebra also underlies the phase space of electrons probing a smeared magnetic monopole in quantum gravity by showing that upon appropriate contractions, the algebra reduces to the noncommutative algebra of a spin foam model of three-dimensional quantum gravity, or to the nonassociative algebra of electrons in a background of uniform magnetic charge. We realise this set-up in M-theory as M-waves probing a delocalised Kaluza-Klein monopole, and show that this system also has a seven-dimensional phase space. We suggest that the smeared Kaluza-Klein monopole is non-geometric because it cannot be described by a local metric. This is the magnetic analogue of the local non-geometry of the R-flux background and arises because the smeared Kaluza-Klein monopole is described by a U(1)-gerbe rather than a U(1)-fibration.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Blumenhagen and E. Plauschinn, Nonassociative Gravity in String Theory?, J. Phys. A 44 (2011) 015401 [arXiv:1010.1263] [INSPIRE].
D. Lüst, T-duality and closed string non-commutative (doubled) geometry, JHEP 12 (2010) 084 [arXiv:1010.1361] [INSPIRE].
R. Blumenhagen, A. Deser, D. Lüst, E. Plauschinn and F. Rennecke, Non-geometric Fluxes, Asymmetric Strings and Nonassociative Geometry, J. Phys. A 44 (2011) 385401 [arXiv:1106.0316] [INSPIRE].
C. Condeescu, I. Florakis and D. Lüst, Asymmetric Orbifolds, Non-Geometric Fluxes and Non-Commutativity in Closed String Theory, JHEP 04 (2012) 121 [arXiv:1202.6366] [INSPIRE].
E. Plauschinn, Non-geometric fluxes and non-associative geometry, PoS(CORFU2011)061 [arXiv:1203.6203] [INSPIRE].
D. Mylonas, P. Schupp and R.J. Szabo, Membrane σ-models and Quantization of Non-Geometric Flux Backgrounds, JHEP 09 (2012) 012 [arXiv:1207.0926] [INSPIRE].
D. Andriot, M. Larfors, D. Lüst and P. Patalong, (Non-)commutative closed string on T-dual toroidal backgrounds, JHEP 06 (2013) 021 [arXiv:1211.6437] [INSPIRE].
C.D.A. Blair, Non-commutativity and non-associativity of the doubled string in non-geometric backgrounds, JHEP 06 (2015) 091 [arXiv:1405.2283] [INSPIRE].
I. Bakas and D. Lüst, T-duality, Quotients and Currents for Non-Geometric Closed Strings, Fortsch. Phys. 63 (2015) 543 [arXiv:1505.04004] [INSPIRE].
R. Jackiw, Three-cocycle in mathematics and physics, Phys. Rev. Lett. 54 (1985) 159 [INSPIRE].
B. Grossman, A 3-cocycle in quantum mechanics, Phys. Lett. B 152 (1985) 93 [INSPIRE].
Y.-S. Wu and A. Zee, Cocycles and Magnetic Monopoles, Phys. Lett. B 152 (1985) 98 [INSPIRE].
R. Jackiw, Magnetic sources and 3-cocycles (comment), Phys. Lett. 154B (1985) 303 [INSPIRE].
J. Mickelsson, Comment on ‘Three-Cocycle in Mathematics Physics’ by R. Jackiw, Phys. Rev. Lett. 54 (1985) 2379 [INSPIRE].
M. Günaydin and B. Zumino, Magnetic charge and nonassociative algebras, in proceedings of the Symposium on Old and New Problems in Fundamental Physics, held in Honor of G.C. Wick, Pisa, Italy, 25 October 1984, pp. 43-53 [INSPIRE].
I. Bakas and D. Lüst, 3-Cocycles, Non-Associative Star-Products and the Magnetic Paradigm of R-Flux String Vacua, JHEP 01 (2014) 171 [arXiv:1309.3172] [INSPIRE].
M. Günaydin, D. Lüst and E. Malek, Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions and missing momentum modes, JHEP 11 (2016) 027 [arXiv:1607.06474] [INSPIRE].
C.D.A. Blair and E. Malek, Geometry and fluxes of SL(5) exceptional field theory, JHEP 03 (2015) 144 [arXiv:1412.0635] [INSPIRE].
B. Wecht, Lectures on Nongeometric Flux Compactifications, Class. Quant. Grav. 24 (2007) S773 [arXiv:0708.3984] [INSPIRE].
D. Mylonas, P. Schupp and R.J. Szabo, Non-Geometric Fluxes, Quasi-Hopf Twist Deformations and Nonassociative Quantum Mechanics, J. Math. Phys. 55 (2014) 122301 [arXiv:1312.1621] [INSPIRE].
V.G. Kupriyanov and R.J. Szabo, G 2 -structures and quantization of non-geometric M-theory backgrounds, JHEP 02 (2017) 099 [arXiv:1701.02574] [INSPIRE].
L. Freidel and E.R. Livine, Effective 3D quantum gravity and non-commutative quantum field theory, Phys. Rev. Lett. 96 (2006) 221301 [hep-th/0512113] [INSPIRE].
P.K. Townsend, The eleven-dimensional supermembrane revisited, Phys. Lett. B 350 (1995) 184 [hep-th/9501068] [INSPIRE].
S.W. Hawking, Gravitational Instantons, Phys. Lett. A 60 (1977) 81 [INSPIRE].
G.W. Gibbons, P. Rychenkova and R. Goto, Hyper-Kähler quotient construction of BPS monopole moduli spaces, Commun. Math. Phys. 186 (1997) 585 [hep-th/9608085] [INSPIRE].
J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Birkhäuser, Boston U.S.A. (1993).
A.S. Pande, Topological T-duality and Kaluza-Klein monopoles, Adv. Theor. Math. Phys. 12 (2008) 185 [math-ph/0612034] [INSPIRE].
P. Bouwknegt, K. Hannabuss and V. Mathai, Nonassociative tori and applications to T-duality, Commun. Math. Phys. 264 (2006) 41 [hep-th/0412092] [INSPIRE].
J. Gaillard and J. Schmude, The Lift of type IIA supergravity with D6 sources: M-theory with torsion, JHEP 02 (2010) 032 [arXiv:0908.0305] [INSPIRE].
U. Danielsson, G. Dibitetto and A. Guarino, KK-monopoles and G-structures in M-theory/type IIA reductions, JHEP 02 (2015) 096 [arXiv:1411.0575] [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: 1705.09639
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
Lüst, D., Malek, E. & Szabo, R.J. Non-geometric Kaluza-Klein monopoles and magnetic duals of M-theory R-flux backgrounds. J. High Energ. Phys. 2017, 144 (2017). https://doi.org/10.1007/JHEP10(2017)144
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2017)144