Abstract
Violating the strong constraint of double field theory, non-geometric fluxes were argued to give rise to noncommutative/nonassociative structures. We derive in a rather pedestrian physicist way a differential geometry on the simplest nonassociative (phase-)space arising for a constant non-geometric R-flux. This provides a complementary presentation to the quasi-Hopf representation categorial one delivered by Barnes, Schenkel, Szabo in arXiv:1409.6331 + arXiv:1507.02792. As there, the notions of tensors, covariant derivative, torsion and curvature find a star-generalization. We continue the construction with the introduction of a star-metric and its star-inverse where, due to the nonassociativity, we encounter major deviations from the familiar structure. Comments on the Levi-Civita connection, a star-Einstein-Hilbert action and the relation to string theory are included, as well.
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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].
P. Bouwknegt, K. Hannabuss and V. Mathai, Nonassociative tori and applications to T-duality, Commun. Math. Phys. 264 (2006) 41 [hep-th/0412092] [INSPIRE].
E. Plauschinn, Non-geometric fluxes and non-associative geometry, PoS(CORFU2011)061 [arXiv:1203.6203] [INSPIRE].
R. Blumenhagen, A Course on Noncommutative Geometry in String Theory, Fortsch. Phys. 62 (2014) 709 [arXiv:1403.4805] [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].
A. Chatzistavrakidis and L. Jonke, Matrix theory origins of non-geometric fluxes, JHEP 02 (2013) 040 [arXiv:1207.6412] [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. Condeescu, I. Florakis, C. Kounnas and D. Lüst, Gauged supergravities and non-geometric Q/R-fluxes from asymmetric orbifold CFT‘s, JHEP 10 (2013) 057 [arXiv:1307.0999] [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].
W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].
G. Aldazabal, D. Marques and C. Núñez, Double Field Theory: A Pedagogical Review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [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].
R. Blumenhagen, M. Fuchs, F. Haßler, D. Lüst and R. Sun, Non-associative Deformations of Geometry in Double Field Theory, JHEP 04 (2014) 141 [arXiv:1312.0719] [INSPIRE].
R. Blumenhagen, F. Hassler and D. Lüst, Double Field Theory on Group Manifolds, JHEP 02 (2015) 001 [arXiv:1410.6374] [INSPIRE].
R. Blumenhagen, P.d. Bosque, F. Hassler and D. Lüst, Generalized Metric Formulation of Double Field Theory on Group Manifolds, JHEP 08 (2015) 056 [arXiv:1502.02428] [INSPIRE].
A. Deser, Star products on graded manifolds and α ′ -corrections to Courant algebroids from string theory, J. Math. Phys. 56 (2015) 092302 [arXiv:1412.5966] [INSPIRE].
A. Deser, Star products on graded manifolds and α ′ -corrections to double field theory, in proceedings of the 34th Workshop on Geometric Methods in Physics (XXXIV WGMP), Bialowieza, Poland, June 28 - July 4 2015, arXiv:1511.03929 [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].
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].
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].
P. Aschieri, C. Blohmann, M. Dimitrijević, F. Meyer, P. Schupp and J. Wess, A gravity theory on noncommutative spaces, Class. Quant. Grav. 22 (2005) 3511 [hep-th/0504183] [INSPIRE].
P. Aschieri, M. Dimitrijević, F. Meyer and J. Wess, Noncommutative geometry and gravity, Class. Quant. Grav. 23 (2006) 1883 [hep-th/0510059] [INSPIRE].
L. Álvarez-Gaumé, F. Meyer and M.A. Vazquez-Mozo, Comments on noncommutative gravity, Nucl. Phys. B 753 (2006) 92 [hep-th/0605113] [INSPIRE].
G.E. Barnes, A. Schenkel and R.J. Szabo, Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms, J. Geom. Phys. 89 (2014) 111 [arXiv:1409.6331] [INSPIRE].
G.E. Barnes, A. Schenkel and R.J. Szabo, Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature, J. Geom. Phys. 106 (2016) 234 [arXiv:1507.02792] [INSPIRE].
G.E. Barnes, A. Schenkel and R.J. Szabo, Working with Nonassociative Geometry and Field Theory, [arXiv:1601.07353] [INSPIRE].
P. Aschieri and R.J. Szabo, Triproducts, nonassociative star products and geometry of R-flux string compactifications, J. Phys. Conf. Ser. 634 (2015) 012004 [arXiv:1504.03915] [INSPIRE].
V.G. Kupriyanov, Alternative multiplications and non-associativity in physics, in proceedings of the 15th Hellenic School and Workshops on Elementary Particle Physics and Gravity (CORFU2015), Corfu, Greece, September 1-26, 2015, arXiv:1603.00218 [INSPIRE].
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Blumenhagen, R., Fuchs, M. Towards a theory of nonassociative gravity. J. High Energ. Phys. 2016, 19 (2016). https://doi.org/10.1007/JHEP07(2016)019
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DOI: https://doi.org/10.1007/JHEP07(2016)019