Abstract
We describe the doubled space of Double Field Theory as a group manifold G with an arbitrary generalized metric. Local information from the latter is not relevant to our discussion and so G only captures the topology of the doubled space. Strong Constraint solutions are maximal isotropic submanifold M in G. We construct them and their Generalized Geometry in Double Field Theory on Group Manifolds. In general, G admits different physical subspace M which are Poisson-Lie T-dual to each other. By studying two examples, we reproduce the topology changes induced by T-duality with non-trivial H-flux which were discussed by Bouwknegt, Evslin and Mathai [1].
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P. Bouwknegt, J. Evslin and V. Mathai, T duality: Topology change from H flux, Commun. Math. Phys. 249 (2004) 383 [hep-th/0306062] [INSPIRE].
C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].
E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].
T.H. Buscher, A Symmetry of the String Background Field Equations, Phys. Lett. B 194 (1987) 59 [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].
C. Hull and B. Zwiebach, The gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].
G. Aldazabal, D. Marques and C. Núñez, Double Field Theory: A Pedagogical Review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [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].
I. Vaisman, On the geometry of double field theory, J. Math. Phys. 53 (2012) 033509 [arXiv:1203.0836] [INSPIRE].
I. Vaisman, Towards a double field theory on para-Hermitian manifolds, J. Math. Phys. 54 (2013) 123507 [arXiv:1209.0152] [INSPIRE].
M. Cederwall, The geometry behind double geometry, JHEP 09 (2014) 070 [arXiv:1402.2513] [INSPIRE].
M. Cederwall, T-duality and non-geometric solutions from double geometry, Fortsch. Phys. 62 (2014) 942 [arXiv:1409.4463] [INSPIRE].
O. Hohm and B. Zwiebach, Large Gauge Transformations in Double Field Theory, JHEP 02 (2013) 075 [arXiv:1207.4198] [INSPIRE].
J.-H. Park, Comments on double field theory and diffeomorphisms, JHEP 06 (2013) 098 [arXiv:1304.5946] [INSPIRE].
D.S. Berman, M. Cederwall and M.J. Perry, Global aspects of double geometry, JHEP 09 (2014) 066 [arXiv:1401.1311] [INSPIRE].
C.M. Hull, Finite Gauge Transformations and Geometry in Double Field Theory, JHEP 04 (2015) 109 [arXiv:1406.7794] [INSPIRE].
U. Naseer, A note on large gauge transformations in double field theory, JHEP 06 (2015) 002 [arXiv:1504.05913] [INSPIRE].
G. Papadopoulos, Seeking the balance: Patching double and exceptional field theories, JHEP 10 (2014) 089 [arXiv:1402.2586] [INSPIRE].
P.S. Howe and G. Papadopoulos, Patching DFT, T-duality and Gerbes, JHEP 04 (2017) 074 [arXiv:1612.07968] [INSPIRE].
E. Alvarez, L. Álvarez-Gaumé, J.L.F. Barbon and Y. Lozano, Some global aspects of duality in string theory, Nucl. Phys. B 415 (1994) 71 [hep-th/9309039] [INSPIRE].
C.M. Hull, Global aspects of T-duality, gauged σ-models and T-folds, JHEP 10 (2007) 057 [hep-th/0604178] [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, Gauge symmetry, T-duality and doubled geometry, JHEP 08 (2008) 043 [arXiv:0711.4818] [INSPIRE].
G. Dall’Agata and N. Prezas, Worldsheet theories for non-geometric string backgrounds, JHEP 08 (2008) 088 [arXiv:0806.2003] [INSPIRE].
C.M. Hull and R.A. Reid-Edwards, Non-geometric backgrounds, doubled geometry and generalised T-duality, JHEP 09 (2009) 014 [arXiv:0902.4032] [INSPIRE].
J. Scherk and J.H. Schwarz, How to Get Masses from Extra Dimensions, Nucl. Phys. B 153 (1979) 61 [INSPIRE].
J. Scherk and J.H. Schwarz, Spontaneous Breaking of Supersymmetry Through Dimensional Reduction, Phys. Lett. B 82 (1979) 60 [INSPIRE].
G. Aldazabal, W. Baron, D. Marques and C. Núñez, The effective action of Double Field Theory, JHEP 11 (2011) 052 [Erratum ibid. 11 (2011) 109] [arXiv:1109.0290] [INSPIRE].
D. Geissbuhler, Double Field Theory and N = 4 Gauged Supergravity, JHEP 11 (2011) 116 [arXiv:1109.4280] [INSPIRE].
D. Geissbuhler, D. Marques, C. Núñez and V. Penas, Exploring Double Field Theory, JHEP 06 (2013) 101 [arXiv:1304.1472] [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. du Bosque, F. Hassler and D. Lüst, Generalized Metric Formulation of Double Field Theory on Group Manifolds, JHEP 08 (2015) 056 [arXiv:1502.02428] [INSPIRE].
P. du Bosque, F. Hassler and D. Lüst, Flux Formulation of DFT on Group Manifolds and Generalized Scherk-Schwarz Compactifications, JHEP 02 (2016) 039 [arXiv:1509.04176] [INSPIRE].
C. Klimčík and P. Ševera, Dual nonAbelian duality and the Drinfeld double, Phys. Lett. B 351 (1995) 455 [hep-th/9502122] [INSPIRE].
C. Klimčík and P. Ševera, NonAbelian momentum winding exchange, Phys. Lett. B 383 (1996) 281 [hep-th/9605212] [INSPIRE].
P. Ševera, Poisson-Lie T-duality as a boundary phenomenon of Chern-Simons theory, JHEP 05 (2016) 044 [arXiv:1602.05126] [INSPIRE].
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].
M. Gualtieri, Generalized complex geometry, Ph.D. thesis, Oxford U., 2003. math/0401221 [INSPIRE].
P. Koerber, Lectures on Generalized Complex Geometry for Physicists, Fortsch. Phys. 59 (2011) 169 [arXiv:1006.1536] [INSPIRE].
J. Shelton, W. Taylor and B. Wecht, Nongeometric flux compactifications, JHEP 10 (2005) 085 [hep-th/0508133] [INSPIRE].
A. Dabholkar and C. Hull, Generalised T-duality and non-geometric backgrounds, JHEP 05 (2006) 009 [hep-th/0512005] [INSPIRE].
M.B. Schulz, T-folds, doubled geometry and the SU(2) WZW model, JHEP 06 (2012) 158 [arXiv:1106.6291] [INSPIRE].
S.B. Giddings, J. Polchinski and A. Strominger, Four-dimensional black holes in string theory, Phys. Rev. D 48 (1993) 5784 [hep-th/9305083] [INSPIRE].
J.M. Maldacena, G.W. Moore and N. Seiberg, Geometrical interpretation of D-branes in gauged WZW models, JHEP 07 (2001) 046 [hep-th/0105038] [INSPIRE].
O. Hohm and D. Marques, Perturbative Double Field Theory on General Backgrounds, Phys. Rev. D 93 (2016) 025032 [arXiv:1512.02658] [INSPIRE].
F. Hassler, Double Field Theory on Group Manifolds (Thesis), Ph.D. thesis, Munich University, Germany, 2015, arXiv:1509.07153 [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [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].
O. Hohm and S.K. Kwak, Frame-like Geometry of Double Field Theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [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].
K. Lee, C. Strickland-Constable and D. Waldram, Spheres, generalised parallelisability and consistent truncations, Fortsch. Phys. 65 (2017) 1700048 [arXiv:1401.3360] [INSPIRE].
G. Dibitetto, J.J. Fernandez-Melgarejo, D. Marques and D. Roest, Duality orbits of non-geometric fluxes, Fortsch. Phys. 60 (2012) 1123 [arXiv:1203.6562] [INSPIRE].
C.M. Hull, Doubled Geometry and T-Folds, JHEP 07 (2007) 080 [hep-th/0605149] [INSPIRE].
J. Shelton, W. Taylor and B. Wecht, Generalized Flux Vacua, JHEP 02 (2007) 095 [hep-th/0607015] [INSPIRE].
E. Plauschinn, T-duality revisited, JHEP 01 (2014) 131 [arXiv:1310.4194] [INSPIRE].
E. Plauschinn, On T-duality transformations for the three-sphere, Nucl. Phys. B 893 (2015) 257 [arXiv:1408.1715] [INSPIRE].
N.J. Hitchin, Lectures on special Lagrangian submanifolds, in Proceedings, Winter School on Mirror Symmetry and Vector Bundles: Cambridge, Massachusetts, January 4-15, 1999, pp. 151–182, math/9907034 [INSPIRE].
D. Lüst, T-duality and closed string non-commutative (doubled) geometry, JHEP 12 (2010) 084 [arXiv:1010.1361] [INSPIRE].
D. Andriot, O. Hohm, M. Larfors, D. Lüst and P. Patalong, Non-Geometric Fluxes in Supergravity and Double Field Theory, Fortsch. Phys. 60 (2012) 1150 [arXiv:1204.1979] [INSPIRE].
R. Blumenhagen, A. Deser, E. Plauschinn and F. Rennecke, Bianchi Identities for Non-Geometric Fluxes — From Quasi-Poisson Structures to Courant Algebroids, Fortsch. Phys. 60 (2012) 1217 [arXiv:1205.1522] [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].
I. Bakas and D. Lüst, T-duality, Quotients and Currents for Non-Geometric Closed Strings, Fortsch. Phys. 63 (2015) 543 [arXiv:1505.04004] [INSPIRE].
V. Mathai and J.M. Rosenberg, T-duality for torus bundles with H-fluxes via noncommutative topology, Commun. Math. Phys. 253 (2004) 705 [hep-th/0401168] [INSPIRE].
A. Deser, Lie algebroids, non-associative structures and non-geometric fluxes, Fortsch. Phys. 61 (2013) 1056 [arXiv:1309.5792] [INSPIRE].
F. Hassler and D. Lüst, Consistent Compactification of Double Field Theory on Non-geometric Flux Backgrounds, JHEP 05 (2014) 085 [arXiv:1401.5068] [INSPIRE].
J. de Boer and M. Shigemori, Exotic Branes in String Theory, Phys. Rept. 532 (2013) 65 [arXiv:1209.6056] [INSPIRE].
F. Hassler and D. Lüst, Non-commutative/non-associative IIA ( IIB) Q- and R-branes and their intersections, JHEP 07 (2013) 048 [arXiv:1303.1413] [INSPIRE].
J. Berkeley, D.S. Berman and F.J. Rudolph, Strings and Branes are Waves, JHEP 06 (2014) 006 [arXiv:1403.7198] [INSPIRE].
D.S. Berman and F.J. Rudolph, Branes are Waves and Monopoles, JHEP 05 (2015) 015 [arXiv:1409.6314] [INSPIRE].
I. Bakhmatov, A. Kleinschmidt and E.T. Musaev, Non-geometric branes are DFT monopoles, JHEP 10 (2016) 076 [arXiv:1607.05450] [INSPIRE].
A. Deser and J. Stasheff, Even symplectic supermanifolds and double field theory, Commun. Math. Phys. 339 (2015) 1003 [arXiv:1406.3601] [INSPIRE].
A. Deser and C. Sämann, Extended Riemannian Geometry I: Local Double Field Theory, arXiv:1611.02772 [INSPIRE].
R. Blumenhagen and E. Plauschinn, Nonassociative Gravity in String Theory?, J. Phys. A 44 (2011) 015401 [arXiv:1010.1263] [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].
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].
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].
S. Ramgoolam, On spherical harmonics for fuzzy spheres in diverse dimensions, Nucl. Phys. B 610 (2001) 461 [hep-th/0105006] [INSPIRE].
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Hassler, F. The topology of Double Field Theory. J. High Energ. Phys. 2018, 128 (2018). https://doi.org/10.1007/JHEP04(2018)128
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DOI: https://doi.org/10.1007/JHEP04(2018)128