Abstract
We consider IIA compactifications on solvmanifolds with O6/D6 branes and study the conditions for obtaining de Sitter vacua in ten dimensions. While this is a popular set-up for searching de Sitter vacua, we propose a new method to include supersymmetry breaking sources. For space-time filling branes preserving bulk supersymmetry, the energy density can easily be extremized with respect to all fields, thanks to the replacement of the DBI action by a pullback of a special form given by a pure spinor. For sources breaking bulk supersymmetry, we propose to replace the DBI action by the pullback of a more general polyform, which is no longer pure. This generalization provides corrections to the energy-momentum tensor which give a positive contribution to the cosmological constant. We find a de Sitter solution to all (bulk and world-volume) equations derived from this action. We argue it solves the equations derived from the standard source action. The paper also contains a review of solvmanifolds.
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References
M. Graña, Flux compactifications in string theory: A comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [SPIRES].
M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [SPIRES].
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional String Compactifications with D-branes, Orientifolds and Fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [SPIRES].
D. Lüst and D. Tsimpis, Supersymmetric AdS 4 compactifications of IIA supergravity, JHEP 02 (2005) 027 [hep-th/0412250] [SPIRES].
J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS 5 solutions of type IIB supergravity, Class. Quant. Grav. 23 (2006) 4693 [hep-th/0510125] [SPIRES].
P. Koerber and D. Tsimpis, Supersymmetric sources, integrability and generalized-structure compactifications, JHEP 08 (2007) 082 [arXiv:0706.1244] [SPIRES].
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. Oxford Ser. 54 (2003) 281 [math/0209099].
M. Gualtieri, Generalized complex geometry, mat h/ 0401221.
M. Graña, R. Minasian, M. Petrini and A. Tomasiello, Supersymmetric backgrounds from generalized Calabi-Yau manifolds, JHEP 08 (2004) 046 [hep-th/0406137] [SPIRES].
M. Graña, R. Minasian, M. Petrini and A. Tomasiello, Generalized structures of \( \mathcal{N} = 1 \) vacua, JHEP 11 (2005) 020 [hep-th/0505212] [SPIRES].
D. Lüst, F. Marchesano, L. Martucci and D. Tsimpis, Generalized non-supersymmetric flux vacua, JHEP 11 (2008) 021 [arXiv:0807.4540] [SPIRES].
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] [SPIRES].
M. Ihl, D. Robbins and T. Wrase, Toroidal Orientifolds in IIA with General NS-NS Fluxes, JHEP 08 (2007) 043 [arXiv:0705.3410] [SPIRES].
M.P. Hertzberg, S. Kachru, W. Taylor and M. Tegmark, Inflationary Constraints on Type IIA String Theory, JHEP 12 (2007) 095 [arXiv:0711.2512] [SPIRES].
E. Silverstein, Simple de Sitter Solutions, Phys. Rev. D 77 (2008) 106006 [arXiv:0712.1196] [SPIRES].
S.S. Haque, G. Shiu, B. Underwood and T. Van Riet, Minimal simple de Sitter solutions, Phys. Rev. D 79 (2009) 086005 [arXiv:0810.5328] [SPIRES].
U.H. Danielsson, S.S. Haque, G. Shiu and T. Van Riet, Towards Classical de Sitter Solutions in String Theory, JHEP 09 (2009) 114 [arXiv:0907.2041] [SPIRES].
C. Caviezel et al., On the Cosmology of Type IIA Compactifications on SU(3)-structure Manifolds, JHEP 04 (2009) 010 [arXiv:0812.3551] [SPIRES].
R. Flauger, S. Paban, D. Robbins and T. Wrase, On Slow-roll Moduli Inflation in Massive IIA Supergravity with Metric Fluxes, Phys. Rev. D 79 (2009) 086011 [arXiv:0812.3886] [SPIRES].
U.H. Danielsson, P. Koerber and T. Van Riet, Universal de Sitter solutions at tree-level, JHEP 05 (2010) 090 [arXiv:1003.3590] [SPIRES].
P. Koerber, Stable D-branes, calibrations and generalized Calabi-Yau geometry, JHEP 08 (2005) 099 [hep-th/0506154] [SPIRES].
L. Martucci and P. Smyth, Supersymmetric D-branes and calibrations on general \( \mathcal{N} = 1 \) backgrounds, JHEP 11 (2005) 048 [hep-th/0507099] [SPIRES].
P. Koerber and L. Martucci, Deformations of calibrated D-branes in flux generalized complex manifolds, JHEP 12 (2006) 062 [hep-th/0610044] [SPIRES].
B. de Carlos, A. Guarino and J.M. Moreno, Complete classification of Minkowski vacua in generalised flux models, JHEP 02 (2010) 076 [arXiv:0911.2876] [SPIRES].
B. de Carlos, A. Guarino and J.M. Moreno, Flux moduli stabilisation, Supergravity algebras and no-go theorems, JHEP 01 (2010) 012 [arXiv:0907.5580] [SPIRES].
E. Palti, G. Tasinato and J. Ward, WEAKLY-coupled IIA Flux Compactifications, JHEP 06 (2008) 084 [arXiv:0804.1248] [SPIRES].
M.R. Douglas and R. Kallosh, Compactification on negatively curved manifolds, JHEP 06 (2010) 004 [arXiv:1001.4008] [SPIRES].
D. Andriot, R. Minasian and M. Petrini, Flux backgrounds from Twists, JHEP 12 (2009) 028 [arXiv:0903.0633] [SPIRES].
G.R. Cavalcanti, M. Gualtieri, Generalized complex structures on nilmanifolds, math/0404451.
A.I. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Transl. 39 (1951) 1.
K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. Math. 59 (1954) 531.
J. Oprea and A. Tralle, Lecture Notes in Mathematics. Vol. 1661: Symplectic Manifolds with no Kähler Structure, Springer, Heidelberg Germany (1997).
Ch. Bock, On Low-Dimensional Solvmanifolds, arXiv:0903.2926.
S. Rollenske, Geometry of nilmanifolds with left-invariant complex structure and deformations in the large, arXiv:0901.3120.
L. Auslander, An exposition of the structure of solvmanifolds. Part 1: Algebraic theory, Bull. Amer. Math. Soc. 79 (1973) 227.
R. Campoamor-Stursberg, Some remarks concerning the invariants of rank one solvable real Lie algebras, Algebra Colloq. 12 (2005) 497.
G. D. Mostow, Factor spaces of solvable groups, Ann. Math. 60 (1954) 1.
G.M. Mubarakzyanov, On solvable Lie algebras, Izv. Vyssh. Uchebn. Zaved. Mat. 32 (1963) 114.
P. Turkowski, Solvable Lie algebras of dimension six, J. Math. Phys. 31 (1990) 1344.
P.G. Camara, A. Font and L.E. Ibáñez, Fluxes, moduli fixing and MSSM-like vacua in a simple IIA orientifold, JHEP 09 (2005) 013 [hep-th/0506066] [SPIRES].
M. Graña, R. Minasian, M. Petrini and A. Tomasiello, A scan for new \( \mathcal{N} = 1 \) vacua on twisted tori, JHEP 05 (2007) 031 [hep-th/0609124] [SPIRES].
D. Andriot, New supersymmetric flux vacua with intermediate SU(2) structure, JHEP 08 (2008) 096 [arXiv:0804.1769] [SPIRES].
M. Graña, R. Minasian, M. Petrini and D. Waldram, T-duality, Generalized Geometry and Non-Geometric Backgrounds, JHEP 04 (2009) 075 [arXiv:0807.4527] [SPIRES].
E. Bergshoeff, R. Kallosh, T. Ortín, D. Roest and A. Van Proeyen, New Formulations of D = 10 Supersymmetry and D8-O8 Domain Walls, Class. Quant. Grav. 18 (2001) 3359 [hep-th/0103233] [SPIRES].
D. Cassani, Reducing democratic type-II supergravity on SU(3) × SU(3) structures, JHEP 06 (2008) 027 [arXiv:0804.0595] [SPIRES].
S. Fidanza, R. Minasian and A. Tomasiello, Mirror symmetric SU(3)-structure manifolds with NS fluxes, Commun. Math. Phys. 254 (2005) 401 [hep-th/0311122] [SPIRES].
S. Kachru and A.-K. Kashani-Poor, Moduli potentials in type IIA compactifications with RR and NS flux, JHEP 03 (2005) 066 [hep-th/0411279] [SPIRES].
O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, Type IIA moduli stabilization, JHEP 07 (2005) 066 [hep-th/0505160] [SPIRES].
S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66 (2002) 106006 [hep-th/0105097] [SPIRES].
K. Skenderis and M. Taylor, Branes in AdS and pp-wave spacetimes, JHEP 06 (2002) 025 [hep-th/0204054] [SPIRES].
S. Console and A. Fino, On the de Rham Cohomology of solvmanifolds, arXiv:0912.2006.
S.M. Salomon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001) 311.
G.M. Mubarakzyanov, Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element, Izv. V yssh. Uchebn. Zaved. Mat. 35 (1963) 104.
G.M. Mubarakzyanov, Certain theorems on solvable Lie algebras, Izv. Vyssh. Uchebn. Zaved. Mat. 55 (1966) 95.
M. Saitô, Sous-groupes discrets des groupes résolubles, Am. J. Math. 83 (1961) 369.
K. Hasegawa, Four dimensional compact solvmanifolds with and without complex analytic structures, math/0401413.
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Andriot, D., Goi, E., Minasian, R. et al. Supersymmetry breaking branes on solvmanifolds and de Sitter vacua in string theory. J. High Energ. Phys. 2011, 28 (2011). https://doi.org/10.1007/JHEP05(2011)028
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DOI: https://doi.org/10.1007/JHEP05(2011)028