Abstract
Gaugino condensation on D-branes wrapping internal cycles gives a mechanism to stabilize the associated moduli. According to the effective field theory, this gives rise, when combined with fluxes, to supersymmetric AdS4 solutions. In this paper we provide a ten-dimensional description of these vacua. We first find the supersymmetry equations for type II AdS4 vacua with gaugino condensates on D-branes, in the framework of generalized complex geometry. We then solve them for type IIB compactifications with gaugino condensates on smeared D7-branes. We show that supersymmetry requires a (conformal) Calabi-Yau manifold and imaginary self-dual three-form fluxes with an additional (0,3) component. The latter is proportional to the cosmological constant, whose magnitude is determined by the expectation value of the gaugino condensate and the stabilized volume of the cycle wrapped by the branes. This confirms, qualitatively and quantitatively, the results obtained using effective field theory. We find that exponential separation between the AdS and the KK scales seems possible as long as the three-form fluxes are such that their (0,3) component is exponentially suppressed. As for the localized solution, it requires going beyond SU(3)-structure internal manifolds. Nevertheless, we show that the action can be evaluated on-shell without relying on the details of such complicated configuration. We find that no “perfect square” structure occurs, and the result is divergent. We compute the four-fermion contributions, including a counterterm, needed to cancel these divergences.
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References
S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68 (2003) 046005 [hep-th/0301240] [INSPIRE].
I. Bena, J. Blåbäck, M. Graña and S. Lüst, The tadpole problem, JHEP 11 (2021) 223 [arXiv:2010.10519] [INSPIRE].
F.F. Gautason, V. Van Hemelryck and T. Van Riet, The Tension between 10D Supergravity and dS Uplifts, Fortsch. Phys. 67 (2019) 1800091 [arXiv:1810.08518] [INSPIRE].
M. Demirtas, M. Kim, L. Mcallister and J. Moritz, Vacua with Small Flux Superpotential, Phys. Rev. Lett. 124 (2020) 211603 [arXiv:1912.10047] [INSPIRE].
M. Demirtas, M. Kim, L. McAllister and J. Moritz, Conifold Vacua with Small Flux Superpotential, Fortsch. Phys. 68 (2020) 2000085 [arXiv:2009.03312] [INSPIRE].
M. Demirtas et al., Small cosmological constants in string theory, JHEP 12 (2021) 136 [arXiv:2107.09064] [INSPIRE].
M. Demirtas et al., Exponentially Small Cosmological Constant in String Theory, Phys. Rev. Lett. 128 (2022) 011602 [arXiv:2107.09065] [INSPIRE].
M. Grana, R. Minasian, M. Petrini and A. Tomasiello, Generalized structures of N=1 vacua, JHEP 11 (2005) 020 [hep-th/0505212] [INSPIRE].
P. Koerber and L. Martucci, Warped generalized geometry compactifications, effective theories and non-perturbative effects, Fortsch. Phys. 56 (2008) 862 [arXiv:0803.3149] [INSPIRE].
A. Dymarsky and L. Martucci, D-brane non-perturbative effects and geometric deformations, JHEP 04 (2011) 061 [arXiv:1012.4018] [INSPIRE].
I. Bena, M. Graña, N. Kovensky and A. Retolaza, Kähler moduli stabilization from ten dimensions, JHEP 10 (2019) 200 [arXiv:1908.01785] [INSPIRE].
S. Kachru, M. Kim, L. Mcallister and M. Zimet, de Sitter vacua from ten dimensions, JHEP 12 (2021) 111 [arXiv:1908.04788] [INSPIRE].
P. Koerber and L. Martucci, D-branes on AdS flux compactifications, JHEP 01 (2008) 047 [arXiv:0710.5530] [INSPIRE].
P. Koerber and L. Martucci, From ten to four and back again: How to generalize the geometry, JHEP 08 (2007) 059 [arXiv:0707.1038] [INSPIRE].
I. Benmachiche and T.W. Grimm, Generalized N=1 orientifold compactifications and the Hitchin functionals, Nucl. Phys. B 748 (2006) 200 [hep-th/0602241] [INSPIRE].
M. Graña, N. Kovensky and A. Retolaza, Gaugino mass term for D-branes and Generalized Complex Geometry, JHEP 06 (2020) 047 [arXiv:2002.01481] [INSPIRE].
F. Apers, M. Montero, T. Van Riet and T. Wrase, Comments on classical AdS flux vacua with scale separation, JHEP 05 (2022) 167 [arXiv:2202.00682] [INSPIRE].
Y. Hamada, A. Hebecker, G. Shiu and P. Soler, On brane gaugino condensates in 10d, JHEP 04 (2019) 008 [arXiv:1812.06097] [INSPIRE].
Y. Hamada, A. Hebecker, G. Shiu and P. Soler, Understanding KKLT from a 10d perspective, JHEP 06 (2019) 019 [arXiv:1902.01410] [INSPIRE].
F.F. Gautason, V. Van Hemelryck, T. Van Riet and G. Venken, A 10d view on the KKLT AdS vacuum and uplifting, JHEP 06 (2020) 074 [arXiv:1902.01415] [INSPIRE].
R. Kallosh, Gaugino Condensation and Geometry of the Perfect Square, Phys. Rev. D 99 (2019) 066003 [arXiv:1901.02023] [INSPIRE].
Y. Hamada, A. Hebecker, G. Shiu and P. Soler, Completing the D7-brane local gaugino action, JHEP 11 (2021) 033 [arXiv:2105.11467] [INSPIRE].
D. Lust, F. Marchesano, L. Martucci and D. Tsimpis, Generalized non-supersymmetric flux vacua, JHEP 11 (2008) 021 [arXiv:0807.4540] [INSPIRE].
P. Koerber, Lectures on Generalized Complex Geometry for Physicists, Fortsch. Phys. 59 (2011) 169 [arXiv:1006.1536] [INSPIRE].
E. Bergshoeff et al., New formulations of D = 10 supersymmetry and D8 - O8 domain walls, Class. Quant. Grav. 18 (2001) 3359 [hep-th/0103233] [INSPIRE].
P. Koerber and D. Tsimpis, Supersymmetric sources, integrability and generalized-structure compactifications, JHEP 08 (2007) 082 [arXiv:0706.1244] [INSPIRE].
M. Grana, J. Louis and D. Waldram, Hitchin functionals in N=2 supergravity, JHEP 01 (2006) 008 [hep-th/0505264] [INSPIRE].
M. Grana, J. Louis and D. Waldram, SU(3) × SU(3) compactification and mirror duals of magnetic fluxes, JHEP 04 (2007) 101 [hep-th/0612237] [INSPIRE].
D. Cassani and A. Bilal, Effective actions and N=1 vacuum conditions from SU(3) × SU(3) compactifications, JHEP 09 (2007) 076 [arXiv:0707.3125] [INSPIRE].
M. Grana, Flux compactifications in string theory: A Comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [INSPIRE].
S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B608 (2001) 477] [hep-th/9906070] [INSPIRE].
M. Grana and J. Polchinski, Supersymmetric three form flux perturbations on AdS(5), Phys. Rev. D 63 (2001) 026001 [hep-th/0009211] [INSPIRE].
G. Veneziano and S. Yankielowicz, An Effective Lagrangian for the Pure N=1 Supersymmetric Yang-Mills Theory, Phys. Lett. B 113 (1982) 231 [INSPIRE].
A.R. Frey and M. Lippert, AdS strings with torsion: Non-complex heterotic compactifications, Phys. Rev. D 72 (2005) 126001 [hep-th/0507202] [INSPIRE].
R. Minasian, M. Petrini and E.E. Svanes, On Heterotic Vacua with Fermionic Expectation Values, Fortsch. Phys. 65 (2017) 1700010 [arXiv:1702.01156] [INSPIRE].
J. Held, D. Lust, F. Marchesano and L. Martucci, DWSB in heterotic flux compactifications, JHEP 06 (2010) 090 [arXiv:1004.0867] [INSPIRE].
L. Martucci, D-branes on general N=1 backgrounds: Superpotentials and D-terms, JHEP 06 (2006) 033 [hep-th/0602129] [INSPIRE].
F. Cachazo, K.A. Intriligator and C. Vafa, A Large N duality via a geometric transition, Nucl. Phys. B 603 (2001) 3 [hep-th/0103067] [INSPIRE].
M. Atiyah, J.M. Maldacena and C. Vafa, An M theory flop as a large N duality, J. Math. Phys. 42 (2001) 3209 [hep-th/0011256] [INSPIRE].
J. Maldacena and D. Martelli, The Unwarped, resolved, deformed conifold: Fivebranes and the baryonic branch of the Klebanov-Strassler theory, JHEP 01 (2010) 104 [arXiv:0906.0591] [INSPIRE].
J.M. Maldacena and C. Nunez, Towards the large N limit of pure N=1 superYang-Mills, Phys. Rev. Lett. 86 (2001) 588 [hep-th/0008001] [INSPIRE].
B. Heidenreich, L. McAllister and G. Torroba, Dynamic SU(2) Structure from Seven-branes, JHEP 05 (2011) 110 [arXiv:1011.3510] [INSPIRE].
D. Baumann et al., D3-brane Potentials from Fluxes in AdS/CFT, JHEP 06 (2010) 072 [arXiv:1001.5028] [INSPIRE].
O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, Type IIA moduli stabilization, JHEP 07 (2005) 066 [hep-th/0505160] [INSPIRE].
D. Junghans, O-Plane Backreaction and Scale Separation in Type IIA Flux Vacua, Fortsch. Phys. 68 (2020) 2000040 [arXiv:2003.06274] [INSPIRE].
F. Marchesano, E. Palti, J. Quirant and A. Tomasiello, On supersymmetric AdS4 orientifold vacua, JHEP 08 (2020) 087 [arXiv:2003.13578] [INSPIRE].
J. Gray, A.S. Haupt and A. Lukas, Topological Invariants and Fibration Structure of Complete Intersection Calabi-Yau Four-Folds, JHEP 09 (2014) 093 [arXiv:1405.2073] [INSPIRE].
P. Candelas, E. Perevalov and G. Rajesh, Toric geometry and enhanced gauge symmetry of F theory / heterotic vacua, Nucl. Phys. B 507 (1997) 445 [hep-th/9704097] [INSPIRE].
S. Lüst, C. Vafa, M. Wiesner and K. Xu, Holography and the KKLT scenario, JHEP 10 (2022) 188 [arXiv:2204.07171] [INSPIRE].
A. Retolaza, J. Rogers, R. Tatar and F. Tonioni, Branes, fermions, and superspace dualities, JHEP 10 (2021) 243 [Erratum ibid. 11 (2021) 124] [arXiv:2106.02090] [INSPIRE].
S. Sethi, Supersymmetry Breaking by Fluxes, JHEP 10 (2018) 022 [arXiv:1709.03554] [INSPIRE].
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Graña, M., Kovensky, N. & Toulikas, D. Smearing and unsmearing KKLT AdS vacua. J. High Energ. Phys. 2023, 15 (2023). https://doi.org/10.1007/JHEP03(2023)015
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DOI: https://doi.org/10.1007/JHEP03(2023)015