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A systematic approach to Kähler moduli stabilisation. (English) Zbl 1454.83150

Summary: Achieving full moduli stabilisation in type IIB string compactifications for generic Calabi-Yau threefolds with hundreds of Kähler moduli is notoriously hard. This is due not just to the very fast increase of the computational complexity with the number of moduli, but also to the fact that the scalar potential depends in general on the supergravity variables only implicitly. In fact, the supergravity chiral coordinates are 4- cycle volume moduli but the Kähler potential is an explicit function of the 2-cycle moduli and inverting between these two variables is in general impossible. In this paper we pro- pose a general method to fix all type IIB Kähler moduli in a systematic way by working directly in terms of 2-cycle moduli: on one side we present a ‘master formula’ for the scalar potential which can depend on an arbitrary number of Kähler moduli, while on the other we perform a computer-based search for critical points, introducing a hybrid Genetic/Clustering/Amoeba algorithm and other computational techniques. This allows us to reproduce several known minima, but also to discover new examples of both KKLT and LVS models, together with novel classes of LVS minima without diagonal del Pezzo divisors and hybrid vacua which share some features with KKLT and other with LVS solutions.

MSC:

83E50 Supergravity
83E30 String and superstring theories in gravitational theory
81T60 Supersymmetric field theories in quantum mechanics
81T33 Dimensional compactification in quantum field theory
53B35 Local differential geometry of Hermitian and Kählerian structures
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[1] P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete intersection Calabi-Yau manifolds, Nucl. Phys. B298 (1988) 493 [INSPIRE].
[2] P. Candelas, M. Lynker and R. Schimmrigk, Calabi-Yau manifolds in weighted P4 , Nucl. Phys. B341 (1990) 383 [INSPIRE]. · Zbl 0962.14029
[3] A. Constantin, J. Gray and A. Lukas, Hodge numbers for all CICY quotients, JHEP01 (2017) 001 [arXiv:1607.01830] [INSPIRE]. · Zbl 1373.14038
[4] M. Kreuzer and H. Skarke, PALP: a package for analyzing lattice polytopes with applications to toric geometry, Comput. Phys. Commun.157 (2004) 87 [math/0204356] [INSPIRE]. · Zbl 1196.14007
[5] M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys.4 (2002) 1209 [hep-th/0002240] [INSPIRE]. · Zbl 1017.52007
[6] M. Cicoli, S. Krippendorf, C. Mayrhofer, F. Quevedo and R. Valandro, D-branes at del Pezzo singularities: global embedding and moduli stabilisation, JHEP09 (2012) 019 [arXiv:1206.5237] [INSPIRE]. · Zbl 1397.81228
[7] R. Altman, J. Gray, Y.-H. He, V. Jejjala and B.D. Nelson, A Calabi-Yau database: threefolds constructed from the Kreuzer-Skarke list, JHEP02 (2015) 158 [arXiv:1411.1418] [INSPIRE]. · Zbl 1388.53071
[8] M. Demirtas, C. Long, L. McAllister and M. Stillman, The Kreuzer-Skarke axiverse, JHEP04 (2020) 138 [arXiv:1808.01282] [INSPIRE].
[9] A.P. Braun, C. Long, L. McAllister, M. Stillman and B. Sung, The Hodge numbers of divisors of Calabi-Yau threefold hypersurfaces, arXiv:1712.04946 [INSPIRE].
[10] S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D68 (2003) 046005 [hep-th/0301240] [INSPIRE]. · Zbl 1244.83036
[11] V. Balasubramanian, P. Berglund, J.P. Conlon and F. Quevedo, Systematics of moduli stabilisation in Calabi-Yau flux compactifications, JHEP03 (2005) 007 [hep-th/0502058] [INSPIRE].
[12] J.P. Conlon, F. Quevedo and K. Suruliz, Large-volume flux compactifications: moduli spectrum and D3/D7 soft supersymmetry breaking, JHEP08 (2005) 007 [hep-th/0505076] [INSPIRE].
[13] M. Cicoli, J.P. Conlon and F. Quevedo, Systematics of string loop corrections in type IIB Calabi-Yau flux compactifications, JHEP01 (2008) 052 [arXiv:0708.1873] [INSPIRE].
[14] V. Balasubramanian and P. Berglund, Stringy corrections to Kähler potentials, SUSY breaking, and the cosmological constant problem, JHEP11 (2004) 085 [hep-th/0408054] [INSPIRE].
[15] A. Westphal, De Sitter string vacua from Kähler uplifting, JHEP03 (2007) 102 [hep-th/0611332] [INSPIRE]. · Zbl 1119.81092
[16] M. Rummel and A. Westphal, A sufficient condition for de Sitter vacua in type IIB string theory, JHEP01 (2012) 020 [arXiv:1107.2115] [INSPIRE]. · Zbl 1306.81273
[17] I. Ben-Dayan, S. Jing, A. Westphal and C. Wieck, Accidental inflation from Kähler uplifting, JCAP03 (2014) 054 [arXiv:1309.0529] [INSPIRE].
[18] K. Becker, M. Becker, M. Haack and J. Louis, Supersymmetry breaking and α^Mcorrections to flux induced potentials, JHEP06 (2002) 060 [hep-th/0204254] [INSPIRE].
[19] M. Berg, M. Haack and B. Körs, String loop corrections to Kähler potentials in orientifolds, JHEP11 (2005) 030 [hep-th/0508043] [INSPIRE].
[20] M. Berg, M. Haack and E. Pajer, Jumping through loops: on soft terms from large volume compactifications, JHEP09 (2007) 031 [arXiv:0704.0737] [INSPIRE].
[21] D. Ciupke, J. Louis and A. Westphal, Higher-derivative supergravity and moduli stabilization, JHEP10 (2015) 094 [arXiv:1505.03092] [INSPIRE]. · Zbl 1388.83776
[22] M. Cicoli, J.P. Conlon and F. Quevedo, General analysis of LARGE volume scenarios with string loop moduli stabilisation, JHEP10 (2008) 105 [arXiv:0805.1029] [INSPIRE]. · Zbl 1245.81155
[23] M. Cicoli, C.P. Burgess and F. Quevedo, Fibre inflation: observable gravity waves from IIB string compactifications, JCAP03 (2009) 013 [arXiv:0808.0691] [INSPIRE].
[24] M. Cicoli, C. Mayrhofer and R. Valandro, Moduli stabilisation for chiral global models, JHEP02 (2012) 062 [arXiv:1110.3333] [INSPIRE]. · Zbl 1309.81208
[25] M. Cicoli, F. Muia and P. Shukla, Global embedding of fibre inflation models, JHEP11 (2016) 182 [arXiv:1611.04612] [INSPIRE]. · Zbl 1390.83441
[26] M. Cicoli, D. Ciupke, V.A. Diaz, V. Guidetti, F. Muia and P. Shukla, Chiral global embedding of fibre inflation models, JHEP11 (2017) 207 [arXiv:1709.01518] [INSPIRE]. · Zbl 1383.83227
[27] C.P. Burgess, M. Cicoli, M. Gomez-Reino, F. Quevedo, G. Tasinato and I. Zavala, Non-standard primordial fluctuations and non-Gaussianity in string inflation, JHEP08 (2010) 045 [arXiv:1005.4840] [INSPIRE]. · Zbl 1291.83207
[28] M. Cicoli, D. Ciupke, S. de Alwis and F. Muia, α′ inflation: moduli stabilisation and observable tensors from higher derivatives, JHEP09 (2016) 026 [arXiv:1607.01395] [INSPIRE]. · Zbl 1390.83440
[29] D. Lüst, S. Reffert, E. Scheidegger, W. Schulgin and S. Stieberger, Moduli stabilization in type IIB orientifolds (II), Nucl. Phys. B766 (2007) 178 [hep-th/0609013] [INSPIRE]. · Zbl 1119.81089
[30] D. Lüst, S. Reffert, E. Scheidegger and S. Stieberger, Resolved toroidal orbifolds and their orientifolds, Adv. Theor. Math. Phys.12 (2008) 67 [hep-th/0609014] [INSPIRE]. · Zbl 1152.81883
[31] R. Blumenhagen, V. Braun, T.W. Grimm and T. Weigand, GUTs in type IIB orientifold compactifications, Nucl. Phys. B815 (2009) 1 [arXiv:0811.2936] [INSPIRE]. · Zbl 1194.81288
[32] X. Gao and P. Shukla, F-term stabilization of odd axions in LARGE volume scenario, Nucl. Phys. B878 (2014) 269 [arXiv:1307.1141] [INSPIRE]. · Zbl 1284.81232
[33] X. Gao and P. Shukla, On classifying the divisor involutions in Calabi-Yau threefolds, JHEP11 (2013) 170 [arXiv:1307.1139] [INSPIRE]. · Zbl 1342.81425
[34] D. Robbins and T. Wrase, D-terms from generalized NS-NS fluxes in type-II, JHEP12 (2007) 058 [arXiv:0709.2186] [INSPIRE]. · Zbl 1246.81296
[35] T.W. Grimm and J. Louis, The effective action of N = 1 Calabi-Yau orientifolds, Nucl. Phys. B699 (2004) 387 [hep-th/0403067] [INSPIRE]. · Zbl 1123.81393
[36] S. Hosono, A. Klemm and S. Theisen, Lectures on mirror symmetry, Lect. Notes Phys.436 (1994) 235 [hep-th/9403096] [INSPIRE]. · Zbl 0812.53061
[37] I. Benmachiche and T.W. Grimm, Generalized N = 1 orientifold compactifications and the Hitchin functionals, Nucl. Phys. B748 (2006) 200 [hep-th/0602241] [INSPIRE]. · Zbl 1186.81101
[38] M. Cicoli, S. De Alwis, A. Maharana, F. Muia and F. Quevedo, De Sitter vs. quintessence in string theory, Fortsch. Phys.67 (2019) 1800079 [arXiv:1808.08967] [INSPIRE]. · Zbl 07762667
[39] K. Bobkov, Volume stabilization via α^Mcorrections in type IIB theory with fluxes, JHEP05 (2005) 010 [hep-th/0412239] [INSPIRE].
[40] M. Bianchi, A. Collinucci and L. Martucci, Magnetized E3-brane instantons in F-theory, JHEP12 (2011) 045 [arXiv:1107.3732] [INSPIRE]. · Zbl 1306.81197
[41] M. Bianchi, A. Collinucci and L. Martucci, Freezing E3-brane instantons with fluxes, Fortsch. Phys.60 (2012) 914 [arXiv:1202.5045] [INSPIRE]. · Zbl 1254.81069
[42] J. Louis, M. Rummel, R. Valandro and A. Westphal, Building an explicit de Sitter, JHEP10 (2012) 163 [arXiv:1208.3208] [INSPIRE]. · Zbl 1397.83201
[43] P. Shukla, A dictionary for the type-II non-geometric flux compactifications, arXiv:1909.07391 [INSPIRE].
[44] F. Denef, M.R. Douglas and B. Florea, Building a better racetrack, JHEP06 (2004) 034 [hep-th/0404257] [INSPIRE].
[45] J.J. Blanco-Pillado et al., Inflating in a better racetrack, JHEP09 (2006) 002 [hep-th/0603129] [INSPIRE].
[46] M. Cicoli, S. Krippendorf, C. Mayrhofer, F. Quevedo and R. Valandro, D3/D7 branes at singularities: constraints from global embedding and moduli stabilisation, JHEP07 (2013) 150 [arXiv:1304.0022] [INSPIRE]. · Zbl 1342.83345
[47] M. Cicoli, D. Klevers, S. Krippendorf, C. Mayrhofer, F. Quevedo and R. Valandro, Explicit de Sitter flux vacua for global string models with chiral matter, JHEP05 (2014) 001 [arXiv:1312.0014] [INSPIRE]. · Zbl 1333.83178
[48] J.P. Conlon and F. Quevedo, Kähler moduli inflation, JHEP01 (2006) 146 [hep-th/0509012] [INSPIRE].
[49] M. Cicoli, I. Garcìa-Etxebarria, C. Mayrhofer, F. Quevedo, P. Shukla and R. Valandro, Global orientifolded quivers with inflation, JHEP11 (2017) 134 [arXiv:1706.06128] [INSPIRE]. · Zbl 1383.83228
[50] C.P. Burgess, M. Cicoli, S. de Alwis and F. Quevedo, Robust inflation from fibrous strings, JCAP05 (2016) 032 [arXiv:1603.06789] [INSPIRE].
[51] M. Cicoli, J.P. Conlon, A. Maharana and F. Quevedo, A note on the magnitude of the flux superpotential, JHEP01 (2014) 027 [arXiv:1310.6694] [INSPIRE].
[52] R. Blumenhagen, B. Jurke, T. Rahn and H. Roschy, Cohomology of line bundles: a computational algorithm, J. Math. Phys.51 (2010) 103525 [arXiv:1003.5217] [INSPIRE]. · Zbl 1314.55012
[53] R. Blumenhagen, B. Jurke and T. Rahn, Computational tools for cohomology of toric varieties, Adv. High Energy Phys.2011 (2011) 152749 [arXiv:1104.1187] [INSPIRE]. · Zbl 1234.81107
[54] K. Oguiso, On algebraic fiber space structures on a Calabi-Yau 3-fold, Int. J. Math.04 (1993) 439. · Zbl 0793.14030
[55] M.B. Schulz, Calabi-Yau duals of torus orientifolds, JHEP05 (2006) 023 [hep-th/0412270] [INSPIRE].
[56] M. Cicoli, D. Ciupke, C. Mayrhofer and P. Shukla, A geometrical upper bound on the inflaton range, JHEP05 (2018) 001 [arXiv:1801.05434] [INSPIRE]. · Zbl 1391.83136
[57] R. Blumenhagen, X. Gao, T. Rahn and P. Shukla, A note on poly-instanton effects in type IIB orientifolds on Calabi-Yau threefolds, JHEP06 (2012) 162 [arXiv:1205.2485] [INSPIRE]. · Zbl 1397.81226
[58] M. Cicoli, F.G. Pedro and G. Tasinato, Poly-instanton inflation, JCAP12 (2011) 022 [arXiv:1110.6182] [INSPIRE].
[59] M. Cicoli, F.G. Pedro and G. Tasinato, Natural quintessence in string theory, JCAP07 (2012) 044 [arXiv:1203.6655] [INSPIRE].
[60] R. Blumenhagen, X. Gao, T. Rahn and P. Shukla, Moduli stabilization and inflationary cosmology with poly-instantons in type IIB orientifolds, JHEP11 (2012) 101 [arXiv:1208.1160] [INSPIRE]. · Zbl 1397.83129
[61] X. Gao and P. Shukla, On non-Gaussianities in two-field poly-instanton inflation, JHEP03 (2013) 061 [arXiv:1301.6076] [INSPIRE].
[62] X. Gao, T. Li and P. Shukla, Cosmological observables in multi-field inflation with a non-flat field space, JCAP10 (2014) 008 [arXiv:1403.0654] [INSPIRE].
[63] T. Kobayashi, S. Uemura and J. Yamamoto, Polyinstanton axion inflation, Phys. Rev. D96 (2017) 026007 [arXiv:1705.04088] [INSPIRE].
[64] J.P. Conlon, Quantum gravity constraints on inflation, JCAP09 (2012) 019 [arXiv:1203.5476] [INSPIRE].
[65] M. Cicoli, A. Maharana, F. Quevedo and C.P. Burgess, De Sitter string vacua from dilaton-dependent non-perturbative effects, JHEP06 (2012) 011 [arXiv:1203.1750] [INSPIRE]. · Zbl 1397.81229
[66] M. Cicoli, F. Quevedo and R. Valandro, De Sitter from T-branes, JHEP03 (2016) 141 [arXiv:1512.04558] [INSPIRE].
[67] T.C. Bachlechner, K. Eckerle, O. Janssen and M. Kleban, Multiple-axion framework, Phys. Rev. D98 (2018) 061301 [arXiv:1703.00453] [INSPIRE].
[68] T.C. Bachlechner, K. Eckerle, O. Janssen and M. Kleban, Systematics of aligned axions, JHEP11 (2017) 036 [arXiv:1709.01080] [INSPIRE]. · Zbl 1383.83151
[69] T.C. Bachlechner, K. Eckerle, O. Janssen and M. Kleban, The axidental universe, arXiv:1902.05952 [INSPIRE].
[70] M. Cicoli, M. Kreuzer and C. Mayrhofer, Toric K 3-fibred Calabi-Yau manifolds with del Pezzo divisors for string compactifications, JHEP02 (2012) 002 [arXiv:1107.0383] [INSPIRE]. · Zbl 1309.81149
[71] A. Neumaier, Complete search in continuous global optimization and constraint satisfaction, Cambridge University Press, Cambridge, U.K. (2004). · Zbl 1113.90124
[72] C. Malherbe and N. Vayatis, Global optimization of Lipschitz functions, in Proceedings of the 34^thInternational Conference on Machine Learning, D. Precup and Y.W. Teh eds., Proc. Machine Learn. Res.70, International Convention Centre, Sydney, NSW, Australia, 06-11 August 2017, pg. 2314.
[73] J. Holland, Adaptation in natural and artificial systems, MIT Press, Cambridge, MA, U.S.A. (1992).
[74] E. David, Genetic algorithms in search, optimization and machine learning, Addison-Wesley Longman Publishing Co., Boston, MA, U.S.A. (1989). · Zbl 0721.68056
[75] J. Holland, The royal road for genetic algorithms: fitness landscapes and GA performance, technical report, (1992).
[76] C.R. Reeves and J.E. Rowe, Genetic algorithms: principles and perspectives, Springer, New York, NY, U.S.A. (2002).
[77] J.A. Nelder and R. Mead, A simplex method for function minimization, Comput. J.7 (1965) 308. · Zbl 0229.65053
[78] J.C. Lagarias, J.A. Reeds, M.H. Wright and P.E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optim.9 (1998) 112. · Zbl 1005.90056
[79] T.G. Kolda, R.M. Lewis and V. Torczon, Optimization by direct search: new perspectives on some classical and modern methods, SIAM Rev.45 (2003) 385. · Zbl 1059.90146
[80] T.S. Metcalfe, R.E. Nather and D.E. Winget, Genetic-algorithm-based asteroseismological analysis of the DBV white dwarf GD 358, Astrophys. J.545 (2000) 974 [astro-ph/0008022] [INSPIRE].
[81] B.C. Allanach, D. Grellscheid and F. Quevedo, Genetic algorithms and experimental discrimination of SUSY models, JHEP07 (2004) 069 [hep-ph/0406277] [INSPIRE].
[82] M. Mokiem, A. de Koter, J. Puls, A. Herrero, F. Najarro and M.R. Villamariz, Spectral analysis of early-type stars using a genetic algorithm based fitting method, Astron. Astrophys.441 (2005) 711 [astro-ph/0506751] [INSPIRE].
[83] Y. Akrami, P. Scott, J. Edsjo, J. Conrad and L. Bergstrom, A profile likelihood analysis of the constrained MSSM with genetic algorithms, JHEP04 (2010) 057 [arXiv:0910.3950] [INSPIRE]. · Zbl 1272.81204
[84] S. Nesseris and J. Garćıa-Bellido, A new perspective on dark energy modeling via genetic algorithms, JCAP11 (2012) 033 [arXiv:1205.0364] [INSPIRE].
[85] J. Blåbäck, U. Danielsson and G. Dibitetto, Fully stable dS vacua from generalised fluxes, JHEP08 (2013) 054 [arXiv:1301.7073] [INSPIRE]. · Zbl 1342.83334
[86] C. Damian, L.R. Diaz-Barron, O. Loaiza-Brito and M. Sabido, Slow-roll inflation in non-geometric flux compactification, JHEP06 (2013) 109 [arXiv:1302.0529] [INSPIRE]. · Zbl 1342.83352
[87] C. Damian and O. Loaiza-Brito, More stable de Sitter vacua from S-dual nongeometric fluxes, Phys. Rev. D88 (2013) 046008 [arXiv:1304.0792] [INSPIRE].
[88] J. Blåbäck, U. Danielsson and G. Dibitetto, Accelerated universes from type IIA compactifications, JCAP03 (2014) 003 [arXiv:1310.8300] [INSPIRE].
[89] J. Blåbäck, D. Roest and I. Zavala, De Sitter vacua from nonperturbative flux compactifications, Phys. Rev. D90 (2014) 024065 [arXiv:1312.5328] [INSPIRE].
[90] S. Abel and J. Rizos, Genetic algorithms and the search for viable string vacua, JHEP08 (2014) 010 [arXiv:1404.7359] [INSPIRE].
[91] R. Hogan, M. Fairbairn and N. Seeburn, GAz: a genetic algorithm for photometric redshift estimation, Mon. Not. Roy. Astron. Soc.449 (2015) 2040 [arXiv:1412.5997] [INSPIRE].
[92] F. Ruehle, Evolving neural networks with genetic algorithms to study the string landscape, JHEP08 (2017) 038 [arXiv:1706.07024] [INSPIRE]. · Zbl 1381.83128
[93] S. Abel, D.G. Cerdeño and S. Robles, The power of genetic algorithms: what remains of the pMSSM?, arXiv:1805.03615 [INSPIRE].
[94] K. Bull, Y.-H. He, V. Jejjala and C. Mishra, Getting CICY high, Phys. Lett. B795 (2019) 700 [arXiv:1903.03113] [INSPIRE]. · Zbl 1420.14002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.