Abstract
We construct non-perturbatively exact four-dimensional Minkowski vacua of type IIB string theory with non-trivial fluxes. These solutions are found by gluing together, consistently with U-duality, local solutions of type IIB supergravity on \( {T}^4\times \mathbb{C} \) with the metric, dilaton and flux potentials varying along \( \mathbb{C} \) and the flux potentials oriented along T 4. We focus on solutions locally related via U-duality to non-compact Ricci-flat geometries. More general solutions and a complete analysis of the supersymmetry equations are presented in the companion paper [1]. We build a precise dictionary between fluxes in the global solutions and the geometry of an auxiliary K3 surface fibered over \( \mathbb{C}{\mathrm{\mathbb{P}}}^1 \). In the spirit of F-theory, the flux potentials are expressed in terms of locally holomorphic functions that parametrize the complex structure moduli space of the K3 fiber in the auxiliary geometry. The brane content is inferred from the monodromy data around the degeneration points of the fiber.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Candelas, A. Constantin, C. Damian, M. Larfors and J.F. Morales, Type IIB flux vacua from G-theory II, arXiv:1411.4786 [INSPIRE].
M. Graña, R. Minasian, M. Petrini and A. Tomasiello, Supersymmetric backgrounds from generalized Calabi-Yau manifolds, JHEP 08 (2004) 046 [hep-th/0406137] [INSPIRE].
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] [INSPIRE].
S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66 (2002) 106006 [hep-th/0105097] [INSPIRE].
L. Martucci, J.F. Morales and D.R. Pacifici, Branes, U-folds and hyperelliptic fibrations, JHEP 01 (2013) 145 [arXiv:1207.6120] [INSPIRE].
A.P. Braun, F. Fucito and J.F. Morales, U-folds as K3 fibrations, JHEP 10 (2013) 154 [arXiv:1308.0553] [INSPIRE].
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
A. Kumar and C. Vafa, U manifolds, Phys. Lett. B 396 (1997) 85 [hep-th/9611007] [INSPIRE].
J.T. Liu and R. Minasian, U-branes and T 3 fibrations, Nucl. Phys. B 510 (1998) 538 [hep-th/9707125] [INSPIRE].
S. Hellerman, J. McGreevy and B. Williams, Geometric constructions of nongeometric string theories, JHEP 01 (2004) 024 [hep-th/0208174] [INSPIRE].
C.M. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].
A. Flournoy, B. Wecht and B. Williams, Constructing nongeometric vacua in string theory, Nucl. Phys. B 706 (2005) 127 [hep-th/0404217] [INSPIRE].
A. Dabholkar and C. Hull, Generalised T-duality and non-geometric backgrounds, JHEP 05 (2006) 009 [hep-th/0512005] [INSPIRE].
J. Gray and E.J. Hackett-Jones, On T-folds, G-structures and supersymmetry, JHEP 05 (2006) 071 [hep-th/0506092] [INSPIRE].
C.M. Hull, Generalised geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].
D. Vegh and J. McGreevy, Semi-flatland, JHEP 10 (2008) 068 [arXiv:0808.1569] [INSPIRE].
P.P. Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [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].
J. McOrist, D.R. Morrison and S. Sethi, Geometries, non-geometries and fluxes, Adv. Theor. Math. Phys. 14 (2010) [arXiv:1004.5447] [INSPIRE].
D. Andriot, M. Larfors, D. Lüst and P. Patalong, A ten-dimensional action for non-geometric fluxes, JHEP 09 (2011) 134 [arXiv:1106.4015] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry I: type II theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].
D.S. Berman, H. Godazgar, M. Godazgar and M.J. Perry, The local symmetries of M-theory and their formulation in generalised geometry, JHEP 01 (2012) 012 [arXiv:1110.3930] [INSPIRE].
D.S. Berman, H. Godazgar, M.J. Perry and P. West, Duality invariant actions and generalised geometry, JHEP 02 (2012) 108 [arXiv:1111.0459] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, \( {E_d}_{(d)}\times {\mathbb{R}}^{+} \) generalised geometry, connections and M-theory, JHEP 02 (2014) 054 [arXiv:1112.3989] [INSPIRE].
O. Hohm and B. Zwiebach, On the Riemann tensor in double field theory, JHEP 05 (2012) 126 [arXiv:1112.5296] [INSPIRE].
D. Andriot, O. Hohm, M. Larfors, D. Lüst and P. Patalong, A geometric action for non-geometric fluxes, Phys. Rev. Lett. 108 (2012) 261602 [arXiv:1202.3060] [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, Non-geometric strings, symplectic gravity and differential geometry of Lie algebroids, JHEP 02 (2013) 122 [arXiv:1211.0030] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry II: \( {E_d}_{(d)}\times {\mathbb{R}}^{+} \) and M-theory, JHEP 03 (2014) 019 [arXiv:1212.1586] [INSPIRE].
G. Aldazabal, M. Graña, D. Marqués and J.A. Rosabal, Extended geometry and gauged maximal supergravity, JHEP 06 (2013) 046 [arXiv:1302.5419] [INSPIRE].
M. Cederwall, J. Edlund and A. Karlsson, Exceptional geometry and tensor fields, JHEP 07 (2013) 028 [arXiv:1302.6736] [INSPIRE].
R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke and C. Schmid, The intriguing structure of non-geometric frames in string theory, Fortsch. Phys. 61 (2013) 893 [arXiv:1304.2784] [INSPIRE].
D. Andriot and A. Betz, β-supergravity: a ten-dimensional theory with non-geometric fluxes and its geometric framework, JHEP 12 (2013) 083 [arXiv:1306.4381] [INSPIRE].
M. Cederwall, T-duality and non-geometric solutions from double geometry, Fortsch. Phys. 62 (2014) 942 [arXiv:1409.4463] [INSPIRE].
J. de Boer and M. Shigemori, Exotic branes in string theory, Phys. Rept. 532 (2013) 65 [arXiv:1209.6056] [INSPIRE].
N.J. Hitchin, The geometry of three-forms in six and seven dimensions, math/0010054 [INSPIRE].
M. Larfors, D. Lüst and D. Tsimpis, Flux compactification on smooth, compact three-dimensional toric varieties, JHEP 07 (2010) 073 [arXiv:1005.2194] [INSPIRE].
T.H. Buscher, A symmetry of the string background field equations, Phys. Lett. B 194 (1987) 59 [INSPIRE].
T.H. Buscher, Path integral derivation of quantum duality in nonlinear σ-models, Phys. Lett. B 201 (1988) 466 [INSPIRE].
O. Lunin and S.D. Mathur, Metric of the multiply wound rotating string, Nucl. Phys. B 610 (2001) 49 [hep-th/0105136] [INSPIRE].
P.S. Aspinwall, K3 surfaces and string duality, hep-th/9611137 [INSPIRE].
B.R. Greene, A.D. Shapere, C. Vafa and S.-T. Yau, Stringy cosmic strings and noncompact Calabi-Yau manifolds, Nucl. Phys. B 337 (1990) 1 [INSPIRE].
V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003] [INSPIRE].
M. Kreuzer and H. Skarke, Classification of reflexive polyhedra in three-dimensions, Adv. Theor. Math. Phys. 2 (1998) 847 [hep-th/9805190] [INSPIRE].
P. Berglund et al., Periods for Calabi-Yau and Landau-Ginzburg vacua, Nucl. Phys. B 419 (1994) 352 [hep-th/9308005] [INSPIRE].
P. Griffiths, On the periods of certain rational integrals. I, Ann. Math. 90 (1969) 460.
P. Griffiths, On the periods of certain rational integrals. II, Ann. Math. 90 (1969) 466.
D.R. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces, hep-th/9111025 [INSPIRE].
P. Candelas, X. De La Ossa, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. 1., Nucl. Phys. B 416 (1994) 481 [hep-th/9308083] [INSPIRE].
P. Candelas, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. 2., Nucl. Phys. B 429 (1994) 626 [hep-th/9403187] [INSPIRE].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys. B 433 (1995) 501 [hep-th/9406055] [INSPIRE].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995) 301 [hep-th/9308122] [INSPIRE].
M. Bianchi, J.F. Morales and G. Pradisi, Discrete torsion in nongeometric orbifolds and their open string descendants, Nucl. Phys. B 573 (2000) 314 [hep-th/9910228] [INSPIRE].
V. Braun, Toric elliptic fibrations and F-theory compactifications, JHEP 01 (2013) 016 [arXiv:1110.4883] [INSPIRE].
A. Malmendier and D.R. Morrison, K3 surfaces, modular forms and non-geometric heterotic compactifications, arXiv:1406.4873 [INSPIRE].
M. Billó et al., Non-perturbative gauge/gravity correspondence in N = 2 theories, JHEP 08 (2012) 166 [arXiv:1206.3914] [INSPIRE].
W. Fulton, Introduction to toric varieties. The 1989 William H. Roever lectures in geometry, Annals of Mathematics Studies volume 131, Princeton University Press, Princeton U.S.A. (1993).
D.A. Cox, J.B. Little and H.K. Schenck, Toric varieties, Graduate Studies In Mathematics volume 124, American Mathematical Society, U.S.A. (2011).
H. Skarke, String dualities and toric geometry: an introduction, Chaos Solitons Fractals 10 (1999) 543 [hep-th/9806059] [INSPIRE].
A.C. Avram, M. Kreuzer, M. Mandelberg and H. Skarke, Searching for K3 fibrations, Nucl. Phys. B 494 (1997) 567 [hep-th/9610154] [INSPIRE].
P. Candelas and A. Font, Duality between the webs of heterotic and type-II vacua, Nucl. Phys. B 511 (1998) 295 [hep-th/9603170] [INSPIRE].
P. Candelas and H. Skarke, F theory, SO(32) and toric geometry, Phys. Lett. B 413 (1997) 63 [hep-th/9706226] [INSPIRE].
P. Candelas, A. Constantin and H. Skarke, An abundance of K3 fibrations from polyhedra with interchangeable parts, Commun. Math. Phys. 324 (2013) 937 [arXiv:1207.4792] [INSPIRE].
M. Kreuzer and H. Skarke, Calabi-Yau four folds and toric fibrations, J. Geom. Phys. 26 (1998) 272 [hep-th/9701175] [INSPIRE].
D. Chialva, U.H. Danielsson, N. Johansson, M. Larfors and M. Vonk, Deforming, revolving and resolving — New paths in the string theory landscape, JHEP 02 (2008) 016 [arXiv:0710.0620] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1411.4785
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Candelas, P., Constantin, A., Damian, C. et al. Type IIB flux vacua from G-theory I. J. High Energ. Phys. 2015, 187 (2015). https://doi.org/10.1007/JHEP02(2015)187
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2015)187