We analyse geometric type IIA flux compactifications leading to N=4 gauged
supergravities in four dimensions. The complete landscape of isotropic vacua is
presented, which turns out to belong to a unique theory. The solutions admit an
uplift to maximal supergravity due to the vanishing of the flux-induced
tadpoles for all the supersymmetry-breaking branes. Such an uplift is sketched
out and the full N=8 mass spectra are discussed. We find the interesting
presence of a non-supersymmetric and nevertheless stable minimum.Comment: 7 pages, contribution to the proceedings of the XVII European
  Workshop on String Theory, Padua 201

Borghese

de Wit

Dibitetto

Gianni

Le Diffon

Schon

Shelton

English

arXiv

We analyse geometric type IIA flux compactifications leading to ?? = 4 gauged supergravities in four dimensions. The complete landscape of isotropic vacua is presented, which turns out to belong to a unique theory. The solutions admit an uplift to maximal supergravity due to the vanishing of the flux-induced tadpoles for all the supersymmetry-breaking branes. Such an uplift is sketched out and the full ?? = 8 mass spectra are discussed. We find the interesting presence of a non-supersymmetric and nevertheless stable minimum.</p

Dibitetto, Giuseppe

Guarino, Adolfo

Roest, Diederik

University of Groningen

University of GroningenVacua analysis in extended supersymmetry compactificationsDibitetto, Giuseppe; Guarino, Adolfo; Roest, DiederikPublished in:Fortschritte der physik-Progress of physicsDOI:10.1002/prop.201200004IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dibitetto, G., Guarino, A., & Roest, D. (2012). Vacua analysis in extended supersymmetrycompactifications. Fortschritte der physik-Progress of physics, 60(9-10), 987-990.https://doi.org/10.1002/prop.201200004CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 30-12-2025Fortschr. Phys. 60, No. 9 – 10, 987 – 990 (2012) / DOI 10.1002/prop.201200004Vacua analysis in extended supersymmetry compactificationsGiuseppe Dibitetto∗, Adolfo Guarino, and Diederik RoestCentre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The NetherlandsReceived 11 January 2012, accepted 11 January 2012Published online 27 January 2012Key words Flux compactifications, extended supersymmetry, supergravity models.We analyse geometric type IIA flux compactifications leading to N = 4 gauged supergravities in fourdimensions. The complete landscape of isotropic vacua is presented, which turns out to belong to a uniquetheory. The solutions admit an uplift to maximal supergravity due to the vanishing of the flux-inducedtadpoles for all the supersymmetry-breaking branes. Such an uplift is sketched out and the full N = 8 massspectra are discussed. We find the interesting presence of a non-supersymmetric and nevertheless stableminimum.© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim1 IntroductionHalf-maximal and maximal gauged supergravities in four dimensions are the low-energy effective theoriesarising from flux compactifications in string theory, provided that only internal manifolds and extendedobjects are included which are compatible with such amounts of supersymmetry. In the last decade theembedding tensor formalism has been used extensively in the context of (half-)maximal supergravity inorder to describe all the deformations of the free theory in a duality-covariant way. Nevertheless, it hasalready been pointed out in the literature that not all the gaugings of supergravity have a higher-dimensionalorigin in terms of a geometric flux compactification in string theory. This indicates that gaugings comingfrom geometric flux compactifications are not a closed set under general duality transformations and thisis the origin of non-geometric fluxes [1]. Gaugings associated with such fluxes might yet have a higher-dimensional description in terms of a double field theory (DFT) [2,3], in which duality covariance becomesthe fundamental principle to start with, independently of the compactification procedure. Because of thisinterpretation, gauged supergravities with extended supersymmetry seem to be suitable frameworks forinvestigating the nature of non-geometric fluxes, as summarised in Table 1.Table 1 Half-maximal and maximal gauged supegravities in four dimensions seem to be suitable play-grounds to understand how to restore duality covariance in flux compactifications.SUSY G stringy interpretationN = 4 SL(2)× SO(6, 6) S- and T-dualityN = 8 E7(7) U-dualityAfter introducing the SO(3) truncation of half-maximal supergravity as the effective theory arisingfrom specific type IIA orientifold reductions with background fluxes, we analyse the landscape of isotropicvacua in geometric backgrounds, i.e. including metric and gauge fluxes. We find a set of anti-de Sitter(AdS) critical points admitting an uplift to N = 8, one of which is remarkably stable without preserving∗ Corresponding author E-mail: g.dibitetto@rug.nl, Phone: +31 (0)50 363 9043, Fax: +31 (0)50 363 4947© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim988 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsany supersymmetry. More details on this work can be found in [4–6]. The goal of finding de Sitter (dS)vacua motivates the analysis of non-geometric flux compactifications as possible future extensions.2 The geometric type IIA with O6/D6 setupUpon the SO(3) truncation, half-maximal supergravity in four dimensions reduces to a three-field STUmodel in the following waySL(2) × SO(6, 6) −→ SL(2) × SO(2, 2) = SL(2)S × SL(2)T × SL(2)U . (1)The truncation allows for forty SO(3)-singlet embedding tensor components that can be written in terms ofan SL(2) × SO(2, 2) tensor ΛαABC = Λα(ABC), with α = ± and A = 1, ..., 4 being SL(2) and SO(2, 2)fundamental indices, respectively. As shown in [4], these forty embedding tensor components correspondexactly with the set of generalised fluxes in type II orientifold reductions on a Z2 × Z2 isotropic orbifold.As a consequence of the truncation, the theory preserves only N = 1 supersymmetry out of the originalN = 4. Therefore, the corresponding scalar potential can be written in terms of a (logarithmic) Kählerpotential K and a holomorphic superpotential W by using the standard N = 1 expressionV (SO(3)) = eK(−3|W |2 + |D W |2) . (2)At the effective level, fluxes appear as arbitrary superpotential couplings up to linear in S and up to cubicin T and U . The scalar potential computed from (2) turns out to coincide with the scalar potential givenin [7] up to terms projected out by a set of N = 4 quadratic constraints on the embedding tensor Λ [4].Table 2 The set of SO(3)-invariant embedding tensor components of Λ admitting a higher-dimensionalorigin as type IIA fluxes: a metric flux ω together with R-R F0,2,4,6 and NS-NS H3 gauge fluxes.W couplings Type IIA fluxes Embedding tensor componentsa0 F6 −Λ+333a1 F4 Λ+334a2 F2 −Λ+344a3 F0 Λ+444b0 H3 −Λ−333b1 ω Λ−334c0 H3 Λ+233c1 , c̃1 ω Λ+234 , Λ+133Restricting to the components of Λ that can be interpreted as metric or gauge fluxes in a type IIArealisation of the model, we are left with the following superpotentialWIIA = a0 − 3 a1 U + 3 a2 U2 − a3 U3 − b0 S + 3 b1 S U + 3 c0 T + (6 c1 − 3 c̃1)T U , (3)consisting of nine flux-induced couplings (see Table 2). These fluxes are demanded to satisfy the set ofN = 4 quadratic constraintsc1 (c1 − c̃1) = 0 , b1 (c1 − c̃1) = 0 (ω2 = 0) ,−a3 c0 − a2 (2 c1 − c̃1) = 0 (N6⊥ = 0) ,(4)where the first two constraints are related to the nilpotency of the twisted exterior derivative operator,whereas the third one imposes the absence of D6-branes wrapping the directions orthogonal to the O6-planes, which would break supersymmetry explicitly down to N = 1. In contrast, D6-branes parallel to© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.orgFortschr. Phys. 60, No. 9 – 10 (2012) 989the O6-planes are compatible with N = 4 supersymmetry, hence being allowed. Their correspondingflux-induced tadpole readsN6|| = 3 a2 b1 − a3 b0 = NO62 − ND6 . (5)After observing that the set of fluxes given in Table 2 is closed under non-compact duality transfor-mations (i.e. real shifts and rescalings of S, T and U ), we can restrict the search for critical points of thescalar potential to the point S0 = T0 = U0 = i without losing generality. The field equations becomethen quadratic conditions in the fluxes that have to be satisfied together with the quadratic constraints in(4). These equations generate a quadratic ideal which we decompose in terms of prime ideals by usingthe Gianni-Trager-Zacharias (GTZ) decomposition [8] with the help of Singular [9]. By solving them, thecomplete set of critical points of the scalar potential is presented in Table 3. They turn out to be (modulothe discrete Z2 symmetry introduced in the caption) different AdS critical points of a unique theory withan underlying gauging given by the gauge group G0 =ISO(3)  U(1)6.Table 3 The set of critical points of the scalar potential for geometric type IIA isotropic flux compactifications. Thesolutions labelled with 1s turn out to preserve N = 1 supersymmetry for s = +1 and to be non-supersymmetric fors = −1, whereas all the others are non-supersymmetric. It is worth noticing that s = ± 1 appears as an accidental Z2symmetry which relates solutions having exactly the same energy and the same mass spectrum. The parameter λ is aglobal scaling parameter such that V ∝ λ2.ID a0 a1 a2 a3 b0 b1 c0 c1 = c̃11s s3√102λ√62λ −s√106λ5√66λ −s√63λ√103λ s√63λ√10λ2s s16√109λ 0 016√29λ 016√1045λ 016√1015λ3s s4√105λ − 4√3015λ s4√1015λ s4√3015λ s4√3015λ4√1015λ −s 4√3015λ4√105λ4s s16√109λ 0 016√29λ 016√29λ 016√29λRegarding stability at the critical points, we computed in [4] the full mass matrix for the 38 physicalscalars in N = 4 making use of the results in [10]. At this point, we can say two things: firstly, thesolutions 1s are fully stable because of (fake-) supersymmetry, and secondly the solutions 2s are alreadyunstable because of the presence of a tachyon whose mass is below the Breitenlohner-Freedman (BF)bound. However, this is not enough since all the critical points turn out to be compatible with the totalabsence of sources, i.e., the flux-induced tadpole (5) does accidentally vanish at these points. As a result,they admit an uplift to N = 8 and hence one should analyse the full mass matrix for the 70 scalars spanningthe coset E7(7)/SU(8) in order to make any final statement about stability. We worked out the uplifting byembedding the R-symmetry group of half-maximal theory, i.e. U(4) = U(1) × SU(4), into that of themaximal, i.e. SU(8), and relating the fermionic shifts given in [7] to those ones in [11] according to thedecompositionE7(7)↗↘SL(2) × SO(6, 6)SU(8)↘↗ U(1) × SU(4)m︸ ︷︷ ︸R-symmetry of the N=4 theory× SU(4)a . (6)After this uplifting, the whole set of critical points are found to satisfy the equations of motion and thequadratic constraints of the maximal theory. Subsequently we computed the mass matrix for the scalars inwww.fp-journal.org © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim990 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsthe N = 8 theory using the results in [12]. The results are summarised in Table 4. We find that the solutions3s are non-supersymmetric and nevertheless stable. Up to our knowledge, this is the second example in theliterature (after the one in [13]) of such a solution in maximal supergravity; as opposed to the first example,though, this solution is completely tachyon-free rather than presenting tachyons although still above theBF bound. A final remark is that now, what used to be an accidental Z2 symmetry in the solutions has aproper interpretation within the N = 8 theory, i.e. it interchanges SU(4)m and SU(4)a in Eq. (6).ID V0 m2(N=4) m2(N=8) Stability1s −λ2 − 23 −23stable2s − 3227 λ2 − 45− 45unstable3s − 815 λ2 0 0 stable4s − 3227 λ2 0 − 43unstableTable 4 The values of the energy and the nor-malised mass for the lightest scalar at the set ofcritical points. One observes that, when lifting fromN = 4 to N = 8, the solutions 4s become unstablebecause of the appearance of an unstable tachyonicdirection within the new scalar modes. We remindthe reader that in four dimensions the BF bound isgiven by m2BF = −3/4 in units of the scalar poten-tial.3 ConclusionsThe study of isotropic type IIA orientifolds including geometric fluxes and preserving half-maximal super-symmetry reveals the presence of only AdS vacua in the landscape. The solutions turn out to be all liftableto maximal gauged supergravity where they appear as four different critical points of a unique theory: oneis supersymmetric and stable, another one is non-supersymmetry and nevertheless stable and the remainingtwo are non-supersymmetry and unstable. The natural extension of this work will be to study the effect ofnon-geometric fluxes in this setup in order to get a richer landscape, maybe even containing dS solutions.Acknowledgements We are grateful to Andrea Borghese for stimulating discussions. The work of the authors is sup-ported by a VIDI grant from the Netherlands Organisation for Scientific Research (NWO). Furthermore, G.D. wouldlike to thank the organisers of the “XVII European Workshop on String Theory”, held in September 2011 in Padua, fora very stimulating experience.References[1] J. Shelton, W. Taylor, and B. Wecht, JHEP 0510, 085 (2005) [hep-th/0508133].[2] G. Aldazabal, W. Baron, D. Marques, and C. Nunez, [arXiv:1109.0290 [hep-th]].[3] D. Geissbuhler, [arXiv:1109.4280 [hep-th]].[4] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1103, 137 (2011) [arXiv:1102.0239 [hep-th]].[5] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1106, 030 (2011) [arXiv:1104.3587 [hep-th]].[6] G. Dibitetto, A. Guarino and D. Roest, to appear.[7] J. Schon and M. Weidner, JHEP 05, 034 (2006) [arXiv:hep-th/0602024].[8] B. T. P. Gianni and G. Zacharias, J. Symb. Comput. 6, (1988) 149–167.[9] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, http://www.singular.uni-kl.de.[10] A. Borghese and D. Roest, JHEP 1105, 102 (2011) [arXiv:1012.3736 [hep-th]].[11] B. de Wit, H. Samtleben, and M. Trigiante, JHEP 0706, 049 (2007) [arXiv:0705.2101 [hep-th]].[12] A. Le Diffon, H. Samtleben, and M. Trigiante, JHEP 1104, 079 (2011) [arXiv:1103.2785 [hep-th]].[13] T. Fischbacher, K. Pilch, and N. P. Warner, [arXiv:1010.4910 [hep-th]].© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.org

Vacua analysis in extended supersymmetry compactifications

   University of GroningenVacua analysis in extended supersymmetry compactificationsDibitetto, Giuseppe; Guarino, Adolfo; Roest, DiederikPublished in:Fortschritte der physik-Progress of physicsDOI:10.1002/prop.201200004IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dibitetto, G., Guarino, A., & Roest, D. (2012). Vacua analysis in extended supersymmetrycompactifications. Fortschritte der physik-Progress of physics, 60(9-10), 987-990.https://doi.org/10.1002/prop.201200004CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 26-04-2023Fortschr. Phys. 60, No. 9 – 10, 987 – 990 (2012) / DOI 10.1002/prop.201200004Vacua analysis in extended supersymmetry compactificationsGiuseppe Dibitetto∗, Adolfo Guarino, and Diederik RoestCentre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The NetherlandsReceived 11 January 2012, accepted 11 January 2012Published online 27 January 2012Key words Flux compactifications, extended supersymmetry, supergravity models.We analyse geometric type IIA flux compactifications leading to N = 4 gauged supergravities in fourdimensions. The complete landscape of isotropic vacua is presented, which turns out to belong to a uniquetheory. The solutions admit an uplift to maximal supergravity due to the vanishing of the flux-inducedtadpoles for all the supersymmetry-breaking branes. Such an uplift is sketched out and the full N = 8 massspectra are discussed. We find the interesting presence of a non-supersymmetric and nevertheless stableminimum.© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim1 IntroductionHalf-maximal and maximal gauged supergravities in four dimensions are the low-energy effective theoriesarising from flux compactifications in string theory, provided that only internal manifolds and extendedobjects are included which are compatible with such amounts of supersymmetry. In the last decade theembedding tensor formalism has been used extensively in the context of (half-)maximal supergravity inorder to describe all the deformations of the free theory in a duality-covariant way. Nevertheless, it hasalready been pointed out in the literature that not all the gaugings of supergravity have a higher-dimensionalorigin in terms of a geometric flux compactification in string theory. This indicates that gaugings comingfrom geometric flux compactifications are not a closed set under general duality transformations and thisis the origin of non-geometric fluxes [1]. Gaugings associated with such fluxes might yet have a higher-dimensional description in terms of a double field theory (DFT) [2,3], in which duality covariance becomesthe fundamental principle to start with, independently of the compactification procedure. Because of thisinterpretation, gauged supergravities with extended supersymmetry seem to be suitable frameworks forinvestigating the nature of non-geometric fluxes, as summarised in Table 1.Table 1 Half-maximal and maximal gauged supegravities in four dimensions seem to be suitable play-grounds to understand how to restore duality covariance in flux compactifications.SUSY G stringy interpretationN = 4 SL(2)× SO(6, 6) S- and T-dualityN = 8 E7(7) U-dualityAfter introducing the SO(3) truncation of half-maximal supergravity as the effective theory arisingfrom specific type IIA orientifold reductions with background fluxes, we analyse the landscape of isotropicvacua in geometric backgrounds, i.e. including metric and gauge fluxes. We find a set of anti-de Sitter(AdS) critical points admitting an uplift to N = 8, one of which is remarkably stable without preserving∗ Corresponding author E-mail: g.dibitetto@rug.nl, Phone: +31 (0)50 363 9043, Fax: +31 (0)50 363 4947© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim988 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsany supersymmetry. More details on this work can be found in [4–6]. The goal of finding de Sitter (dS)vacua motivates the analysis of non-geometric flux compactifications as possible future extensions.2 The geometric type IIA with O6/D6 setupUpon the SO(3) truncation, half-maximal supergravity in four dimensions reduces to a three-field STUmodel in the following waySL(2) × SO(6, 6) −→ SL(2) × SO(2, 2) = SL(2)S × SL(2)T × SL(2)U . (1)The truncation allows for forty SO(3)-singlet embedding tensor components that can be written in terms ofan SL(2) × SO(2, 2) tensor ΛαABC = Λα(ABC), with α = ± and A = 1, ..., 4 being SL(2) and SO(2, 2)fundamental indices, respectively. As shown in [4], these forty embedding tensor components correspondexactly with the set of generalised fluxes in type II orientifold reductions on a Z2 × Z2 isotropic orbifold.As a consequence of the truncation, the theory preserves only N = 1 supersymmetry out of the originalN = 4. Therefore, the corresponding scalar potential can be written in terms of a (logarithmic) Kählerpotential K and a holomorphic superpotential W by using the standard N = 1 expressionV (SO(3)) = eK(−3|W |2 + |D W |2) . (2)At the effective level, fluxes appear as arbitrary superpotential couplings up to linear in S and up to cubicin T and U . The scalar potential computed from (2) turns out to coincide with the scalar potential givenin [7] up to terms projected out by a set of N = 4 quadratic constraints on the embedding tensor Λ [4].Table 2 The set of SO(3)-invariant embedding tensor components of Λ admitting a higher-dimensionalorigin as type IIA fluxes: a metric flux ω together with R-R F0,2,4,6 and NS-NS H3 gauge fluxes.W couplings Type IIA fluxes Embedding tensor componentsa0 F6 −Λ+333a1 F4 Λ+334a2 F2 −Λ+344a3 F0 Λ+444b0 H3 −Λ−333b1 ω Λ−334c0 H3 Λ+233c1 , c̃1 ω Λ+234 , Λ+133Restricting to the components of Λ that can be interpreted as metric or gauge fluxes in a type IIArealisation of the model, we are left with the following superpotentialWIIA = a0 − 3 a1 U + 3 a2 U2 − a3 U3 − b0 S + 3 b1 S U + 3 c0 T + (6 c1 − 3 c̃1)T U , (3)consisting of nine flux-induced couplings (see Table 2). These fluxes are demanded to satisfy the set ofN = 4 quadratic constraintsc1 (c1 − c̃1) = 0 , b1 (c1 − c̃1) = 0 (ω2 = 0) ,−a3 c0 − a2 (2 c1 − c̃1) = 0 (N6⊥ = 0) ,(4)where the first two constraints are related to the nilpotency of the twisted exterior derivative operator,whereas the third one imposes the absence of D6-branes wrapping the directions orthogonal to the O6-planes, which would break supersymmetry explicitly down to N = 1. In contrast, D6-branes parallel to© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.orgFortschr. Phys. 60, No. 9 – 10 (2012) 989the O6-planes are compatible with N = 4 supersymmetry, hence being allowed. Their correspondingflux-induced tadpole readsN6|| = 3 a2 b1 − a3 b0 = NO62 − ND6 . (5)After observing that the set of fluxes given in Table 2 is closed under non-compact duality transfor-mations (i.e. real shifts and rescalings of S, T and U ), we can restrict the search for critical points of thescalar potential to the point S0 = T0 = U0 = i without losing generality. The field equations becomethen quadratic conditions in the fluxes that have to be satisfied together with the quadratic constraints in(4). These equations generate a quadratic ideal which we decompose in terms of prime ideals by usingthe Gianni-Trager-Zacharias (GTZ) decomposition [8] with the help of Singular [9]. By solving them, thecomplete set of critical points of the scalar potential is presented in Table 3. They turn out to be (modulothe discrete Z2 symmetry introduced in the caption) different AdS critical points of a unique theory withan underlying gauging given by the gauge group G0 =ISO(3)  U(1)6.Table 3 The set of critical points of the scalar potential for geometric type IIA isotropic flux compactifications. Thesolutions labelled with 1s turn out to preserve N = 1 supersymmetry for s = +1 and to be non-supersymmetric fors = −1, whereas all the others are non-supersymmetric. It is worth noticing that s = ± 1 appears as an accidental Z2symmetry which relates solutions having exactly the same energy and the same mass spectrum. The parameter λ is aglobal scaling parameter such that V ∝ λ2.ID a0 a1 a2 a3 b0 b1 c0 c1 = c̃11s s3√102λ√62λ −s√106λ5√66λ −s√63λ√103λ s√63λ√10λ2s s16√109λ 0 016√29λ 016√1045λ 016√1015λ3s s4√105λ − 4√3015λ s4√1015λ s4√3015λ s4√3015λ4√1015λ −s 4√3015λ4√105λ4s s16√109λ 0 016√29λ 016√29λ 016√29λRegarding stability at the critical points, we computed in [4] the full mass matrix for the 38 physicalscalars in N = 4 making use of the results in [10]. At this point, we can say two things: firstly, thesolutions 1s are fully stable because of (fake-) supersymmetry, and secondly the solutions 2s are alreadyunstable because of the presence of a tachyon whose mass is below the Breitenlohner-Freedman (BF)bound. However, this is not enough since all the critical points turn out to be compatible with the totalabsence of sources, i.e., the flux-induced tadpole (5) does accidentally vanish at these points. As a result,they admit an uplift to N = 8 and hence one should analyse the full mass matrix for the 70 scalars spanningthe coset E7(7)/SU(8) in order to make any final statement about stability. We worked out the uplifting byembedding the R-symmetry group of half-maximal theory, i.e. U(4) = U(1) × SU(4), into that of themaximal, i.e. SU(8), and relating the fermionic shifts given in [7] to those ones in [11] according to thedecompositionE7(7)↗↘SL(2) × SO(6, 6)SU(8)↘↗ U(1) × SU(4)m︸ ︷︷ ︸R-symmetry of the N=4 theory× SU(4)a . (6)After this uplifting, the whole set of critical points are found to satisfy the equations of motion and thequadratic constraints of the maximal theory. Subsequently we computed the mass matrix for the scalars inwww.fp-journal.org © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim990 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsthe N = 8 theory using the results in [12]. The results are summarised in Table 4. We find that the solutions3s are non-supersymmetric and nevertheless stable. Up to our knowledge, this is the second example in theliterature (after the one in [13]) of such a solution in maximal supergravity; as opposed to the first example,though, this solution is completely tachyon-free rather than presenting tachyons although still above theBF bound. A final remark is that now, what used to be an accidental Z2 symmetry in the solutions has aproper interpretation within the N = 8 theory, i.e. it interchanges SU(4)m and SU(4)a in Eq. (6).ID V0 m2(N=4) m2(N=8) Stability1s −λ2 − 23 −23stable2s − 3227 λ2 − 45− 45unstable3s − 815 λ2 0 0 stable4s − 3227 λ2 0 − 43unstableTable 4 The values of the energy and the nor-malised mass for the lightest scalar at the set ofcritical points. One observes that, when lifting fromN = 4 to N = 8, the solutions 4s become unstablebecause of the appearance of an unstable tachyonicdirection within the new scalar modes. We remindthe reader that in four dimensions the BF bound isgiven by m2BF = −3/4 in units of the scalar poten-tial.3 ConclusionsThe study of isotropic type IIA orientifolds including geometric fluxes and preserving half-maximal super-symmetry reveals the presence of only AdS vacua in the landscape. The solutions turn out to be all liftableto maximal gauged supergravity where they appear as four different critical points of a unique theory: oneis supersymmetric and stable, another one is non-supersymmetry and nevertheless stable and the remainingtwo are non-supersymmetry and unstable. The natural extension of this work will be to study the effect ofnon-geometric fluxes in this setup in order to get a richer landscape, maybe even containing dS solutions.Acknowledgements We are grateful to Andrea Borghese for stimulating discussions. The work of the authors is sup-ported by a VIDI grant from the Netherlands Organisation for Scientific Research (NWO). Furthermore, G.D. wouldlike to thank the organisers of the “XVII European Workshop on String Theory”, held in September 2011 in Padua, fora very stimulating experience.References[1] J. Shelton, W. Taylor, and B. Wecht, JHEP 0510, 085 (2005) [hep-th/0508133].[2] G. Aldazabal, W. Baron, D. Marques, and C. Nunez, [arXiv:1109.0290 [hep-th]].[3] D. Geissbuhler, [arXiv:1109.4280 [hep-th]].[4] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1103, 137 (2011) [arXiv:1102.0239 [hep-th]].[5] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1106, 030 (2011) [arXiv:1104.3587 [hep-th]].[6] G. Dibitetto, A. Guarino and D. Roest, to appear.[7] J. Schon and M. Weidner, JHEP 05, 034 (2006) [arXiv:hep-th/0602024].[8] B. T. P. Gianni and G. Zacharias, J. Symb. Comput. 6, (1988) 149–167.[9] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, http://www.singular.uni-kl.de.[10] A. Borghese and D. Roest, JHEP 1105, 102 (2011) [arXiv:1012.3736 [hep-th]].[11] B. de Wit, H. Samtleben, and M. Trigiante, JHEP 0706, 049 (2007) [arXiv:0705.2101 [hep-th]].[12] A. Le Diffon, H. Samtleben, and M. Trigiante, JHEP 1104, 079 (2011) [arXiv:1103.2785 [hep-th]].[13] T. Fischbacher, K. Pilch, and N. P. Warner, [arXiv:1010.4910 [hep-th]].© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.org

ARTS repository - University of Groningen

University of GroningenVacua analysis in extended supersymmetry compactificationsDibitetto, Giuseppe; Guarino, Adolfo; Roest, DiederikPublished in:Fortschritte der physik-Progress of physicsDOI:10.1002/prop.201200004IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dibitetto, G., Guarino, A., & Roest, D. (2012). Vacua analysis in extended supersymmetrycompactifications. Fortschritte der physik-Progress of physics, 60(9-10), 987-990.https://doi.org/10.1002/prop.201200004CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 24-12-2025Fortschr. Phys. 60, No. 9 – 10, 987 – 990 (2012) / DOI 10.1002/prop.201200004Vacua analysis in extended supersymmetry compactificationsGiuseppe Dibitetto∗, Adolfo Guarino, and Diederik RoestCentre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The NetherlandsReceived 11 January 2012, accepted 11 January 2012Published online 27 January 2012Key words Flux compactifications, extended supersymmetry, supergravity models.We analyse geometric type IIA flux compactifications leading to N = 4 gauged supergravities in fourdimensions. The complete landscape of isotropic vacua is presented, which turns out to belong to a uniquetheory. The solutions admit an uplift to maximal supergravity due to the vanishing of the flux-inducedtadpoles for all the supersymmetry-breaking branes. Such an uplift is sketched out and the full N = 8 massspectra are discussed. We find the interesting presence of a non-supersymmetric and nevertheless stableminimum.© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim1 IntroductionHalf-maximal and maximal gauged supergravities in four dimensions are the low-energy effective theoriesarising from flux compactifications in string theory, provided that only internal manifolds and extendedobjects are included which are compatible with such amounts of supersymmetry. In the last decade theembedding tensor formalism has been used extensively in the context of (half-)maximal supergravity inorder to describe all the deformations of the free theory in a duality-covariant way. Nevertheless, it hasalready been pointed out in the literature that not all the gaugings of supergravity have a higher-dimensionalorigin in terms of a geometric flux compactification in string theory. This indicates that gaugings comingfrom geometric flux compactifications are not a closed set under general duality transformations and thisis the origin of non-geometric fluxes [1]. Gaugings associated with such fluxes might yet have a higher-dimensional description in terms of a double field theory (DFT) [2,3], in which duality covariance becomesthe fundamental principle to start with, independently of the compactification procedure. Because of thisinterpretation, gauged supergravities with extended supersymmetry seem to be suitable frameworks forinvestigating the nature of non-geometric fluxes, as summarised in Table 1.Table 1 Half-maximal and maximal gauged supegravities in four dimensions seem to be suitable play-grounds to understand how to restore duality covariance in flux compactifications.SUSY G stringy interpretationN = 4 SL(2)× SO(6, 6) S- and T-dualityN = 8 E7(7) U-dualityAfter introducing the SO(3) truncation of half-maximal supergravity as the effective theory arisingfrom specific type IIA orientifold reductions with background fluxes, we analyse the landscape of isotropicvacua in geometric backgrounds, i.e. including metric and gauge fluxes. We find a set of anti-de Sitter(AdS) critical points admitting an uplift to N = 8, one of which is remarkably stable without preserving∗ Corresponding author E-mail: g.dibitetto@rug.nl, Phone: +31 (0)50 363 9043, Fax: +31 (0)50 363 4947© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim988 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsany supersymmetry. More details on this work can be found in [4–6]. The goal of finding de Sitter (dS)vacua motivates the analysis of non-geometric flux compactifications as possible future extensions.2 The geometric type IIA with O6/D6 setupUpon the SO(3) truncation, half-maximal supergravity in four dimensions reduces to a three-field STUmodel in the following waySL(2) × SO(6, 6) −→ SL(2) × SO(2, 2) = SL(2)S × SL(2)T × SL(2)U . (1)The truncation allows for forty SO(3)-singlet embedding tensor components that can be written in terms ofan SL(2) × SO(2, 2) tensor ΛαABC = Λα(ABC), with α = ± and A = 1, ..., 4 being SL(2) and SO(2, 2)fundamental indices, respectively. As shown in [4], these forty embedding tensor components correspondexactly with the set of generalised fluxes in type II orientifold reductions on a Z2 × Z2 isotropic orbifold.As a consequence of the truncation, the theory preserves only N = 1 supersymmetry out of the originalN = 4. Therefore, the corresponding scalar potential can be written in terms of a (logarithmic) Kählerpotential K and a holomorphic superpotential W by using the standard N = 1 expressionV (SO(3)) = eK(−3|W |2 + |D W |2) . (2)At the effective level, fluxes appear as arbitrary superpotential couplings up to linear in S and up to cubicin T and U . The scalar potential computed from (2) turns out to coincide with the scalar potential givenin [7] up to terms projected out by a set of N = 4 quadratic constraints on the embedding tensor Λ [4].Table 2 The set of SO(3)-invariant embedding tensor components of Λ admitting a higher-dimensionalorigin as type IIA fluxes: a metric flux ω together with R-R F0,2,4,6 and NS-NS H3 gauge fluxes.W couplings Type IIA fluxes Embedding tensor componentsa0 F6 −Λ+333a1 F4 Λ+334a2 F2 −Λ+344a3 F0 Λ+444b0 H3 −Λ−333b1 ω Λ−334c0 H3 Λ+233c1 , c̃1 ω Λ+234 , Λ+133Restricting to the components of Λ that can be interpreted as metric or gauge fluxes in a type IIArealisation of the model, we are left with the following superpotentialWIIA = a0 − 3 a1 U + 3 a2 U2 − a3 U3 − b0 S + 3 b1 S U + 3 c0 T + (6 c1 − 3 c̃1)T U , (3)consisting of nine flux-induced couplings (see Table 2). These fluxes are demanded to satisfy the set ofN = 4 quadratic constraintsc1 (c1 − c̃1) = 0 , b1 (c1 − c̃1) = 0 (ω2 = 0) ,−a3 c0 − a2 (2 c1 − c̃1) = 0 (N6⊥ = 0) ,(4)where the first two constraints are related to the nilpotency of the twisted exterior derivative operator,whereas the third one imposes the absence of D6-branes wrapping the directions orthogonal to the O6-planes, which would break supersymmetry explicitly down to N = 1. In contrast, D6-branes parallel to© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.orgFortschr. Phys. 60, No. 9 – 10 (2012) 989the O6-planes are compatible with N = 4 supersymmetry, hence being allowed. Their correspondingflux-induced tadpole readsN6|| = 3 a2 b1 − a3 b0 = NO62 − ND6 . (5)After observing that the set of fluxes given in Table 2 is closed under non-compact duality transfor-mations (i.e. real shifts and rescalings of S, T and U ), we can restrict the search for critical points of thescalar potential to the point S0 = T0 = U0 = i without losing generality. The field equations becomethen quadratic conditions in the fluxes that have to be satisfied together with the quadratic constraints in(4). These equations generate a quadratic ideal which we decompose in terms of prime ideals by usingthe Gianni-Trager-Zacharias (GTZ) decomposition [8] with the help of Singular [9]. By solving them, thecomplete set of critical points of the scalar potential is presented in Table 3. They turn out to be (modulothe discrete Z2 symmetry introduced in the caption) different AdS critical points of a unique theory withan underlying gauging given by the gauge group G0 =ISO(3)  U(1)6.Table 3 The set of critical points of the scalar potential for geometric type IIA isotropic flux compactifications. Thesolutions labelled with 1s turn out to preserve N = 1 supersymmetry for s = +1 and to be non-supersymmetric fors = −1, whereas all the others are non-supersymmetric. It is worth noticing that s = ± 1 appears as an accidental Z2symmetry which relates solutions having exactly the same energy and the same mass spectrum. The parameter λ is aglobal scaling parameter such that V ∝ λ2.ID a0 a1 a2 a3 b0 b1 c0 c1 = c̃11s s3√102λ√62λ −s√106λ5√66λ −s√63λ√103λ s√63λ√10λ2s s16√109λ 0 016√29λ 016√1045λ 016√1015λ3s s4√105λ − 4√3015λ s4√1015λ s4√3015λ s4√3015λ4√1015λ −s 4√3015λ4√105λ4s s16√109λ 0 016√29λ 016√29λ 016√29λRegarding stability at the critical points, we computed in [4] the full mass matrix for the 38 physicalscalars in N = 4 making use of the results in [10]. At this point, we can say two things: firstly, thesolutions 1s are fully stable because of (fake-) supersymmetry, and secondly the solutions 2s are alreadyunstable because of the presence of a tachyon whose mass is below the Breitenlohner-Freedman (BF)bound. However, this is not enough since all the critical points turn out to be compatible with the totalabsence of sources, i.e., the flux-induced tadpole (5) does accidentally vanish at these points. As a result,they admit an uplift to N = 8 and hence one should analyse the full mass matrix for the 70 scalars spanningthe coset E7(7)/SU(8) in order to make any final statement about stability. We worked out the uplifting byembedding the R-symmetry group of half-maximal theory, i.e. U(4) = U(1) × SU(4), into that of themaximal, i.e. SU(8), and relating the fermionic shifts given in [7] to those ones in [11] according to thedecompositionE7(7)↗↘SL(2) × SO(6, 6)SU(8)↘↗ U(1) × SU(4)m︸ ︷︷ ︸R-symmetry of the N=4 theory× SU(4)a . (6)After this uplifting, the whole set of critical points are found to satisfy the equations of motion and thequadratic constraints of the maximal theory. Subsequently we computed the mass matrix for the scalars inwww.fp-journal.org © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim990 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsthe N = 8 theory using the results in [12]. The results are summarised in Table 4. We find that the solutions3s are non-supersymmetric and nevertheless stable. Up to our knowledge, this is the second example in theliterature (after the one in [13]) of such a solution in maximal supergravity; as opposed to the first example,though, this solution is completely tachyon-free rather than presenting tachyons although still above theBF bound. A final remark is that now, what used to be an accidental Z2 symmetry in the solutions has aproper interpretation within the N = 8 theory, i.e. it interchanges SU(4)m and SU(4)a in Eq. (6).ID V0 m2(N=4) m2(N=8) Stability1s −λ2 − 23 −23stable2s − 3227 λ2 − 45− 45unstable3s − 815 λ2 0 0 stable4s − 3227 λ2 0 − 43unstableTable 4 The values of the energy and the nor-malised mass for the lightest scalar at the set ofcritical points. One observes that, when lifting fromN = 4 to N = 8, the solutions 4s become unstablebecause of the appearance of an unstable tachyonicdirection within the new scalar modes. We remindthe reader that in four dimensions the BF bound isgiven by m2BF = −3/4 in units of the scalar poten-tial.3 ConclusionsThe study of isotropic type IIA orientifolds including geometric fluxes and preserving half-maximal super-symmetry reveals the presence of only AdS vacua in the landscape. The solutions turn out to be all liftableto maximal gauged supergravity where they appear as four different critical points of a unique theory: oneis supersymmetric and stable, another one is non-supersymmetry and nevertheless stable and the remainingtwo are non-supersymmetry and unstable. The natural extension of this work will be to study the effect ofnon-geometric fluxes in this setup in order to get a richer landscape, maybe even containing dS solutions.Acknowledgements We are grateful to Andrea Borghese for stimulating discussions. The work of the authors is sup-ported by a VIDI grant from the Netherlands Organisation for Scientific Research (NWO). Furthermore, G.D. wouldlike to thank the organisers of the “XVII European Workshop on String Theory”, held in September 2011 in Padua, fora very stimulating experience.References[1] J. Shelton, W. Taylor, and B. Wecht, JHEP 0510, 085 (2005) [hep-th/0508133].[2] G. Aldazabal, W. Baron, D. Marques, and C. Nunez, [arXiv:1109.0290 [hep-th]].[3] D. Geissbuhler, [arXiv:1109.4280 [hep-th]].[4] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1103, 137 (2011) [arXiv:1102.0239 [hep-th]].[5] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1106, 030 (2011) [arXiv:1104.3587 [hep-th]].[6] G. Dibitetto, A. Guarino and D. Roest, to appear.[7] J. Schon and M. Weidner, JHEP 05, 034 (2006) [arXiv:hep-th/0602024].[8] B. T. P. Gianni and G. Zacharias, J. Symb. Comput. 6, (1988) 149–167.[9] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, http://www.singular.uni-kl.de.[10] A. Borghese and D. Roest, JHEP 1105, 102 (2011) [arXiv:1012.3736 [hep-th]].[11] B. de Wit, H. Samtleben, and M. Trigiante, JHEP 0706, 049 (2007) [arXiv:0705.2101 [hep-th]].[12] A. Le Diffon, H. Samtleben, and M. Trigiante, JHEP 1104, 079 (2011) [arXiv:1103.2785 [hep-th]].[13] T. Fischbacher, K. Pilch, and N. P. Warner, [arXiv:1010.4910 [hep-th]].© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.org

   University of GroningenVacua analysis in extended supersymmetry compactificationsDibitetto, Giuseppe; Guarino, Adolfo; Roest, DiederikPublished in:Fortschritte der physik-Progress of physicsDOI:10.1002/prop.201200004IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dibitetto, G., Guarino, A., & Roest, D. (2012). Vacua analysis in extended supersymmetrycompactifications. Fortschritte der physik-Progress of physics, 60(9-10), 987-990.https://doi.org/10.1002/prop.201200004CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 30-10-2023Fortschr. Phys. 60, No. 9 – 10, 987 – 990 (2012) / DOI 10.1002/prop.201200004Vacua analysis in extended supersymmetry compactificationsGiuseppe Dibitetto∗, Adolfo Guarino, and Diederik RoestCentre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The NetherlandsReceived 11 January 2012, accepted 11 January 2012Published online 27 January 2012Key words Flux compactifications, extended supersymmetry, supergravity models.We analyse geometric type IIA flux compactifications leading to N = 4 gauged supergravities in fourdimensions. The complete landscape of isotropic vacua is presented, which turns out to belong to a uniquetheory. The solutions admit an uplift to maximal supergravity due to the vanishing of the flux-inducedtadpoles for all the supersymmetry-breaking branes. Such an uplift is sketched out and the full N = 8 massspectra are discussed. We find the interesting presence of a non-supersymmetric and nevertheless stableminimum.© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim1 IntroductionHalf-maximal and maximal gauged supergravities in four dimensions are the low-energy effective theoriesarising from flux compactifications in string theory, provided that only internal manifolds and extendedobjects are included which are compatible with such amounts of supersymmetry. In the last decade theembedding tensor formalism has been used extensively in the context of (half-)maximal supergravity inorder to describe all the deformations of the free theory in a duality-covariant way. Nevertheless, it hasalready been pointed out in the literature that not all the gaugings of supergravity have a higher-dimensionalorigin in terms of a geometric flux compactification in string theory. This indicates that gaugings comingfrom geometric flux compactifications are not a closed set under general duality transformations and thisis the origin of non-geometric fluxes [1]. Gaugings associated with such fluxes might yet have a higher-dimensional description in terms of a double field theory (DFT) [2,3], in which duality covariance becomesthe fundamental principle to start with, independently of the compactification procedure. Because of thisinterpretation, gauged supergravities with extended supersymmetry seem to be suitable frameworks forinvestigating the nature of non-geometric fluxes, as summarised in Table 1.Table 1 Half-maximal and maximal gauged supegravities in four dimensions seem to be suitable play-grounds to understand how to restore duality covariance in flux compactifications.SUSY G stringy interpretationN = 4 SL(2)× SO(6, 6) S- and T-dualityN = 8 E7(7) U-dualityAfter introducing the SO(3) truncation of half-maximal supergravity as the effective theory arisingfrom specific type IIA orientifold reductions with background fluxes, we analyse the landscape of isotropicvacua in geometric backgrounds, i.e. including metric and gauge fluxes. We find a set of anti-de Sitter(AdS) critical points admitting an uplift to N = 8, one of which is remarkably stable without preserving∗ Corresponding author E-mail: g.dibitetto@rug.nl, Phone: +31 (0)50 363 9043, Fax: +31 (0)50 363 4947© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim988 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsany supersymmetry. More details on this work can be found in [4–6]. The goal of finding de Sitter (dS)vacua motivates the analysis of non-geometric flux compactifications as possible future extensions.2 The geometric type IIA with O6/D6 setupUpon the SO(3) truncation, half-maximal supergravity in four dimensions reduces to a three-field STUmodel in the following waySL(2) × SO(6, 6) −→ SL(2) × SO(2, 2) = SL(2)S × SL(2)T × SL(2)U . (1)The truncation allows for forty SO(3)-singlet embedding tensor components that can be written in terms ofan SL(2) × SO(2, 2) tensor ΛαABC = Λα(ABC), with α = ± and A = 1, ..., 4 being SL(2) and SO(2, 2)fundamental indices, respectively. As shown in [4], these forty embedding tensor components correspondexactly with the set of generalised fluxes in type II orientifold reductions on a Z2 × Z2 isotropic orbifold.As a consequence of the truncation, the theory preserves only N = 1 supersymmetry out of the originalN = 4. Therefore, the corresponding scalar potential can be written in terms of a (logarithmic) Kählerpotential K and a holomorphic superpotential W by using the standard N = 1 expressionV (SO(3)) = eK(−3|W |2 + |D W |2) . (2)At the effective level, fluxes appear as arbitrary superpotential couplings up to linear in S and up to cubicin T and U . The scalar potential computed from (2) turns out to coincide with the scalar potential givenin [7] up to terms projected out by a set of N = 4 quadratic constraints on the embedding tensor Λ [4].Table 2 The set of SO(3)-invariant embedding tensor components of Λ admitting a higher-dimensionalorigin as type IIA fluxes: a metric flux ω together with R-R F0,2,4,6 and NS-NS H3 gauge fluxes.W couplings Type IIA fluxes Embedding tensor componentsa0 F6 −Λ+333a1 F4 Λ+334a2 F2 −Λ+344a3 F0 Λ+444b0 H3 −Λ−333b1 ω Λ−334c0 H3 Λ+233c1 , c̃1 ω Λ+234 , Λ+133Restricting to the components of Λ that can be interpreted as metric or gauge fluxes in a type IIArealisation of the model, we are left with the following superpotentialWIIA = a0 − 3 a1 U + 3 a2 U2 − a3 U3 − b0 S + 3 b1 S U + 3 c0 T + (6 c1 − 3 c̃1)T U , (3)consisting of nine flux-induced couplings (see Table 2). These fluxes are demanded to satisfy the set ofN = 4 quadratic constraintsc1 (c1 − c̃1) = 0 , b1 (c1 − c̃1) = 0 (ω2 = 0) ,−a3 c0 − a2 (2 c1 − c̃1) = 0 (N6⊥ = 0) ,(4)where the first two constraints are related to the nilpotency of the twisted exterior derivative operator,whereas the third one imposes the absence of D6-branes wrapping the directions orthogonal to the O6-planes, which would break supersymmetry explicitly down to N = 1. In contrast, D6-branes parallel to© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.orgFortschr. Phys. 60, No. 9 – 10 (2012) 989the O6-planes are compatible with N = 4 supersymmetry, hence being allowed. Their correspondingflux-induced tadpole readsN6|| = 3 a2 b1 − a3 b0 = NO62 − ND6 . (5)After observing that the set of fluxes given in Table 2 is closed under non-compact duality transfor-mations (i.e. real shifts and rescalings of S, T and U ), we can restrict the search for critical points of thescalar potential to the point S0 = T0 = U0 = i without losing generality. The field equations becomethen quadratic conditions in the fluxes that have to be satisfied together with the quadratic constraints in(4). These equations generate a quadratic ideal which we decompose in terms of prime ideals by usingthe Gianni-Trager-Zacharias (GTZ) decomposition [8] with the help of Singular [9]. By solving them, thecomplete set of critical points of the scalar potential is presented in Table 3. They turn out to be (modulothe discrete Z2 symmetry introduced in the caption) different AdS critical points of a unique theory withan underlying gauging given by the gauge group G0 =ISO(3)  U(1)6.Table 3 The set of critical points of the scalar potential for geometric type IIA isotropic flux compactifications. Thesolutions labelled with 1s turn out to preserve N = 1 supersymmetry for s = +1 and to be non-supersymmetric fors = −1, whereas all the others are non-supersymmetric. It is worth noticing that s = ± 1 appears as an accidental Z2symmetry which relates solutions having exactly the same energy and the same mass spectrum. The parameter λ is aglobal scaling parameter such that V ∝ λ2.ID a0 a1 a2 a3 b0 b1 c0 c1 = c̃11s s3√102λ√62λ −s√106λ5√66λ −s√63λ√103λ s√63λ√10λ2s s16√109λ 0 016√29λ 016√1045λ 016√1015λ3s s4√105λ − 4√3015λ s4√1015λ s4√3015λ s4√3015λ4√1015λ −s 4√3015λ4√105λ4s s16√109λ 0 016√29λ 016√29λ 016√29λRegarding stability at the critical points, we computed in [4] the full mass matrix for the 38 physicalscalars in N = 4 making use of the results in [10]. At this point, we can say two things: firstly, thesolutions 1s are fully stable because of (fake-) supersymmetry, and secondly the solutions 2s are alreadyunstable because of the presence of a tachyon whose mass is below the Breitenlohner-Freedman (BF)bound. However, this is not enough since all the critical points turn out to be compatible with the totalabsence of sources, i.e., the flux-induced tadpole (5) does accidentally vanish at these points. As a result,they admit an uplift to N = 8 and hence one should analyse the full mass matrix for the 70 scalars spanningthe coset E7(7)/SU(8) in order to make any final statement about stability. We worked out the uplifting byembedding the R-symmetry group of half-maximal theory, i.e. U(4) = U(1) × SU(4), into that of themaximal, i.e. SU(8), and relating the fermionic shifts given in [7] to those ones in [11] according to thedecompositionE7(7)↗↘SL(2) × SO(6, 6)SU(8)↘↗ U(1) × SU(4)m︸ ︷︷ ︸R-symmetry of the N=4 theory× SU(4)a . (6)After this uplifting, the whole set of critical points are found to satisfy the equations of motion and thequadratic constraints of the maximal theory. Subsequently we computed the mass matrix for the scalars inwww.fp-journal.org © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim990 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsthe N = 8 theory using the results in [12]. The results are summarised in Table 4. We find that the solutions3s are non-supersymmetric and nevertheless stable. Up to our knowledge, this is the second example in theliterature (after the one in [13]) of such a solution in maximal supergravity; as opposed to the first example,though, this solution is completely tachyon-free rather than presenting tachyons although still above theBF bound. A final remark is that now, what used to be an accidental Z2 symmetry in the solutions has aproper interpretation within the N = 8 theory, i.e. it interchanges SU(4)m and SU(4)a in Eq. (6).ID V0 m2(N=4) m2(N=8) Stability1s −λ2 − 23 −23stable2s − 3227 λ2 − 45− 45unstable3s − 815 λ2 0 0 stable4s − 3227 λ2 0 − 43unstableTable 4 The values of the energy and the nor-malised mass for the lightest scalar at the set ofcritical points. One observes that, when lifting fromN = 4 to N = 8, the solutions 4s become unstablebecause of the appearance of an unstable tachyonicdirection within the new scalar modes. We remindthe reader that in four dimensions the BF bound isgiven by m2BF = −3/4 in units of the scalar poten-tial.3 ConclusionsThe study of isotropic type IIA orientifolds including geometric fluxes and preserving half-maximal super-symmetry reveals the presence of only AdS vacua in the landscape. The solutions turn out to be all liftableto maximal gauged supergravity where they appear as four different critical points of a unique theory: oneis supersymmetric and stable, another one is non-supersymmetry and nevertheless stable and the remainingtwo are non-supersymmetry and unstable. The natural extension of this work will be to study the effect ofnon-geometric fluxes in this setup in order to get a richer landscape, maybe even containing dS solutions.Acknowledgements We are grateful to Andrea Borghese for stimulating discussions. The work of the authors is sup-ported by a VIDI grant from the Netherlands Organisation for Scientific Research (NWO). Furthermore, G.D. wouldlike to thank the organisers of the “XVII European Workshop on String Theory”, held in September 2011 in Padua, fora very stimulating experience.References[1] J. Shelton, W. Taylor, and B. Wecht, JHEP 0510, 085 (2005) [hep-th/0508133].[2] G. Aldazabal, W. Baron, D. Marques, and C. Nunez, [arXiv:1109.0290 [hep-th]].[3] D. Geissbuhler, [arXiv:1109.4280 [hep-th]].[4] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1103, 137 (2011) [arXiv:1102.0239 [hep-th]].[5] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1106, 030 (2011) [arXiv:1104.3587 [hep-th]].[6] G. Dibitetto, A. Guarino and D. Roest, to appear.[7] J. Schon and M. Weidner, JHEP 05, 034 (2006) [arXiv:hep-th/0602024].[8] B. T. P. Gianni and G. Zacharias, J. Symb. Comput. 6, (1988) 149–167.[9] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, http://www.singular.uni-kl.de.[10] A. Borghese and D. Roest, JHEP 1105, 102 (2011) [arXiv:1012.3736 [hep-th]].[11] B. de Wit, H. Samtleben, and M. Trigiante, JHEP 0706, 049 (2007) [arXiv:0705.2101 [hep-th]].[12] A. Le Diffon, H. Samtleben, and M. Trigiante, JHEP 1104, 079 (2011) [arXiv:1103.2785 [hep-th]].[13] T. Fischbacher, K. Pilch, and N. P. Warner, [arXiv:1010.4910 [hep-th]].© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.org

   University of GroningenVacua analysis in extended supersymmetry compactificationsDibitetto, Giuseppe; Guarino, Adolfo; Roest, DiederikPublished in:Fortschritte der physik-Progress of physicsDOI:10.1002/prop.201200004IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dibitetto, G., Guarino, A., & Roest, D. (2012). Vacua analysis in extended supersymmetrycompactifications. Fortschritte der physik-Progress of physics, 60(9-10), 987-990.https://doi.org/10.1002/prop.201200004CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 20-06-2022Fortschr. Phys. 60, No. 9 – 10, 987 – 990 (2012) / DOI 10.1002/prop.201200004Vacua analysis in extended supersymmetry compactificationsGiuseppe Dibitetto∗, Adolfo Guarino, and Diederik RoestCentre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The NetherlandsReceived 11 January 2012, accepted 11 January 2012Published online 27 January 2012Key words Flux compactifications, extended supersymmetry, supergravity models.We analyse geometric type IIA flux compactifications leading to N = 4 gauged supergravities in fourdimensions. The complete landscape of isotropic vacua is presented, which turns out to belong to a uniquetheory. The solutions admit an uplift to maximal supergravity due to the vanishing of the flux-inducedtadpoles for all the supersymmetry-breaking branes. Such an uplift is sketched out and the full N = 8 massspectra are discussed. We find the interesting presence of a non-supersymmetric and nevertheless stableminimum.© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim1 IntroductionHalf-maximal and maximal gauged supergravities in four dimensions are the low-energy effective theoriesarising from flux compactifications in string theory, provided that only internal manifolds and extendedobjects are included which are compatible with such amounts of supersymmetry. In the last decade theembedding tensor formalism has been used extensively in the context of (half-)maximal supergravity inorder to describe all the deformations of the free theory in a duality-covariant way. Nevertheless, it hasalready been pointed out in the literature that not all the gaugings of supergravity have a higher-dimensionalorigin in terms of a geometric flux compactification in string theory. This indicates that gaugings comingfrom geometric flux compactifications are not a closed set under general duality transformations and thisis the origin of non-geometric fluxes [1]. Gaugings associated with such fluxes might yet have a higher-dimensional description in terms of a double field theory (DFT) [2,3], in which duality covariance becomesthe fundamental principle to start with, independently of the compactification procedure. Because of thisinterpretation, gauged supergravities with extended supersymmetry seem to be suitable frameworks forinvestigating the nature of non-geometric fluxes, as summarised in Table 1.Table 1 Half-maximal and maximal gauged supegravities in four dimensions seem to be suitable play-grounds to understand how to restore duality covariance in flux compactifications.SUSY G stringy interpretationN = 4 SL(2)× SO(6, 6) S- and T-dualityN = 8 E7(7) U-dualityAfter introducing the SO(3) truncation of half-maximal supergravity as the effective theory arisingfrom specific type IIA orientifold reductions with background fluxes, we analyse the landscape of isotropicvacua in geometric backgrounds, i.e. including metric and gauge fluxes. We find a set of anti-de Sitter(AdS) critical points admitting an uplift to N = 8, one of which is remarkably stable without preserving∗ Corresponding author E-mail: g.dibitetto@rug.nl, Phone: +31 (0)50 363 9043, Fax: +31 (0)50 363 4947© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim988 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsany supersymmetry. More details on this work can be found in [4–6]. The goal of finding de Sitter (dS)vacua motivates the analysis of non-geometric flux compactifications as possible future extensions.2 The geometric type IIA with O6/D6 setupUpon the SO(3) truncation, half-maximal supergravity in four dimensions reduces to a three-field STUmodel in the following waySL(2) × SO(6, 6) −→ SL(2) × SO(2, 2) = SL(2)S × SL(2)T × SL(2)U . (1)The truncation allows for forty SO(3)-singlet embedding tensor components that can be written in terms ofan SL(2) × SO(2, 2) tensor ΛαABC = Λα(ABC), with α = ± and A = 1, ..., 4 being SL(2) and SO(2, 2)fundamental indices, respectively. As shown in [4], these forty embedding tensor components correspondexactly with the set of generalised fluxes in type II orientifold reductions on a Z2 × Z2 isotropic orbifold.As a consequence of the truncation, the theory preserves only N = 1 supersymmetry out of the originalN = 4. Therefore, the corresponding scalar potential can be written in terms of a (logarithmic) Kählerpotential K and a holomorphic superpotential W by using the standard N = 1 expressionV (SO(3)) = eK(−3|W |2 + |D W |2) . (2)At the effective level, fluxes appear as arbitrary superpotential couplings up to linear in S and up to cubicin T and U . The scalar potential computed from (2) turns out to coincide with the scalar potential givenin [7] up to terms projected out by a set of N = 4 quadratic constraints on the embedding tensor Λ [4].Table 2 The set of SO(3)-invariant embedding tensor components of Λ admitting a higher-dimensionalorigin as type IIA fluxes: a metric flux ω together with R-R F0,2,4,6 and NS-NS H3 gauge fluxes.W couplings Type IIA fluxes Embedding tensor componentsa0 F6 −Λ+333a1 F4 Λ+334a2 F2 −Λ+344a3 F0 Λ+444b0 H3 −Λ−333b1 ω Λ−334c0 H3 Λ+233c1 , c̃1 ω Λ+234 , Λ+133Restricting to the components of Λ that can be interpreted as metric or gauge fluxes in a type IIArealisation of the model, we are left with the following superpotentialWIIA = a0 − 3 a1 U + 3 a2 U2 − a3 U3 − b0 S + 3 b1 S U + 3 c0 T + (6 c1 − 3 c̃1)T U , (3)consisting of nine flux-induced couplings (see Table 2). These fluxes are demanded to satisfy the set ofN = 4 quadratic constraintsc1 (c1 − c̃1) = 0 , b1 (c1 − c̃1) = 0 (ω2 = 0) ,−a3 c0 − a2 (2 c1 − c̃1) = 0 (N6⊥ = 0) ,(4)where the first two constraints are related to the nilpotency of the twisted exterior derivative operator,whereas the third one imposes the absence of D6-branes wrapping the directions orthogonal to the O6-planes, which would break supersymmetry explicitly down to N = 1. In contrast, D6-branes parallel to© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.orgFortschr. Phys. 60, No. 9 – 10 (2012) 989the O6-planes are compatible with N = 4 supersymmetry, hence being allowed. Their correspondingflux-induced tadpole readsN6|| = 3 a2 b1 − a3 b0 = NO62 − ND6 . (5)After observing that the set of fluxes given in Table 2 is closed under non-compact duality transfor-mations (i.e. real shifts and rescalings of S, T and U ), we can restrict the search for critical points of thescalar potential to the point S0 = T0 = U0 = i without losing generality. The field equations becomethen quadratic conditions in the fluxes that have to be satisfied together with the quadratic constraints in(4). These equations generate a quadratic ideal which we decompose in terms of prime ideals by usingthe Gianni-Trager-Zacharias (GTZ) decomposition [8] with the help of Singular [9]. By solving them, thecomplete set of critical points of the scalar potential is presented in Table 3. They turn out to be (modulothe discrete Z2 symmetry introduced in the caption) different AdS critical points of a unique theory withan underlying gauging given by the gauge group G0 =ISO(3)  U(1)6.Table 3 The set of critical points of the scalar potential for geometric type IIA isotropic flux compactifications. Thesolutions labelled with 1s turn out to preserve N = 1 supersymmetry for s = +1 and to be non-supersymmetric fors = −1, whereas all the others are non-supersymmetric. It is worth noticing that s = ± 1 appears as an accidental Z2symmetry which relates solutions having exactly the same energy and the same mass spectrum. The parameter λ is aglobal scaling parameter such that V ∝ λ2.ID a0 a1 a2 a3 b0 b1 c0 c1 = c̃11s s3√102λ√62λ −s√106λ5√66λ −s√63λ√103λ s√63λ√10λ2s s16√109λ 0 016√29λ 016√1045λ 016√1015λ3s s4√105λ − 4√3015λ s4√1015λ s4√3015λ s4√3015λ4√1015λ −s 4√3015λ4√105λ4s s16√109λ 0 016√29λ 016√29λ 016√29λRegarding stability at the critical points, we computed in [4] the full mass matrix for the 38 physicalscalars in N = 4 making use of the results in [10]. At this point, we can say two things: firstly, thesolutions 1s are fully stable because of (fake-) supersymmetry, and secondly the solutions 2s are alreadyunstable because of the presence of a tachyon whose mass is below the Breitenlohner-Freedman (BF)bound. However, this is not enough since all the critical points turn out to be compatible with the totalabsence of sources, i.e., the flux-induced tadpole (5) does accidentally vanish at these points. As a result,they admit an uplift to N = 8 and hence one should analyse the full mass matrix for the 70 scalars spanningthe coset E7(7)/SU(8) in order to make any final statement about stability. We worked out the uplifting byembedding the R-symmetry group of half-maximal theory, i.e. U(4) = U(1) × SU(4), into that of themaximal, i.e. SU(8), and relating the fermionic shifts given in [7] to those ones in [11] according to thedecompositionE7(7)↗↘SL(2) × SO(6, 6)SU(8)↘↗ U(1) × SU(4)m︸ ︷︷ ︸R-symmetry of the N=4 theory× SU(4)a . (6)After this uplifting, the whole set of critical points are found to satisfy the equations of motion and thequadratic constraints of the maximal theory. Subsequently we computed the mass matrix for the scalars inwww.fp-journal.org © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim990 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsthe N = 8 theory using the results in [12]. The results are summarised in Table 4. We find that the solutions3s are non-supersymmetric and nevertheless stable. Up to our knowledge, this is the second example in theliterature (after the one in [13]) of such a solution in maximal supergravity; as opposed to the first example,though, this solution is completely tachyon-free rather than presenting tachyons although still above theBF bound. A final remark is that now, what used to be an accidental Z2 symmetry in the solutions has aproper interpretation within the N = 8 theory, i.e. it interchanges SU(4)m and SU(4)a in Eq. (6).ID V0 m2(N=4) m2(N=8) Stability1s −λ2 − 23 −23stable2s − 3227 λ2 − 45− 45unstable3s − 815 λ2 0 0 stable4s − 3227 λ2 0 − 43unstableTable 4 The values of the energy and the nor-malised mass for the lightest scalar at the set ofcritical points. One observes that, when lifting fromN = 4 to N = 8, the solutions 4s become unstablebecause of the appearance of an unstable tachyonicdirection within the new scalar modes. We remindthe reader that in four dimensions the BF bound isgiven by m2BF = −3/4 in units of the scalar poten-tial.3 ConclusionsThe study of isotropic type IIA orientifolds including geometric fluxes and preserving half-maximal super-symmetry reveals the presence of only AdS vacua in the landscape. The solutions turn out to be all liftableto maximal gauged supergravity where they appear as four different critical points of a unique theory: oneis supersymmetric and stable, another one is non-supersymmetry and nevertheless stable and the remainingtwo are non-supersymmetry and unstable. The natural extension of this work will be to study the effect ofnon-geometric fluxes in this setup in order to get a richer landscape, maybe even containing dS solutions.Acknowledgements We are grateful to Andrea Borghese for stimulating discussions. The work of the authors is sup-ported by a VIDI grant from the Netherlands Organisation for Scientific Research (NWO). Furthermore, G.D. wouldlike to thank the organisers of the “XVII European Workshop on String Theory”, held in September 2011 in Padua, fora very stimulating experience.References[1] J. Shelton, W. Taylor, and B. Wecht, JHEP 0510, 085 (2005) [hep-th/0508133].[2] G. Aldazabal, W. Baron, D. Marques, and C. Nunez, [arXiv:1109.0290 [hep-th]].[3] D. Geissbuhler, [arXiv:1109.4280 [hep-th]].[4] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1103, 137 (2011) [arXiv:1102.0239 [hep-th]].[5] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1106, 030 (2011) [arXiv:1104.3587 [hep-th]].[6] G. Dibitetto, A. Guarino and D. Roest, to appear.[7] J. Schon and M. Weidner, JHEP 05, 034 (2006) [arXiv:hep-th/0602024].[8] B. T. P. Gianni and G. Zacharias, J. Symb. Comput. 6, (1988) 149–167.[9] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, http://www.singular.uni-kl.de.[10] A. Borghese and D. Roest, JHEP 1105, 102 (2011) [arXiv:1012.3736 [hep-th]].[11] B. de Wit, H. Samtleben, and M. Trigiante, JHEP 0706, 049 (2007) [arXiv:0705.2101 [hep-th]].[12] A. Le Diffon, H. Samtleben, and M. Trigiante, JHEP 1104, 079 (2011) [arXiv:1103.2785 [hep-th]].[13] T. Fischbacher, K. Pilch, and N. P. Warner, [arXiv:1010.4910 [hep-th]].© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.org

   University of GroningenVacua analysis in extended supersymmetry compactificationsDibitetto, Giuseppe; Guarino, Adolfo; Roest, DiederikPublished in:Fortschritte der physik-Progress of physicsDOI:10.1002/prop.201200004IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dibitetto, G., Guarino, A., & Roest, D. (2012). Vacua analysis in extended supersymmetrycompactifications. Fortschritte der physik-Progress of physics, 60(9-10), 987-990.https://doi.org/10.1002/prop.201200004CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Fortschr. Phys. 60, No. 9 – 10, 987 – 990 (2012) / DOI 10.1002/prop.201200004Vacua analysis in extended supersymmetry compactificationsGiuseppe Dibitetto∗, Adolfo Guarino, and Diederik RoestCentre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The NetherlandsReceived 11 January 2012, accepted 11 January 2012Published online 27 January 2012Key words Flux compactifications, extended supersymmetry, supergravity models.We analyse geometric type IIA flux compactifications leading to N = 4 gauged supergravities in fourdimensions. The complete landscape of isotropic vacua is presented, which turns out to belong to a uniquetheory. The solutions admit an uplift to maximal supergravity due to the vanishing of the flux-inducedtadpoles for all the supersymmetry-breaking branes. Such an uplift is sketched out and the fullN = 8 massspectra are discussed. We find the interesting presence of a non-supersymmetric and nevertheless stableminimum.© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim1 IntroductionHalf-maximal and maximal gauged supergravities in four dimensions are the low-energy effective theoriesarising from flux compactifications in string theory, provided that only internal manifolds and extendedobjects are included which are compatible with such amounts of supersymmetry. In the last decade theembedding tensor formalism has been used extensively in the context of (half-)maximal supergravity inorder to describe all the deformations of the free theory in a duality-covariant way. Nevertheless, it hasalready been pointed out in the literature that not all the gaugings of supergravity have a higher-dimensionalorigin in terms of a geometric flux compactification in string theory. This indicates that gaugings comingfrom geometric flux compactifications are not a closed set under general duality transformations and thisis the origin of non-geometric fluxes [1]. Gaugings associated with such fluxes might yet have a higher-dimensional description in terms of a double field theory (DFT) [2,3], in which duality covariance becomesthe fundamental principle to start with, independently of the compactification procedure. Because of thisinterpretation, gauged supergravities with extended supersymmetry seem to be suitable frameworks forinvestigating the nature of non-geometric fluxes, as summarised in Table 1.Table 1 Half-maximal and maximal gauged supegravities in four dimensions seem to be suitable play-grounds to understand how to restore duality covariance in flux compactifications.SUSY G stringy interpretationN = 4 SL(2)× SO(6, 6) S- and T-dualityN = 8 E7(7) U-dualityAfter introducing the SO(3) truncation of half-maximal supergravity as the effective theory arisingfrom specific type IIA orientifold reductions with background fluxes, we analyse the landscape of isotropicvacua in geometric backgrounds, i.e. including metric and gauge fluxes. We find a set of anti-de Sitter(AdS) critical points admitting an uplift to N = 8, one of which is remarkably stable without preserving∗ Corresponding author E-mail: g.dibitetto@rug.nl, Phone: +31 (0)50 363 9043, Fax: +31 (0)50 363 4947© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim988 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationsany supersymmetry. More details on this work can be found in [4–6]. The goal of finding de Sitter (dS)vacua motivates the analysis of non-geometric flux compactifications as possible future extensions.2 The geometric type IIA with O6/D6 setupUpon the SO(3) truncation, half-maximal supergravity in four dimensions reduces to a three-field STUmodel in the following waySL(2) × SO(6, 6) −→ SL(2) × SO(2, 2) = SL(2)S × SL(2)T × SL(2)U . (1)The truncation allows for forty SO(3)-singlet embedding tensor components that can be written in terms ofan SL(2) × SO(2, 2) tensor ΛαABC = Λα(ABC), with α = ± and A = 1, ..., 4 being SL(2) and SO(2, 2)fundamental indices, respectively. As shown in [4], these forty embedding tensor components correspondexactly with the set of generalised fluxes in type II orientifold reductions on a Z2 × Z2 isotropic orbifold.As a consequence of the truncation, the theory preserves only N = 1 supersymmetry out of the originalN = 4. Therefore, the corresponding scalar potential can be written in terms of a (logarithmic) Ka¨hlerpotential K and a holomorphic superpotential W by using the standardN = 1 expressionV (SO(3)) = eK(−3|W |2 + |D W |2) . (2)At the effective level, fluxes appear as arbitrary superpotential couplings up to linear in S and up to cubicin T and U . The scalar potential computed from (2) turns out to coincide with the scalar potential givenin [7] up to terms projected out by a set ofN = 4 quadratic constraints on the embedding tensor Λ [4].Table 2 The set of SO(3)-invariant embedding tensor components of Λ admitting a higher-dimensionalorigin as type IIA fluxes: a metric flux ω together with R-R F0,2,4,6 and NS-NS H3 gauge fluxes.W couplings Type IIA fluxes Embedding tensor componentsa0 F6 −Λ+333a1 F4 Λ+334a2 F2 −Λ+344a3 F0 Λ+444b0 H3 −Λ−333b1 ω Λ−334c0 H3 Λ+233c1 , c˜1 ω Λ+234 , Λ+133Restricting to the components of Λ that can be interpreted as metric or gauge fluxes in a type IIArealisation of the model, we are left with the following superpotentialWIIA = a0 − 3 a1 U + 3 a2 U2 − a3 U3 − b0 S + 3 b1 S U + 3 c0 T + (6 c1 − 3 c˜1)T U , (3)consisting of nine flux-induced couplings (see Table 2). These fluxes are demanded to satisfy the set ofN = 4 quadratic constraintsc1 (c1 − c˜1) = 0 , b1 (c1 − c˜1) = 0 (ω2 = 0) ,−a3 c0 − a2 (2 c1 − c˜1) = 0 (N6⊥ = 0) ,(4)where the first two constraints are related to the nilpotency of the twisted exterior derivative operator,whereas the third one imposes the absence of D6-branes wrapping the directions orthogonal to the O6-planes, which would break supersymmetry explicitly down to N = 1. In contrast, D6-branes parallel to© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.orgFortschr. Phys. 60, No. 9 – 10 (2012) 989the O6-planes are compatible with N = 4 supersymmetry, hence being allowed. Their correspondingflux-induced tadpole readsN6|| = 3 a2 b1 − a3 b0 = NO62 −ND6 . (5)After observing that the set of fluxes given in Table 2 is closed under non-compact duality transfor-mations (i.e. real shifts and rescalings of S, T and U ), we can restrict the search for critical points of thescalar potential to the point S0 = T0 = U0 = i without losing generality. The field equations becomethen quadratic conditions in the fluxes that have to be satisfied together with the quadratic constraints in(4). These equations generate a quadratic ideal which we decompose in terms of prime ideals by usingthe Gianni-Trager-Zacharias (GTZ) decomposition [8] with the help of Singular [9]. By solving them, thecomplete set of critical points of the scalar potential is presented in Table 3. They turn out to be (modulothe discrete Z2 symmetry introduced in the caption) different AdS critical points of a unique theory withan underlying gauging given by the gauge group G0 =ISO(3)  U(1)6.Table 3 The set of critical points of the scalar potential for geometric type IIA isotropic flux compactifications. Thesolutions labelled with 1s turn out to preserve N = 1 supersymmetry for s = +1 and to be non-supersymmetric fors = −1, whereas all the others are non-supersymmetric. It is worth noticing that s = ± 1 appears as an accidental Z2symmetry which relates solutions having exactly the same energy and the same mass spectrum. The parameter λ is aglobal scaling parameter such that V ∝ λ2.ID a0 a1 a2 a3 b0 b1 c0 c1 = c˜11s s3√102λ√62λ −s√106λ5√66λ −s√63λ√103λ s√63λ√10λ2s s16√109λ 0 016√29λ 016√1045λ 016√1015λ3s s4√105λ − 4√3015λ s4√1015λ s4√3015λ s4√3015λ4√1015λ −s 4√3015λ4√105λ4s s16√109λ 0 016√29λ 016√29λ 016√29λRegarding stability at the critical points, we computed in [4] the full mass matrix for the 38 physicalscalars in N = 4 making use of the results in [10]. At this point, we can say two things: firstly, thesolutions 1s are fully stable because of (fake-) supersymmetry, and secondly the solutions 2s are alreadyunstable because of the presence of a tachyon whose mass is below the Breitenlohner-Freedman (BF)bound. However, this is not enough since all the critical points turn out to be compatible with the totalabsence of sources, i.e., the flux-induced tadpole (5) does accidentally vanish at these points. As a result,they admit an uplift toN = 8 and hence one should analyse the full mass matrix for the 70 scalars spanningthe coset E7(7)/SU(8) in order to make any final statement about stability. We worked out the uplifting byembedding the R-symmetry group of half-maximal theory, i.e. U(4) = U(1) × SU(4), into that of themaximal, i.e. SU(8), and relating the fermionic shifts given in [7] to those ones in [11] according to thedecompositionE7(7)↗↘SL(2)× SO(6, 6)SU(8)↘↗ U(1) × SU(4)m︸ ︷︷ ︸R-symmetry of the N=4 theory× SU(4)a . (6)After this uplifting, the whole set of critical points are found to satisfy the equations of motion and thequadratic constraints of the maximal theory. Subsequently we computed the mass matrix for the scalars inwww.fp-journal.org © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim990 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactificationstheN = 8 theory using the results in [12]. The results are summarised in Table 4. We find that the solutions3s are non-supersymmetric and nevertheless stable. Up to our knowledge, this is the second example in theliterature (after the one in [13]) of such a solution in maximal supergravity; as opposed to the first example,though, this solution is completely tachyon-free rather than presenting tachyons although still above theBF bound. A final remark is that now, what used to be an accidental Z2 symmetry in the solutions has aproper interpretation within theN = 8 theory, i.e. it interchanges SU(4)m and SU(4)a in Eq. (6).ID V0 m2(N=4) m2(N=8) Stability1s −λ2 − 23 −23stable2s − 3227 λ2 − 45− 45unstable3s − 815 λ2 0 0 stable4s − 3227 λ2 0 − 43unstableTable 4 The values of the energy and the nor-malised mass for the lightest scalar at the set ofcritical points. One observes that, when lifting fromN = 4 to N = 8, the solutions 4s become unstablebecause of the appearance of an unstable tachyonicdirection within the new scalar modes. We remindthe reader that in four dimensions the BF bound isgiven by m2BF = −3/4 in units of the scalar poten-tial.3 ConclusionsThe study of isotropic type IIA orientifolds including geometric fluxes and preserving half-maximal super-symmetry reveals the presence of only AdS vacua in the landscape. The solutions turn out to be all liftableto maximal gauged supergravity where they appear as four different critical points of a unique theory: oneis supersymmetric and stable, another one is non-supersymmetry and nevertheless stable and the remainingtwo are non-supersymmetry and unstable. The natural extension of this work will be to study the effect ofnon-geometric fluxes in this setup in order to get a richer landscape, maybe even containing dS solutions.Acknowledgements We are grateful to Andrea Borghese for stimulating discussions. The work of the authors is sup-ported by a VIDI grant from the Netherlands Organisation for Scientific Research (NWO). Furthermore, G.D. wouldlike to thank the organisers of the “XVII European Workshop on String Theory”, held in September 2011 in Padua, fora very stimulating experience.References[1] J. Shelton, W. Taylor, and B. Wecht, JHEP 0510, 085 (2005) [hep-th/0508133].[2] G. Aldazabal, W. Baron, D. Marques, and C. Nunez, [arXiv:1109.0290 [hep-th]].[3] D. Geissbuhler, [arXiv:1109.4280 [hep-th]].[4] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1103, 137 (2011) [arXiv:1102.0239 [hep-th]].[5] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1106, 030 (2011) [arXiv:1104.3587 [hep-th]].[6] G. Dibitetto, A. Guarino and D. Roest, to appear.[7] J. Schon and M. Weidner, JHEP 05, 034 (2006) [arXiv:hep-th/0602024].[8] B. T. P. Gianni and G. Zacharias, J. Symb. Comput. 6, (1988) 149–167.[9] W. Decker, G.-M. Greuel, G. Pfister, and H. Scho¨nemann, http://www.singular.uni-kl.de.[10] A. Borghese and D. Roest, JHEP 1105, 102 (2011) [arXiv:1012.3736 [hep-th]].[11] B. de Wit, H. Samtleben, and M. Trigiante, JHEP 0706, 049 (2007) [arXiv:0705.2101 [hep-th]].[12] A. Le Diffon, H. Samtleben, and M. Trigiante, JHEP 1104, 079 (2011) [arXiv:1103.2785 [hep-th]].[13] T. Fischbacher, K. Pilch, and N. P. Warner, [arXiv:1010.4910 [hep-th]].© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.org

We analyse geometric type IIA flux compactifications leading to N = 4 gauged supergravities in four dimensions. The complete landscape of isotropic vacua is presented, which turns out to belong to a unique theory. The solutions admit an uplift to maximal supergravity due to the vanishing of the flux-induced tadpoles for all the supersymmetry-breaking branes. Such an uplift is sketched out and the full N = 8 mass spectra are discussed. We find the interesting presence of a non-supersymmetric and nevertheless stable minimum.

Dibitetto, Giuseppe,

Guarino, Adolfo,

Roest, Diederik,

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Vacua analysis in extended supersymmetry compactifications
Dibitetto, Giuseppe; Guarino, Adolfo; Roest, Diederik
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Fortschritte der physik-Progress of physics
DOI:
10.1002/prop.201200004
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10.1002/prop.201200004
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Fortschr. Phys. 60, No. 9 – 10, 987 – 990 (2012) / DOI 10.1002/prop.201200004
Vacua analysis in extended supersymmetry compactifications
Giuseppe Dibitetto∗, Adolfo Guarino, and Diederik Roest
Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen,
The Netherlands
Received 11 January 2012, accepted 11 January 2012
Published online 27 January 2012
Key words Flux compactifications, extended supersymmetry, supergravity models.
We analyse geometric type IIA flux compactifications leading to N = 4 gauged supergravities in four
dimensions. The complete landscape of isotropic vacua is presented, which turns out to belong to a unique
theory. The solutions admit an uplift to maximal supergravity due to the vanishing of the flux-induced
tadpoles for all the supersymmetry-breaking branes. Such an uplift is sketched out and the fullN = 8 mass
spectra are discussed. We find the interesting presence of a non-supersymmetric and nevertheless stable
minimum.
© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
Half-maximal and maximal gauged supergravities in four dimensions are the low-energy effective theories
arising from flux compactifications in string theory, provided that only internal manifolds and extended
objects are included which are compatible with such amounts of supersymmetry. In the last decade the
embedding tensor formalism has been used extensively in the context of (half-)maximal supergravity in
order to describe all the deformations of the free theory in a duality-covariant way. Nevertheless, it has
already been pointed out in the literature that not all the gaugings of supergravity have a higher-dimensional
origin in terms of a geometric flux compactification in string theory. This indicates that gaugings coming
from geometric flux compactifications are not a closed set under general duality transformations and this
is the origin of non-geometric fluxes [1]. Gaugings associated with such fluxes might yet have a higher-
dimensional description in terms of a double field theory (DFT) [2,3], in which duality covariance becomes
the fundamental principle to start with, independently of the compactification procedure. Because of this
interpretation, gauged supergravities with extended supersymmetry seem to be suitable frameworks for
investigating the nature of non-geometric fluxes, as summarised in Table 1.
Table 1 Half-maximal and maximal gauged supegravities in four dimensions seem to be suitable play-
grounds to understand how to restore duality covariance in flux compactifications.
SUSY G stringy interpretation
N = 4 SL(2)× SO(6, 6) S- and T-duality
N = 8 E7(7) U-duality
After introducing the SO(3) truncation of half-maximal supergravity as the effective theory arising
from specific type IIA orientifold reductions with background fluxes, we analyse the landscape of isotropic
vacua in geometric backgrounds, i.e. including metric and gauge fluxes. We find a set of anti-de Sitter
(AdS) critical points admitting an uplift to N = 8, one of which is remarkably stable without preserving
∗ Corresponding author E-mail: g.dibitetto@rug.nl, Phone: +31 (0)50 363 9043, Fax: +31 (0)50 363 4947
© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
988 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactifications
any supersymmetry. More details on this work can be found in [4–6]. The goal of finding de Sitter (dS)
vacua motivates the analysis of non-geometric flux compactifications as possible future extensions.
2 The geometric type IIA with O6/D6 setup
Upon the SO(3) truncation, half-maximal supergravity in four dimensions reduces to a three-field STU
model in the following way
SL(2) × SO(6, 6) −→ SL(2) × SO(2, 2) = SL(2)S × SL(2)T × SL(2)U . (1)
The truncation allows for forty SO(3)-singlet embedding tensor components that can be written in terms of
an SL(2) × SO(2, 2) tensor ΛαABC = Λα(ABC), with α = ± and A = 1, ..., 4 being SL(2) and SO(2, 2)
fundamental indices, respectively. As shown in [4], these forty embedding tensor components correspond
exactly with the set of generalised fluxes in type II orientifold reductions on a Z2 × Z2 isotropic orbifold.
As a consequence of the truncation, the theory preserves only N = 1 supersymmetry out of the original
N = 4. Therefore, the corresponding scalar potential can be written in terms of a (logarithmic) Ka¨hler
potential K and a holomorphic superpotential W by using the standardN = 1 expression
V (SO(3)) = eK
(−3|W |2 + |D W |2) . (2)
At the effective level, fluxes appear as arbitrary superpotential couplings up to linear in S and up to cubic
in T and U . The scalar potential computed from (2) turns out to coincide with the scalar potential given
in [7] up to terms projected out by a set ofN = 4 quadratic constraints on the embedding tensor Λ [4].
Table 2 The set of SO(3)-invariant embedding tensor components of Λ admitting a higher-dimensional
origin as type IIA fluxes: a metric flux ω together with R-R F0,2,4,6 and NS-NS H3 gauge fluxes.
W couplings Type IIA fluxes Embedding tensor components
a0 F6 −Λ+333
a1 F4 Λ+334
a2 F2 −Λ+344
a3 F0 Λ+444
b0 H3 −Λ−333
b1 ω Λ−334
c0 H3 Λ+233
c1 , c˜1 ω Λ+234 , Λ+133
Restricting to the components of Λ that can be interpreted as metric or gauge fluxes in a type IIA
realisation of the model, we are left with the following superpotential
WIIA = a0 − 3 a1 U + 3 a2 U2 − a3 U3 − b0 S + 3 b1 S U + 3 c0 T + (6 c1 − 3 c˜1)T U , (3)
consisting of nine flux-induced couplings (see Table 2). These fluxes are demanded to satisfy the set of
N = 4 quadratic constraints
c1 (c1 − c˜1) = 0 , b1 (c1 − c˜1) = 0 (ω2 = 0) ,
−a3 c0 − a2 (2 c1 − c˜1) = 0 (N6⊥ = 0) ,
(4)
where the first two constraints are related to the nilpotency of the twisted exterior derivative operator,
whereas the third one imposes the absence of D6-branes wrapping the directions orthogonal to the O6-
planes, which would break supersymmetry explicitly down to N = 1. In contrast, D6-branes parallel to
© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.org
Fortschr. Phys. 60, No. 9 – 10 (2012) 989
the O6-planes are compatible with N = 4 supersymmetry, hence being allowed. Their corresponding
flux-induced tadpole reads
N6
|| = 3 a2 b1 − a3 b0 = NO62 −ND6 . (5)
After observing that the set of fluxes given in Table 2 is closed under non-compact duality transfor-
mations (i.e. real shifts and rescalings of S, T and U ), we can restrict the search for critical points of the
scalar potential to the point S0 = T0 = U0 = i without losing generality. The field equations become
then quadratic conditions in the fluxes that have to be satisfied together with the quadratic constraints in
(4). These equations generate a quadratic ideal which we decompose in terms of prime ideals by using
the Gianni-Trager-Zacharias (GTZ) decomposition [8] with the help of Singular [9]. By solving them, the
complete set of critical points of the scalar potential is presented in Table 3. They turn out to be (modulo
the discrete Z2 symmetry introduced in the caption) different AdS critical points of a unique theory with
an underlying gauging given by the gauge group G0 =ISO(3)  U(1)6.
Table 3 The set of critical points of the scalar potential for geometric type IIA isotropic flux compactifications. The
solutions labelled with 1s turn out to preserve N = 1 supersymmetry for s = +1 and to be non-supersymmetric for
s = −1, whereas all the others are non-supersymmetric. It is worth noticing that s = ± 1 appears as an accidental Z2
symmetry which relates solutions having exactly the same energy and the same mass spectrum. The parameter λ is a
global scaling parameter such that V ∝ λ2.
ID a0 a1 a2 a3 b0 b1 c0 c1 = c˜1
1s s
3
√
10
2
λ
√
6
2
λ −s
√
10
6
λ
5
√
6
6
λ −s
√
6
3
λ
√
10
3
λ s
√
6
3
λ
√
10λ
2s s
16
√
10
9
λ 0 0
16
√
2
9
λ 0
16
√
10
45
λ 0
16
√
10
15
λ
3s s
4
√
10
5
λ − 4
√
30
15
λ s
4
√
10
15
λ s
4
√
30
15
λ s
4
√
30
15
λ
4
√
10
15
λ −s 4
√
30
15
λ
4
√
10
5
λ
4s s
16
√
10
9
λ 0 0
16
√
2
9
λ 0
16
√
2
9
λ 0
16
√
2
9
λ
Regarding stability at the critical points, we computed in [4] the full mass matrix for the 38 physical
scalars in N = 4 making use of the results in [10]. At this point, we can say two things: firstly, the
solutions 1s are fully stable because of (fake-) supersymmetry, and secondly the solutions 2s are already
unstable because of the presence of a tachyon whose mass is below the Breitenlohner-Freedman (BF)
bound. However, this is not enough since all the critical points turn out to be compatible with the total
absence of sources, i.e., the flux-induced tadpole (5) does accidentally vanish at these points. As a result,
they admit an uplift toN = 8 and hence one should analyse the full mass matrix for the 70 scalars spanning
the coset E7(7)/SU(8) in order to make any final statement about stability. We worked out the uplifting by
embedding the R-symmetry group of half-maximal theory, i.e. U(4) = U(1) × SU(4), into that of the
maximal, i.e. SU(8), and relating the fermionic shifts given in [7] to those ones in [11] according to the
decomposition
E7(7)
↗
↘
SL(2)× SO(6, 6)
SU(8)
↘
↗ U(1) × SU(4)m︸ ︷︷ ︸
R-symmetry of the N=4 theory
× SU(4)a . (6)
After this uplifting, the whole set of critical points are found to satisfy the equations of motion and the
quadratic constraints of the maximal theory. Subsequently we computed the mass matrix for the scalars in
www.fp-journal.org © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
990 G. Dibitetto et al.: Vacua analysis in extended supersymmetry compactifications
theN = 8 theory using the results in [12]. The results are summarised in Table 4. We find that the solutions
3s are non-supersymmetric and nevertheless stable. Up to our knowledge, this is the second example in the
literature (after the one in [13]) of such a solution in maximal supergravity; as opposed to the first example,
though, this solution is completely tachyon-free rather than presenting tachyons although still above the
BF bound. A final remark is that now, what used to be an accidental Z2 symmetry in the solutions has a
proper interpretation within theN = 8 theory, i.e. it interchanges SU(4)m and SU(4)a in Eq. (6).
ID V0 m2(N=4) m2(N=8) Stability
1s −λ2 − 23 −
2
3
stable
2s − 3227 λ
2 − 4
5
− 4
5
unstable
3s − 815 λ
2 0 0 stable
4s − 3227 λ
2 0 − 4
3
unstable
Table 4 The values of the energy and the nor-
malised mass for the lightest scalar at the set of
critical points. One observes that, when lifting from
N = 4 to N = 8, the solutions 4s become unstable
because of the appearance of an unstable tachyonic
direction within the new scalar modes. We remind
the reader that in four dimensions the BF bound is
given by m2BF = −3/4 in units of the scalar poten-
tial.
3 Conclusions
The study of isotropic type IIA orientifolds including geometric fluxes and preserving half-maximal super-
symmetry reveals the presence of only AdS vacua in the landscape. The solutions turn out to be all liftable
to maximal gauged supergravity where they appear as four different critical points of a unique theory: one
is supersymmetric and stable, another one is non-supersymmetry and nevertheless stable and the remaining
two are non-supersymmetry and unstable. The natural extension of this work will be to study the effect of
non-geometric fluxes in this setup in order to get a richer landscape, maybe even containing dS solutions.
Acknowledgements We are grateful to Andrea Borghese for stimulating discussions. The work of the authors is sup-
ported by a VIDI grant from the Netherlands Organisation for Scientific Research (NWO). Furthermore, G.D. would
like to thank the organisers of the “XVII European Workshop on String Theory”, held in September 2011 in Padua, for
a very stimulating experience.
References
[1] J. Shelton, W. Taylor, and B. Wecht, JHEP 0510, 085 (2005) [hep-th/0508133].
[2] G. Aldazabal, W. Baron, D. Marques, and C. Nunez, [arXiv:1109.0290 [hep-th]].
[3] D. Geissbuhler, [arXiv:1109.4280 [hep-th]].
[4] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1103, 137 (2011) [arXiv:1102.0239 [hep-th]].
[5] G. Dibitetto, A. Guarino, and D. Roest, JHEP 1106, 030 (2011) [arXiv:1104.3587 [hep-th]].
[6] G. Dibitetto, A. Guarino and D. Roest, to appear.
[7] J. Schon and M. Weidner, JHEP 05, 034 (2006) [arXiv:hep-th/0602024].
[8] B. T. P. Gianni and G. Zacharias, J. Symb. Comput. 6, (1988) 149–167.
[9] W. Decker, G.-M. Greuel, G. Pfister, and H. Scho¨nemann, http://www.singular.uni-kl.de.
[10] A. Borghese and D. Roest, JHEP 1105, 102 (2011) [arXiv:1012.3736 [hep-th]].
[11] B. de Wit, H. Samtleben, and M. Trigiante, JHEP 0706, 049 (2007) [arXiv:0705.2101 [hep-th]].
[12] A. Le Diffon, H. Samtleben, and M. Trigiante, JHEP 1104, 079 (2011) [arXiv:1103.2785 [hep-th]].
[13] T. Fischbacher, K. Pilch, and N. P. Warner, [arXiv:1010.4910 [hep-th]].
© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.fp-journal.org


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