Physics Letters B 768 (2017) 92–96

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Physics Letters B

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AdS vacua from tadpoles and form fluxes ∗ J. Mourad a, A. Sagnotti b, a APC, UMR 7164-CNRS, Université Paris Diderot – Paris 7, 10 rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France b Scuola Normale Superiore and INFN, Piazza dei Cavalieri, 7, 56126 Pisa, Italy a r t i c l e i n f o a b s t r a c t

Article history: We describe how unbounded three-form fluxes can lead to families of AdS3 × S7 vacua, with constant Received 27 December 2016 dilaton profiles, in the USp(32) model with “ breaking” and in the U (32) 0’B model, Received in revised form 14 February 2017 if their (projective-)disk dilaton tadpoles are taken into account. We also describe how, in the SO(16) × Accepted 21 February 2017 SO(16) heterotic model, if the torus vacuum energy is taken into account, unbounded seven-form Available online 28 February 2017 fluxes can support similar AdS × S vacua, while unbounded three-form fluxes, when combined with Editor: N. Lambert 7 3 internal gauge fields, can support AdS3 × S7 vacua, which continue to be available even if is neglected. In addition, special gauge field fluxes can support, in the SO(16) × SO(16) heterotic model, a set of  AdSn × S10−n vacua, for all n = 2, .., 8. loop and α corrections appear under control when large form fluxes are allowed. © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction and setup tions of BPS and orientifolds. The simplest example of this type is Sugimoto’s model in [7]. Its low-lying closed sector includes Supersymmetry breaking appears typically accompanied, in the , a dilaton, a RR two-form potential, a Majorana–Weyl [1], by the emergence of runaway potentials. The gravitino and a Majorana–Weyl spinor of opposite chiralities, as vacuum is deeply affected by their presence and current string in the type-I superstring. However, in the low-lying open sector tools, which are very powerful when combined with supergrav- there are adjoint vectors of a USp(32) gauge and fermions ity [2] in the supersymmetric case, become ineffective in front of in the reducible antisymmetric representation, whose singlet plays the resulting redefinitions. These reflect an untamed problem that the role of a goldstino. There is also another non-tachyonic ori- started to surface long ago [3]. entifold, which is not supersymmetric to begin with but whose As in other systems, internal fluxes can make a difference in features are rather similar. It is the 0’B model of [9], whose closed this context, and interesting progress was recently made in [4], sector is purely bosonic and contains, in addition to the bosonic while [5] contains notable earlier results. What we shall present modes of the type-I superstring, an axion and a self-dual four form here bears some similarities to these works, although it originates potential, and whose open sector hosts the vectors of a U (32) from a different line of thought, as we are about to describe. In this gauge group and Fermi modes in the antisymmetric representa- letter, relying on the low-energy effective field theory, we explore tion and its conjugate. Both settings afford a rich family of lower- vacuum solutions in ten-dimensional non-tachyonic models, taking dimensional counterparts [7,10], but here we shall content our- into account the low-lying potentials that are induced when su- selves with the simplest 10D examples. Finally, in the third non- persymmetry is broken or absent altogether. The cases at stake in- tachyonic 10D non-supersymmetric model, of the heterotic type clude two types of orientifold models [6] and the SO(16) × SO(16) [11], the bosonic spectrum describes again gravity, a dilaton and heterotic model. In the first orientifold model, an exponential po- a two-form potential that comes from the NS–NS sector, together × tential results from the (projective-)disk tadpoles that accompany with SO(16) SO(16) gauge fields. These theories are “brane supersymmetry breaking” [7]. This string-scale mechanism free, but the 0’B model rests on a more complicated, non-factorized reflects the presence, in vacua that host a non-linear realization of Green–Schwarz mechanism, as a wide class of lower-dimensional supersymmetry [8], of the residual tension from non-BPS combina- examples [13,6,9]. A 9D vacuum of Sugimoto’s model was exhibited long ago by Dudas and one of us [12], but it contains singularities and regions of strong coupling. In this note we shall see that form fluxes can * Corresponding author. E-mail addresses: [email protected] (J. Mourad), [email protected] lead to some smooth symmetric vacua for 10D orientifolds even (A. Sagnotti). in the presence of dilaton tadpoles. Moreover, when large form http://dx.doi.org/10.1016/j.physletb.2017.02.053 0370-2693/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. J. Mourad, A. Sagnotti / Physics Letters B 768 (2017) 92–96 93

 fluxes are allowed the resulting α and string loop corrections ap- scalar, by arbitrary scale factors A(r) and C(r), and by form-fields pear small. There are AdS3 × S7 solutions of this type for both the proportional to the (p + 1)-dimensional volume element, up to a USp(32) model of [7] and the U (32) 0’B model of [9], and more- second scalar function b(r). Here, however, we shall restrict our over there are similar AdS3 × S7 and AdS7 × S3 solutions for the attention to highly symmetric configurations where both φ and C heterotic SO(16) × SO(16) model. are constant. For gauge fields, non-trivial symmetric profiles exist  The low-energy dynamics of the systems of interest is de- in this case for k = 1. The resulting internal spaces are spheres, scribed, in the Einstein frame, by effective Lagrangians of the type whose SO(9 − p) isometries can be identified with correspond-   ing orthogonal subgroups of the available gauge groups, USp(32), 1 √ 1 S = d10x −g − R − (∂φ)2 U (32) or SO(16) × SO(16), up to a third scalar function a(r). In 2 2 ˜ i 2k10 detail, given the embedding coordinates y on the internal spheres, T (p) such that y˜ y˜ = 1, the so(9 − p)-valued gauge field configurations 1 −2 β φ 2 − e E H + read 2 (p + 2)! p 2    1 − A = ˜ ˜ T − ˜ ˜ T − e 2 αE φ tr F 2 − TeγE φ , (1.1) ia(r) ydy dy y . (1.4) 4  As we shall see shortly, the solutions resulting from this set of where 2 k2 = (2π)7 and we have set α = 1. These originate from 10 profiles for the different available fields, when they exist, describe the string-frame actions ×   vacua of the AdS S type. √   A common signature of the 10D USp(32) and U (32) orientifold 1 10 −2φ 2 S = d x − G e − R + 4(∂φ) models is the special value γ = 3 , which reflects the (projective- 2k2 E 2 10 )disk origin of the dilaton potential. In both cases one can turn on 1 −2 β φ 2 Hp+2 form fluxes with p = 1or p = 5, corresponding to two-form − e S H + 2(p + 2)! p 2 potentials or to dual six-form ones, and the preceding discussion  implies that 1 − − e 2 αS φ tr F 2 − TeγS φ , (1.2) 4 =−1 (1) =−1 (5) = 1 αE ,βE ,βE . (1.5) 1 4 2 2 and Hp+2 is the field strength of a (p + 1)-form potential Bp+1 that presents itself, in various incarnations, in both heterotic and Moreover, the U (32) model also allows profiles of a four-form orientifold 10D models. Moreover γS =−1for the orientifold mod- potential with self-dual field strength and of an eight-form dual els, where the contribution arises from (projective) disk amplitudes, potential, for which while γS = 0for the SO(16) × SO(16) heterotic model, where the modification arises from the torus amplitude. Hence, γ = 3 for (3) = (7) = E 2 βE 0 ,βE 1 . (1.6) 5 orientifold models and γE = for the SO(16) × SO(16) heterotic 2 Finally, as we have anticipated, in the heterotic SO(16) × SO(16) model. In this last case a symbol would be more appropriate, model γ = 5 , since the dilaton potential originates from the but in eqs. (1.1) and (1.2) we have used T in all cases for brevity. E 2 (p) torus level, and one can turn on form fluxes with p = 1or p = 5, The parameters β that enter eqs. (1.1) and (1.2) are related by E with Weyl rescalings to their string-frame counterparts, and βS = 1for the heterotic NS–NS two-form potential of ordinary 10D supergrav- = 1 (1) = 1 (5) =−1 = αE ,β ,β . (1.7) ity [14] while βS 0for the R–R two-form potential present in 4 E 2 E 2 =−1 = 1 orientifold models [15]. Moreover, αE 4 (αS 2 ) for the ori- = 1 = × 2. AdS × S solutions of 10D non-tachyonic models entifold models and αE 4 (αS 1) for the SO(16) SO(16) model. Our aim here is to exhibit for the three 10D models solutions The class of solutions that we would like to illustrate rests on of the type constant values for C and φ, on special constant values for the gauge field functions a in eq. (1.4) that solve the corresponding 2 2A(r) 2 2C(r) ds = e g(Lk) + dr + e g(Ek ), (1.3) non-linear field equations, and on the explicit non-constant solu- tions where Lk is a maximally symmetric (p +1)-dimensional Lorentzian  (p) of curvature k, while Ek is a maximally symmetric (p+1) A(r) + 2 β φ − (8−p) C  H + = he E (p + 1) dr (2.1) (8 − p)-dimensional Euclidean manifold of curvature k . These p 2 × − space times possess a manifest ISO(1, p) SO(9 p) symme- for the relevant form field strengths, where h is a constant and =  = try when k 0 and k 1, which will be the case of main interest (p + 1) denotes the (p + 1)-dimensional volume form. With these 2 for us, and rest, in general, on two scalar functions, A(r) and C(r). types of profiles the field equations arising from (1.1) reduce to This is a convenient setting to explore brane configurations, if one allows for non-trivial profiles of the dilaton and the available form- ξ αE − − TeγE φ = (8 − p)(7 − p) e 4 C 2 αE φ fields. Non-abelian gauge fields valued in suitable subalgebras of γE the gauge algebra can also be compatible with the manifest isome- (p) 2 (p) βE h − 2 (8 − p) C + 2 β φ tries. These properties are granted by any radial profile φ(r) for the − e E , (2.2) γ E  − 2 α − − 1 H 2 C E 4 C 2 αE φ The definition of the p+2’s involves, in general, Chern–Simons terms related to 16k e = ξ 8 + p + (8 − p) e the Green–Schwarz cancellation mechanism, or its generalization, at work in these γE systems. (p) (p) 2  2 β − 2 8 − p C + 2 If k =−1, the manifest internal isometry would be SO(1, 8 − p). Moreover, the 2 + − E ( ) βE φ h p 1 γ e space–time portion would have a manifest SO(1, p + 1) isometry if k = 1and a E =− + , (2.3) manifest SO(2, p) isometry if k 1. (7 − p) 94 J. Mourad, A. Sagnotti / Physics Letters B 768 (2017) 92–96

φ  − (8 − p)(7 − p) 2αE − − 2 = 2 A + − 4 C 2 αE φ 2 ξ e 2 (A ) ke ξ 1 e e 2 C =  16(p + 1) γE  ± − ξ T 2 φ (p) 1 1 3 e 2 (p) h 2 βE − 2 (8 − p) C + 2 β φ + 7 − p + e E . 2 7 4 φ + h ξ e 16(p 1) γE =  32 7 (2.4) ± − ξ T 2 φ 1 1 3 e Here ξ = 0, 1define two distinct choices for the internal gauge     42 ξ T field strength, × 1 ± 1 − e 2 φ + 5 Te2 φ . (2.7) ξ 3 F = i ξ dyd˜ y˜ T , (2.5) For the reader’s convenience, we have kept all powers of ξ , despite which correspond to a = ξ , both of which satisfy the correspond- 2 the fact that ξ = 0, 1, in order that the smooth limiting behavior of F ing field equations, with vanishing in the first case. In three these and other solutions as ξ → 0be manifest. There is a branch = dimensions the choice ξ 1would identify the Wu–Yang solution of solutions, corresponding to the “minus” sign above, which con- [16]. nects smoothly to the ξ = 0case, where large AdS3 and S7 radii Notice that eq. (2.2) implies strong constraints, due to the pos- accompany small couplings. In this case the preceding relations × itivity of T (or , in the SO(16) SO(16) model). Unbounded imply the ξ -independent limiting large-h behavior values of h are manifestly possible, in the absence of internal gauge (p) fields, when β < 0, and thus for three-form fluxes in the orien- φ 12 4 3 144 E gs ≡ e ∼   , R g ∼ , tifold models or seven-form fluxes in the heterotic model. In ad- 1 s T 2 2 hT3 4 dition we shall see that, surprisingly, unbounded three-form fluxes   ×  2 − 6 are also allowed in the SO(16) SO(16) model, in the presence A ∼ ke 2 A + . (2.8) of non-trivial internal gauge fields. R2 H As is well known, the field strength p+2 involves, in general, Notice the crucial role played by the tension T in granting the ex- Chern–Simons couplings. These, in their turn, bring about a sub- istence of these solutions. For large values of h these expressions tlety related to the Bianchi identities, which are modified into receive small corrections also for ξ = 1, and thus within an inter- esting corner of parameter space the complete equations of String d H + ∼ X + F R (2.6) p 2 p 2 ( , ) , Theory appear reliably captured by our approximations. Large val- where the X’s are invariant polynomials involving space–time and ues of h imply large values for R, which sets the scales for the gauge-field curvatures. However, in our highly symmetric back- AdS3 and S7 factors, and also small values for the string coupling.  grounds these Chern–Simons terms vanish, and the X’s with them, These, in their turn, are expected to translate into small α and so that there are no further conditions coming from this end. string loop corrections. Finally, for ξ = 0the USp(32) gauge group For the internal gauge fields this is manifest from eq. (2.5), since is unbroken, while for ξ = 1it is broken to a USp(24) subgroup. dy˜ T dy˜ = 0, and a similar link holds between the vielbein one-form The second branch of solutions, corresponding to the “plus” and the Riemann curvature. sign in eq. (2.7), is only available for ξ = 1, and thus in the pres- With constant φ and C profiles, the space–time manifold ac- ence of internal gauge fields. It associates large radii to strong quires additional isometries, and its AdS nature is evident from couplings (whose upper bound is determined by T ) and, surpris- eqs. (1.3) and (2.4) for k = 0, since in this case one recovers a stan- ingly, has a smooth limit for vanishing tension T , when it is also dard presentation of the symmetric space in Poincaré coordinates. a solution for the SO(32) type-I superstring [13]. Although it lies outside the perturbative reach, we find this option intriguing, es- Actually, when combined with the radial direction, Lk is always de- scribing an AdS space, independently of the value of k: this only pecially in view of a similar weak-coupling heterotic counterpart affects the slicing, which is Minkowski for k = 0, dS for k = 1 and that we are about to describe. AdS for k =−1. These considerations also apply to the 0’B orientifold, where if We can now analyze in detail the solutions of eqs. (2.2), (2.3) ξ = 0the original U (32) gauge group is unbroken, while if ξ = 1it and (2.4). is broken to a U (24) subgroup. In this case there would be also, in principle, the two other options of eq. (1.6), which correspond to = 2.1. AdS3 × S7 Solutions in 10D Orientifolds p 3, 7. However, they do not lead to consistent solutions of this type, since the corresponding values for βE are not positive, while Taking into account the corresponding vacuum configurations, the corresponding αE of eq. (1.5) are negative. one can see that for Sugimoto’s model T = 16 , while the 0’B π 2 × × model the total tension is half of this value. For their RR two- 2.2. AdS3 S7 and AdS7 S3 solutions in the 10D heterotic × form potentials βE is negative, which allows unbounded values SO(16) SO(16) for h. Since for these models αE < 0, as we have seen in eq. (1.5), × the very consistency of eq. (2.2) requires a non-vanishing h. As In the heterotic SO(16) SO(16) model the sign of αE is pos- aresult, the vacua that we are about to describe are sustained by itive, as we have seen in eq. (1.7). As a result, internal non-abelian three-form fluxes that, strictly speaking, are quantized but can be gauge fields can sustain by themselves the class of vacua under 2 unbounded. In addition, the r.h.s. of eq. (2.2) is manifestly non- scrutiny. Let us also recall that for this model  4π [11].  25 negative, and thus k = 1, so that the internal spaces of these solu- If only internal gauge fields are retained, eq. (2.3) reveals that  tions are seven-dimensional spheres S7. One has also the option of k = 1, so that the internal spaces are again, consistently, spheres. turning on internal gauge fields, provided h is not vanishing. The In the absence of form fluxes there is thus a whole family of solutions that we have identified are thus AdS3 × S7 vacua. AdS p+2 × S8−p solutions, for p = 1, ..., 6, where the original gauge The key features of this class of solutions become quite trans- group is broken accordingly. The problem with these solutions is parent if ξ = 0, but in general eqs. (2.2) and (2.3) imply that, for that the value of a that solves the gauge field equations is fixed, so the two non-tachyonic orientifold models, that one looses the large deformation parameter available in the J. Mourad, A. Sagnotti / Physics Letters B 768 (2017) 92–96 95

− φ preceding examples. As a result, these solutions are expected to ξ e 2  2 C =  suffer from sizable α and string loop corrections. e , (2.12) 2 1 − 1 − ξe 2 φ For p = 1, however, one can also turn on a three-form flux, but     17 2 φ 1 βE is now positive and naively this would seem to imply bounded 2 − − − 2 φ h 24 e ξ 1 1 ξe values for h. However, the actual range for h is the result of com- = ξ 3    , 3 3 peting effects, and definite assessments can be made only after 1 − 1 − ξe 2 φ combining eqs. (2.2) and (2.3). In this fashion one can show that − φ and there is again a single consistent branch, which now connects 2 2 C 2 ξ e = e =  , (2.9) smoothly to the ξ 0case. These expressions clearly show that 1 + 1 + ξe 2 φ this last class of solutions rests on , and ceases to exist in its 3 absence. Finally, in the large-h limit since only one branch is compatible with the positivity of the ex- 1 ponential of C. Notice that these solutions are supported by the 5 4 1 ≡ φ ∼ 5 4 ∼ internal gauge fields and disappear for ξ = 0, and letting ξ = 1the gs e , gs R , h2 2 2 corresponding link between h and φ reads    2 − 2 A 1 2 − 4 φ A ∼ ke + . (2.13) h e 2 =  4 R 32 7 + + 2 φ Once more, large values for h result in large AdS7 and S3 radii, 1 1 3 e and also in small values for the string coupling gs. These results      are expected to translate into small α and string loop corrections. × 42 1 + 1 + e 2 φ − 13 e 2 φ . (2.10) 3 3. Conclusions The preceding relations encode the limiting large-h behavior We have described how runaway exponential potentials aris- 1 4 ing from (projective-)disk dilaton tadpoles can combine with form φ 21 4 gs ≡ e ∼ , gs R ∼ 1 , × h2 fluxes to yield a family of AdS3 S7 vacua for Sugimoto’s ori-   entifold model of [7] and for the non-tachyonic 0’B orientifold  2 − 21 A ∼ ke 2 A + , (2.11) of [9], with unbroken gauge groups. Alternatively, if internal non- 2 4 R abelian profiles are also turned on the gauge groups break, in the  so that even this class of solutions affords a region where α and two cases, to USp(24) and U (24). We have shown that there are string loop corrections appear under control. Notice that these actually two branches of solutions. The first branch is a more con- weak-coupling solutions of eqs. (2.9) and (2.10) have a smooth ventional weak-coupling one, which can be supported by a three- limit for vanishing , which is not needed to grant their exis- form flux alone and whose existence rests on the presence of the tence. The marginal role of in this case is clearly reflected in the tadpole tension T . The second branch is a strong-coupling one, limiting behavior of eqs. (2.11), and indeed this class of solutions which rests on the presence of internal gauge fields and contin- would be available also in the supersymmetric 10D heterotic mod- ues to exist in the limit of vanishing tension, when it applies to els of [17]. This contrasts with what we found for first branch of the SO(32) type-I superstring. In the SO(16) × SO(16) heterotic AdS3 × S7 solutions in the orientifold models, where a (projective- model [11] internal gauge fields alone can sustain AdSn × S10−n )disk tadpole was essential, but it resonates with what we found vacua for n = 2, .., 8, but perturbative large-field limits exist only for the second branch, whose strong-coupling solutions also con- when form fields can be included, i.e. for n = 3, 7. In the first case tinue to exist in the limit of vanishing T . On the other hand, a the vacuum is sustained by the essential contribution of internal strong-weak coupling link is expected to hold when both T and gauge fields, combined with an H3 flux, and continues to exist vanish, between solutions of the SO(32) type-I superstring and in the limit of vanishing , when the solution would also apply to of the Spin(32)/Z2 heterotic [17] model, which are weak-strong supersymmetric heterotic strings, consistently with the string dual- coupling partners in the general picture of [15]. An extension of ity link between the Spin(32)/Z2 heterotic string and the SO(32) the link to the case of non-vanishing T and , which the solu- type-I superstring. If a link transcends the case of vanishing T and tions somehow suggest, could provide interesting clues on non- , as the solutions would seem to suggest, it encodes potentially supersymmetric string dualities.3 interesting information of string dualities beyond the supersym- This is not all for this heterotic SO(16) × SO(16) model, metric case, on the par with cases explored long ago in [19]. We since there is another potentially interesting flux, corresponding have also found a second class of weak-coupling solutions of the = to p 5, and thus to the six-form potential present in the dual SO(16) × SO(16) heterotic model, supported by H7 fluxes, which formulation of [20]. The duality connecting this case to the more exists only in the presence of a non-vanishing . = familiar one with p 1 entails a reversal of the sign of βE , as we Although our results rest on the low-energy field equations cap- have seen, so that the resulting vacua can again involve unbounded tured by (super)gravity, in the AdS3 × S7 orientifold solutions and tensor profiles, possibly accompanied by internal gauge fields as in in both the AdS3 × S7 and AdS7 × S3 heterotic solutions one can eqs. (1.4) and (2.5). The gauge group is again unbroken in their turn on large form fluxes, which result in large manifold sizes and ×  absence, and otherwise it is broken to SO(16) SO(12). small string couplings, so that both α and string loop corrections Let us take a closer look at this last class of solutions. Combin- are expected to be small in these regions of parameter space. ing eqs. (2.2) and (2.3), one can now derive the two relations We were led to the present considerations by our interest in the general problem of back-reactions to broken supersymmetry in String Theory, and in particular to the string-scale phenomenon 3 The link that is surfacing here is related to heterotic-type I duality along the of “brane supersymmetry breaking” [7]. The simple vacua that we lines of [15]. Other dualities relating heterotic strings with broken and unbroken su- persymmetry, suggested by Scherk–Schwarz deformations [18], were first explored have found have the virtue of lacking spatial regions of strong cou- long ago in [19]. pling, which were present in the nine-dimensional one found in 96 J. Mourad, A. Sagnotti / Physics Letters B 768 (2017) 92–96

[12], and a variety of options of this type can be available in lower- [6] A. Sagnotti, in: G. Mack, et al. (Eds.), Cargese ’87, Non-Perturbative Quantum dimensional cases. Field Theory, Pergamon Press, 1988, p. 521, arXiv:hep-th/0208020; The 0’B model was explored by Armoni and others [21], over G. Pradisi, A. Sagnotti, Phys. Lett. B 216 (1989) 59; P. Horava, Nucl. Phys. B 327 (1989) 461; the years, with an eye to large-N limits of QCD, in view of its P. Horava, Phys. Lett. B 231 (1989) 251; formal proximity to a supersymmetric spectrum. The vacua that M. Bianchi, A. Sagnotti, Phys. Lett. B 247 (1990) 517; we have exhibited might prove useful for this program, or in other M. Bianchi, A. Sagnotti, Nucl. Phys. B 361 (1991) 519; directions related to the AdS/CFT correspondence [22]. In addition, M. Bianchi, G. Pradisi, A. Sagnotti, Nucl. Phys. B 376 (1992) 365; A. Sagnotti, Phys. Lett. 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