Physics Letters B 768 (2017) 92–96 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb AdS vacua from dilaton 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 “brane supersymmetry 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 AdS7 × S3 vacua, while unbounded three-form fluxes, when combined with Editor: N. Lambert 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. String 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 branes 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 graviton, a dilaton, a RR two-form potential, a Majorana–Weyl String Theory [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 group 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 anomaly “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 3 (3) (7) modification arises from the torus amplitude. Hence, γ = for = = E 2 βE 0 ,βE 1 . (1.6) = 5 × orientifold models and γE 2 for the SO(16) SO(16) heterotic Finally, as we have anticipated, in the heterotic SO(16) × SO(16) model. In this last case a symbol would be more appropriate, 5 model γE = , since the dilaton potential originates from the but in eqs. (1.1) and (1.2) we have used T in all cases for brevity. 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.