UUITP-01/15 Geometric non-geometry Ulf Danielsson and Giuseppe Dibitetto 5 Institutionen f¨or fysik och astronomi, University of Uppsala, 1 0 Box 803, SE-751 08 Uppsala, Sweden 2 n a J 6 1 ABSTRACT ] h t We consider a class of (orbifolds of) M-theory compactifications on Sd×T7−d - p with gauge fluxes yielding minimally supersymmetric STU-models in 4D. We e h presentagroup-theoreticalderivationofthecorrespondingflux-inducedsuper- [ potentialsandarguethattheaforementionedbackgroundsprovidea(globally) 1 v geometric origin for 4D theories that only look locally geometric from the per- 4 4 spective of twisted tori. In particular, we show that Q-flux can be used to 9 generate compactifications on S4 ×T3. We thus conclude that the effect of 3 0 turning on non-geometric fluxes, at least when the section condition is solved, . 1 may be recovered by considering reductions on different topologies other than 0 5 toroidal. 1 : v i X r a [email protected], [email protected] Contents 1 Introduction 1 2 M-theory on Different Geometries and Topologies 2 2.1 An SO(3)×Z truncation of N = 8 supergravity . . . . . . . . . . . . . . . 3 2 2.2 Compactifications on a twisted T7 . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Compactifications on S7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Compactifications on S4 ×T3 . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Discussion 12 A Relevant Branching Rules 14 1 Introduction The issue of studying compactifications of string theory producing satisfactory phenomenol- ogy has always been of utmost importance from several different perspectives. In particular, dimensional reductions of type IIA string theory and the possibility of generating a pertur- bative moduli potential induced by gauge fluxes and geometry has been widely explored in the literature over the last decade. More specifically, type II reductions on twisted tori with gauge fluxes have received a lot of attention over the years owing to the possibility of analysing them in terms of their underlying lower-dimensional supergravity descriptions. In this context, a central role is played by those string backgrounds that can be described by a class of minimal supergravity theories a.k.a. STU-models in four dimensions due to their remarkable simplicity. However, the search for (meta)stable de Sitter (dS) extrema within the above class of STU-models has turned out to be unsuccessful [1–4]. A possible further development of this research line includes the possibility of taking some strongly-coupled effects into account. Therefore, a very natural framework is that of M-theory compactifications. The correspond- ing flux-induced superpotentials present a complete set of quadratic couplings induced by the curvature [5]. Still, in such a context, reductions on twisted tori are known not to allow for any dS solutions either [6]. Within those STU-models describing M-theory on twisted tori, all the couplings higher than quadratic are still judged as non-geometric [7], i.e. they do not admit any eleven- dimensional origin. Nevertheless, by moving to topologies other than toroidal, it is actually possible to find examples of flux superpotentials with homogenous degree higher than two. 1 A particularly enlightening case is that of reduction on S7 yielding maximal SO(8) gauged supergravity in four dimensions admitting a truncation to an STU-model featuring quartic superpotential couplings. Analytic continuations thereof describe non-compact gaugings exhibiting unstable dS extrema where, however, the internal space is non-compact [8]. The goal of our work is to investigate which STU-models containing non-geometric fluxes can be understood as M-theory reductions on internal spaces with non-trivial topologies. It is worth mentioning that, by construction, all our models will admit a locally geometric de- scription in the sense that they rely on an eleven-dimensional formulation correctly satisfying the section condition [9] in the language of Exceptional Field Theory (EFT) [10–12]. This is in the spirit of ref. [13] and does not lead to non-geometric duality orbits in the sense of ref. [14]. However, such a formulation will in general only be equivalent to the traditional one up to total derivative terms [15] that might play an important role upon reductions on non-toroidal topologies. Even though our present work aims at shedding further light on the meaning of non- geometric fluxes, one cannot conclude anything about those non-geometric STU-models that were found to allow for stable dS critical points [16–18]. Whether it is possible to find novel examples of stable dS vacua satisfying the section condition still remains to be seen. Even so we expect that there will be compactness issues due to the no-go result in ref. [19]. The paper is organised as follows. We first present a group-theoretical truncation of max- imal supergravity in four dimensions leading to isotropic STU-models with three complex scalars. We then employ some group theory arguments applied to the embedding tensor formalism in order to derive the flux-induced superpotentials describing M-theory compact- ifications on a twisted T7, S7 and S4×T3. The result of this procedure will be a quadratic, quartic and cubic superpotential, respectively. We then discuss our results as well as some possibleimplicationsandfutureresearchdirections. Finally, wecollectsometechnicaldetails concerning group theory in appendix A. 2 M-theory on Different Geometries and Topologies The low-energy M-theory action in its democratic formulation reads (cid:90) (cid:18) (cid:19) (cid:90) 1 (cid:112) 1 1 1 S = d11x −g(11) R(11) − |G |2 − |G |2 − C ∧G ∧G , (2.1) 2κ2 2 (4) 2 (7) 6 (3) (4) (4) 11 where |G |2 ≡ 1 G G M1···M4 and |G |2 ≡ 1 G G M1···M7 with (4) 4! (4)M1···M4 (4) (7) 7! (7)M1···M7 (7) M = 0,...,10. We choose the following reduction Ansatz ds2 = τ−2ds2 + ρds2 , (2.2) (11) (4) (7) 2 where ρ represents the volume of the internal space X and τ is suitably determined, 7 τ = ρ7/4 , (2.3) such that the Ansatz in (2.2) yield a 4D Lagrangian in the Einstein frame. In the second part of this section we will be considering different choices for X within 7 the class of Sd×T7−d leading to STU-models within N = 1 supergravity in 4D. We will start out revisiting the case of a twisted T7, and we will derive the flux-induced superpotential for this class of compactifications through group-theoretical considerations. This will help us construct our working conventions, which will be used in the analogous derivations carried out for different choices of X other than twisted tori. Before we do this, we first need to 7 introduce a particular group-theoretical truncation of maximal supergravity in 4D leading to the isotropic STU-models that we are interested in. 2.1 An SO(3) × Z truncation of N = 8 supergravity 2 Maximal supergravity in 4D [20] enjoys E global symmetry and all its fields and deforma- 7(7) tions (i.e. gaugings) transform into irrep’s of such a global symmetry group. Vector fields transform in the 56 though only half of them are physically independent due to electromag- netic duality, while scalar fields transform in the 133, though only 70 of them are physically propagating due to the presence of a local SU(8) symmetry. A group-theoretical truncation consists in branching all fields and deformations of the theory into irrep’s of a suitable sub- group G ⊂ E and retaining only the G -singlets. Such a truncation is guaranteed to be 0 7(7) 0 mathematically consistent due the E covariance of the eom’s of maximal supergravity. 7(7) A first discrete Z truncation reads 2 E ⊃ SL(2) ×SO(6,6) , 7(7) S Z 56 →2 (2,12) ⊕(1,32) , (+) (−) whereonlytheZ -evenirrep’sareretainedinthetruncation2. Thisprocedureyields(gauged) 2 N = 4 supergravity in D = 4 [21]. In the second step, we perform a truncation to the SO(3)-invariant sector in the following way (cid:89) SL(2) ×SO(6,6) ⊃ SL(2) ×SO(2,2)×SO(3) ∼ SL(2) × SO(3) . (2.4) S S Φ Φ=S,T,U 2From a more physical perspective, such a Z can be understood as an orientifold involution for those 2 supergravities coming from reductions of type II theories. 3 This step breaks half-maximal to minimal N = 1 supergravity due to the decomposition 4 → 1 ⊕ 3 of the fundamental representation of the SU(4) R-symmetry group in N = 4 supergravity under the SO(3) subgroup SU(4) ⊃ SU(3) ⊃ SO(3) . (2.5) The resulting theory does not contain vectors since there are no SO(3)-singlets in the decom- position 12 → (4,3) of the fundamental representation of SO(6,6) under SO(2,2)×SO(3). The physical scalar fields span the coset space (cid:18) (cid:19) (cid:89) SL(2) M = , (2.6) scalar SO(2) Φ=S,T,U Φ involving three SL(2)/SO(2) factors each of which can be parameterised by the complex scalars Φ = (S,T,U). Such scalars can be obtained by decomposing the adjoint represen- tation 133 of E according to the chain in (2.4) to find nine real SO(3)-singlets, out of 7(7) which only six correspond to physical dof’s. The K¨ahler potential of the theory reads (cid:0) ¯ (cid:1) (cid:0) ¯ (cid:1) (cid:0) ¯ (cid:1) K = −log −i(S −S) − 3log −i(T −T) − 3log −i(U −U) . (2.7) In addition, the embedding tensor of the theory contains 40 independent components (com- ingthistimefromthedecompositionofthe912ofE accordingtothechainin(2.4))which 7(7) can be viewed as the superpotential couplings3 representing a complete duality-inviariant set of generalised fluxes [7]. This yields the following duality-covariant flux-induced superpoten- tial W = (P −P S)+3T (P −P S)+3T2(P −P S)+T3(P −P S) , (2.8) F H Q P Q(cid:48) P(cid:48) F(cid:48) H(cid:48) involving the three complex moduli S, T and U surviving the SO(3)-truncation introduced ealier in this section. P = a −3a U +3a U2 −a U3 , P = b −3b U +3b U2 −b U3 , F 0 1 2 3 H 0 1 2 3 (2.9) P = c +C U −C U2 −c U3 , P = d +D U −D U2 −d U3 , Q 0 1 2 3 P 0 1 2 3 as well as those induced by their primed counterparts (F(cid:48),H(cid:48)) and (Q(cid:48),P(cid:48)) fluxes [24], P = a(cid:48) +3a(cid:48) U +3a(cid:48) U2 +a(cid:48) U3 , P = b(cid:48) +3b(cid:48) U +3b(cid:48) U2 +b(cid:48) U3 , F(cid:48) 3 2 1 0 H(cid:48) 3 2 1 0 (2.10) P = −c(cid:48) +C(cid:48) U +C(cid:48) U2 −c(cid:48) U3 , P = −d(cid:48) +D(cid:48) U +D(cid:48) U2 −d(cid:48) U3 . Q(cid:48) 3 2 1 0 P(cid:48) 3 2 1 0 3The connection between the N = 1 and N = 4 theory was extensively investigated in refs [22,23]. However, the explicit agreement between the scalar potentials up to quadratic constraints was first shown in ref. [4]. 4 For the sake of simplicity, we have introduced the flux combinations C ≡ 2c −c˜ , D ≡ i i i i 2d −d˜ , C(cid:48) ≡ 2c(cid:48) −c˜(cid:48) and D(cid:48) ≡ 2d(cid:48) −d˜(cid:48) entering the superpotential (2.8), and hence also i i i i i i i i the scalar potential. In order to relate our 4D deformed supergravity models to M-theory reductions on dif- ferent geometries and topologies, one needs to fix some conventions for assigning a Z parity 2 to the seven physical coordinates on X . We adopt a set of conventions that is inherited 7 from the link with type IIA compactifications with O6-planes [25], where such a parity transformation can be viewed as orientifold involution. xM −→ xµ ⊕ xa ⊕ xi ⊕ x7 , (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (2.11) 4D (+) (−) where xm ≡ (xa, xi, x7) realise the compact geometry of X . Retaining only even fields and 7 fluxes w.r.t. the action of the above Z will automatically restrict the supergravity theory 2 obtained through an M-theory reduction to the framework of N = 1 STU-models. The metric (2.2) splits accordingly into (cid:16) (cid:17) ds2 = ρ−7/2ds2 + ρ σ−1κ−1M ηa ⊗ηb + σκ−1M ηi ⊗ηj + κ6(cid:0)η7(cid:1)2 , (2.12) 11 4 ab ij where {ηm} ≡ {ηa, ηi, η7} represents a basis of one-forms carrying information about the dependenceofthemetricontheinternalcoordinates. TheR+ scalarsσ andκparametrisethe relative size between the a and i coordinates, which acquire opposite involution-parity when adopting the type IIA picture [26] and the relative size between the type IIA directions and the M-theory circle, respectively. Moreover, M and M contain in general SL(3) × SL(3) ab ij a i scalar excitations. However, such degrees of freedom are frozen due to the requirement of SO(3)-invariance, i.e. M = δ and M = δ . ab ab ij ij The relationship between the STU scalars and the above geometric moduli reads Im(S) = ρ3/2 (cid:0)κ(cid:1)3/2 , Im(T) = ρ3/2 σ1/2 , Im(U) = ρ3/2κ2 . (2.13) σ κ3/2 2.2 Compactifications on a twisted T7 The seven compact coordinates of the torus transform in the fundamental representation of the SL(7) subgroup of E , which can be viewed as the group of diffeomorphisms on T7 7(7) with twist. The relevant chain of decompositions is E ⊃ SL(8) ⊃ R+ ×SL(7) ⊃ R+ ×R+ ×SL(6) , 7(7) M M B and finally down to R+ ×R+ ×R+ ×SL(3) ×SL(3) , M B A a i 5 STU couplings M-theory fluxes Flux labels R+ ×R+ ×R+ ×SL(3) ×SL(3) irrep’s S T U a i 1 G a (1,1) aibjck7 0 (+1;+3;+3) 2 2 2 S G b (1,1) ijk7 0 (−1;+3;+3) 2 2 2 T G c (3,3(cid:48)) ibc7 0 (+1;+1;+3) 2 2 2 U G a (3,3) aibj 1 (+1;+3;+1) 2 2 2 ST ω a d (3,3(cid:48)) 7i 0 (−1;+1;+3) 2 2 2 T2 ω i c(cid:48) (3(cid:48),3) a7 3 (+1;−1;+3) 2 2 2 T U ω k, ω a c , c˜ ((3(cid:48),8)⊕(6,1)) aj bc 1 1 (+1;+1;+1) 2 2 2 SU ω a b (3,3) jk 1 (−1;+3;+1) 2 2 2 U2 ω 7 a (3(cid:48),3(cid:48)) ai 2 (+1;+3;−1) 2 2 2 Table 1: Summary of M-theory fluxes and superpotential couplings on a twisted T7. Isotropy (i.e. SO(3)-invariance) only allows for flux components that can be constructed by using (cid:15) ’s and δ ’s. These symmetries also induce a natural splitting ηA = (ηa, ηi, η7) where (3) (3) a = 1,3,5 and i = 2,4,6. where one should, furthermore, only restrict to isotropic objects. Following the philosophy of ref. [25], one can match the na¨ıve scaling behaviour coming from dimensional reductions of the various terms in the action (2.1) with the correct STU-charges by using the relations in (2.13). This results in the following mapping  q = 1 q − 1 q − 1q ,  S 28 M 28 B 4 A   q = 3 q − 3 q + 1q , (2.14) T 28 M 28 B 4 A     q = 3 q + 1q , U 28 M 7 B between the group-theoretical R+ charges obtained from the above decomposition and the STU-charges realised in N = 1 supergravity. Such a mapping allows one to derive a dictio- nary between fluxes and superpotential couplings. From the decomposition of the fundamental representation of E (see appendix A for 7(7) the details) E ⊃ R+ ×R+ ×R+ ×SL(3) ×SL(3) , 7(7) M B A a i (2.15) 56 → (1,1) ⊕(3(cid:48),1) ⊕(1,3(cid:48)) ⊕ ... , (+6;+6;0) (+6;−1;−1) (+6;−1;+1) one can exactly and unambiguously identify the physical derivative operators along the seven M-theory internal directions as ∂ ∈ (3(cid:48),1) , ∂ ∈ (1,3(cid:48)) , ∂ ∈ (1,1) (2.16) a (+1;+1;+1) i (0;+1;+1) 7 (0;0;+3) 2 2 2 2 2 6 w.r.t. R+ ×R+ ×R+ ×SL(3) ×SL(3) . S T U a i Asfarasthefluxesareconcerned,onehastodecomposetheembeddingtensorofmaximal supergravity E ⊃ SL(8) , 7(7) (2.17) 912 → 36⊕36(cid:48) ⊕420⊕420(cid:48) , to find SL(8) ⊃ R+ ×R+ ×R+ ×SL(3) ×SL(3) , M B A a i 36(cid:48) → (1,1) ⊕ ... , (+14;0;0) (2.18) 420 → (1,1) ⊕(3,3) ⊕(3,3(cid:48)) ⊕ ... , (+10;+3;+3) (+10;−4;0) (+10;+3;−1) 420(cid:48) → (3,3) ⊕(6,1) ⊕(3(cid:48),8) ⊕ (+6;−1;+3) (+6;−1;−1) (+6;−1;−1) (3(cid:48),3(cid:48)) ⊕(3(cid:48),3) ⊕(3,3(cid:48)) ⊕ ... , (+6;−8;0) (+6;+6;−2) (+6;+6;+2) where we have used the decomposition in (A.1). By using the dictionary (2.14), we were able to reproduce all the correct STU scalings of the fluxes on a twisted T7. The results of this procedure are collected in table 1 and agree with those already found earlier in refs [5,6]. As a consequence, the flux-induced superpotential in this case reads W = a − b S + 3c T − 3a U + 3a U2 + 3(2c −c˜ )TU + 3b SU − 3c(cid:48)T2 − 3d ST . (T7) 0 0 0 1 2 1 1 1 3 0 (2.19) One should note that the underlying gauging for this class of compactifications is expected to be non-semisimple, its semisimple part being the group realised by the components of ω-flux as structure constants. The non-semisimple extension is given by the presence of 4- and 7-form gauge fluxes. This is in line with what already observed in refs [23,25,27] in the context of massive type IIA compactifications on a twisted T6 in the absence of local sources, where the corresponding effective 4D description turned out to be N = 8 supergravity with gauge group SO(4)(cid:110)Nil . 22 2.3 Compactifications on S7 Let us now consider the compactification of M-theory on S7. In refs [28,29] it was already noted that such a compactification is described by an SO(8) gauging in 4D maximal super- gravity. The components of the embedding tensor are parametrised by a symmetric 8 × 8 matrix Θ transforming4 in the 36(cid:48) of SL(8). AB 4We adopt the following conventions XA ≡ (cid:0)Xa, Xi, X7, X8(cid:1) for the fundamental representation of SL(8). 7 Gaugings in the 36 ⊕ 36(cid:48) are in general identified by Θ˜AB ⊕ Θ satisfying the following AB Quadratic Constraints [30] 1 (cid:16) (cid:17) Θ Θ˜CB − Θ Θ˜CD δB = 0 . (2.20) AC 8 CD A Such theories have a subgroup of SL(8) as gauge group and their scalar potential can be written in terms of a complex pseudo-superpotential [31] 3 1 V = − |W|2 + |∂W|2 , (2.21) 8 4 (cid:16) (cid:17) where W ≡ 1 Θ MAB − iΘ˜ABM , MAB being the SL(8)/SO(8) coset representa- 2 AB AB tive and M its inverse. AB In the relevant S7 example, the embedding tensor reads  −c˜(cid:48) 1  1 3    −d˜ 1  Θ =  2 3  = 1 , Θ˜AB = 0 , (2.22) AB 8 8  −b(cid:48)   3    a 0 which belongs to the semisimple branch of solutions to the constraints in (2.20). The corre- sponding scalar potential in (2.21) simplifies to 1 (cid:0) (cid:1) V = Θ Θ 2MACMBD − MABMCD . (2.23) AB CD 8 We will now interpret this theory as an STU-model and rederive the corresponding flux- induced superpotential by means of group theory arguments. In this case the relevant de- composition is still the same as the one in the twisted T7 case SL(8) ⊃ R+ ×R+ ×R+ ×SL(3) ×SL(3) , M B A a i (2.24) 36(cid:48) → (1,1) ⊕(1,1) ⊕(6(cid:48),1) ⊕(1,6) ⊕ ... . (+14;0;0) (−2;+12;0) (−2;−2;−2) (−2;−2;+2) This gives the STU-couplings collected in table 2 upon using the dictionary (2.14). The associated flux-induced superpotential is given by W = a − b(cid:48)ST3 − 3c˜(cid:48)T2U2 − 3d˜STU2 , (2.25) (S7) 0 3 1 2 which matches what was found in refs [27,32] in the context of STU-models. The N = 1 scalar potential computed from (2.25) coincides with (2.23) upon using the correct identifi- cation of the STU scalars inside the coset representative MAB. 8 STU couplings M-theory fluxes Flux labels R+ ×R+ ×R+ ×SL(3) ×SL(3) irrep’s S T U a i 1 G a (1,1) aibjck7 0 (+1;+3;+3) 2 2 2 ST3 Θ b(cid:48) (1,1) 77 3 (−1;−3;+3) 2 2 2 T2U2 Θ c˜(cid:48) (6(cid:48),1) ab 1 (+1;−1;−1) 2 2 2 ST U2 Θ d˜ (1,6(cid:48)) ij 2 (−1;+1;−1) 2 2 2 T U Θ c (3(cid:48),1) (non-isotropic) i7 1 (+1;+1;+1) 2 2 2 Table 2: Summary of M-theory fluxes and superpotential couplings on S7. Isotropy (i.e. SO(3)-invariance) only allows for flux components that can be constructed by using (cid:15) ’s (3) and δ ’s. In the frame we have chosen one of the objects in the 36(cid:48) is G flux, whereas (3) (7) the quartic couplings describe the S7 geometry. 2.4 Compactifications on S4 × T3 We have seen how for S7 the superpotential contains only the constant part and some quartic parts. We will now analyse the flux-induced superpotential for S4 × T3 to find that cubic terms will appear, thus mimicking the effect of the presence of Q-flux. GiventhenaturalfactorisationthatX hasinthiscase, therelevantbranchingoneshould 7 analyse goes through E ⊃ SL(8) ⊃ R+ ×SL(3) ×SL(5) ⊃ R+ ×R+ ×SL(3) ×SL(4) , 7(7) Q a Q 1 a and finally down to R+ ×R+ ×R+ ×SL(3) ×SL(3) , Q 1 2 a i where, as usual, only isotropic objects should be retained within our STU-model. By following the new branching of the fundamental representation of E 7(7) E ⊃ R+ ×R+ ×R+ ×SL(3) ×SL(3) , 7(7) Q 1 2 a i (2.26) 56 → (1,1) ⊕(3(cid:48),1) ⊕(1,3(cid:48)) ⊕ ... , (−6;+3;+3) (−10;0;0) (−6;+3;−1) and demanding that the physical derivative operators identified in (2.16) be the same, one finds  q = − 1 q − 1 q ,  S 20 Q 10 1   q = − 1 q + 3 q − 1q , (2.27) T 20 Q 20 1 4 2     q = − 1 q + 3 q + 1q , U 20 Q 20 1 4 2 as a new dictionary between STU-scaling weights and group-theoretical R+ × R+ × R+ Q 1 2 charges. Note that this procedure of identifying the seven physical derivative operators 9