JHEP11(2010)083 - β Springer ctifications, October 26, 2010 September 3, 2010 : November 18, 2010 : : , rganizing principle. In 10.1007/JHEP11(2010)083 Accepted Received R counterparts of the Published ch structure, involving new doi: all the allowed fluxes. mit an uplift to 10d heterotic lysis also provides some clues e the natural extension of the d tood gaugings of 4d supergrav- gings of maximal 4d supergrav- tive superpotential in arbitrary Published for SISSA by and M. Gra˜na c [email protected] , P.G. C´amara b [email protected] , 1007.5509 E. Andr´es, a,b Flux compactifications, Supergravity Models We perform a systematic analysis of generic string flux compa [email protected] = 1 heterotic or type IIB orientifold compactifications, for CERN, PH-TH Division, CH-1211 23, Gen`eve Switzerland Institut de Physique CEA/ Th´eorique, Saclay, 91191 Gif-sur-Yvette Cedex, France E-mail: Centro Bariloche, At´omico 8400 S.C. de Bariloche, Argentina Instituto Balseiro (CNEA-UNC) and8400 CONICET, S.C. de Bariloche, Argentina [email protected] -deformation to full-fledged F-theory backgrounds. Our ana c b d a Open Access Abstract: G. Aldazabal, U-dual fluxes and Generalized Geometry ity and EGG, identifying the completeor set type of gaugings IIB that supegravity ad deformations backgrounds. of Our 10d results supergravity reveal backgrounds, a such ri as the R making use of Exceptionalparticular, we Generalized establish Geometry the precise (EGG) map as between fluxes, an gau o ArXiv ePrint: Keywords: deformation. These new deformations areβ expected to provid ity. Finally, we deriveN the explicit expression for the effec on the 10d origin of some of the particularly less well unders JHEP11(2010)083 4 7 8 1 4 8 25 27 29 30 36 38 10 10 19 22 23 26 34 36 39 13 19 theory of particles. s and exchanges compact sense that stringy aspects isometries is considered, T- d pactified on a circle of a given 7 E ) group that, among other features, mixes the – 1 – Z ) d, d, R ( , = 1 backgrounds O N SL(2 = 1 backgrounds × N 6) , (6 O to representation of the gauge fields representation of the gauge parameters = 1 compactifications 7 ) structure E 6) formulation of 56 133 N , formulation of root and weight system d, d (6 ( 7 7 E O E 5.2.1 Type5.2.2 IIB orientifold compactifications Heterotic orbifold compactifications O A.1 A.2 Some relevant formulas 6.1 6.2 6.3 General U-duality covariant superpotential 5.2 Toroidal orbifolds and flux induced superpotentials 3.3 From 5.1 U-dual gauge algebra and constraints 2.1 2.2 T-dual field strengths 3.1 The 3.2 The come into play. In itsradius simplest form with it a identifies amomenta theory theory com and compactified winding on modes,When a compactification circle which on of simply aduality inverse more do action general radiu not background is exist with enhanced in to a an field 1 Introduction T-duality is a distinctive symmetry of String Theory, in the A Summary of group theory results on B Tensor structure of U-dual fluxes 7 Conclusions 6 General 4 U-dual field strengths 5 Toroidal compactifications and dual fluxes 3 Exceptional Generalized Geometry and U-dual gauge fields Contents 1 Introduction 2 Generalized Geometry JHEP11(2010)083 – 5 [ orm (1.1) m unda- . The new . ansformation mn m np β ential couplings ω p ∂ being independent rnal direction 2 = B field are taken now on ( ifold in the usual sense 2 mn p flux, corresponding to a uxes”, following the rules m B Q ion of the concept of Rie- fferent intersecting coordi- . 3 ive superpotentials derived r dimensional origin of these le, given by new fluxes must be included fluxes, we would arrive to a eometric transition functions H ckground, but as a 4d theory d supergravity actions. Com- , referred to as twisted torus. ver, after a global transforma- n the presence of a flux of the mnp ng type IIA and IIB string the- ection mn p ions are also allowed for as part rounds lead to the same CFT. gravity should not be thought of R Q p T actually not allowed in these cases since the , a new 10d supergravity background ←→ n interpretation in terms of asymmetric orbifolds ) is more obscure. In fact, it has been ese rules even in these “obstructed” cases seems field components. In a 2d sigma model is also called a non-geometric flux. mn p 2 1.1 Q B mn p Q n ) is well understood from Buscher rules. If a T- T – 2 – ←→ 1.1 m np ω ], the situation becomes richer and more intricate. A 4 m for which not even a local 10d supergravity description , T 3 ←→ mnp ] indicate the precise way in which target space fields transf mnp R 2 H , 1 . It has been suggested that, taking T-duality symmetry as a f ]). 2 9 is performed in a background with dB p = 3 ) are related to the various coefficients of effective superpot ] from the 4d effective action viewpoint that if a T-duality tr which can be locally characterized by the tensor H 5 ] (see also [ 1.1 8 1 denotes a T-duality transformation performed along an inte m T From the above chain of T-dualities a “stringy” generalizat An obvious question arising at this stage concerns the highe When fluxes are turned on [ By performing a new duality, let us say along The last T-duality transformation in eq. ( Strictly speaking, the usual derivation of Buscher rules is 1 -deformed) background, however, does not define a global man ]. These rules manifest, for instance, by comparing 4d effect β mannian geometry seems to emerge, on which symmetries of the is available. In thisas sense, a the dimensional resulting reduction 4d of effectivethat super a incorporates 10d information effective of supergravity the ba full string theory. approach, Buscher rules [ “truly non-geometric” flux isometry is not globally wellto defined. be meaningful. However, following For th instance,can to be some of given these [ solutions an 7 metric modes with the antisymmetric NSNS along a direction conjectured [ new fluxes. The first transformationduality in transformation ( is performed along the isometry dir from orientifold compactifications of theponents type in IIA ( and IIBin 10 the dimensionally reduced theory.ories Since are the supposed correspondi to bein connected order by for mirror symmetry, the these superpotentials to match. mental symmetry of String Theory would lead to include new “fl under T-duality, highlighting the fact that different backg clear illustration is providedNSNS by 3-form a torus compactification i compactification of 10d supergravityThis on is a characterized different by manifold the first Chern class of the spin bund ( of this direction), we end up with a background with no is obtained, involving diffeomorphisms and gauge transformations.tion Howe solutions are connectedof only the if transition T-duality functions. transformat For that reason of Riemannian geometry. Innate patches, order local to solutions link can such be solutions connected in by the di usual g where JHEP11(2010)083 s d d = 2 ], the N ver the 11 , 2 10 be accounted field. These are sum- SL(2) subgroup. 2 zing principle for 3 × B voking other duality shing spinor requires 6) , rgravity, metric fluxes (6 tended) Generalized Ge- e gaugings of 4d gauged f 4d gauged supergravity O mations related to the RR ctifications with fluxes. In rgravity. paper we focus on aspects ) for compactification on of F-term and D-term den- tive 4d gauged supergravity aling with general 4d rs starting from [ ntwisted modes can be then ork of Generalized Geometry l one. The generalized metric uilt up from the tangent and ciated to obstructed dualities, d, d th general gauged supergravity ( eory or heterotic/type I S-duality O ] for an introduction to the topic) nd on six dimensional coordinates of the 6) ( 16 oints towards a geometric description where , SU(3) structure, characterized in terms and a holomorphic 3-form Ω. The 10d (6 ] in twelve dimensional space. J O 21 structure to some – global 7 17 structure of the compactification. In particu- – 3 – E ]. field. Patching in overlapping regions requires, 7 . Apart from already known deformations, such ] (see also [ 2 34 4 E – 15 B – 28 12 6) action involving, in particular, the , ) dimensional manifold. Namely, vectors in this generalize (6 d , so that symmetries of the RR fields are also naturally in- O ) of section 7 ]. Again, such new fluxes can be inferred by matching coupling 7 E 4.18 ]. In EGG the structure group of the generalized bundle is now and Ω, which after Kaluza-Klein reduction and integration o )–( 27 J , 4.14 26 ] for a review) gauge symmetries and structure constants can 6) structure to be reduced to a ]. 25 , 24 (6 – O 22 = 1 compactifications. The existence of a single nowhere vani N The natural generalization appears to be Exceptional (or Ex Hence, Generalized Geometry is particularly powerful in de The above picture, however, is far from being complete. By in The main aim of this paper is to use the tools of EGG as an organi -deformed backgrounds, our analysis also reveals new defor It is worth noticing that, in this description, fields do depe Related partial results can be found in [ β 2 3 besides diffeomorphisms, an dimensions) is the structurecotangent group bundles of of a the generalized six bundle ( b bundle are built up fromcombines vectors the and one-forms original of metric the origina and the original compactification space. Afields more depend ambitious on program coordinates p on a “doubled torus” [ equal footing than diffeomorphisms. Thus, in a series of pape as backgrounds for theor RR and NSNS field-strengths of 10d supe marized in eqs. ( the full U-duality group, for in a string theory frameworkare only incorporated. if these In new this fluxes,is regard, asso called an for. extension of the framew ometry (EGG) [ in 4d effective superpotentials.theories (see Similarly, [ when dealing wi concept of Generalized Geometrywas [ proposed as aGeneralized natural Geometry framework the to full describe T-duality string group compa transformations of String Theory, like IIBextra S-duality, fluxes M-th are suggested [ lar, we perform aand complete EGG. mapping This between allows fluxes,supergravity gaugings for that o admit a an systematic uplift determination to of backgrounds all of th 10d supe corporated. The effectunderstood of from the the orientifold breaking projection of on the local the u which are directly related to the local generic string flux compactifications. In the first part of the and compact space give rise totheory the F-terms [ and D-terms of the effec the local of a globally SU(3) invariant (1,1)-form K¨ahler supergravity equations of motionsities can depending be on then recast in terms JHEP11(2010)083 6 6 we ies. = 4 (2.1) 3 ] (see tter to (where ction N 35 bundles ξ is a two- + s of EGG, M 6) [ ∗ , x αβ gauge trans- b T (6 2 = we consider the O B ts related to the X 5 we introduce the 2 ed to the global SU(3) cture compactifications. . The structure group is field besides the usual dif- , 2 o such transformations and tood gaugings of 4d and cotangent ) scuss the action of different ) is the structure group of a B . In section at U-dual fluxes must satisfy, 4 αβ compactifications. The relation presentation of b d, d U-dual fluxes allowed in general ions, in the presence of general n ′ β ( the TM x ι O . This patching corresponds to the − β β x ξ T αβ , the generalized vectors a − αβ β ]). Moreover, our analysis also sheds light a U ≡ 33 + ( ′ β field of the manifold. Matching of overlapping – 4 – x β and 2 ) is a conventional diffeomorphism, x α T B ) ) representation is also presented. In section αβ U = 1 backgrounds in EGG and to the construction 1 a − field is incorporated to the geometry of the generalized d, d a = N ( ( 2 α ], the T-duality group O B ξ . A generalized metric on this generalized bundle encodes ≡ 15 + – T M transformation, associated to non globally defined geometr U-duality group and the generalized tangent space becomes 5 − α 12 a x 7 β E = 4 gauged supergravity is also discussed in that section. Se -deformation. These provide the natural extension of the la β N a one-form), are identified as follows ) (and 6) structure group in connection with diffeomorphisms, ξ R , = 1 compactifications. More precisely, making use of the tool (6 d, ] for related work). ( means a contraction along structure O N 37 ) ′ β = 8 and , GL x ι provides some final comments. Some notation and useful resul 36 ∈ d, d N 7 ( , a O -dimensional manifold = 1 untwisted sector superpotentials for general SU(3) stru 34 d At the intersection of two patches In the second part of the paper then we consider aspects relat A more detailed outline of the paper is as follows. In section algebra are summarized in the appendices. = 1 heterotic or type IIB orientifold compactifications. N is a vector and 7 The interpretation of NSNShow field they strength combine fluxes to associated fulfill t an feomorphisms. In that way the x generalized bundle constructed by combining the tangent Section where bundle. patches is achieved by allowing transformations involving form and of dimensional. U-dual field strengths are discussed in sectio of a structure of information about the metric and the In Generalized Geometry [ is devoted to the formulation of counterparts of the formations and the extra 2 Generalized Geometry 2.1 E simplest compactifications, that isdual fluxes. toroidal We compactificat discuss theand derive algebra the and flux global induced effective constraintswith superpotential th 4d in these address the inclusion ofthus enlarged RR to fields the full into geometry, defining EGG we write explicitly the effectiveN superpotential for all the basic concepts of Generalizedgenerators Geometry. of In particular we di on the 10d origin of some of thealso [ particularly less well unders full-fledged F-theory backgrounds (see also [ supergravity, such as the ones transforming in the vector re JHEP11(2010)083 is αβ αβ 2 . not β B (2.2) (2.5) (2.6) (2.4) (2.3) G s because , they form 3 ). One could l 1-forms Λ a ξ ) that is a semi- 6) define another is no gauge trans- ) + β -dimensional, even , ) symmetry to the b d, d d ξ X ( (6 ab d, d O O αβ ( β h O + = 6, this subgroup has 51 = a d x , ! ( ld is still β 0 β ξ x ! 7→ ns as well. This is perfectly fine β β ction such as that outlined above αβγ U(1). The gauge parameters Λ a ξ x −→ ξ is required to be locally exact, i.e. ! ∈ dg 2 T , + b iθ 0 ! , − αβ a e αβγ TM a T . ). As we will see in section αβ x g 0 ! − αβ B Λ = 2 a I π d = αβ 0 G β TM −→ a ) form a subgroup of + I 0 whose patching is αβ 0 αβγ γα

E β g a

α 2.2 -transform” action on a generalized vector is – 5 – B β B = + Λ but is an extension of the tangent bundle by the −→ ! = 2 I 0 β B βγ α M e e M ∗ in the patching implies that the generalized tangent B , where I ∗ ). Specializing to the case ) αβ determines the quantized global curvature of the gerbe. T γ b d + Λ 2 T 2 ( b -dimensional tangent bundle on it. Hence, whenever U

. The “ ⊕ d, d d ) generator corresponding to an Abelian subgroup, is fibered over 2 αβ ( ∩ dB β 6). The remaining 15 generators of GL = ≡ Λ −→ O β , d, d = M 0 ⋊ U TM ( (6) and 15 make up ! -transforms in ( ∈ ∗ (in components, this is (6 3 b α B α O ∩ T ξ h O ξ x H G α GL

U ) + = ξ ) has an index down, and even when combined with a vector it can = x is the ) that allows to define it patchwise. Roughly speaking, this i α β 2.5 ! geom X + 0 2.5 G x 2 ( 0 0 is a globally defined bi-vector. If that is not the case, there b and (in an analogous way as one patches a U(1) bundle) the loca 2

describes how 7→ β αβ ξ = αβ Λ + b B . It is therefore the derivative operator which breaks the d 2 x Diffeomorphisms and The presence of the 2-form = B = αβ not globally well-defined, there isfor no “gerbe-like” constru generators (36 generate if we have defined an extended 2 direct product following transformation the derivative in ( define a bi-vector. This is related to the fact that the manifo a parabolic subgroup of where Abelian subgroup allow to define a local 2-form gauge field formation like ( and whose field strength characterized by a bi-vector on triple intersections bundle is not just the sum X cotangent one should satisfy the cocycle condition where b a priori allow foras patchings long involving as these transformatio JHEP11(2010)083 . ], − as − C , or 15 C E – (2.7) (2.9) (2.8) field. up of ∈ (6) 2 ∈ 12 6) there is O − , B ξ × 2 (6 X is patched β + O (6) 2 x ), while that O (6). One can B a ) = O 2 . This structure for some matrix X × E g, B ( , while t does not seem pos- (6) M x 6)-invariant metric T β , O a g + (6 x = → O α ) angent bundle, and encodes as 2 g t bundle into two orthogonal field at a point on the mani- quired to be globally defined. 4d effective supergravity. The + . d the Courant bracket [ 2 C g, B ! B of a background with no 1 implies that an element (6) structure on ∈ 1 − B (i.e., a transformation that does not O − . g + − − 2 2 ) implies g C − × X and a negative-definite metric on B B C to be locally exact, while for = | 2.1 2 + field parameterize the coset -transform rotates into a diffeomorphism ! η 2 b + (6) I b 0 and β C B g 2 2 -transforms and T-dualities along any two O 1 − I 0 b B + − B . However, it is always allowed to perform an + . One can therefore think about a background

1 g 2 C 2 C 2 – 6 – − | b such that the natural B = g η B we find that the generalized metric takes the form 6) transformations. The effect of diffeomorphisms − − η − , x 2 − = C . When we write a generic element ) are related by T-duality. In fact, there is another (6 B g cannot be purely a vector or a one-form, since these g x H ⊕ O 2

) − ± → β g + 2 C = 2 -transform with C − B 2 field accordingly (namely ( B 2 b H ), this constrains or = and 2 B and the new +( + B E 2.1 2 -transforms can be locally “gauged away”. We will come back field) such that the x g b C + ( . We can therefore write β 2 combine to define an = η x , the patching condition ( as the B 2 6) that preserves each metric separately is orbit can be reached by acting only with the geometric subgro g ± , = B B X + (6 − (6) 2 6) O O , X B and ). Orthogonality between × (6 field ) given by diffeomorphisms, g O = (6) 2 , with 2.5 O B − d, d M ( X O (6) transformation in the stabilizer of + -transformation leads to a more complicated action so that i -transform. In other words, given a metric and O + β b X × 6) that implies that The 36-dimensional space of metric and The transformations A A generic element of , -transformations is to shift = b . Taking (6 (6) geometric subgroup. The derivative defines a bracket, calle O which is the extensionthe of differential the properties Lie of bracketFor the to patchings bundle. of the the generalized Such form t bracket ( is re in other words according to ( X no analogous condition. can also be defined6-dimensional by the sub-bundles splitting of the generalized tangen can be written as to this point in the next section. are null with respect to decomposes into a positive-definite metric on The sub-group of sible to tell apart the new basis for and parameterizes the space ofgeneralized scalar metric can fields be of acted the by resulting now define a positive definite generalized metric is changing the metric and with a given of and a change the metric and M O fold, the full JHEP11(2010)083 . are 6 2 (2.10) (2.12) (2.11) β and 5 subscripts and b 2 s) in the Lie . b ≡               and b b and b b bb bb b fb fb f b f f ˜ ˜ ˆ a β β ˆ T a β β ˆ ˜ a . ˆ ˜ a a f f f f bf b np ff bf f ff b f ˜ b ˜ b a β a β β ˆ ˜ a ˆ ˜ a ≡ − b m b b b b bb b a f e generated by T-dualizing bb fb ∂ f fb wo basis vectors is realized b b b a ˜ b ˜ a a ˜ ˜ a that the action of T-duality ; ), where = sm is exchanged with a fiber- ed ˆ b f f f f bf indices down, and each of them b ff b bf f f ff , v np m b ˆ ˜ a b ˜ a β ˆ ˜ ations purely on the base are not a ˜ a β d f ξ               , ; b l be mostly discussed in sections = = ,Q        , ξ p 4 ˜ f 0 ] f I A n have 90 components each. This gives a v 2 a 0 0 I AT purely along the fiber are interchanged. A Q m [ 4 T 0 0 0 f ∂ 2 I 6) representation containing 200 elements is T , β 2 0 0 0 0 0 0 = . We see that the effect of T-duality is, roughly I – 7 – and indices up and ) into ( (6 b bf d        O ω p mn −→ and β , v , we can also introduce some field strengths for all 3 f = 2 − v        b H b ; b 12 = bb b b fb f , ω T 4 ] ˆ a β ˆ a β , ξ fb , while f np 4 identity matrices. Hence, T-duality transforms each fun- f f 3 β ξ b bf b ff f -indices and move the block to the corresponding new place , × H ˆ a β ˆ m a β is split into [ f bf ∂ b = ( b d b b bb f fb 2 field with respectively one and two legs along the fiber. b b a A − a = 2 )-field, while X f f = b 2 b bf f ff 2 and 4 6) transformation b β ,..., b a mnp a , fb ( × b H        (6 2 = 1 b O = ) transformations, A on a generic element (split among its base and fiber component 6) form B , ) is f A d, d (6 ( A are the 2 O d, d O ( 4 , o 2 I Note that these are all local definitions. Global aspects wil 4 -transform with one or two legs along the fiber can therefore b algebra where There are 20 components of damental by the following directions on the tangent space. A T-duality along the first t Here the index indicate fiber and base,along respectively. a fiber It is not hard to check total of 200 components. The smallest is further split into fiber and base indices. We have also defin antisymmetric, i.e. speaking, to raise andaccording lower to the structure of thetouched indices. by Hence, T-duality. transform A fiber-basebase (base-fiber) (base-fiber) diffeomorphi the other 2.2 T-dual fieldAnalogous strengths to the local definition of β a diffeomorphism or a JHEP11(2010)083 . e of 6) , geom (6 along (2.13) mnp ). G O H mnp 2.11 H etry, similarly iated that a flux of the fundamental A field, one needs to ex- ) transformation needed 2 ) which are not in d roup is the full U-duality B , which is therefore not an ) transformations without ( vative with an index up, or s section, non-geometric el- d m l not be single-valued if the O oordinates. This is the spirit d, d ( ), we can see that the ll introduce a “derivative up” x ( acting on the element ree indices up. This generates re T-duality covariance of the ) are geometric, i.e. are (stan- O × 6) index O ω not be the field strengths of any , d, d ) × ( d (6 ( ) mnp O d O and O . ( . The locally geometric fluxes can be ] 3 On the contrary, locally non-geometric O R ] is forbidden since in order to get a flux H 6) element as introduced in ( 2 5 BC , , down and six indices up (i.e. a form and A 1 ]. It is the candidate T-dual of . The (6 5 A . o [ E 7 m ∂ E – 8 – on = , such that it combines with the latter to form a ]. Even though we will stick to a six-dimensional -flux [ 6) transformation since the derivative has an index representation an object with three indices up is m , 21 R ∂ – . The other 110 are divided into 90 “locally geomet- (6 ABC ABC representation of the fluxes is therefore obtained by 17 F F O 220 is a generic ), half of them ( geom field. However, the G 220 C 2 ]. D 2.13 B 38 A . The A in the previous section. With that aim we extend the structur and ∂ 2 C B D A ) can be locally gauged away by ). As for the local non-geometricity, it can be easily apprec , the latter should depend on the coordinate BD η d, d np and 20 “locally non-geometric” fluxes ], in this section we incorporate also the RR fields to the geom 1.1 ( = B (though such T-duality `ala Buscher [ 39 O Q p BC , which consists of 3-forms A from ) gauge field. cannot be the derivative of an and Out of the 220 fluxes in ( 220 Examples of this are given in [ , T-dual to the standard derivative d, d 5 mnp ( mnp m the 220 representation gives indeedthe another chain in element ( with th down. A tri-vector wouldin be other generated words, if if we weof doubled introduced the the a coordinates double deri by torus adding construction dual of c [ missing. Such “locally non-geometric”4d flux, low energy needed action, to has resto been termed representation can be splita into vectorial six index). indices To fill out the transformation corresponding to three T-dualities along dard) derivatives of elements in ric” fluxes the 3 Exceptional Generalized GeometryFollowing and [ U-dual gauge fields tend the generalized tangent bundle to one whose structure g to what we did for fluxes require a non-standard derivative,O and therefore can manifold, from a purely group-theoretic∂ point of view we sha H R isometry). From the point of view of representations of group of the generalized tangent bundle to m, n 3.1 The 56In representation order of to the gauge geometrize parameters the RR fields at the same time as the where They are locally geometricements since, of as we argued in the previou 12-dimensional 1-form to rotate the non-geometricnon-geometric element element is into not a globally geometric well defined. one wil built using standard derivatives, but acting on elements of changing the metric and the JHEP11(2010)083 ), of 6). , The 2 e di- 3.1 (3.2) (3.3) (3.4) (3.1) (6 6 O associated U-duality), µ a ) it contains a representation i ω ), and the duals 6) (made out of odd 2 with one external 12 , M. decomposes under ∗ parameters. These B 6 (6 d get extended to a led 7 b T O (whose corresponding E , , the odd i iven by odd forms in the s. Six come from gauge an uneven assignment of n the NS sector, namely M representation of Λ v  space-time index. ∗ of the notation in eq. ( − µ B, where the RR potentials rms). However, in order to ⊕ T ˜ nsional exceptional general- c 12 6 ,
l i Λ ω M are a sum of odd multi-vectors . (the dual of ⊗ ) ∗ p , 6 T 1) ) mµ 6 M , ...i B representation of 1 ˜ ∗ k 1 Λ ′ , T oice is required. µi + ⊗ c 32 representation of = (0 . 6 µ 32’ i m M ...i
representation. This is the appropriate choice for ) picks out a vector that breaks the 56 b ∗ ˜ 1 i − R T ǫ ) + ( , ! 1 + ( p 2 , λ 133 is the magnetic dual of the vector i – 9 – , ⊕ , ω v Ai +1 ) λ contains the 12 SL(2 n 0) M , ∗ mµ × =2 ] for details). Without loss of generality one can take 56 b 123456 = ( = p T 5 39 6) λ + mµ P Λ = (1 , 56 µ k i (6 ⊕ ≡ m v ≡ (i.e. it is the Hodge dual of the 6-form a O ( − µ M c  ∗ mµ ⊂ ]. The fundamental field, given by one-forms, and six from vectors pointing in th ˜ T k = 2 27 as = 1. nopqrµ ⊕ µ (6) , b B 7 i λ E ω 26 such that in one direction that we will call j GL TM v ⊂ (even forms) appears in the 1 ij ) = mnopqr ǫ + ǫ representation (or in other words, to have a closed set under 32 R , 1 E 1 5! ≡ 2, and a minus denotes a sum of odd forms on the internal space. , 56 ≡ vω SL(2 µ . The 12-dimensional generalized tangent space hence shoul = 1 7 m 6) is built out of a vector and a 1-form, while in the other (cal × ) into i b E ˜ , R 6) , (6 There are 4d gauge fields associated to each of the above gauge representation (in this paper we will concentrate on type II , We are choosing conventions where the (6) weights (see appendix A of [ O 6 (6 5-form and a 1-formGL tensor a volume form. This is the result of The embedding of SL(2 of gauge charges are theized Kaluza-Klein tangent monopoles). space The is 56-dime therefore given by We have to add to32’ them gauge transformations of the RR fields, g rections of the diffeomorphisms. Those build the fundamenta are even, and theirfill gauge out transformations are the givenwe by need odd another fo 12a parameters. 5-form, These corresponding are to the gauge magnetic transformations duals of i of the diffeomorphism vectors, given by elements of type IIB compactifications, while in type IIA the opposite ch Note that come from 10d gauge fields with one space-time index. In terms 56-dimensional one [ where group the 4d vectors are 56 degrees of freedomtransformations can of the be accounted for by gauge parameter forms), while the O and five internal indices), with diffeomorphisms, and ˜ corresponding to the Hodge dual of the RR potentials with one where JHEP11(2010)083 . 6 B b E and ) we (3.6) (3.7) (3.5) (3.8) (3.9) ). In 6 repre- ] b R 3.5 , , 39 2 b . 133 ] SL(2 action on 39 ! i 7 × ω ) decomposition E ) as + 6) R R , γ , , TM. (6 6) (diffeomorphisms, + ). The full , of the complex axion- i O even R v SL(2 SL(2 case where they are not (6 , Λ + 6) case: they can locally × O × . The remaining elements , p is a semi-direct product ⊕ , c + 6) j (6 ! 6) , i ω C used to patch the exceptional , , M O i . Diffeomorphisms, v ∗ (6 ) 7 ω + (6 + 2 T on-trivial flux that signals a non- s, O E 6 , C O and , c β j ⊂ in the following way [ 6 ]). Using the notation in ( v 32 even 6 i . 7 ⊕ Λ v and 39 ) + ,B E j
vol )-dual to the RR gauge fields), a scalar 6 2 , i 6 ⊕ geom v φ ) + ( i R B + 2 TM B , b G , 3 27 ω 2 − , representation of ! , the non-geometric gauge fields are a sum of Λ ,A 1 ie + TM 2 j 6 T i j into representations of β ⊕ + 66 a 0 Λ form the triplet of SL(2 ω . 7 i 6 j − – 10 – 6 ) + ( ,A M v ⊕ E v b B 1 ( ∗ B 2 a ij , 0 -form elements on the second line correspond to b T A ǫ c = M decomposes under p 2 -transformation in the ) embedding, these further decompose into [ ∗ A 66 ) 7 = Λ + ˆ β H R T R

i E j S , 6 , ⊕ v v = = ( is the nilpotent subgroup corresponding to the shift sym- Λ (which are SL(2 i = ) transformations that do not change the metric, dilaton, v A -form gauge fields A and Generalized Geometry, we have decomposed the fun- + ⊕ 6 7 6) subgroup is the one of Generalized Geometry, whereas M p SL(2 SL(2 γ and G , ∗ 133 7 E R , b geom 6 × T E × (6 c ⊂ A ⊕ ! 6) action on the generalized tangent space is an O ⊗ 6) , 6) T + , 2 , a 4 (6 for the that together with (6 c (6 − p O 6 TM ), where O O a 2 + β -transforms). The d a β b ( 2 β ⊂ + representation of = ( c to p 0 GL + 7

A (6) A 0 E ⋊ 133 c = → ) factor corresponds to fractional linear transformations GL A = p R A G , A transformations, which shift the value of + c = -fields (which together define an SU(8) structure), but in the play an analogous role to the + c C As in the generalized geometric case, the geometric subgrou 7 transformations form the geometric subgroup E , and a six-vector geom + 0 ˆ globally well defined, they might carry somegeometric topologically background. n Besides the bi-vector We recognize the first line to be the adjoint The adjoint even vectors that we will call The analogue of the 3.2 The 133 representation of the gauge fields embed the generators of the geometric subgroup of generalized tangent space (see more details in [ and c where and b-transforms and sentation arranges in the following way with respect to the Under the metries G of c be gauged away by SU(8) 3.3 From this decomposition, the the SL(2 To make contact between dilaton which appears in heterotic string compactification damental and adjoint representations of JHEP11(2010)083 t 1 H H | | ) ) R R (3.10) (3.11) (3.14) (3.12) , , ), while . Thus, 2 . We can ˜ ω b A SL(2 6). SL(2 are positive , × × for which the (6 + → − 6) 7 c form a doublet 6) O , E , v 2 (6 decompositions of c , ˜ (6 ⊂ , 8, the orthonormal ˜ v O j ! B O 6 (3.13) ), type IIB S-duality i acts on | ˜ B ω d to themselves. For | ) ) and i ω ) and → . ˜ R ω ) ˜ R 3.8 ,..., R 1 2 , 2 6 , 0 s as ˜ , which are contained in ω , b c ,..., γ 2 n eight dimensional vector c t s in the + = 1 er to reproduce the algebra SL(2 2 . = 1 a ) + SL(2 ionary between the two bases b j 8 , ± ≡ ± × ˜ v and ) is reexpressed as, × e a i + i, j e ]. A convenient way to write the 6) 2 ω 6 6) − , b 3.8 , β (which is itself T-dual of 30 i 7 + ˜ (6 ˜ e e representation and every gauge field 2 (6 j + O φ β ˜ . ω 2 O − = i − γ 56 ) j i v ˜ 8 e ie (˜ 8, are orthonormal basis vectors. The full ˜ e and . Calling ˜ 0 + ( c + T 6 is, however, not unique. For instance, while =1 i ). This implies that S-duality is not in the − H j a X 0 | v 7 + ˆ R 7 ) ,..., C 1 2 ) ˜ , j e E , ω 6 → – 11 – R (˜ ˜ v b 8 , i = − 1 4 a e ˜ = 1 v ) + B 0 i 8 SL(2 such that the roots corresponding to + + 2 ˜ S e , i j , c c i 7 SL(2 ) into × ˜ is the S-dual of e e e ω . This requires − R ! × + ). We summarize the weight of each gauge field in table 2 , − 7 6) 6 6)) acts straightforwardly on eq. ( =1 a 6 γ 4 e , j 6) X , A.1 (˜ γ γ = , − and A.4 (6 4 2 1 1 (6 i i + SL(2 (6 4 T v O v O c 2 a , where = = O × β 8 i ( 8 e . To relate fields in one decomposition to fields in the other, i e representations we will deal with are given in appendix e 6) B ,

+ | 7 ) − 7 E (6 = 7 e R e , O ) vectors B | R A , ), given in eq. ( SL(2 ) we observe that 3.8 × a weight vector, in the same spirit as e.g. [ are also related by S-duality. 3.11 6) 6 , γ ) subgroup is the one that contains type IIB S-duality, which ) piece, but it is a combination of generators in both SL(2 133 (6 R R , , O -duality in the new basis, and that diffeomorphisms are mappe We can therefore select another decomposition Weight vectors for the In what follows, we will refer to the above two decomposition roots is as vectors lying in a seven dimensional subspace of a are related by a change of basis. Given the assignment of root and S 7 7 6 by fractional linear transformations. In this basis, eq. ( different representations of SL(2 in the conventions of appendix below, where we have made use of the shorthand notation The embedding of SL(2 E space orthogonal to choose the SL(2 seems more complicated. The latter should exchange from eq. ( T-duality (in the adjoint of Note that type IIB S-duality now corresponds to the exchange β Given that choice, thesatisfied assignment by of ( weights is unique in ord E set of weights for the basis, in order to find such change, it is enough to require tha and is useful to assignin every the gauge parameter in the of convenience, we take the case on which basis vectors in the type IIB basis, we get the following dict JHEP11(2010)083 B | ) the R 1 , lgebra pens. P SL(2 + − − + + − − + − − − − + + + ) which are Ω contains all × L 6) F geom , , i < j mations. j + − − + + 1) + + − − + + − − − − G i (6 a − O B ( | ) B | − ) ) ) ) ) bsection, it is possible to , basis) , generated by all positive − +) − +) − +) +) − − 0) B 7 in the Borel subgroup (the , | , , , , , , , , 0) 1) 1) , H ; + ) 0) tal scalars corresponding to , , | for the E − − − − − , ) 1 − R ; + ; + ; ; , , , 0; 0 R e the Borel subgroup by means 0 ponding to − 0; 0 , +; + − +; − +; +; + − +; + , , − 1; 0 ). We present also in table , , , , , , , , , 0 0 0; 0; 1 133 , , 0 , , , , + − + − + + − + 1 , 0 − 0 SL(2 0 0 , , , , , , , , , 0 , , , , , 3.11 SL(2 0 , 0 + − + − + − − + 0 0 0 × , 0 − , , , , , , , , , , , , is its unipotent radical. The geometric 0 , , × 1 1 Cartan 0 0 0 , 6 Cartans 1 + − + − + − − + 6) , SU(8). This is also the space of scalar , , − 0 − , , , , , , , , + , 6) 1 , / − 0 0 , , c , 7 , , − − + + , − − − + 1 (6 − − , , , , , , , , (0 ( ( E (6 (1 (0 (0 − O − − − − − ( O (+ ( (+ ( ( (+ ( ( and → → – 12 – 6 b ) ) ) ) H H fields) define a SU(8) structure on the generalized | , | ) +) +) +) − +) − − − 2 0) H + , , , , , , , , | b 1) 0) 1) 1) , − ) 0) , , , C − − − − − − , 1 R ; + ; ; + ; ; ; + , , 0; 0 , − 0; 0 ; + , − − − − − +; +; + − 0; 0 , , , , , , , , , 0 0 and 0; 1 0; 0; 1 , 0 , , , , , − − − − − + + − 0 , 0 0 SL(2 0 0 0 , , , , , , , , , 0 , B , , , , 0 , 0 0 basis one can change conventions such that the same thing hap − − − − + + + + 0 0 0 × , 0 , , , , , , , , , , , , , 0 , 1 0 B 1 Cartan 0 0 0 | , 6 Cartans 1 − − − − + + + + , 6) ) , , , − 1 , , , , , , , , , , 0 This is referred to as a parabolic subgroup. The further suba 0 0 R 0 , , 1 , , , , − − + + + + + + , 1 (6 7 , , , , , , , , − (1 (1 ( − (0 (0 (0 O − − we observe that (in the ( (+ ( ( (+ (+ (+ (+ (+ SL(2 . Weights of the geometric and non-geometric gauge transfor contains in particular the Borel subgroup of 1 × 6) , i 2 0 geom 0 0 i 12 56 1 12 12 (6 ˆ c c γ 3456 a 1234 c b a γ β G 123456 123456 field 123456 123456 c Table 1 O γ c b γ β From table This change of basis dictates the form of the full In the 7 minus a simple root. positive roots, the Cartans and a few negative roots (corres Thus, for instance subgroup containing just the shift symmetries assignment of weights in this type IIB basis. fields of the resulting 4d effective supergravity (7 fundamen tangent bundle and parameterize the coset of SU(8) transformations. Thedilaton, 70 the metric dimensional and space the of fields roots and Cartans. Aslocally it gauge has away been all commented the in transformations the which previous lie su outsid decomposition, which has the structure given in eq. ( JHEP11(2010)083 , 7 8 ), 4 ⊂ 6) c E , R ) , , type such 0 (6 R c , P O , heterotic. 0 SL(2 , whereas c ). In this the parity ere the 4 → hereas the , ˆ × SL(2 c R a 1 ) is broken to , n × we see indeed 7 1 SL(2), for both type I E 6) (or alternatively 6+ = 8 supergravity. , SL(2 , 1 × → (6 × (6 N 6) O decomposition of the O , 6) 0) (3.15) , , an orientifold involution (6 epresentations of a heterotic compactifica- (6 O σ basis, whereas 1; 0 O to the positive roots, which , s in table come from truncation of the bgroup H 1 ith the space-time fermionic nced to | , reserved, and sulting 4d ) tifolding’, referring also to the non- e introduce an operator that the gauge fields surviving 1 variant gauge potentials follows nt of , R , 1 , ) corresponds to T-dualizing along . These were obtained by tensoring 1 , on the states in table is mapped to the type IIB SL(2 2.2 3.14 2 P × b the duality chain type IIB = (1 )–( l sense, the heterotic string would be for instance 6) u basis. We have stated in table ]). , . We will see more details on toroidal com- obtained in this way. . 0 (6 B 41 3.13 = 4 supergravity, c | 5 , ) P O – 13 – R N 40 with , and applying type I - SO(32) heterotic duality, thus extra vector multiplets in the theory, as required by ) subgroup is the result of orientifolding the theory. and Ω 6 is the worldsheet parity and SL(2 . Acting with n k , R 6 L T , u · P F SU(8) [ × T k / 1) 7 , whereas in the type IIB basis corresponds to the orien- 1) 6) as L , − E SL(2 − F k (6 are kept in the × 1) O 6 − 6) β , where Ω ) = ( , k σ ( (6 L F ). The simplest examples are toroidal compactifications, wh P and O )) we can see how gauge transformations are mapped under this R 2 , etc.) have negative parity under the orientifold action, w 1) , β − is mapped to the type IIB , ( 3.11 6 6 352 P b b SL(2 , , ′ 2 × b are kept in the ) 32 , 4 ) and ( n a , γ , 0 3.8 32 c 6+ One may easily check that states transforming in spinorial r From the point of view of string theory compactifications the , Note that we are making a rather general use of the term ‘orien structure into a and 8 (6 7 0 pactifications with general fluxes in section here we choose to focusparent on unorientifolded the set theory. of This (untwisted) states set which is closed under a su (e.g. the heterotic perturbative description of anconsidered orientifold. an orientifold In of this type IIB genera String Theory through transform non-linearly under Whereas in general thisanomaly introduces cancellation, and the global symmetry group is enha the Cartans, 48 axions and 15 diffeomorphisms corresponding that ˆ for the different fields under ( IIB - heterotic duality. For instance, the heterotic all the coordinates of the internal eqs. ( E remaining states have positivethe parity. orientifold In projection particular, are note precisely those in the adjoi γ The discussion of field strengthsclosely associated the to one U-duality for co T-dual field strengths in section 4 U-dual field strengths decompositions. In terms of weight vectors, we can therefor that it acts on a given state This operator isnumber identified for in left-movers heterotic ( compactifications w dualizing from ation. type By IIB comparing compactification the assignment with of O3-planes weights for to the two basi O which reverses all coordinates of tifold action Ω context, the change of basis vectors in eqs. ( After orientifolding only half of the supersymmetries are p structure group corresponds to the U-duality group of the re the U-duality group of the resulting 4d JHEP11(2010)083 6 and ′ ]), (4.3) (4.1) (4.2) F 42 , ↔ ′ H ed from their 123456 ǫ  ]. , similarly to what abc = 0, but it has the ′ 43 1 [ H + 7 , A ) engths we can compute E + , 2 f j , ]. In particular, A ab ω ′ c 7 charges of the fluxes. Moreover, i ω Writing a generic element in L or. We summarize in table ω 220 F 2 . st line, the absence of a 6-form the standard derivative which nt context, the same procedure + ) decomposes as (see e.g. [ 1) ds in table that we have already introduced. a gths were obtained by projecting ′ bc 6 − hat of [ 7 ) + ( of the fluxes in slightly different ways by sible ambiguity. Q 6) factor is identified in this basis ...i E etc. . 1 1 , 6480 123456 i , = 0 that has to be imposed on this + ǫ ǫ ,
(6 e, ( + 4 i  O abc 3 ′− 6 352 i ˜ representation of 2 H F 123456 ...i ABCi 912 , m ǫ  6 1 ′ 123456 mi i i ǫ does not satisfy Γ 16 ǫ ) + ( + , 1 P , f ) + ′ ω 1 912 ,...,i 3 j m ˆ + + P F 1 , ˆ v i F – 14 – + A A 56 representation (which corresponds to a traceless i 123456 9 ǫ ≡ ≡ f ǫ 1 ω  6! ) = 32’ , f  a 4 ), ′ + − i abc + 2345 352 1 representations of j weights for the field strengths in both basis and their 3 j basis ω i i -basis, as the 4 R P 123456 i 2 4.3 , ω 7 133 ) + ( ]. i 3 1 i 6), containing the T-duality covariant gauge potentials, H + m ′ H i 1 7 | ′ + 2 , , f | E v i 2 ˆ ) × ′ a i F ) , 133 ( P 1 (6 Ai c ab R R Q , 1 f , 12 4! ( O 10 56 Q i m,i . and ω P P + + 123456 = ( = ǫ 1 2 4! SL(2 i SL(2 56 c ab − f m 1 ′ ) + i ω × + × a P P 912 2 and Ω + i 1 2 Q 6) 6) + 1 with respect to [ , , L j + + F abc ω v (6 m,i (6 i a H 1) m P v ′ O O ω the assignment of  2 1 ( ˆ − − 6) tensorial structure of the field strengths can be determin → − F i i , B F v  v ω  (6 = = = = O as and are exchanged. We find this notation more suited to the ( − + Ai j ′ A i Q f Q f f ABCi To shed light on the 10d supergravity uplift of these field str The 912 Our notation for some of the fluxes is slightly different from t Depending on the context, we represent the tensor structure f 9 ↔ → − 10 ′ In terms of irreducible representations the tensor product same number of degrees of freedom of a traceless vector-spin of appendix the adjoint representation of weights in the parity under ( making use of the 6d antisymmetric tensor. Thus, for instanc their explicit expression in terms of derivatives of the fiel with the vector representation,also containing accounts for an states extension with non-zero of to winding. the antisymmetric Field part stren of theamounts tensor to product. tensoring the In the prese with the structure groupthe of the generalized tangent bundle. The notation has been chosen in such a way that there is not pos P Q where the traceless condition on the vector-spinor) is encoded in the absencein of the a 0-form second in line, the fir and the extra condition Field strengths are then identified with the we have in the flux. Note that as defined in eq. ( JHEP11(2010)083 56 (4.6) (4.5) algebra , i.e. an +) , 7 +) +) +) +) 1)] on theory, , , , , − E , 1 − − − − 123456 +) (4.4) ; ; ; ; +; ǫ − , , representation − − − − − , , , , + 0; ; , , + + + − , 0 − , , , , + 912 , ) , , 0) 0) 0) means ) B 0 + + − + | , , , +) − + , , , , , + ) ǫ +) − 133 , , , 0 , , , ; 0 + − + + R , − + × , , , , + , 0 − +; 0 +; 0 − , , , , , , 0; 0; + − + + + that the r.h.s. of the equa- erms correspond to locally  0; 1; + + , , , , , , 56 terior derivative in the + + − (0 , , , (+ 0 0 − , , , 0 0 , , SL(2 ts for the ˆ λ + representation expressed as a (+ (+ (+ (+ × ct , , 123456 0 0 × + + − , , b 0 0 , , i , , , × 6 × × × × , , 0 0 − 0) ω ˆ + − − 0 0 λ , , ( , +) ) 6) , , , , , 0 0 912 , , ) reads, 0 0 m +) +) +) +) , , + for further details on the × ; 0 + − − , , , , , , − ˜ 1 1 ∂ , , , (6 , as corresponds to the highest weight − A 4.4 − − − − 7 (1 (0 − − , 1; − − O +) + ( ( ) corresponds to the standard definition ( (+ ( E , + 0; 0; 0; 0; − . . m , , , , , , 123456 − 2 ˜ ∂ 0 0 0 1 0 c 4.5 + , , , , , 6 ) , eq. ( 0; , 0) 0) 0) 133 ) 0 0 1 0 0 H ∂ , 6 , , , | +) − + ( , , , , , + 0 ) +) − , , 0 1 0 0 0 i , , − , , ; 0 ; 0 ; 0 , , , , , R v − – 15 – 0 + and , ) 1 0 0 0 0 − , − − − and , , , , , , , , , 0; 0; + 0 m 0; 0; + , , , 1 2345] ∂ (0 (0 (0 + − − , , 56 0 0 0 c , , , 0 0 , , , SL(2 + [1 , , 0 0 + − − + (+ − + (0 − + (0 − 0 0 ∂ , , , , , × [(1 m , , 0 0 + + − ∂ 0 0 , , 7 6) ( , , , , , 1 0 0 = 5 , 0 0 √ , ,  + + − , , 1 1 , , , (6 ≡ − (1 (1 − O ( ( (+ (+ (+ 12345 1) = D , F 1 − ǫ ǫ 1 6 1 ; 1 1 456 ) for T-dual field strengths. In the language of representati D ˆ ∂ ∂ λ ˜ ˜ ∂ 23456 ∂ − ˆ λ , ˆ λ + 2.12 , + , + , . Assignment of weights in the 56 representation. The symbol + , With that aim, we take the highest weight of the Note that the first term in the r.h.s. of eq. ( In terms of the elements in tables (+ linear combination of weights belonging to the tensor produ The numerical coefficients have beention determined vanishes in when such acted as with way any positive root of of a representation. We refer the reader to appendix expressed in terms of elements of the Table 2 of the 5-form RR fieldnon-geometric strength, contributions. whereas the second and third t this is equivalent to computing the Clebsh-Gordan coefficien inverse volume factor. we did in eqs. ( and root system. The assignment of weights is presented in table where, for convenience,representation we have introduced a generalized ex JHEP11(2010)083 ) ) 7 n . 3 R p , , i ′ (4.8) (4.9) (4.7) dx 32 (4.14) (4.10) (4.15) (4.11) (4.13) ,..., SL(2 ∧ 2 × i = 1 6) − dx i ) and ( , In terms of p 1 , D r elements in , (6 (i.e. we set to i ...i α O 2 1 − 11 0 352 mi c r. MNP E . This is the origin of i A Γ ∂ 1)! 912 1 − = NP p 6 is also a basis of negative ( 12 i A = 1 leads to, 10 esentations of p ,..., ), or in the ( i 5 2 . anize them as follows: + , α dx = 1 ,F p − 1. − ]] i 12 ∧ , ] tentials in table − 123456 E 2 , 1 6 D b i 4 i , ... jk i χ α 3 however. In particular, notice that there α c i N ]] , i dx − [ 1 ] − + 456 4 Γ ∧ p α ∂ E , ˆ ) and ( λ ) (4.12) E jk A p N 5 − , 2 ...i 6 b , m , 1 ψ i A + in the same weight of α M i 6 [ M ,E = 3 , 2] i , ι − ∂ Γ 2 A 4 A / ! , α ∧ , 1 220 E p − 3 123 − ijk − m , c m = 3 +1) E ι i 2 ) is given by the combination j [ D p , , h ∂ R dx [( + 7 i 3 , ( ) ijk i , α – 16 – A + + 6 + − − ) with H )), and which therefore admit an uplift to locally , j j ( M ↔ ,F 4 i E i A ) we get, [ ). For simplicity we only present here explicitly the , ] 23] N p 4.4 A A A 3 c 4.6 , X Γ ... Γ − [1 4.2 5 4.6 [ , , Mj ∂ jklm MNP Ni = D 4 7 c , i α Γ D i [ D 6 + , − = 3 , ∂ ij , which acts like ) − E ǫ [ k , expressed in terms of generalized derivatives of the gauge + 2 m ) and ( ) and ( ψ, χ 7 D + 1 123 h = 5 h 11 E A F ) encode the expression of all the field strengths contained i 3.5 4.3 NP + − η M ] j 6) Gamma matrices, which act on forms by P Γ ), ( representation. We can recast them in a more standard form by ] ), ( ijklm i , ) with the generators associated to negative roots + 4.11 ( NP F D M (6 A 3.8 M A 4.4 A )–( O → 133 D Mi M N [ [ i i 4.8 λ jk D ǫ D D ) in the appendix) we can build similar relations for the othe ij ). In order to disentangle weights appearing in various repr ǫ − A 6), whereas the negative root of SL(2 = 3 = 2 = = are the A.2 , representation of + − (6 ) (with M . For instance, acting on eq. ( j Mi i O M f means a contraction along f f 3.1 MNP i Acting on eq. ( The procedure can be systematized with the aid of the compute Equations ( m 912 912 There are some subtleties that have to be taken into account, ι f 2. RR fluxes: 1. NSNS fluxes: 11 12 roots of can be independent sequences ofthe negative multiple roots copies which of result the same weight appearing in the ( (c.f. appendix making use of eqs. ( potentials in the geometric solutions of 10d type IIB supergravity. We can org the field strengths which involve standardzero derivatives of all the the po exotic derivatives in eq. ( eqs. ( the where Γ and the bracket denotes the Mukai pairing defined by where the sequence of subindices is 7 (c.f. eqs. ( it is important to stress that in the conventions of appendix JHEP11(2010)083 , ) ]. H 6 2 | , ) 46 R , (4.18) (4.17) (4.19) (4.16) 12 , 0 γ i ∂ SL(2 ) + ( 2 = × , i orldvolume of ˆ F 6) , 220 (6 123456 , O β i . On the other hand, ∂ − fined in the next section, o be an anti-holomorphic at have a globaly defined , where we write in paren- interpretation in terms of e field strengths are in one ent”, in the sense that for should admit a consistent, ijklmq geom iklmrs 4 ravity. From the algebraic = e in the ( γ = 4 super Yang-Mills [ γ p G i q i ). As we will see in section y geometric 10d background. ∂ ∂ rgravity. Indeed, the uplift to a i N and ∂ = = -deformations) in general require 3 3.10 123456 ′ i γ = , for both type IIB with O3-planes Q 6 -deformations have been considered p β -deformation. klmrs ω i,jklm ′ β P - and -deformations provide the complexifica- 2 , β γ γ ip , B ] β -deformations, on the other hand, have been S j i ,P , γ k ∂ − a – 17 – i [ 2 = iklm jklm ∂ β p γ γ i i Q ∂ ∂ = 2 ] for some partial results). As it has been commented, k = = ij 33 ω , klm ], important in the context of the AdS/CFT correspondence ij jklm i P β 45 k , ∂ , etc. are not uniquely defined, and by means of SU(8) transor- γ 44 = ,P , , basis) and heterotic string compactifications ( ij k β ik jk Q H | γ γ i i ) ∂ ∂ R , = = parameter of the deformed 4d super Yang-Mills theory in the w k jk i β P SL(2 P is the type IIB complex dilaton defined in eq. ( × B -deformations (NSNS): -deformations (RR): S γ β 6) , For type IIB orientifold compactifications, The orientifold projection selects field strengths which li Backgrounds of 10d supergravity which involve = 1 supersymmetry equations require the above combination t 5. 3. Metric fluxes: 4. (6 -deformations provide the RR counterpart of the O Any vacuum of the 4dlocally theory geometric, involving only description these in backgrounds 10d terms is of almost 10d automatic type forto NSNS, IIB RR one supe and correspondence metric fluxes, withpoint as field thes of strengths view of they type correspond IIB to superg derivatives of elements in N ( thesis the components which10d do supergravity. not admit a locally geometric Backgrounds of 10d supergravity involving γ for understanding some of the marginal deformations of D3-branes, where elements which lie outside the Borel subalgebra ( also local SU(8) transformationsNote to however that be our described definitiongiven as of a a the background, locall fluxes is “gauge depend tion of the much less studied (see, however, [ representation. These can be read directly from table when we deal withbasis backgrounds admitting for a the generalized torus tangent action bundle. and th mations they can even be gauged away. They will be uniquely de in the recent literature [ basis). We summarize the surviving set of fields in tables JHEP11(2010)083 , i ω ) trans- ). Com- R ). Hence, basis ( ). Compo- , R 4.18 H , | ) SL(2 )–( 123456 R ′ i ) , × SL(2 ) ) Q ) × 4.16 23 ) ) 6) , ) , m m m SL(2 6) 123456 3456 3456] 23 1 m 3 12 , ) 1 m 123 123 m 123 ′ , , and 1 (6 1 , 3 1 1 1 ω ype of backgrounds to ˜ × Q ′ ′ [2 ω ω Q O R H H ′ 23456 (6 ′ ( 123 ( ( ω 123456 Q ′ ω Q ′ 1 6) Q O heterotic ω ( ( , ( ( ijkrs H Q heterotic ( (6 P hs in eqs. ( rgravity backgrounds. These , O ) representation admit a priori i 2 P ) tric description. , ) ic description. , ) ) ) i = 4 gauged supergravity, as it will 12 Q 1 1 m m 123456 1456 3456] 3456 1456 123456 12 3 12 N , 1 m 3 1 456 , , , , , 456 m 23456 Q 1 123456 2 P 1 P Q ′ ′ 1 ′ ′ [2 -deformations. These field strengths will F Q P H ′ ) representation of 456 P 456 γ ′ P ′ P Q Q 2 Q ( ( ) representation of basis ( , ( F H 2 ( ( , B | IIB with D3/D7 IIB with D3/D7 220 ) - and 12 – 18 – R β , ) ) ) ) ) ) ) ) ) ) 1 2 2 1 ) ) 1 1 2 2 1 2 2 1 2 2 1 1 ) ) 1 2 2 1 2 2 1 1 , , , ) ) − 2 2 1 1 − , , − 2 1 , 2 1 1 2 − 1 2 2 1 , − , − 2 1 , , 2 1 SL(2 1 2 , , − 2 1 , , 2 1 − 1 1 2 2 − − − 2 1 , 2 1 − 1 1 2 2 − × − , 1 2 − 1 2 − 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 6) 0 ; 0 ; , , 0 ; 0 ; , , 0 ; 0 ; , , , , , , , 0 ; 0 ; 0 0 , , 0 0 0 0 0 ; 0 ; 0 0 , , , , 0 0 , , 0 0 , , (6 , , , , , , 0 0 0 0 , , 0 0 0 0 0 0 0 weight 0 , , , , 0 0 O , , 0 0 , , , , , , , , 0 0 0 0 , , 0 0 1 1 0 weight 0 0 0 , , , , 0 0 , , 1 1 , , , , , , 1 1 − − 0 0 0 0 1 1 − − 0 0 , , , , 0 0 , , − − , , , , , , , , 1 1 1 1 1 1 , , 1 1 1 1 , most of the field strengths in the ( 1 1 − − (1 − (1 − , , (1 (1 − − − − ( ( 4 , , , , ( ( , , ) representation, either with respect to the (1 (1 1 1 × × (1 (1 2 × × ) or to the − (1 − (1 5 5 , ( ( 5 5 12 123456 ′ i . Field strengths transforming in the ( Q . Field strengths transforming in the ( It is also interesting to stress that some of the field strengt and 2) complex bi-vector. We postpone further comments on this t , i according to table become manifest in next section. Q ponents between parenthesis do not admit a 10d locally geome Table 4 Table 3 form in the ( an uplift to 10dare locally related geometric to non-traceless type metric IIB fluxes, or heterotic supe (0 that section. nents between parenthesis do not admit a 10d locally geometr turn out particularly relevant in the context of 4d JHEP11(2010)083 t to rep- that , are ) and 7 4 4 ions of . E 6 912 ]. This is gaugings. ), whereas and 37 = 8 gauged R , 3 N structure of the n toroidal com- ]. 7 SL(2 35 E type IIB orientifold × 6) , . This is also the global m have been identified as (6 reveal that all gaugings in implest possible situation, lusion of topologically non- rientifold compactifications 3.3 dently”, as there is a global O structure group. ing case of (not necessarily the local 4 gings are known to be related lation), whereas the ng effective 4d oint of view, Bianchi identities ed a set of gauge parameters, relates to the 56 vector fields of (tori, or “twisted tori”). Since erotic supergravity [ servation found in section ll be treated in section ), the structure group is reduced e of the tangent bundle therefore ure groups in this case coincide. 8 es. Also, more recently it has been 3.1 ) subgroup which survives after taking , is nothing but the embedding tensor of R 4 , ) representations (c.f. tables ]. Their higher dimensional origin, however, 2 -deformations. = 4 gauged supergravity [ , γ SL(2 36 representation of the U-duality group ) representation of N 12 × – 19 – 2 , 6) = 4 gauged supergravity. In this case, vector fields , - and 12 912 ) + ( β (6 N 2 , O 220 ) representation admit an uplift to locally geometric solut 2 , , where we observe that gaugings in the first row admit an uplif ]. In this regard, the results of section ], thus establishing a precise dictionary between fluxes and representation discussed in section 4 12 34 ), accordingly to the discussion in section 47 R 56 , structure group of the generalized tangent bundle becomes i SL(2 7 × ) representation can be also understood as originating from E 2 6) , , 12 (6 In this context, it results particularly interesting the ob After taking the orientifold projection (see footnote The In the present (and forthcoming) section we consider the inc = 8 supergravity (in its electric-magnetic covariant formu O field strengths arrange in the ( 10d supergravity. From the 4d pointto of view, twists some by of an these axionic gau rescaling symmetry [ supergravity. The in the 4d theory are grouped in the ( Up to here we have been discussing various aspects related to 5 Toroidal compactifications and dual fluxes heterotic compactifications with non-traceless metric flux shown that some of these gaugingswith admit dilaton an fluxes uplift to [ typethe IIA ( o all gaugings in the ( has been a more obscurearising and on longstanding particular problem. Scherk-Schwarz reductions Someconsistent of of with 10d the table het symmetry group of the resulting 4d correspond to the embedding tensor of 4d N the gauged version [ to expected to obey diverse constraints.as In well fact, as from tadpole the 10d cancelation p equations must be satisfied. pactifications also the global symmetry group of the resulti 5.1 U-dual gauge algebraThe and different constraints fluxes, grouped in the generalized tangent bundle, or the resentation of field strengths, discussed in section basis for the generalizedtoroidal) compactifications tangent with bundle. global SU(3) The structure wi more interest trivial fluxes for these fieldbecomes strengths. relevant The also. global structur namely In compactifications that on regard, manifoldstori here of are trivial we structure parallelizable consider manifolds, theThe fluxes global s are and in local this case struct defined globally and “gauge indepen the orientifold projection.gauge We fields have and in field strengths particular which introduc are covariant under the compactifications with non-traceless JHEP11(2010)083 ). ⊂ ave ) 3.4 (5.4) (5.1) (5.5) R 1539 , nstraints generators ]. SL(2 7 48 × E ) (5.6) 6) 1 , etc. are indices , , decomposes as B 1 (6 7 , O E A expressed in terms of ]. In other words, by onstants of the gauge ) + ( ese Jacobi identities are 33 2 P s belonging to the AB , S er an ) .[ constants, contracted with F rom the Jacobi identities of ic and antisymmetric repre- ion of the structure constants 8645 352 cture constants and fluxes is f structure constants this new ), we can apply equations of the higher dimen- + 1463 5.1 + ) + ( 1 and therefore does not define a con- , ), we see that in terms of irreducible = 0 (5.3) P ing the scalars of the theory [ B 133 representation of 40755 X 5.1 , L PC 495 + A P + ( F AB = 0 (5.2) = 8 gauged supergravity, given in eq. ( ]), the function F A P D ) BA 1539 ] ) + ( N ). If Jacobi identities are satisfied, then fluxes 25 encode the fluxes. Here L 133 C F 3 ] = , F + + 3.1 – 20 – B P AB 1539 66 AB X P F [ L , PC F + A F 1539 1 X ) + ( = 0. Jacobi identities are therefore satisfied when a [ P AB 6= 0, in order to covariantize the Lagrangian 4d gauge bosons h 2 = , ) F P = ( X AB 32 P 56 ( ) (see section 56 F 7 × AB P × ( E ) + ( F 1 56 , 912 77 , the left hand side leads to 7 E In order to ensure antisymmetry of the commutators, extra co = ( 13 1539 are the gauge generators and representation of A X 56 ] it was shown, for a particular type IIB orientifold, that th We start considering the sector of the algebra invariant und By looking at both sides of the gauge algebra ( In a generic situation (see for instance [ From a 4d perspective fluxes are associated to the structure c 33 In the general situation where subgroup. It is possible to check that the 13 7 to be supplemented with the 2-forms which result from dualiz sentations. Similarly, the right hand side decomposes as Thus, in order forantisymmetric representation both must sides be kept. to match, only products of state where we have made explicit the distinction between symmetr to it in order to construct the entire algebra. representations of starting with a previously known sector of the algebra ( must be imposed. Namely, thegenerators symmetric must part vanish, of the structure sional theory. Therefore,known once we should the be dictionary ablethe between to 4d read stru gauge the algebra. flux constraints directly f are constrained to obey actually identified with the Bianchi identities and tadpole algebra satisfied by the vectors of theFormally, we effective expect to have an algebra In what follows we sketch ain possible terms way to of obtain the fluxes. express The idea is to follow similar steps as in ref the fluxes is not antisymmetric in the subindices of the In [ sistent algebra. where contraction with a third generatorcondition is reads considered. In terms o E JHEP11(2010)083 ) ), ± 2 , , (5.7) (5.8) (5.9) ∓ (5.13) (5.10) ) 12 , can be ) fluxes ) repre- 0 R = 2 and 2 2 , , ) content − , , 0 j ′ S  R , ) ), ] , 0 k j X ). Inspection ] , δ ) times gauge 220 220 SL(2 5.9 3.1 2 0 i P [ = 1, , 35 , f × . 1 i SL(2 , 5.10 12 6) ( AB × 6 (5.14) , η m × ′ (6 mnopq 6) − X ′′− , hoose ′ O X 1 ) S 912 m ) (5.12) ,..., (6 P f representations, the left B 1 X )] X 12 ⊕ δ , O 2 2 efined ˜ lk ′′− , H = 1 ǫ P S 77 l AB | ′ iven in eq. ( mno + ordingly to eq. ( 12 F − A ( X m 2 f s is known to be [ + ) + ( ⊕ ij )+( ), with weights (0 X X 1 ǫ 1 2 2 P k , P k ′ , m 12 m p , , 1 X P X + ) contains fluxes in the ( − P X 2 ′ = 0 (5.11) S 12 P B P k AB − +6)2 = 220 δ − P k AiBj F ) + ( 5.13 p m 1 − ) representation of ( Ai ) fluxes. ⊕ X ) comes from products with ( ˆ 3 F ′ X representation. Alternatively, S ˆ f 2 2 , 3 X P DLj k , j , , X m 2 Ai f ] = δ i ≡ X 66 X 12 ] = Q ′− 12 P 66 – 21 – 32’ − ˜ ′ S 220 F m = Bj = ( P A AB + 2 ). To start with, we choose the subsector given by X + δ A f A , 2 ⊕ X ) ′− ), while the antisymmetry constraint is X ij m Bi ) + ( ˆ Ai 2 ǫ X f 2 ˆ , 1 X Bj k j , X 4.2 [ ,X δ X 12 p Q , while the second row corresponds to fluxes in the ( 12  , 12 Bj, S 3 ( Q + − 1 2 S +6)1 × = ( + ) m ( ] = i 2 B 2 ⊕ X , P ) representation, namely [( ≡ X Ai ,X 12 AB 2 ( 1 X f , given in eq. ( [ m ˆ -th entry). The first row of ( . k j X ) and ) leads to ˆ [ 4 δ X 12 m − = 5.9 ABCi f = 2+6. Then, in terms of type IIB fluxes, we read from eq. ( Ai, S B kP is a weight belonging to the AiBj and F = ( − S A Ai f For the sake of clarity let us present a concrete example and c Notice that performing a similar analysis as above in terms o The general structure of the algebra, written in terms of = 1 + 6, while the rest comes from products with ( presented in table sentation, summarized in table of the structure constants indicates that ( (where 1 is in the A where, in order to avoid confusion with the notation we have d Similarly, the generators of the algebra are decomposed acc member of eq. ( with generators in the ( to denote the two elements of the SL(2) doublet in ( where while the right hand side contains the structure constants g expressed in terms of a set of multi-vectors, hence reads where In this case the explicit expression for structure constant the fluxes transforming in the ( JHEP11(2010)083 ] ) ) ]. n 2 2 , 32’ ) we ,X 5.13 ,X actifi- (5.18) (5.16) (5.17) (5.15) 12 1 − m ′ , ˆ X X [ ; + − → m , 0) of the , ] ˆ = 1 theory. X − 2 , n ; 0 ′ N ) and taking into − , keeping in mind − ω ,X ) symmetry of the , which arise from , , hus, by acting with 1 R − + ˆ − , X 5.13 , , n − ˆ + 1)[ X , , espite the small amount t hand side of eq. ( SL(2 m − ⊗ ′ − ( , orbifolded 7 × ω ors. For instance, by acting way for the other [ α he tangent bundle is still the − ≡ dal) case will be considered in , E 6) . The presentation of the full + 2 , ′ − µ = 4 theories in 4d, one can easily + p p (6 = 4 theory and transform under a untwisted states ˆ 32 7 is reduced to SU(3). Hence, we can X 12 O α ) N N × m P E ′ ′ 2 1 mnp X ⊗ localized at orbifold singularities, which √ n 32 P Q 2 − 1 √ 12 = 1 by orbifolding the theory with a discrete + 2 p, i 12 ′ ′ + ′ p n – 22 – ′ N ω P ω ] in the X 2 ′ p,mn m → → → ′ i p generators the full algebra can be reconstructed. For Q ,X P twisted states 12 . p orientifold leads to = 4 theory and are invariant under a global symmetry 1 Q 12 ′ 7 7 + P 6 ˜ + ( E E X H p -th position. N ′ [ T p , the generator corresponding to the positive simple root 6⊂ m X ˆ , we observe that (1 7 X ), and → α ab p m ′ R ). Hence, acting in the left hand side of ( ] E +), we obtain the commutator between a vector in the ( mn , Q ′ p twist 2 , 1 X , ω + ˆ ′ G − X ; , SU(3). Toroidal orbifolds are the simplest examples of comp → SL(2 1 32 ′ − ] = ⊂ , m × n X ˆ − ˆ X , X :[ 6) 7 , , α − µ m , ′ E (6 E − O X , is the generator corresponding to the weight ( [ = 4 theory to describe the effective action of untwisted fields − ⊂ . m , ′ 6 N − G X ( on the fluxes we see that In this way, we can easily make use of the global As mentioned, by applying The feature that makes toroidal orbifolds quite treatable d In the same way we can obtain the terms on the right hand side. T ≡ 7 α 7 cations with global SU(3) structure.section The general (non toroi symmetry group Γ group larger symmetry group reduce the amount of supersymmetry to would have Whereas compactification on a algebra and constraints is beyond the scope5.2 of this work. Toroidal orbifolds and flux induced superpotentials direct truncation of the parent are not directly related to a truncation of the parent parent with the generator corresponding to the negative root Following similar steps we could derive the rest of commutat vanishes. Putting all together, and proceeding in a similar account that of supersymmetry preserved, istrivial that one, the except structure at group thedistinguish orbifold of singularities, two t where types it of states in the effective theory: whereas the action on the other terms which appear in the righ and a spinor in the ( Here, commutators, we get that only the subset of fields invariant under Γ survive in the representation, with + in the E instance, by acting with α JHEP11(2010)083 , ˜ q m U (5.21) (5.20) (5.23) (5.19) (5.24) (5.25) (5.22) , can be 6 (6) and 3 6 T Ω. There O , T 2 − × , 15 ˜  ) below is gener- . Consistently i (6) as, = 1 6 U 6.1 O ˜  6 T ˜ q (Ω) = T σ m lution, U ˜ p ˜ p l and Γ], where the orientifold , i, U 0 dx ˜ q i J ˜ p . For that one has to note β pactification to cancel the , ˜ × p ithin the ) ˜ 1  ˜  m ˜ q ǫ dx σ , U L U eeping elements in the Borel alrstructure¨ahler of ). The complex scalars ) = dx ∧ ˜ F m dx nding on the particular choice ilm l J + ǫ ∧ ˜ ( m ) survive the orientifold projec- ω 1) ∧ σ 1 3.10 3! ˜ m R m m − ] in order to match the usual conventions . Their weight vectors can be ob- l m dx , ( + (det . ˜  33 dx T = dx P rst of the equations in ( ∧ l ˜ 5 ˜  dx (6) , δ i l ˜ − 7 6) ∧ O and Ω define an SU(3) structure. = . Subtracting out the , U k [Ω ∧ β i SL(2 l × ˜ /  (6 dx = ˜ J δ p i 6 m ) 3.3 ˜ O × m (6) dx J ˜ l T det = dx ˜  U O ǫ ∧ 6) k ilm ∧ l ˜ 1 2 , ǫ × l , summarized table J β 2 1 7 − (6 φ (cof – 23 – , B ∧ . In particular, | dx , dz E − O ˜ ˜ q 6 q ˜  = = − e 3 14 ˜ i p ˜  m ˜ ˜   T ˜ i  2 with respect to [ U(1) α i i i of ǫ dz c U SL(2) α ˜ p − ˜  ∧ Z l ilm i 4 ǫ 2 U U 133 C ˜ 4 1 q → −J ˜ p has been defined in eq. ( dz ˜  + c = ⊂ ǫ = ∧ 0 J B c ˜ , β , α  ) 1 i α S J ˜ 3 3 3 ilm ω ǫ , dz and 1 1 2 dx dx 4 Ω = C ∧ ∧ = ˜ 2 2 Ω = ˜  ) + ( i ) → − 1 dx dx , 4 U ] we can introduce a basis for 3-forms in the covering correspond to moduli of the complex structure defined by Ω in ∧ ∧ C ˜ q ˜ 1 1 66 49 m parameterize the coset (cof dx dx reverses all coordinates of U is a basis of integer 4-forms even under the orientifold invo B σ = = S ˜ 0 0 m l β α ω and Following [ We have taken Notice that before applying the orbifold projection Γ, the fi ˜ m 14 15 l where, In terms of these, the holomorphic 3-form can be expanded as, with tion, according to what was described in section Similarly, the moduli parameterizing deformations of the K tained from the adjoint representation of T extracted from the complexified 4-form, We focus here on type IIB compactifications on 5.2.1 Type IIB orientifold compactifications The complex axion-dilaton where in Generalized Complex Geometry. ically not satisfied, and it is only after the projection that that only the ( The scalars involution U(1) pieces associated tosubalgebra, local then leads gauge to transformations the and weight k vectors in table total RR charge. with this, D3 and/or D7-branes may be also required in the com are O3-planes spanning theof space-time directions Γ, and, there depe can be also O7-planes wrapping complex 4-cycles w JHEP11(2010)083 . . al B 2 | ) C eory R . The it was (5.26) trans- , c 4 for 3 J ], O 1 the fluxes, ˆ Λ SL(2 50 W ×  ], we act with 6) associated to the , J 33 s in the compact B (6 ∧ | ) , O J 7 R , , ∧ -brane. Under global r potential [ 5 J SL(2 es. The ones that survive orm fluxes along three cy- ), and similarly Z ial in cohomology, such that × 6 1 ded in a non-trivial effective complex structure moduli for 2.5  6) , ust. However, in section ) (5.27) 0) 1) 3 (6 0) , , , 1 2log O H ; 0 − B ; 0 − 0 S 0 ,  ) representation of 0 ; , 0 2 ¯ − Ω , 0 , , ). For that we observe that , as in eq. ( 0 , 3 1 2 ∧ , 1 F ]. − 0 , B weight ( Ω , , 220 0 ), or rather, on the invariant superpotential, 5.26 with one or more copies of the 4-form 0 0 49 , ∧ , , 0 Z 3 ) to incorporate all the remaining fluxes in the – 24 – 0 0 , Ω i , , 5.27 (6)] covariant 3-forms linear in the fluxes by con- − (1 shift, and one has to patch the background by using Z O (0 (1  2 ) and moduli transform with weights summarized in 5.27 ) representation of 2 1 × = B 3 , log , 3 ˜ given in ( q 1 2 ˜ 1 m is parallelizable, there is a basis of globally defined one- O l (6) m − 3 S − and 6 T O  W O U 2 [ scalar ) 0; ˆ T ], K / ¯ ) + ( ) is at the core of many of the recent phenomenological ap- , S C 1 0 51 6) , , − , 0 generators on ( 5.27 , 66 S ) does not vanish. Locally, these fluxes are introduced by loc (6 ( 1 i B , O , both . The deformation induced in the 4d effective supergravity th which are required is dictated by the tensorial structure of , with | are only a piece of the ( 1 6 6 3 ) − c , 3 5.27  O T T R J ˆ , H K log + − and 2 SL(2 | can be found, for instance, in [ = 3 3 . × 6 F 3 O 6 we summarized all the elements in the representation. T O T 6) W 3 ˆ | , . Weights in the ( K ) representation. Following the same strategy than in [ . We can build (6 2 5 , We can therefore generalize eq. ( Three-form fluxes generically induce an overall charge of D3 We can consider now the effect of switching on background fluxe The superpotential ( The metric of this moduli space is given in terms of the K¨ahle O = log 220 sidering contractions of the fluxes in table forms that can be usedthe to define integral global in fluxes that eq. are ( non-triv number of copies of gauge transformations with parameters Λ for An explicit expression of thea second general integral in terms of the Table 5 Note that, since the covering forms with weight (1 G table proaches to string theory, where modulirecast stabilization that is a m ( the gauge fields as discussed in previous sections. monodromies of the the orientifold and orbifoldcles projections of are the RR internal and NSNS 3-f manifold. Let us start turning on standard supergravity flux is particularly well-known insuperpotential this of the case. form [ Their effect is enco moduli of In table JHEP11(2010)083 , , ] - 7 × ) 33 R 6.3 , 6) , , ). On 7 (5.29) (5.30) gener- on orbifold. (6 y of the ] (5.28) 7 3 2 c O 5.21 E Z 6 . Based on ·J × ...i B ] for SL(2 ) | 1 2 ′ 7 i 3 Z ǫ , F ] 2 4 of the flux com- , B p 3 S ]. p 2 = 1 ting with − 55 p ′ 1 – p H ) , 52 c 6 , i J + ( mplexified 2-form, ( ) ...i rst two pieces in the above 2 6 uperpotential involving the 1 c selected by the p i has to be always of the form . +3 ) representation of erived in the broader context 5 i ǫ c 3-form, as in eq. ( 7 or heterotic compactifications ] ˜ c p e 2 ·J [ 3 ), as desired. 3.3 E , 6 ) J preceding subsection. Complex p hic 3-form Ω carries no charge of 1 − 2 i ′ | 5 i ⊂ , i m 12 Q e ) 2 1 ˜ 6 7 (˜ m i c B ) l 2 1 5 − i R S ˆ J ω ) , ( In addition, the quadratic constraints = ˜ c 4 m 0; − l i , ′ J 3 T ) factor. In the conventions of [ i +3 0 ( 16 i P 2 4 . , R i i 7 , = 0 1 3 i ) i , + ( ) 2 1 , σ i R c . The reader may check that all terms in this c iJ , 1 ) , 3 i J c 1 ) ( – 25 – + ·J , +3 c mn 6 i ] ) 2 p ·J e 3 J 5 ′ ( p P B p | + ˜ 2 4 2 B F p i p p ) = SL(2 ′ = e S 1 basis, introduced in section (˜ c m p , one per SL(2 c ′ [ 1 2 H i J − G J B ( σ P and = have been carried out recently in [ Q 1 2 4! i p ) are projected out by the orbifold action Γ. In particular, c 128 ) are actually related by 4d electric-magnetic duality. Sys mn [ 2 5 · · 1 and Z Q 2 ·J )+ ( 2 1 2 4 3! , σ ′ 5.28 3 5.28 × ) reproduces the results derived by similar arguments in [ ) Z 7 Q 2 H = = = ˜ e , × Z B 3 3 3 c − require in general further components to vanish. Hence, man p p p 2 5.28 S 2 2 2 8 ) the highest (lowest) component in the doublet of SL(2) Z p p p e ·J 1 − 1 1 1 (˜ f ). We postpone however the discussion of these fluxes to secti p p 3 p 5.1 2 1 ( P ) ) ) F c 3 2 2 c c = [( f 5.28 0 ·J , the combination of fluxes which couples to ∧ ·J ·J σ ′ ′ B Ω Q | ) representation of fluxes is given by, ( , since they survive to the orientifold projection too, by ac P H 2 ( ( B , Z , with | ) 1 U(1) the orthonormal vectors of the = R / i 220 , 3 Sf e We could consider also fluxes transforming in the ( Of course, one has to bear in mind that in concrete models many For completeness we present here the embedding of SL(2 O − 16 W 2 with ˜ f SL(2) derived in previous sections. Moreover, since the holomorp on toroidal orbifolds followsstructure closely moduli the are discussion still in definedthe in other the terms hand, moduli of K¨ahler are the now holomorphi given in terms of the co where the complete flux inducedof effective general superpotential SU(3) is structure d compactifications. 5.2.2 Heterotic orbifoldThe compactifications derivation of the flux induced effective superpotential f vacua which result fromtematic ( analysis of thesuperpotential the for vacuum structure Γ = induced by the fi ponents which appear inone ( may check thatfor ( the case ofderived Γ in = section where, the above observations, itfull is ( possible to show then that the s equation indeed transform with weight (1 these are given by, and similarly for This is given by 7 coordinate vectors, ators on eq. ( SL(2 JHEP11(2010)083 , 6) Ai i , 3 f c (6 J (5.31) (5.34) (5.33) (5.35) (5.36) · O s of the ) and ′ H ). H is the  S ABCi · J . The heterotic f 4.12 − ] ˜ 1 m ∧ . All together, the R A 61 J 5 f dx – ∧ H 6 + ( ∧ p 56 S l 2 5 J c p . These are summarized J 4 − dx ogical point of view this · p H Z | 2 3 = 1 supersymmetry in 4d, wn effective superpotential ) ntial to the full set of fluxes ) p 6 ′ 1 . A = sion [ 2 f the tensors f R  ] ω p ) representation whose super- ˜ N , c 1 m l 2 H J p = , ) e ω S log c ) (5.32) A · J ) 12 − ), ˆ − SL(2 mn 1 ] 1 ∧ 3  idJ 15 Q × G p G c ), and the contraction ¯ ( 2 Ω i are defined as + J p 6) + with metric, K¨ahler ) ω + ( ∧ , 3 1 c ∧ ( c c J J ). m G H c (6 Ω i ] J e ( (6) 3 v J , · ∧ O · 6) p ( 10 O ]. We find the compact expression , 1 Z ∧ ) ) 6 c 3 ′ × c i p (6 and J 35 5 Ω J G ( Q O − basis are still given by table p [ (6) – 26 – ( 3 4 1 ABC  2 H O p p Z G ∧ p f mn [ H S 1 | ). These definitions are such that the corresponding R . p H ) × Q Ω m = [ log − 1 S 2 3! 1 R . ω 3.9 H , 6.3 − | Z ω − het = = = )] 2 = U(1) 3 3 3 W (c.f. footnote H . p p p SL(2) SL(2 ¯ 2 2 2 ) + ( ′ S , with the gamma matrices acting as in eq. ( p p p 3 het ABC 1 1 1 × . After some algebra we arrive to the expression, H A ˜ − p p p f H 3 ) representation of ) ) ) W Γ 6) c 3 2 H H c 2 c = , Ai J , S J J S and f · ( · · (6 i ′ − ω O ω R Q − 220 ( 3 ABC ( ( , ) ), which in the present context are just the embedding tensor and ′ H 3 = 1 compactifications ( was defined in eq. ( Q h G 4.3 2log[ ( H N ∧ − ABC S Γ Ω = is a basis of integer 2-forms (see footnote . = 4 gauged supergravity theory [ Z ˜ m het l ABCi = N ˆ f ω K . To get the full superpotential, the starting point is the kno It is illuminating to express this superpotential in terms o het W parent potential will be derived in section for heterotic compactifications with 3-form NSNS flux and tor where ˆ axion-dilaton 6 General In this section we consider compactifications that preserve weight vectors in the scalars parameterize the coset and which are not necessarily toroidal. From the phenomenol in the last column of table We can now proceed as beforetransforming in in order the to ( extend this superpote defined in eqs. ( and similarly for where the contractions are defined as, where we have included also fluxes transforming in the ( where a subindex 2 (1) refers to the direction action JHEP11(2010)083 e onal (6.4) (6.2) (6.5) (6.3) , and J ng inter- the com- = 1 back- 6.3 N 6) , 1)-form , (6 6). In the special , O (6 f is obtained from the = 0) and its complex O ε ± 0 . coincide, the two SU(3) Φ 1 nowhere vanishing spinor ± i . There is no requirement rs are the same, they give ¯ Ω (6.1) structure. The latter can be as sums of spinor bilinears, ...m e manifold where the spinors ) invariant (1 ∧ p . ocal one. ± 0 ructure compactifications with m † Ω , γ 2 − 4 3 , , Ω η i  i 2 1 + + 1 + 1 + − η η − η η . Each of them is invariant under an p 2 = , ⊗ ⊗ ) and these can equivalently be thought covariant formulations of = = 1 6 such that Γ representations of − J − − η 7 ...m − 0 − 0 ε ε 1 , the pure spinors read E ∧ Φ (Φ Φ 2 ]. Then we address in section m + + η 32’ J γ ,..., + † 39 2 1 − − ∧ = 2 ± η η η – 27 – , 1 = 1 , and  ⊗ ⊗ η 6) and † ± iJ ! , 2 + i + + 1 p backgrounds − 32 ] and [ η ε ε (6 e 1 ). + ,J p 24 O η = = = – X = 1 2 1 6.1 = ǫ ǫ 8 1 + 0 22 Φ N + 0 contribute to Φ = Ω = 0 Φ p † ∧ , splitting the generalized tangent bundle into a 6-dimensi 2 + A η J are pure, i.e. each of them is annihilated by half of the twelv 1 + η ± 0 such that the 4d supersymmetry parameter = 1 supersymmetry requires the existence of nowhere vanishi 2 η 0)-form Ω, which satisfy the relations N by , 2 , and 1 formulation of ǫ 1 η 6) , and Ω are those of eqs. ( (6 J O 6) gamma matrices Γ By construction Φ The two spinors can be combined to form complex pure spinors o , (6 By chirality, only even (odd) reduces the structure of thecompletely tangent characterized bundle in to terms a of global a SU(3) globally defined SU(3 is perhaps the most appealing case. The existence of a single case where the two spinors coincide, i.e. complex holomorphic bundle (given by of as sums of even or odd forms in the Four-dimensional nal spinors O We have chosen the chiralitya of priori on the the 10d relative spinors orientation to of be the negative spinors 6.1 putation of the effectivearbitrary superpotential fluxes. in general SU(3) st two 10d ones rise to a singlestructures intersect SU(3) into structure. anare SU(2). Whenever parallel, the If there is there spinors no are do global points not SU(2) on structure, th but only a l SU(3) structure on the 6-dimensional space. If the two spino grounds, following respectively [ In what follows we first review the where namely a holomorphic (3 Making use of Fierz identities, it is possible to express Φ JHEP11(2010)083 - 2 B B = 1 = 0, (6.7) (6.9) (6.6) (6.8) drops θ N 2 B field is an 2 ). Note that B 5.24 . , the dilaton and -transform on the + ) and the represen-  B i terms of the is the complex axion- -dimensional holomorphic + efinite) metric on the A.4 field, one needs to ¯ Φ + C φ 2 − ), it gives a block diagonal B , e = 1 theory in the context of . 6.5 + ) (6.10) mplex structures associated to Ω Φ N + 2 defined in eq. ( φ , , corresponding to a single SU(3) define each a generalized (almost) B Φ − 2 i C e 6) invariant, and the η iθ ie ± 0 ± 0 h ] , i J e ¯ − Φ . = ± 0 (6 Z 24 B 1 ¯ i 6) spinors. Φ = − O A η Re ( ,  , 6) transformation corresponding to a φ J Γ − , ± 0 (6 and the component forms. Exponentiating ). Note that the contribution of , − + Φ Φ (6 2 ± 0 O , that can be obtained from the spinors by h )). Making use of eq. ( ie ). J I 2 log ¯ ), and the bracket is the Mukai pairing defined  O Φ B i 5.26 η h is obtained in the standard way from an SU(3) − + − 6.3 i – 28 – iJ − g , +  = 4.12 − i = = C 2 iJ ), it is not hard to see that the = 1 orientifold projection. For O3/O7 planes − 2 ) = − In other words, Φ ¯ B Φ ± ). For the pure spinors ( H 2 B 0 ¯ ]  e , N i B A J 4.12 g 17 − e 24 ± 2.9 = 0 Φ J = h + C which for the case ). Hence the 0-form component of Φ + Φ Z = 0), where ± 0 6 i Φ 2 Φ C  2. In the case of type IIB compactifications with O3-planes, 2 B + B π/ e log 2 , satisfying ( C − = = . These can be read off from a combination of Φ ± 0 + B ± = θ J are by construction compatible, which implies that the two 6 S C ˆ ± 0 K J ) (or more generally to eq. ( + and 6) action. 6.5 B , 6) gamma matrices act as in ( ), which is the natural bilinear on C , while the 4-form component is precisely S , J ( (6 B 2 is an angle that defines the (6 S O B 4.13 θ O e Two compatible pure spinors define a generalized (positive d The second term in the potential K¨ahler should be written in In order to obtain a generalized metric that contains a = Furthermore, Φ 17 + C generalized tangent space. Usingthe the pure generalized spinors, almost the co generalized metric is given by This is the same metric as in eq. ( conjugate antiholomorphic bundle. dilaton pure spinors amounts to a wedge product of Φ bundles have a 3-dimensional intersection. tation for gamma matrices in eq. ( matrix (corresponding to in eq. ( while for O5/O9, It is easy to see that this is equivalent to ( structure (in complex coordinates complex structure where adjoint out, as it should be since the Mukai pairing is an where the action we get Φ transform the pure spinors, i.e. to apply an structure, implies field to eq. ( In terms of the pure spinors, the potential K¨ahler for the variables type IIB compactifications reads [ the RR potentials into the following complex form [ JHEP11(2010)083 + C ), 6) by R- ds, , Φ ] = . It 0 2 0 b R (6 H 6.11 (6.13) (6.12) (6.11) | triplet ) O iK ,K )). This R 0 R a + , 0 1 K is built out . Similarly, is therefore r, going be- ). 3.5 7 K 0 6) covariance − 0 ) E , = + C 5.27 (6 -transformation, 0 + can also be easily β O build up, together ) and ( K ij k s, which implies in to an SU(2) + = 2 theory: those in Q 3.1 . The easiest way is to + 0 7 (2) algebra [ basis, where the N E ng the , where (Φ su ially under the SU(2) H 0 s of Φ | te Φ ) ) the + C 133 not be obtained from ( R , ), we define ∈ representation of (Φ representation are locally non- 2 (c.f. eqs. ( ) . The pure spinor Φ B 3.5 6) that appear respectively in the 7 + 0 6). The reason for this is that, as 56 ). e , SL(2 H , . . E | Φ 220 ) = 1 backgrounds that descend from (6 = (6 i ×  R 5.28 O i − 56 O + C , N 6) ω Φ , ∈ , of -transform one acts by a + + C ) (6 in the 6) to the B − 0 Φ H , O ′ 12 d ). S Φ , satisfying the real h (6 i , H into representations of – 29 – v O 3.1 ( Z + 0 φ 133 term in eq. ( = − = (0 and ∈ 0 C , e ′ W representations of λ 0 0 a and Φ Q backgrounds , ·J 0 K representations of 0 −  Q 133 . If instead of a = 1 = 32 6)-covariant superpotential is given by . Using the notation in eq. ( + 0 , 6 0 + N are encoded in their local definition in terms of the gauge fiel ]. We work in the , combines with the dilaton to form a doublet of SL(2 (6 and ) involving the locally geometric NSNS flux K 3 ), with vol + 0 . It is not hard to check that this reproduces eq. ( and covariant formulation of O 0 39 3 2 φ H to an element in the are projected out of the spectrum in this case, we keep them in 2 7 56 gives the , these fluxes require extending the derivatives even furthe 5.28 − 6 E d ,K dB 4 32’ the e 2 0 should be a singlet under SL(2 2 C and β = + C = − 0 3 ,K ). For that, note that in general Φ 3 F κ 0 on 1 and H d K 2 , 6.11 2 one. C 7 , formulation of dC , with 2 E 0 c 7 B = and the pure spinor Φ E K 3 + The moduli space is a direct product of these two moduli space The other spinor, Φ The remaining NSNS fluxes The term in eq. ( In terms of Φ The first step is to embed Φ are the scalars of vector multiplets, while the deformation F abc = 2 ones, following [ C − κǫ explained in section i.e. geometric and their contribution to the superpotential can of elements ( even if 2 symmetry. The way to realize all these conditions is to promo We review now the with the dilaton and the RR axions, the hypermultiplets. particular that Φ assignment is also consistent with the degrees of freedom in not even promoting yond the fundamental representations of non-geometric RR fluxes should also only appear when promoti to a full should also combine with the RR fields and transform non-triv as their fluxes are not. 6.2 embedded as follows of encoded in ( where we have used the notation in eq. ( Φ decomposition of the the action of Here the fluxes subgroup is that of Generalized Geometry. embed them in the N JHEP11(2010)083 C K = 1 = 1 = 0, is the 2 (6.16) (6.14) (6.15) (6.17) (6.18) trans- N N r and S + represen- c . = λ . 1 7  r 56 1 E and , obtained by , , as it should. 0 3 6 iθ ), by the corre- R b , proportional to − K 3 , or equivalently a , e , as expected. The ) decomposition is 2 , i A.4 R 2 K 6 = 1 supersymmetry is indeed the b R b , , 0) C iθ N , + e 0 K covariant superpotential ] i , 2 H is the generalized almost , while SU(2) ent in SL(2 S 0 a 0 39 +  B ⊂ × K K  A → 2 R 0 b + 0 ) = (1 ) = , he geometric objects 6) e H ) plane. The singlet and doublet 3 3 , J 6 be parameterized as is the heterotic complex dilaton B 2 S b ) piece of A , z , z e (6 = 0 and the , r H ), transforming in the + . − − + 0 1 , we can define O c S 3 and r ) J e ± r , z , z 4.6 2,32 C 6 ¯ u shifts + + , = z z action, given in eq. ( a and ( ( + ) forms a triplet of SU(2) a generic group element of a vol , iu 7 φ K + j K 2 A a in the ( E u , 3.1 e λ, DK z i − θ in the ( ,C ( ). u ie i + S = 0) = – 30 – , ω + ¯ ,K 6.6 C g Z j iθ 0 = 2 decomposes as a singlet and a doublet. The ¯ ) are selected in each case by the vectors u − K λ Im Φ i 3 = 2 , -transform of Φ H action on R ie u b φ ) b e  6 − − ¯ ,K 1) S W 6 b , b , in eq. ( 4 1 − , with , e e 0 iθ 0 − + , representation, whose + 0 + = c ie ,K S gλ e as 0 3 covariant formulation is given by [ + 56 0 7 = K = ). The K Re Φ is just the complex conjugate of = ( ) = (0 ) = ( K E φ λ λ ¯ 3 3 u 0 − 3.7 − stand for the adjoint u e = 1 field as , r , r ) for type II compactifications. K 2 and − − b ), i and N e 0 , r , r ). ω 6.10 λ (2) commutation relations, is given by 0 C + + r r + 3.9 su and gK H 6 are those introduced in section S b i = i e v = 1 supersymmetry is selected by choosing a U(1) ω , ( such that a triplet of SU(2) C , + is the generalized derivative defined in eq. ( a i N c K ≡ r e v D u type II : ( In an analogous way as for the An heterotic : ( action generates, among other terms, a non-zero last compon = 1, while for type II orientifold compactifications . This implies that ( 3 + + Hence, defining the symplectic invariant in the where with complex structure defined by Φ formed objects of where 6.3 General U-dualityWe covariant are superpotential now ready tofor compute general with SU(3) the structure above compactifications. tools In the terms U-duality of t sponding generators in ( we have that the spinor component along Here c c tation, demanding the the superpotential in the defined in eq. ( chiral field in eq. ( components in the triplet ( vector supersymmetry selected in heterotic compactifications can r is parameterized in this case by a single angle JHEP11(2010)083 is 0), ], , A i gives ) 62 v (6.22) (6.21) (6.23) (6.24) (6.20) , K m 56 ∂ 39 D containing . They act . ) is actually = ( H 18 |  ) D iJ R 6.21 e . , , obtained by ex- A 4 ). For consistency 7 , etc. were defined in ( )Γ E ) part of ( etric subgroup, here R 2 components of the fluxes SL(2 C 6.16 , A ∈ i . Fluxes surviving the × f ω Q AB g ), leading to iJ H , 1,32’ representation. pactifications with global 6) F e derived for toroidal het- e . lized connection [ S , ω , e ( A.6
(6 − 56 = ents 3 uperindex 0, while derivatives fications, where the RR fields 1 O CE H and projected onto the + 0 A K . f , . E C C i ) (6.19) 133 we have actually used only the first + ( ) D AB 0 AB C g 4 F iJ ) piece (equal to Ω for SU(3) structure A F K e ), and Φ + 1,32’ D D + ( C , in the D and C 0 K CA B ABC 1,32’ B 6.13 λ ) ( δ CD D 1 K )Γ 2.2 , S A − 2 B K – 31 – Ω g ), except that we are missing the complexification ) and the full set of 133 gauge fields. Hence, the D h D is the vector introduced in ( Z ) representations of a and making use of the symplectic invariance, the ABC = Z = ( 2 4.6 , is projected onto the = z , f 5.33 with inverse volume factors in the a BC C C g = S H K H W 12 S AB S AB = W D F appears because the connection − ), has only a ( A 1 2 . In sections ) B A , defined in eq. ( K ) and ( 6.12 + 2 ABC D factor in , ( f K 6 ( defined in eq. (  = 220 ∧ D involves all the field strengths introduced in section C Ω . The difference between these two definitions involves terms φ K C ] the derivative was restricted to the standard derivative, − B ) of the appendix and 912 e A 39 ∈ A , given in eq. ( A.5 Z 0 C D λ = )). . AB = 4.3 F C , which has only a spinor component, as shown in eq. ( het 0 + , the generalized derivative of W Extracting the group elements The generalized connection applied to According to our above discussion, for heterotic string com AB . A bare gauge field like a We have canceled the vol K f J 18 where the derivative is now acting onof the the bare gauge objects fields with a (the s fluxes) have been encoded in the genera superpotential above can be recast as Note that While in ref. [ given in eq. ( ponentiating the generators on order term in the exponentials to label the fluxes, namely of defined in terms of generalized derivatives of the group elem tensor DK kept. Thus we get (see eq. ( SU(3) structure, This is almost exactlyerotic the orbifold same compactifications, expression eq. ( than the one that w as bare gauge fields. For the simpler case of heterotic compacti projection are in the ( and at the samewe time consider the the gauge full fields were taken to be in the geom with compactifications) and therefore only the 3-form piece in th JHEP11(2010)083 ).  × to 6 i 3 c 6) iJ J , vol 6.24 (6.28) e (6.26) (6.27) (6.25) (6.29) · φ (6 4 ) ′ − O ). e H H + basis are S 5.31  J 3 B φ − | J 3 ), leading to the ) . φ − R ) in the 3 R factor in eq. ( 6.9 , ) representations of ie −  φ . e + ( − − i 6 6.14 . e 2 c  220 O3 SL(2 J ) , B compactifications with hat the flux induced su- + 123456 · K 2 ǫ × ) ′ tion of ( 123456 ≡ mn , which combine with 123456 ǫ abc ω ) ), they read 6) projection sets in this case ′ 2 ǫ 3 123456 ) and ( φ , + H ǫ B F − K ′ a e 4.2 S (6 abc e J ′ ) and ( tions, given in eq. ( ( + Q O 6.13 pqrm basis. This can be done with the −  φ H + ) ) + 2 2 12 c c + , Q − B − J + J ), ( | a e 2 mn · ) K ) 2 ∧ ) 123456 P + ( R ′ + ǫ ( c , + i c − 6.12 a ω J ) is an extra overall ′ bc J J ω ( 123456 + ( H · Q ǫ φ m S a K ) ), 2 ′ ′ SL(2 bc ( 5.33 + Q ) + ˜ − i − P 123456 2 Q e – 32 – × ǫ bc a = 2 H 5.34 + Q = P  + S 6) i a , ab c 123456 pqr + ˜ v ǫ + ( − ) K Q ), for general heterotic compactifications with global a (6 2 c c , a ω abc J J P O z 0 + 123456 · ǫ H ∧ ). In the notation of eq. ( , 6.18 + , given in eqs. ( 2  4 ). 0 ) abc Q n a ) + ( ′ J i ( 3 F m K φ ⇒ ω Q Q ˜ 6.6 )  ( H − i i H + e + ˜ H v v , S i 2 J S and =˜ =˜ ( . − − 2 φ c 0 − 2 . The resulting expressions in the = 0  iA J λ (see table ω 3 2 − i · f ( θ e  ′ ω H iABC B 1 2 | ( ω f ) ∧ h + ˜ and − , R ∧ Ω ,  1 Ω) 6 Ω Z , basis, in terms of the is defined in eq. ( Z + SL(2 H + . | = (0 = vol c × ) = J J . R e B B 6) | | , The other difference with respect to eq. ( In a similar way we can compute the superpotential for type II The fluxes surviving the projection are in the ( Hence, taking all these observations into account, we have t , 0 het λ O3 3-planes. This requires a little more work. The orientifold (6 W K where form are set to zero, these fluxes differ precisely by factors of O and similarly for O perpotential, derived from eq.SU(3) ( structure is given by, This factor can becanonical metric actually K¨ahler for absorbed heterotic by string a compactifica transforma K¨ahler SL(2 We need to express where contractions were defined in eqs. ( help of tables JHEP11(2010)083 . – 4 ] act 3 C 12 c 133 a (6.31) (6.30) (6.32) ·J ) to F- P defined ) and , ′ c 0  F J 6.32 J C and B φ S 2 , which build ) gets a shift m 3 combines with 2 − and . − c c e Q J  3 ′ 3 B J ∧ H J S H t the flux induced ) 0 φ c 3 and ′ m + ( − Q e J 2 are respectively propor- c φ 6 i 6 perpotential ( ) tells us that the group − ...i ·J − 1 mn ]. From this point of view, ie . For instance, notice that i r, when all fields in the ) ) ′ ǫ  A.4   ] + 51 − 2 2 c 6 Q · ) fluxes, while p J J ) B m 5 , 2 ′ ot only apply to toroidal orbifold = 1 variables φ p ·J S , ds [ appears linearly, quadratically or P 6 4 − F ) p e , eq. ( 4 φ 3 N − ...i m 2 i p 2 1 ′ + ( A − (whose generator is and ( C i 220 J e P ǫ ) have been derived in the context of P i ie 2 and the factor of  − 2 B p v 2 p mn 1 3 − 1 S ) p  J + ( p [ ′ · H 6.32 φ 6 6 + c − i i 0 ) − H 5 5 c 2 2 J i i ( e m P ·J J J i 2 φ − 4 4 ) − Q ) for the ( i i − – 33 – 3 3 3 P − i i = 1 compactifications of string theory. J ) by the derivative of the Hitchin functional [ ) and ( . When considering only geometric gauge fields, F 2 2 ie B φ + ( i i + 0 3  2 1 1 5.34 2 2 S c N i i ) for general type IIB orientifold compactifications − · − H ) in terms of the J J ) e 6.25 − ) has a well-known interpretation in terms of F-theory B ) = Q · 2 m ∧ J ). S 5 1 factors to build up different powers of 6.18 ) b Q ′ ) 6.30 384 128 0 P 2 5.27 c φ ′ m Q ) + ( J = 6.26 = φ − 3 Q − ) + ( − 6 2 ie 3 B H p 2 1 ie φ c S ...p H p − ( 1 − 6 and the fact that ( 3 J − d p B − / m ′ ie ) is also missing the contribution from the RR axions J S 3 , we obtain m P Q J − − P mn ) to build up 3 ( 6.30 3 ) respectively. For that aim, we observe that = + ( + (  J F F 6 ( ∧ [(  6.30 5.24 . Hence, ∧ Ω ∧ and 2 b Ω Ω 0 Z c mn ), are given in terms of Re(Φ ]. This leads to the useful relations Z + Z J + + 0 in eq. ( ) and ( 23 = 3-planes and global SU(3) structure is, 2 = 3 c , 3 O 3 H Taking into account these two facts, it is not hard to show tha It is also interesting to comment on the possible uplift of su Note that eq. ( We proceed now to express ( Let us stress that expressions ( O O 3.10 22 φ → , W − W 2 cubically, which combine with superpotential derived from eq. ( when considering higher orders in the exponential The reason is the same than for the heterotic superpotential are taken into account, there are extra terms where this is the only contribution of the RR axions appear. Howeve in ( 15 Making use of vol by a single contraction, as in ( tional to general SU(3) structure compactifications, and thereforecompactifications, n but also to general theory. Indeed, superpotential ( compactifications on elliptically fibered Calabi-Yau 4-fol with element corresponding to the 2-form part along ˜ up Im(Φ where the contractions are as in eqs. ( c ie JHEP11(2010)083 ). ij k Q and = 1 into 7 4.19 (6.35) (6.33) (6.34) B N E d, generalizing ave complexified in dz s guaranteed by the s with a holomorphic dz ) becomes, ∧ ) ) 3 pq f the M-theory/F-theory mplex parameter ( tion on the 10d origin of 5.27 ¯ -deformations (that is, Sγ ¯ SH 2 1)-form. This fact, together , γ − − ng principle for generic string n Exceptional Generalized Ge- 3 rphisms and shifts of the ctionary between weights of the pq ] (6.36) ent ways of decomposing F β c and 2 + ( = 0 for all moduli, K¨ahler implies + ( β ¯ z ¯ z B·J 4 d d d W G ) ˜ ∧ q + ) 7-brane charge. Thus, generic transition pq , which turns out to be also the structure ∧ ) m 7 4 are different components of a 1-form along 3 2) bi-vector in order to satisfy the T 4 , E rotations. ∂ Sγ G p, q 2 Ω [ γ SH , the superpotential ( B − ∧ ) also admits a natural uplift to F-theory. This | Z 4 – 34 – to be a primitive (2 − 4 pq G c = 3 and on one side, and fluxes, gauge fields and gauge pa- Ω β 6.32 ) are recast as, F = 0 (in absence of 3-form fluxes), for all complex 2 7 ·J Z W β ) E = ( = ( i P W = 5.28 i θ B θ = S pq i W ∧ F − i 2 W , Q ˜ H q ], topologically non-trivial 2 =1 m X , i 33 U =1 X ∂ = i ) suggest that pq = B 4 6.32 representations of ), we have identified different orientifold projections in 10 G ]. R , 30 56 ) has to be an anti-holomorphic (0 and is the holomorphic 4-form, whose existence and uniqueness i SL(2 is a basis of integral 1-forms in the elliptic fiber, which we h fluxes) correspond to locally geometric type IIB background 4 i × θ ij 4.19 k 133 6) P The form of ( Similarly, the second piece in eq. ( , , fields, but also T-dualities and/or SL(2) (6 so that the first two terms in eq. ( rameters, respectively, on the other. Moreover, from differ 4d gaugings. In particular, we912 have established a precise di group of the generalized bundle in EGG, encodes much informa C 4-form, NSNS and RR 3-form fluxes correspond to different components o Note also that taking the r.h.s. of this expression. In terms of 7 Conclusions We have performed a detailed analysisometry of and the relation 4d betwee gaugedflux supergravities, compactifications. providing an The organizi U-duality group, supersymmetry equations. term is related toAs it a has background been for noted the in [ field strength of the co where and the F-theory fiber, structure moduli, requires ( with the condition obtained from imposing that ( where Ω SU(4) holonomy of the Calabi-Yau 4-fold. O the results of [ complex dilaton on which there isfunctions a gluing deficit different of patches ( contain not only diffeomo JHEP11(2010)083 s ]. = 1 46 N ], where 37 h [ , correspond- parameter [ β ) representation of = 4 supersymmetry, rigin of some of the 2 , field strengths in terms superpotential induced N 12 f 4d supergravity which lated to the RR counter- trengths, metric fluxes or group structure allows us i Institute for Theoretical ave derived here in a 10d hese backgrounds were al- ld or orientifold compacti- o interpreted as particular w deformations provide the e and D. Waldram for use- a higher dimensional super- 7 hat even if in generic entangle fluxes which admit er types of 10d supergravity rly stages of this project. We should be thought directly in E -geometric fluxes, whose field /CFT correspondence, for the = 8 or ). Hence, in the last part of this ated to general backgrounds, in U(3), the structure group of the ergravity. eometric compactifications. ity. In particular, we have shown R N , ould be understood as 4d gaugings -deformations. We have formulated γ SL(2 representation onto the tensor product × 6) 912 , (6 O – 35 – -deformations apart from standard fluxes. We hope γ = 4 Super Yang-Mills with complex N and -deformation to full-fledged F-theory backgrounds. These β β = 1 heterotic compactifications or type IIB compactification = 4 supergravity transforming in the ( N N -deformed backgrounds. Our results are also consistent wit γ ) can be uplifted to backgrounds of 10d type IIB supergravity Z -deformation, which we have dubbed , β has provided us also with explicit local expressions for the SL(2 and/or × β 133 6) Whereas the above results directly apply to toroidal orbifo The above analysis gives also some clues on the possible 10d o The Clebsh-Gordan decomposition of the In this way, we have identified systematically all gaugings o , × (6 -deformed backgrounds. Our analysis, however, reveals oth -deformations in a precise way in the context of EGG. These ne 56 they are also relevant incompactifications more the global generic structure setups. group isgeneralized The reduced tangent reason to bundle S is is still t given by gravity limit. Hence, theterms uplift of of these String backgrounds Theory to compactifications. 10d In addition, the of the gauge potentials.an uplift This to in 10d particularstrengths supergravity involve allows exotic backgrounds, exterior one from derivatives and to locally whichresulting dis non sh from String Theory backgrounds which do not admit fications, where in absence of fluxes the bulk preserves it was noticed thatScherk-Schwarz a reductions small (metric subset fluxes) of of these heterotic sup gaugings can be als particularly less well understood gaugingsthat of 4d 4d supergrav gaugings of β also to distinguish globally geometric from globally non-g admit an uplift toready backgrounds known, of corresponding 10d to fluxes supergravity. of Some the NSNS of and t RR field-s backgrounds apart from the aboveparts ones. of These the turn out to be re O γ by the fluxes in general work we have made use of the tools of EGG to derive the effective ing to natural generalization of the results may have alsostudy interesting of implications, marginal via deformations of the AdS with O3 and/or O7-planes.context, allow These for the superpotentials, study whichparticular of we those the which F-term h involve conditions associ Acknowledgments We are very grateful toalso M. thank Petrini S. for collaboration Capriotti,ful in H. the discussions Montani, ea and H. comments. Samtleben, P. P.G.C. Vanhov and G.A. thank the Kavl to come back to this point in future work. JHEP11(2010)083 ). 2 , (A.1) 32 , ∆, with ∆ ∆ ∆ ]). Namely, ∈ ∈ = 0 ∈ 63 , and similarly, α β β β / 7 64 roots E + + + , α α α e Cartan-Weyl basis, I β en-dimensional subspace elements in the ( + H α the last set of roots there I ed by MINCYT (Ministe- 7 non-zero entries are at any to the root itute for hospitality during α E . The first 60 roots together E by systematic application of = even I i α,β -th Cartan of entina) and ECOS-Sud France (see for instance [ ǫ P 0 ν I anticommuting them. 8 signs in first 6        e =1 i X + − ] = 7 7 β e E 0) (A.2) 0) ,E , ,  α i 6 e E i [ 0; 0 0; 0 ν ≤ , , 0 0 1) , , – 36 – 1 can be explicitly computed, we will not need 6). The next two roots, together with a Cartan − 0 0 , ( , , ± , 0) 60 roots 1) even # of 1) 2 roots i < j 0 1 , (6 α 6 − − , =1 = − i , , X ≤ 1 ; 0 E O , I 0 1 − 1 + , α , , 1; 1 0; 1 α,β 0 7 , ǫ , e ± 0 0 , , ] = , 1 7 − j 0 = (1 = (0 α 0 e , e 8 , ± 1 2 0 e , 1 ), while the last 64 are the adjoint ,E − , ± α α 1  I R 0 ± i 8 , , , 1 2 ± e e H , 1 [ (0 . Even if 1 ± 7 ± ( ± 7, denote matrix representations of the E , 1 , ... ± , 1 = 1 ± ] = 0 ( I J root and weight system 2 1 , I 7 ± ,H E I H H Positive roots are defined as A basis of simple roots is then given by A convenient way to write the roots is as vectors lying in a sev denote matrix representations of the generator associated [ has rank seven (7 Cartan generators) and dimension 133. In th α 7 where in the first settwo of positions roots in the underline theshould means first be that six an the even components two number ofwith of the minus 6 signs vector, Cartan in and the generators, in first generate six entries E A.1 the completion of thisrio work. de Ciencia, e This Tecnolog´ıa Productiva Innovaci´on binational work of collaboration was Arg project partially A08E06. support A Summary of group theory results on Physics and together with M.G thank the Galileo Galilei Inst generator generate SL(2 the root lattice of them in our computations.generators (associated Representations are to simple constructed roots) to states, without of an eight-dimensional vector space, orthogonal to the commutation relations of the algebra read, E where JHEP11(2010)083 highest according to 912 and 133 , 1) 912 , : 27664 56 1 56 8645 : : L 2 − : 133 365750 1539 ω L L 5 7 : : : 1; L 3 ω ω 0) odd # of - ≡ ω − , L L L 1 4 6 , ≡ ≡ ω ω ω 1 ≡ 2) 1; 0 , 1) even #1) of - in first 6 even # of - in first 6   − ≡ ≡ ≡ , 0) 0) 0) 2   , ± 2 2 3 1 3) − 0) , , , 1 1 , 2 1 2 1 , , , − , , 1) 2) 1) 1 ; 3 − , , , − , 2 2 3 1 − , ; 0 0; 0 0; 0 1; 0 1 2 , 1 2 1 ± − , , 1 1; 1; 1 , , the corresponding weight in the notation used − − 1 , 56 − 1 1 2 − − − − 0 1 912 133 1 − 1; ± ± , , − ; ; , L – 37 – 0; 0; , , , , − 1 1 , 0; 0; 0; ± 1 , 0 , 0 1 1 1 1 , , , , 0) + 6 Cartans , 1 , , 1) + 1 Cartan , , 0 0 , , − 1 0 0 , 0 0 0 1 ± ± , 0 , 0 1 1 − − , , , , , , , , , , , 0 0 , , , , ± 1 0 0 0 ; 0 0 1 0 1 1 , 0 , 0 1 1 1 , , , , , , , , 0 , , 1 0 , , 1 0 0 0 0 ± ± , 0 1 0 , 0 , 0 − 1 1 0; 1 , , , , , , , , , 0 ( , , 1 0 , , ± , 1 1 , 0 1 1 , 0 , 0 , 2 1 0 , , , 0 , , 1 1 (1 (1 1 ± ± , , 1 1 , , 1 2 2 1 0   0 ± = (0 = (0 = (0 = (0 = 1 1 , ± ± ( , 0 3 4 5 6 7 1 ± ± 2 1 = (0 = = (1 = = = (1 =   , , , α α α α α 0 ± 7 5 4 3 6 2 1 1 1 , , ) ) ω ω ω ω ω ω ω 1 ± ± 1 2 ) assignments are (0 ( ( ] and we have indicated for which representation they are the , , ± ′ R 1 1 2 2 ± , 64 12 32 ( ) )( ) ( 3 1 2 , , , SL(2 1 ( 66 32 × ( ( 6) , (6 O The corresponding dual fundamental weights are The weights of all the elements in the representations We have indicated with a superscript their weight. in the Lie software [ JHEP11(2010)083 ] ,A ′ BC (A.3) (A.4) (A.5) (A.6) A A = [ iABC δA f + . j 5 , n A + i  ′ + i A i + ABC k + ′ i Γ AB Γ ,A ,A , , + ) is given by + i i ′ AB i iABC A + , A A f − A j f i . λ h 0) odd # of - ij 1 4 A + ǫ , A 4 is given by ′ + ) on the fundamental represen- B 9 j − n Γ i − h A A n , , λ A 1; 0 + + Γ + + i k +   A i , + ± 133 i λ A , 2 2 1 1 Γ + A h + i + ,A 1 , ′ × h j − ) is iA C 2) odd#of-infirst62) odd#of-infirst6 + 2 1 AB + ± A i , A , λ ′ + f R , h 0) even # of - Γ − 0) odd#of-infirst6 2 1 2 − , A ij 1 A , , , A ij ; ; 912 ǫ h AB Γ iB   − ǫ Γ ± 0 0 ′ AB , ; 0 η  λ 2 2 1 1 , , , + 1; 2 1; 1; 0 1 + − A 1 , 0 0 B AC SL(2 jB j − jk   ′ , , 4 1 − ± ± ± ± i 1 2 A ǫ B ± , A 0 0 , , , , 1 2 2 1 λ λ f × , – 38 – 1 2 C , , A 1 1 1 1 8 − , h ′ + 1 + − 0 0 ; ; iA n 1 2 3 ± ± ± ± 6) , AB A , , j + + 0 0 ± λ , , , , , n k j 0 0 , , , 1 2 Γ C ′ ′ − + ij 1 1 1 1 , , 0 0 1 ; ; ǫ A i (6 jA + 1 1 A A , , ± ± ± ± 0 0 j ± j AB λ k 0 0 A , , , , O , , i ± ± i ( j , , A 1 1 1 1 i A 0 0 2 1 A BC 1 1   A ) = − , , − 1 4 ′ A ± ± ± ± η j 0 0 ± ± , , , , i representation (with parameter − B − , , , , 1 1 1 1 = = f 0 0 C j 1 1 CA + λ, λ , , 7 ± ± ± ± k j ( A − A 0 ± ± 0 , , , , iA n A , , , , 133 A in the tensor product S C 3 1 1 1 k j iB 1 1 1 1 δλ + i A i δλ ± ± ± ± ′ ′ ′ f ± ± ± ± ( ( ( ( 56 + 2 A A A 1 1 1 2 2 2 2 1 j 5 copies of  5 copies of  5 copies of   n A = = = ) ) ) ) + A j + B 3 2 1 2 i j i Γ , , , , i A ′ are some constant coefficients which we leave undetermined. iA A δA δA 12 9 f δA 32 ( 220 352 jA ( ij ( ( f ǫ 1 6 . . . n n n symplectic invariant is 1 n 7 = = E − iA λ λ The adjoint action on the tation, decomposed in elements of where The The action of the adjoint representation (with parameter A.2 Some relevant formulas The projection on the where JHEP11(2010)083 . ) 1 2 × 6) basis , ijklm basis), 6 6 6 6 6 6 20 60 60 60 60 20 20 20 30 30 20 20 60 60 B (6 P | ± ≡ ± 6+30 6+30 6+30 6+30 6+30 6+30 ) B O Total # | R 20+40+60 ) , and R , P i + + + − − + + + − − + + + + − − − + − + − + − + − − + Ω P SL(2 rmionic number , L × SL(2 representations of F i 6) + + + − − + + + + − − − + + + − − − − − − − − − + + − 1) × Q , − , ( (6 6) , O ) ) 0) 0) 0) ) ) (6 fluxes transforming in the − +) , , , 1) 1) 1) 1) 1) 1) +) ) 0) 0) 0) − 123456 +) − , , ) − +) , , , 0) 0) 0) , , , , ′ i O , B − − − , , − +) , , , , , ; 0 1 1 1 | − +) − , , , , , − ) ; 0 ; 0 − Q , , − − − − e total number of components ; 0 ; 0 + ; 0 − + ; 0 − ( R ; ; 0; + − 0 ; + , , , 0 ; , 1 ; − − + ; 0 2 3 , 0 ; 0 ; + i , , 1 ; 0 ; + − + ; 0 , , , ′ + − + + ; − + ; 1 + ; 1 − + ; 1 , , , − 0 ; + 0 ; 0 , , , 0 0 , , , − 1 ; 1 ; + , , , , , , , , , , + − + 0 0 , , 0 ω + , , , − + basis ( , , , , , 1 0 0 0 , 0 0 + − + , + − + + − − , , SL(2 1 1 1 , , , 0 0 3 2 , , 0 , , , , , , , , , + − − , , , , 1 0 0 + , 0 0 − + − , , , 1 1 × H − , , 0 + − − 0 , + − + + − − , , 1 , , , | 1 − , , . We use the shorthand notation , , , , , , 1 1 , , , , , , , + − 0 0 and 0 ) , + 0 0 7 , − − 0 0 , , , , 1 6) − , − + − − , − + + − + − − 1 , , , , , , 1 i , 1 1 , R , 0 , , , E , , , , , , , + − 0 0 − , 2 3 + 0 − , , + 1 , , , , , − (1 (1 − ω 1 (6 − − − , , , + + − + − − , 0 , , ( ( 2 2 3 3 1 ( (+ ( , , , , , , , of 0 1 , , 2 3 − (0 (0 O − × × , i ( (1 − − ( ( − (+ − × × − − − − − × × × (1 ( 6 6 ( ( ( ( (+ ( ( (+ ( ( (0 6 6 Q SL(2 6 6 6 912 , – 39 – × ) ) 0) 0) 0) ) ) +) − , , , 1) 1) 1) 1) 1) 1) − +) ) 0) 0) 0) − +) , , ) +) − , , , 0) 0) 0) 6) , , , , , H − − − , , +) − , , , , , ; 0 ; 0 ; 0 1 1 1 , | − +) − , , , 123456 , , − ) ; 0 ′ i − , , − − − − − − − − (6 R ; ; ; ; 1 ; 1 ; 1 − 0 ; + 0 ; , , , Q 0 ; 0 ; + + ; 0 − + ; 0 , 0 ; + 0 ; , , 1 ; + 0 ; + ; 0 + ; 0 + ; 0 O , , , , , + − − − − − − − − , , 0 ; 0 ; + , , , , 0 0 0 0 , , , 0 ; 0 ; + − , , , , , , , , + − + 0 0 , , 0 , , , , , − + + , , , , , 0 0 , 0 0 0 0 + − − + − − + − − , , , SL(2 0 0 1 , , 0 0 , , 0 , , , , , , , , , , , − − + , , , , 0 0 , 0 0 1 1 − + + − , , , 0 0 × , , 0 0 + + − + − − + − − , , 1 , , , , , , , , , , , 1 − 1 − , , , , , , , − − + 0 0 1 1 1 , , , − + − 0 0 , , , , , 1 6) , + + − − + + − + + − , , , , , , , 1 1 1 1 1 , , as it can be read from their weight assignments in the 0 , , , , , , , , , , − − − 1 − 1 , ) in the 7 − + − − − 1 − , , , , , , − (1 − (1 (6 + + − + + − 0 , , , , , , , ( ( 3 2 3 2 3 2 1 2 (+ (+ (+ , , , , , , E , 3 2 3 2 3 2 (1 1 (1 1 O , × × (1 − (1 − ( − ( ( − × × × × × ( − (0 − 6 6 ( ( ( (+ (+ (+ (+ (+ (+ ( ( 6 6 6 6 6 12 ). In particular, ), their transformation properties under the space-time fe R 456 , m m 3456] , 3.3 3456 , m [2 56 m m 2 ,P m ′ , basis. We present also the corresponding weight in the 1 ′ 6 1 m m ′ m 1 m 12 123456 , SL(2 , 56 1 1 123456 1234 3456 123456 23 12 3 2 13 , 1 , 123 123 1 , 123 , , 123 H ˆ ,Q 123456 123456 ,P 2345 ′ , ω , ω ,P 4 ,Q F F 1 1 3 , , | 12345 ω ′ ˜ P Q flux ′ ′ . Field strengths transforming in the F 1 1 6 1 R 1 H 6 H × 12356 ) ′ P P ′ ′ ′ F 123 456 ′ P Q ′ ω 23456 ˆ P Q R ω ω ,P F F ′ 6) F 23456 , representation of H States which come in 6 copies are distributed among different 123456 F , 1 ′ 456 P Q (6 P Table 6 (c.f. section In the following tables912 we summarize the tensor structure of B Tensor structure of U-dual fluxes for left-movers and thefor worldsheet each parity flux. operators, and th O transform in the ( SL(2 JHEP11(2010)083 ijkl i , ) in P 3 55 , ′ ommons ]. and 32 , , (2007) 057 ijk (2003) 281 , L , , 10 54 SPIRES i ′ ji , ][ (2007) 733 , ω ) transform in the J. Diff. Geom. ] , , view i 79 ji JHEP ω , , i,jklm [ ′ P mmercial use, distribution, r(s) and source are credited. ]. ) transform in the ( and ijk hep-th/0406137 [ P ij ). Supersymmetric backgrounds from j More dual fluxes and moduli fixing ]. ]. 1 P SPIRES Rev. Mod. Phys. , , , basis), whereas ][ and ij j Global Differential Geometry: The ]. ]. ]. ]. i B 352 ′ -models and T-folds | Q ) Quart. J. Math. Oxford Ser. σ ω SPIRES SPIRES , , in , ] R , (2004) 046 ][ ][ , i ω Target space duality in string theory 08 ( SPIRES SPIRES SPIRES SPIRES – 40 – ]. i,jklm -structures and type IIB superstrings [ SL(2 ][ ][ ][ ][ 2 ′ ]. Nongeometric flux compactifications G Q × ( , M. Fernandez and J.A. Wolf eds., Contemporary ijklm JHEP 6) i Non-geometric fluxes as supergravity backgrounds ′ ij P arXiv:0704.3272 , , [ SPIRES ω (6 Flux compactification ][ SPIRES [ O ]. ]. and hep-th/9401139 hep-th/0509003 ]. ]. Generalised T-duality and non-geometric backgrounds [ [ ) transform in the ( and Generalised ijk ijkl i i im hep-th/0602089 hep-th/0412280 hep-th/0508133 hep-th/0512005 P basis ( P [ [ [ [ ω SPIRES SPIRES , i SPIRES SPIRES , H (1987) 59 ][ ][ | ij j (2007) 041901 , American Mathematical Society, Providence U.S.A. (2001) ) ][ ][ P (1994) 77 (2006) 91 and Q R A symmetry of the string background field equations This article is distributed under the terms of the Creative C , , 288 The geometry of three-forms in six and seven dimensions Stable forms and special metrics ijk ] 244 423 math.DG/0010054 Generalized Calabi-Yau manifolds L B 194 Global aspects of T-duality, gauged D 76 Flux compactifications in string theory: a comprehensive re (2006) 070 (2005) 053 (2005) 085 (2006) 009 , ] SL(2 i,jklm [ × 05 03 10 05 ′ Q 6) i,jklm [ , ′ ). Similarly, 2 (6 P math.DG/0107101 math.DG/0209099 hep-th/0604178 hep-th/0610102 , Mathematical Legacy of Alfred Gray Mathematics [ (2000) 547 [ [ JHEP [ Phys. Rev. generalized Calabi-Yau manifolds JHEP JHEP Phys. Rept. [ JHEP Phys. Lett. Phys. Rept. , O ji i [1] T.H. Buscher, [8] C.M. Hull, [9] F. Marchesano and W. Schulgin, [6] A. Dabholkar and C. Hull, [7] G. Aldazabal, P.G. Camara, A. Font and L.E. Ib´a˜nez, [4] M.R. Douglas and S. Kachru, [5] J. Shelton, W. Taylor and B. Wecht, [2] A. Giveon, M. Porrati and E. Rabinovici, [3] M. Gra˜na, 220 P [13] N.J. Hitchin, [14] N. Hitchin, [11] C. Jeschek and F. Witt, [12] N.J. Hitchin, [10] M. R. Gra˜na, Minasian, M. Petrini and A. Tomasiello, ( the Attribution Noncommercial License whichand permits reproduction any in any nonco medium, provided the original autho References ( Open Access. whereas JHEP11(2010)083 ]. = 4 ] ] , N , ]. SPIRES (2003) 045 ][ , ]. 06 SPIRES (2009) 21 ][ ]. , (2005) 065 JHEP (2008) 80 supergravity SPIRES , hep-th/0701203 [ 10 arXiv:1006.1536 ][ arXiv:0904.4664 [ supergravity ]. , B 814 = 4 ]. ]. SPIRES hep-th/0605149 [ N B 799 -branes = 2 ][ 3 JHEP ]. N , D SPIRES Gauged supergravities from twisted (2007) 079 SPIRES SPIRES gauged supergravities from ][ , Vaula, hep-th/0310136 (2009) 099 ][ ]. [ ][ 07 Nucl. Phys. orientifold compactifications and the compactification and mirror duals of , = 4 09 (2007) 080 SPIRES hep-th/0602241 Nucl. Phys. [ D ][ ]. , = 1 07 ]. SPIRES JHEP N Flux algebra, Bianchi identities and , SU(3) hep-th/0306185 JHEP ][ [ New Unusual gauged supergravities from type IIA (2004) 263 , gauged supergravity and a IIB orientifold with × , D.Phil. thesis, Oxford University, Oxford U.K. JHEP SPIRES , SPIRES – 41 – ][ (2006) 200 = 4 arXiv:0808.4076 ][ hep-th/0612237 [ [ arXiv:0804.1362 Hitchin functionals in [ B 582 N Gauge symmetry, T-duality and doubled geometry ]. Generalized in presence of fluxes and (2003) 015 hep-th/0206241 An orientifold with fluxes and branes via T-duality [ 2 B 748 M-theory, Exceptional Generalised Geometry and Z 10 / 6 hep-th/0305183 ]. ]. [ SPIRES T ]. ]. (2007) 101 (2008) 123 Double field theory Phys. Lett. ][ , (2008) 214002 JHEP 04 (2002) 71 09 , hep-th/0505264 arXiv:0711.4818 [ [ 4 25 SPIRES SPIRES Nucl. Phys. (2003) 3 SPIRES SPIRES , ][ ][ ][ ][ JHEP JHEP , Lectures on gauged supergravity and flux compactifications , Generalized complex geometry Lectures on generalized complex geometry for physicists B 669 Generalised Geometry for M-theory A geometry for non-geometric string backgrounds Doubled geometry and T-folds (2006) 008 (2008) 043 ]. ]. ]. math.DG/0401221 New J. Phys. 01 08 , arXiv:0811.2900 hep-th/0303049 SPIRES SPIRES arXiv:0712.1026 SPIRES hep-th/0406102 fluxes orientifolds with fluxes magnetic fluxes Nucl. Phys. Freed-Witten anomalies in F-theory compactifications [ and type IIB orientifolds [ Class. Quant. Grav. [ superpotentials JHEP [ JHEP [ [ [ doubled tori and non-geometric string backgrounds (2004) [ Lagrangian for type IIB on Hitchin functionals [33] G. Aldazabal, P.G. Camara and J.A. Rosabal, [30] C. Angelantonj, S. Ferrara and M. Trigiante, [31] C. Angelantonj, S. Ferrara and M. Trigiante, [32] M. Berg, M. Haack and B. K¨ors, [28] R. D’Auria, S. Ferrara and S. Vaula, [29] R. D’Auria, S. Ferrara, F. Gargiulo, M. Trigiante and S. [26] C.M. Hull, [27] P.P. Pacheco and D. Waldram, [23] M. J. Gra˜na, Louis and D. Waldram, SU(3) [24] I. Benmachiche and T.W. Grimm, [25] H. Samtleben, [21] C. Hull and B. Zwiebach, [22] M. J. Gra˜na, Louis and D. Waldram, [18] C.M. Hull, [19] G. Dall’Agata, N. Prezas, H. Samtleben and M. Trigiante [20] C.M. Hull and R.A. Reid-Edwards, [16] P. Koerber, [17] C.M. Hull, [15] M. Gualtieri, JHEP11(2010)083 , ]. l = 4 , ]. ] = 4 SPIRES N ][ N , , , ]. ]. ]. , , SPIRES [ backgrounds global symmetry ]. ]. ]. ]. SPIRES SPIRES (2007) 115 SPIRES ][ = 2 ][ U(1) , ][ (1981) 1 hep-th/9503121 N (2006) 034 [ × R-R scalars, U-duality and SPIRES hep-th/9906070 79 supergravities SPIRES SPIRES SPIRES 05 B 785 ][ ][ ][ ][ U(1) ]. = 4 ]. ]. Calabi-Yau orientifolds D rigiante, (1995) 95 JHEP , T-duality, Generalized Geometry and = 1 hep-th/9611014 hep-th/0606257 SPIRES [ Phys. Rept. Axionic symmetry gaugings in (2001) 477] [ [ formulation of , Nucl. Phys. N ][ arXiv:0807.4527 ]. ]. SPIRES ]. ]. SPIRES no-scale model from generalized [ B 447 , ]. ][ ][ 7(7) E hep-th/0502086 = 4 [ hep-th/0406185 hep-th/0212239 Type-IIA flux compactifications and [ B 608 [ arXiv:0906.0370 The maximal Magnetic charges in local field theory On Lagrangians and gaugings of maximal [ N Gravity duals to deformed SYM theories and SPIRES SPIRES SPIRES SPIRES supergravities (1997) 617 SPIRES (2006) 055 – 42 – ][ ][ ][ ][ Moduli stabilization from fluxes in a simple IIB Algebras and non-geometric flux vacua ][ (2009) 075 = 4 12 Nucl. Phys. , 04 N CFT’s from Calabi-Yau four-folds (2003) 93 (2005) 033 hep-th/0201028 B 496 (2004) 055 [ Non-geometric flux vacua, S-duality and algebraic geometry (2009) 018 hep-th/0403067 hep-th/9612202 Deforming field theories with Exactly marginal operators and duality in four-dimensiona [ [ 05 Erratum ibid. JHEP 07 [ The minimal 08 , ]. JHEP The effective action of ]. B 655 Gauged , JHEP JHEP Solvable Lie algebras in type IIA, type IIB and M theories , arXiv:0809.3748 arXiv:0811.2190 arXiv:0705.2101 hep-th/0507289 arXiv:0904.2333 JHEP (2003) 007 , [ [ [ [ [ Nucl. Phys. , SPIRES , (2004) 387 (2000) 69 (1997) 249 SPIRES 10 ][ ][ Nucl. Phys. , B 699 B 584 B 493 Group theory for unified model building JHEP (2008) 050 (2009) 042 (2007) 049 (2005) 016 (2009) 104 , ]. supersymmetric gauge theory 12 02 06 09 07 = 1 SPIRES arXiv:0705.0008 hep-th/0602024 dimensional reduction and their gravity duals supergravities and their higher-dimensional origin Nucl. Phys. JHEP JHEP orientifold Nucl. Phys. [ JHEP JHEP generalized complex geometry N Nucl. Phys. supergravities JHEP solvable Lie algebras [ non-geometric backgrounds gauged supergravities [ [52] A. Font, A. Guarino and J.M. Moreno, [53] A. Guarino and G.J. Weatherill, [49] S. Kachru, M.B. Schulz and S. Trivedi, [50] T.W. Grimm and J. Louis, [51] S. Gukov, C. Vafa and E. Witten, [47] B. de Wit, H. Samtleben and M. Trigiante, [48] B. de Wit, H. Samtleben and M. Trigiante, [45] R. Minasian, M. Petrini and A. Zaffaroni, [46] R.G. Leigh and M.J. Strassler, [42] R. Slansky, [43] B. de Wit, H. Samtleben and M. Trigiante, [44] O. Lunin and J.M. Maldacena, [39] M. J. Gra˜na, Louis, A. Sim and D. Waldram, [40] L. Andrianopoli, R. D’Auria, S. Ferrara, P. Fr´eand M.[41] T L. Andrianopoli et al., [37] J.-P. Derendinger, P.M. Petropoulos and N. Prezas, [38] M. R. Gra˜na, Minasian, M. Petrini and D. Waldram, [35] J. Schon and M. Weidner, [36] G. Villadoro and F. Zwirner, [34] G. Dall’Agata, G. Villadoro and F. Zwirner, JHEP11(2010)083 ] ]. (2008) 135006 (2004) 483 25 (2004) 126009 SPIRES ]. 52 ]. ][ ]. D 70 hep-th/0306088 [ SPIRES SPIRES ][ ][ SPIRES ][ Fortsch. Phys. , Phys. Rev. BPS action and superpotential for Heterotic string theory on , hep-th/0301161 Properties of heterotic vacua from (2003) 004 [ Class. Quant. Grav. , Compactifications of heterotic theory on 10 The effective action of the heterotic hep-th/0304001 [ arXiv:0911.2876 [ JHEP structure Complete classification of Minkowski vacua in Flux moduli stabilisation, supergravity algebras (2003) 007 , arXiv:0907.5580 [ LiE, a computer algebra package for Lie group 04 – 43 – SU(3) , chapter V.5.III, Hermann, Paris France (1968). (2003) 144 Heterotic on half-flat (2010) 076 JHEP , (2010) 012 02 ]. B 666 01 ]. ]. JHEP , JHEP SPIRES , SPIRES SPIRES ][ ][ ][ Nucl. Phys. , Groups and Lie algebras , Computer Algebra Nederland, The Netherlands (1992). ]. arXiv:0802.0410 hep-th/0310021 hep-th/0408121 SPIRES computations string compactified on manifolds with [ o-¨he manifoldsnon-K¨ahler with H-flux and gaugino[ condensate [ superpotentials heterotic string compactifications with fluxes [ generalised flux models complexnon-K¨ahler manifolds. I and no-go theorems [64] M. van Leeuwen, A. Cohen and B. Lisser, [61] I. Benmachiche, J. Louis and D. Martinez-Pedrera, [62] M. and Gra˜na F. Orsi, to[63] appear. N. Bourbaki, [59] G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. L¨ust, [60] S. Gurrieri, A. Lukas and A. Micu, [57] K. Becker, M. Becker, K. Dasgupta and S. Prokushkin, [58] G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. L¨ust, [55] B. de Carlos, A. Guarino and J.M. Moreno, [56] K. Becker, M. Becker, K. Dasgupta and P.S. Green, [54] B. de Carlos, A. Guarino and J.M. Moreno,