SSeptember 11, 2013 EFI-13-20, DMUS-MP-13/18
New Examples of Flux Vacua
Travis Maxfield a , Jock McOrist b , Daniel Robbins c and Savdeep Sethi aa Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA b Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK c Institute for Theoretical Physics, University of Amsterdam,Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands
Abstract
Type IIB toroidal orientifolds are among the earliest examples of flux vacua. Byapplying T-duality, we construct the first examples of massive IIA flux vacua withMinkowski space-times, along with new examples of type IIA flux vacua. The back-grounds are surprisingly simple with no four-form flux at all. They serve as illustra-tions of the ingredients needed to build type IIA and massive IIA solutions with scaleseparation. To check that these backgrounds are actually solutions, we formulate thecomplete set of type II supergravity equations of motion in a very useful form thattreats the R-R fields democratically. [email protected], maxfi[email protected], [email protected], [email protected]. a r X i v : . [ h e p - t h ] N ov Introduction
The first and simplest examples of four-dimensional type IIB flux vacua that evade theclassic supergravity no-go theorem of [1] involve quotients of T [2]. The easiest fluxesto consider descend from classes on T to the quotient space. The aim of this project isto examine the possible geometries that can be obtained from this starting point usingT-duality on the ambient T . This approach was originally employed to find torsionalsolutions of the type I/heterotic strings [2], and subsequently used to construct N=2 typeIIA flux vacua with dual Calabi-Yau descriptions [3]. However, a complete classificationof the possible geometric backgrounds that can be found from this method has not beendescribed. A scan of supersymmetric twisted tori vacua in the framework of generalizedgeometry can be found in [4].The reason for our interest in this exercise is that new examples of flux vacua for bothtype IIA and massive IIA can be found this way. For models with a non-zero Romans pa-rameter m , this is particularly interesting because we have no prior examples of solutions tomassive IIA with either Minkowski space-times, or AdS space-times with the AdS lengthscale significantly larger than the Kaluza-Klein scale. Indeed, massive IIA supergravity isone of the more mysterious corners of string theory [5]. Classical vacua with AdS space-times of Freund-Rubin type were described in the original work of Romans. Those solutionsdo not possess scale separation, and should not really be viewed as compactifications tofour dimensions in the usual sense, but are really fully ten-dimensional backgrounds. Moregeneral classical AdS solutions were described in subsequent work; see, for example [6–9].The possibility of classical IIA solutions with scale separation was discussed in [10], butno examples are currently known. The status of various attempts to build type IIA andmassive IIA flux vacua is summarized in [11].Much like the torsional solutions of the heterotic/type I strings found in [2], the newexamples of type IIA and massive IIA flux vacua teach us about the topology of solutionsthat can satisfy the equations of motion and Bianchi identities, as well as the structure ofnecessary additional ingredients like orientifold planes and D-branes. The Romans massparameter, m , of massive IIA is equivalent to a constant expectation value for the R-R F -flux in type IIA string theory. Because this flux does not dilute as we scale upthe compactification volume, it is quite difficult to get a handle on solutions with scaleseparation from a direct analysis of the equations of motion expanded around a large volumesolution. For example, a large volume Calabi-Yau space is not a good starting point forconstructing a solution of massive IIA. Indeed, it does not even appear to be a good starting1oint for constructing flux solutions of conventional type IIA string theory [11].One might imagine that four-dimensional type IIA and M-theory flux compactificationsare on better footing than massive IIA, since at least type IIA string theory has a per-turbative quantum definition. Yet nothing could be further from the truth! Unlike typeIIB string theory, the crucial ingredients needed to evade the classic supergravity no-gotheorem have only recently been explored [11], and are still very poorly understood. Thecentral ingredient appears to be the ability of orientifold planes and D-branes to generatebrane charge in directions normal to their world volume. A systematic understanding of thecouplings supported on D-branes and orientifold planes, and the charges they can produce,is sorely needed information. Recent work on higher derivative interactions, including thosesupported on branes, can be found in [12–21, 11, 22–33].Let us summarize our main observations as we outline this paper. In section 2, we beginby ignoring the quotient action entirely. Instead, we work with a torus geometry for whichwe classify all possible fluxes compatible with the equations of motion. This includes fluxesthat both preserve and break supersymmetry. Inclusion of SUSY breaking fluxes turns outto be crucial if one wishes to generate a non-zero Romans mass from T-duality. We thendescribe the constraints that need to be imposed on the choice of fluxes so that T-dualitygives back an honest geometry, rather than a quantum geometry requiring T-duality topatch open sets. The basic condition is that we T-dualize only once along any 3-cyclesupporting an H -flux [34]. It would be interesting to relax this condition and considermore general T-folds, but we will restrict to geometry in this work.We then classify the distinct ways of performing 3 T-dualities on T . The resultingvacua are remarkably nice with the simplest massive IIA example still a topological 6-toruswith a metric of schematic form: ds IIA = e − w g µν dx µ dx ν + e w ds T + e − w ds (cid:101) T . (1.1)Here w is the original type IIB warp factor. Up to an additive constant, the type IIA dilatonis varying and also determined by the warp factor φ A = − w . There is a precisely correlatedpair of fluxes ( F , H ), and an additional F -flux, but no F -flux. Indeed, none of theexamples we find in either type IIA or massive IIA has 4-form flux. Related backgroundshave been studied at the level of supergravity or effective field theory, without explicitsolutions, in past work [35, 36]. However, those analyses do not include the ingredientsneeded to evade the usual supergravity no-go theorem for flux compactifications in type Orientifold examples of type IIA and M-theory vacua with 4-form flux can be found in [11]. Theseexamples are constructed using 1 T-duality rather than the 3 T-dualities studied here. O O O O Section 4 is devoted to a formulation of the type II supergravity equations of motionin a way that treats the R-R fields democratically. This is a very convenient form forchecking that our backgrounds actually solve the equations of motion. This check can beperformed at the level of supergravity, without worrying about the contribution of anybeyond supergravity ingredients like orientifolds, branes or higher derivative interactionsfor two reasons: first, one can consider non-compact solutions for which such ingredientsare not needed, and these backgrounds solve the SUGRA equations of motion on the nose.Second, for compact solutions such ingredients contribute in a known way to the type IIBequations of motion. The type IIA image of those contributions can therefore be determinedby T-duality.In section 5, we take a closer look at these backgrounds. Specifically, we examine the Some potentially smooth geometries in massive IIA that resemble O AdS solutions be found with scale separation?Can a more general IIA metric ansatz be formulated that allows all fluxes to appear?How do O One of the nicest examples of a type IIB flux solution is based on the geometry T × K G -flux preserve either N = 0 , K T and track the quotient action later. We can also consider more generalquotient actions on T . Indeed, we might as well make the geometry as simple as possibleand consider T × T × T as our starting point. Let us take complex coordinates ( x, y, z )for this space with x = x + ix and x i ∼ x i + 1 etc. Choose the square complex structurefor each T factor. The starting type IIB metric takes the form: e − φB ds IIB = e − w g µν dx µ dx ν + e w ( dxd ¯ x + dyd ¯ y + dzd ¯ z ) . (2.1)The warp factor, w , is only a function of the internal coordinates ( x, y, z ). The type IIBstring coupling, τ B = C + ie − φ B , is independent of ( x, y, z ). We will set C = 0 forsimplicity.The type IIB background also includes a self-dual F -flux. The mostly space-timecomponents of F are given by F space = − d (cid:0) e − w (cid:1) ∧ vol , (2.2)where vol is the volume form of the unwarped space-time metric. See Appendix A for signconventions, and ( A.
19) for a definition of vol . In order to correctly T-dualize the R-Rfluxes, it is going to be simpler to work with the R-R field strengths rather than the R-R4otentials. The Bianchi identity for the self-dual F leads to a Laplace type equation forthe warp factor: d ∗ F = dF = − H ∧ F + X + (cid:88) i δ ( x − x i ) δ ( y − y i ) δ ( z − z i ) . (2.3)The gravitational source, X , arises from higher derivative interactions supported on D -branes and orientifold planes; for example, one source is the (cid:82) C ∧ tr ( R ∧ R ) couplingsupported on ( p, q ) 7-branes. Without this higher derivative contribution, no compactsolution is possible. The last term in (2 .
3) is the contribution from any space-filling D3-branes present in the background.From the perspective of M-theory, we are considering T × T × T × T with complexcoordinates ( u, x, y, z ), and a square complex structure for the ( x, y, z ) tori. The u -torus isdistinguished with a complex structure determining the type IIB string coupling, τ u = τ B = ie − φ B and u = u + ie − φ B u . The type IIB limit requires taking the area of the u -torus tozero. We could generalize this starting point by allowing a more general complex structure,but it will obscure the main issues we want to explore.Now we want to parametrize the most general choice of G -flux that can solve theequations of motion. A nice approach is to start in M-theory on T × T × T × T andconsider all G -flux that is self-dual: G = ∗ G [43]. The cohomology H ( T , Z ) contains70 elements. Of these classes, 35 are self-dual and can be decomposed by Hodge type: theyare combinations of the (4 ,
0) and (0 ,
4) elements, 12 non-primitive combinations of (3 , , ,
2) classes together with the square of the K¨ahler form, which is(2 ,
2) and non-primitive.Of these 35 fluxes, the only choices that are compatible with the lift to a Lorentzinvariant type IIB background are those with one leg along the u -torus [2]. There are 20such choices. We can provide a basis for these forms, dudxdydz, d ¯ ud ¯ xdydz, d ¯ udxd ¯ ydz, d ¯ udxdyd ¯ z,d ¯ udx ( dyd ¯ y − dzd ¯ z ) , d ¯ udy ( dzd ¯ z − dxd ¯ x ) , d ¯ udz ( dyd ¯ y − dxd ¯ x ) ,dudx ( dyd ¯ y + dzd ¯ z ) , du ( dxd ¯ x + dyd ¯ y ) dz, du ( dxd ¯ x + dzd ¯ z ) dy, (2.4)together with their complex conjugates. Not all of these choices will be compatible with thedesired quotient action, or the tadpole condition, but we will worry about those constraintslater.The most general G we will consider is a real linear combination of the forms (2 .
4) andtheir complex conjugates together with some integrality constraint. Denote the ordered list5f 4-forms appearing in (2 .
4) by { ω , . . . , ω } . We can express G as follows, G π = (cid:88) i =1 a i ( ω i + ¯ ω i ) + ib i ( ω i − ¯ ω i ) , (2.5)where the choice of ( a i , b i ) determines a choice of flux. From G , we can read off thecorresponding type IIB H and F fluxes using G π = H du + F du . (2.6)Space-time supersymmetry is broken by either turning on fluxes with a (4 ,
0) component,or a non-primitive (3 ,
1) component. These fluxes correspond to the ( a , b ) coefficient,and to the ( a , a , a ) and ( b , b , b ) coefficients. These two distinct ways of breakingsupersymmetry correspond to F -term or D -term breaking, respectively, in four dimensions.If we want to restrict to F -term breaking in four dimensions, we need to impose aprimitivity condition with respect to the canonical K¨ahler form: J = i dud ¯ u + dxd ¯ x + dyd ¯ y + dzd ¯ z ) . (2.7)This constraint is not needed to solve the flux equations of motion. Imposing J ∧ ω = 0 fora self-dual ω ∈ H ( T , Z ) with a leg along the u -torus leaves a total of 14 possibilities. Inthe expansion (2 . a = a = a = 0 , b = b = b = 0 . (2.8)Equivalently, the last line of (2 .
4) is set to zero.
To determine the metric that results from applying T-duality, we only need to worry aboutthe H -flux. We can ignore the R-R flux and the dilaton. The main constraint is thatwe only want to generate geometric backgrounds that do not require patching conditionsinvolving T-duality. This means we can only T-dualize once along any 3-cycle supporting H -flux. Without specifying a quotient action, the directions ( x, y, z ) are completely sym-metric so we are free to T-dualize along x as a first step. Let us examine the additionalT-dualities that we might consider in the search for a combination that can generate an F -flux. We will need at least 3 T-dualities if we have any hope of converting an F -fluxinto an F -flux. More than three T-dualities leads to a non-geometric background so wewill consider three T-dualities. 6ome general comments about T-dualizing R-R fluxes are appropriate at this point: theR-R field strength is a self-dual object. Our starting type IIB configuration has ( F , F , F )fluxes, where F is Hodge dual to F . The F flux is self-dual with two components: thefirst is space-time filling with one internal leg given in (2 . F space , F int ).T-duality preserves self-duality of the R-R field strength. After three T-dualities, theresulting IIA R-R field strength will contain contributions from the F and F fluxes whichare Hodge dual, and contributions from ( F space , F int ), which are also Hodge dual. We onlyneed to specify one half of the components of the resulting R-R field strength. We willchoose to specify ( F , F , F ). To find these fluxes, we can T-dualize the original type IIB F and F space fluxes; the F space flux will generate fluxes of the generic form ( F , F ) whichcan be Hodge dualized to ( F , F , F ). The F -flux can generate ( F , F , F , F ), which canagain be dualized to ( F , F , F ). It is much simpler to consider field strengths rather thanpotentials in implementing T-duality. ( x , x )Let us start by imposing the constraint (2 .
8) that all supersymmetry breaking is by an F -term. If we choose to perform a second T-duality along x , we must supplement (2 . a = b = 0 , a = b = 0 . (2.9)This leaves 12 potential choices of H -flux. However, T-duality along ( x , x ) turns all ofthese choices of H -flux into metric. It also converts the R-R F -flux into F -flux.This is the case first considered in [2] which leads to a torsional solution of the typeI string. The x -torus is fibered over the ( y, z ) directions with a fibration determined bythe H -flux. We cannot generate an F -flux this way so we must either allow D -termsupersymmetry breaking, or consider a second T-duality along the y or z directions. ( x , x , y )Let us first relax the constraint (2 .
8) and allow D -term breaking fluxes. Symmetry singlesout no special direction for the third T-duality so we may as well choose y . Imposing thecondition that we end up with an honest geometry produces 10 linear constraints on the Precise relations between R-R fluxes and their duals can be found in Appendix A. a i , b i ) given below: a + a = a + a = a + a = a − a = a − a = 0 ,b + b = b − b = b + b = b − b = b − b = 0 . (2.10)The flux is then determined by 10 parameters, which we can take to be ( a , a , a , a , a )and ( b , b , b , b , b ). The H -flux is given explicitly by, H = 4 a ( dx dy dz + dx dy dz ) + 4 a ( dx dy dz − dx dy dz ) + 8 a dx dz dz − a dy dz dz − a dy dy dz + 4 b ( dx dy dz − dx dy dz )+4 b ( dx dy dz + dx dy dz ) + 8 b dx dz dz − b dy dz dz − b dy dy dz . (2.11)This now has components that survive the 3 T-dualities as H -flux. The F -flux is givenby, e φ B F = 4 a ( dx dy dz + dx dy dz ) + 4 a ( dx dy dz − dx dy dz ) + 8 a dx dy dy − a dx dx dy − a dx dx dz + 4 b ( dx dy dz − dx dy dz ) − b ( dx dy dz + dx dy dz ) − b dx dy dy + 8 b dx dx dy +8 b dx dx dz . (2.12)The coefficient a turns on a component of F proportional to dx dx dy , which can dualizeto an F -flux. Notice that this is correlated with a component of H -flux in (2 .
11) whichsurvives T-duality. However, a = a turns on a non-primitive flux, as we expected fromsection 2.2.1, breaking supersymmetry via a D -term. ( x , y , z )Without some choice of quotient action singling out particular directions, the only otherdistinct possibility is dualizing ( x , y , z ). This is the case we will focus on for the remainderof this paper.Imposing the constraint that we end up with a geometry, we find that the H -flux isdetermined by 10 parameters. Let us first specify the M-theory G -flux, b = − b = − b = − b , b = b = b = b = b = b = 0 ,a + a + a + a = 0 . (2.13)We can choose the 10 flux parameters to be b and ( a , . . . , a ). Writing out the H -fluxexplicitly gives: H = − a ( dx dy dz + dx dy dz ) − a ( dx dy dz + dx dy dz ) − a ( dx dy dz dx dy dz ) + 4 a ( dx dz dz − dx dy dy ) + 4 a ( dx dx dy − dy dz dz )+4 a ( dx dx dz − dy dy dz ) − a ( dx dz dz + dx dy dy ) − a ( dx dx dz + dy dy dz ) − a ( dx dx dy + dy dz dz ) − b dx dy dz . (2.14)In a similar way, we can write out the starting F -flux: e φ B F = − a ( dx dy dz + dx dy dz ) − a ( dx dy dz + dx dy dz ) − a ( dx dy dz + dx dy dz ) + 4 a ( dx dy dy − dx dz dz ) + 4 a ( dy dz dz − dx dx dy )+4 a ( dy dy dz − dx dx dz ) − a ( dx dy dy + dx dz dz ) − a ( dx dx dz + dy dy dz ) − a ( dx dx dy + dy dz dz )+8 b dx dy dz . (2.15)Notice that only b gives a term in F of the form dx dy dz , which can become F -flux.However, b corresponds to turning on a (4 ,
0) flux in M-theory or a (3 ,
0) flux in type IIB.Such a flux breaks space-time supersymmetry spontaneously via an F -term. Before looking at the details of the metric that emerges from dualizing ( x , y , z ) in thegeneral case, let us note that all of the H -flux, other than the coefficient of b , turns intometric. The only surviving H -flux is again precisely the component correlated with theappearance of F -flux.The simplest example is therefore to set a = . . . = a = 0 and consider only b (cid:54) = 0,with only F -term supersymmetry breaking. In this case, we find a type IIA string framemetric of the form: ds IIA = e − w + φB g µν dx µ dx ν + e w + φB (cid:8) ( dx ) + ( dy ) + ( dz ) (cid:9) + e − w − φB (cid:8) ( dx ) + ( dy ) + ( dz ) (cid:9) . (2.16)The H -flux and type IIA dilaton are given by: H = − b dx dy dz , φ A = φ B − w. (2.17)Notice that φ A is varying because the warp factor depends on the internal coordinates.This space is topologically T . It is metrically T × T .Now we can turn to the R-R fluxes. We will follow the strategy described in section 2.2.The first contribution comes from dualizing F of (2 .
15) using the rules in Appendix B. The9 potential associated to the H -flux has no legs in the ( x , y , z ) directions so T-dualityacts straightforwardly giving: F = − b e − φ B . (2.18)There is one other contribution which comes from the mostly space-time filling F space fieldstrength of (2 . F = − d (cid:0) e − w (cid:1) ∧ vol ∧ dx ∧ dy ∧ dz . (2.19)This F field strength dualizes to an F field strength supported on the internal space: F = − ∗ F = ∗ d (cid:0) e w (cid:1) . (2.20)The Hodge star, ∗ , is with respect to the unwarped metric, while d acts only on thethree-torus given by ( x , y , z ). It is rather surprising that there is such a simple solutionto the Romans theory with only ( F , F , H ) fluxes. We will return to this example later insection 5 with the aim of examining how the equations of motion are being solved. The general case involves “geometric flux” or non-trivial circle bundles. For a review oftwisted tori vacua, including very closely related metrics to those we find below, see [37].We first need to choose a trivialization of H . There are many gauge-equivalent choices,and one such choice appears below: B = − a x ( dy dz + dy dz ) + 4 a y ( dx dz + dx dz ) − a z ( dx dy + dx dy ) + 4 a ( z dx dz − y dx dy ) + 4 a ( y dx dx − z dy dz )+4 a z ( dx dx − dy dy ) − a x ( dz dz + dy dy ) − a z ( dx dx + dy dy ) − a y ( dx dx + dz dz ) − b x dy dz , (2.21)= A x dx + A y dy + A z dz − b x dy dz . (2.22)The potentials ( A x , A y , A z ) determine the twisting of the ( x , y , z ) circles over the base T with coordinates ( x , y , z ). The dual metric follows from the rules of Appendix B, ds IIA = e − w + φB g µν dx µ dx ν + e w + φB (cid:8) ( dx ) + ( dy ) + ( dz ) (cid:9) + e − w − φB (cid:8) ( dx + A x ) + ( dy + A y ) + ( dz + A z ) (cid:9) . (2.23)The H -flux and dilaton are still given by (2 . T as long as at least one of the potentials ( A x , A y , A z ) is non-zero.10he R-R fluxes dualize to F = − b e − φ B , as we saw in the simplest case described insection 2.2.4. In addition we now find R-R 2-form flux from dualizing (2 . e φ B F flux = 4 [( a + a ) dx + ( a − a ) dy + ( − a + a ) dz ] ∧ ( dx + A x )+4 [( a + a ) dx + ( a + a ) dy − ( a + a ) dz ] ∧ ( dy + A y ) (2.24)+4 [( − a − a ) dx + ( − a + a ) dy + ( a + a ) dz ] ∧ ( dz + A z ) . Surprisingly, there is no F generated by dualizing (2 . F space of (2 .
2) becomes an 8-form field strength, F = F space ∧ ( dx + A x ) ∧ ( dy + A y ) ∧ ( dz + A z ) . (2.25)Taking the Hodge dual gives, (cid:101) F = − ∗ F = ∗ d (cid:0) e w (cid:1) , (2.26)where ∗ is again the Hodge star with respect to the unwarped metric, while d again actsonly on the three-torus given by ( x , y , z ).Let us summarize the final data for this general case. The metric, which is of twistedtorus type, appears in (2 . H = − b dx dy dz , F = − b e − φ B , (2.27) F = F flux + ∗ d (cid:0) e w (cid:1) , φ A = φ B − w. (2.28)We can list our major observations: • The lack of any F -flux. • The Romans parameter is completely correlated with H . • Supersymmetry is spontaneously broken.
While the simplest example of section 2.2.4 is still topologically T , this is not true for thegeneral case with metric (2 . H , but worth iterating here [2,44,45].The Betti numbers for T , satisfying b n = b − n , are given below: b = 1 , b = 6 , b = 15 , b = 20 . (2.29)11hen a non-trivial gauge potential like A x appears in (2 . dx is no longerglobally defined if the metric is to be well-defined. Rather the periodicity of x is modifiedso that the combination, dx + A x , (2.30)is globally defined, but not closed. Therefore b is reduced by 1. This is sufficient to seetopology change. For similar reasons, b and b change. For example, the field strength dA x is an exact 2-form supported along the ( x , y , z ) directions. This reduces b by 1.Similarly, ( dx + A x ) ∧ dy , ( dx + A x ) ∧ dz , are no longer closed, reducing b further by 2. The field strength dA x defines a linear map, dA x : Ω ( S x × S y × S z , R ) → Ω ( S x × S y × S z , R ) , (2.31)via the wedge product. The kernel of this map is 2-dimensional. Let α denote a class inthe one-dimensional complement of the kernel then( dx + A x ) ∧ α is also no longer closed. In total, b decreases from 15 to 11 for this example. A similaranalysis can be performed for b , and extended to the case of any three specific gaugepotentials ( A x , A y , A z ). The central observation for us is that the space is only topologically T for the special case of the simplest example with a non-vanishing Romans parameterdescribed in section 2.2.4. To this point, we have worked with the ambient T without worrying about the choice ofquotient action, satisfying tadpole constraints, or tracking branes and orientifold planes.Let us examine each of these issues. All the solutions with a non-vanishing Romans parameter are non-supersymmetric. We cantherefore consider various possible quotient actions that give orbifolds of T in M-theory,or orientifolds of T in type IIB, which preserve different amounts of supersymmetry. Let M = T /G denote the M-theory orbifold obtained by quotienting T by the finite group G .12e would like the geometry to preserve some supersymmetry so the breaking by the fluxlooks spontaneous from the perspective of three or four-dimensional effective field theory.When formulated in the language of M-theory, the M D (cid:90) G π ∧ G π = χ ( M )24 . (3.1)For models with type IIB limits, which are the cases of interest to us, the class G π ∈ H ( M , Z ) [47]. Computing the flux contribution to (3 .
1) from the our general expres-sion (2 .
5) for the models involving dualization along ( x , y , z ) discussed in section 2.2.3gives,12 (cid:90) G π ∧ G π = 32 e − φ B (cid:34) b ) + a a + a a + a a + (cid:88) i =2 ( a i ) (cid:35) × . (3.2)Let us now examine two specific examples found in [2]. T / Z This is a Z quotient denoted I that acts as follows: I : ( u, x, y, z ) → ( − u, − x, − y, − z ) . (3.3)This is a space with terminal singularities. Nevertheless, the charge tadpole can be com-puted from the string orbifold definition of the Euler characteristic giving χ ( M )24 = 16. Allthe fluxes appearing in (2 .
5) are invariant under this quotient action. This leaves manyways to satisfy the tadpole condition.Let us look at the simplest example with b (cid:54) = 0 and a = . . . = a = 0. For simplicity,set e φ B = 1. The flux is given by, G π = 8 b ( du dx dy dz − du dx dy dz ) . (3.4)The volume of 4-cycles on the quotient space is reduced by a factor of 2. The flux quanti-zation condition becomes 4 b ∈ Z while the tadpole condition becomes:2(4 b ) + . (3.5)For this simplest case, we can choose 4 b = 2 then tadpole cancelation requires an additional8 M a parametersin (2 . a and a gives a 4-form flux: G π = 8 b ( du dx dy dz − du dx dy dz )+4 a ( du dx dy dz + du dx dy dz + du dx dy dz + du dx dy dz )+4 a ( du dx dy dy + du dx dz dz + du dx dy dy + du dx dz dz ) . (3.6)For these coefficients 2 a i ∈ Z . The tadpole condition becomes,2(4 b ) + 4(2 a ) + 4(2 a ) = 16 . (3.7)Choosing 4 b = 2 , a = 2 a = 1 solves the tadpole condition completely by flux withoutthe need for additional M T space.This example lifts to type IIB on T / (Ω( − F L Z ) where the Z involution inverts( x, y, z ) producing 64 O x , y , z ), each O O − F L I (cid:48) , where I (cid:48) is the following Z involution: I (cid:48) : ( x , y , z ) → ( − x , − y , − z ) . (3.8)This is a nice clean example but it suffers from the presence of naked orientifold planesin string theory, or equivalently, terminal singularities in M-theory. Neither description isreally well controlled in supergravity; see [11] for a discussion of this issue. K × K
3A nicer example that does not involve naked orientifold planes is obtained by considering K × K
3. Generically, this is a smooth compactification of M-theory. We can replace thefirst K T / Z , where the Z action inverts ( u, x ), in order to obtain the perturbativetype IIB orientifold, T Ω( − F L Z × K , (3.9)where the Z action inverts the x -torus. This generates 4 fixed points, each supportinga single O O K T / Z . InM-theory, this means we consider the quotient of T by the group with generators: (cid:101) g : ( u, x ) → ( − u, − x ) , g : ( y, z ) → ( − y, − z ) . (3.10)14rom the perspective of type IIB, we are then quotienting T by the group with twogenerators, g : x → − x, g : ( y, z ) → ( − y, − z ) , (3.11)where the geometric action generated by g is combined with Ω( − F L . We can view thebackground created by the quotient (3 .
11) as a limit of the smooth case with only O χ ( M )24 = 24 units of brane charge.There is another way to view this orbifold limit as a background with O O O g g . In this case, thereis a puzzle concerning the computation of the D3-brane tadpole. On the one hand, the 64 O −
16 units of D3-brane charge. On the other hand, we expect each O − T / Z for each of the O Z projects out all the odd cohomology giving χ ( T / Z ) = b + b + b = 1 + 6 + 1 = 8 . (3.12)Each O − χ ( T / Z ) / − −
16 units from the O − O −
24 units of D3-brane charge. We should stress that this method of counting shouldbe applied with care to the perturbative orientifold locus. A perturbative orbifold actiongenerates a B -field at each orbifold point. The B -field correlates with a gauge bundle ofa topological type that admits no vector structure [48, 49]. A T-dual description of thisparticular model with D9 and D5-branes was nicely analyzed in [50], where gauge-bundleinstantons provided 16 units of D5-brane charge leaving a deficit of 8 units of D5-branecharge. The method of computing the D3-brane tadpole described above really applies tothe geometric quotient.Now let us consider the choice of fluxes. Restricting to 4-form fluxes compatible withthe quotient action (3 .
10) requires the following additional constraints on the coefficientsappearing in (2 . a = a = a = a = 0 . (3.13)Let us again set e φ B = 1. There are again many ways to solve the tadpole constraint. Letus first look at the simplest example with only b (cid:54) = 0. In this case, inspecting (3 .
4) gives15 1 2 3 x x y y z z O O O O b ∈ Z . The tadpole constraint becomes,4(2 b ) + . (3.14)Choosing 2 b = 2 and adding 8 D3-branes solves the tadpole constraint.If we want to solve the tadpole constraint without branes, we again need to turn onsome a coefficients in (2 . .
6) with a and a non-vanishing.Conveniently, both these fluxes are invariant under the quotient action. We see that fluxquantization requires a ∈ Z and a ∈ Z . The tadpole constraint becomes,4(2 b ) + 8( a ) + 8( a ) = 24 . (3.15)Choosing 2 b = 2 and a = 1 , a = 0 solves the tadpole constraint with no brane sources.Now we dualize along ( x , y , z ). The resulting orientifold action is generated by,ˆ g : ( x , y , z ) → ( − x , − y , − z ) , ˆ g : ( y, z ) → ( − y, − z ) , (3.16)where the geometric action generated by ˆ g is again combined with Ω( − F L . Each O O x , y , z ) directions. Each O O x , y , z ) directions. The resulting background only has O x , y , z ) directions. These D6-branes cancel the charge of the O x , y , z ) directions. This reflects the pointwise cancelation of D7-brane charge in theoriginal IIB theory. Any D3-branes present in type IIB also become D6-branes transverseto the ( x , y , z ) directions. These D6-branes do not cancel the charge of the O x , y , z ) directions. We know there must be an additional contribution16rom flux sources and from the O x , y , z ) directions. We will take a closerlook at tadpole cancelation in this massive IIA background in the following section.Again it is natural to suspect that one can remove the ˆ g quotient action appearingin (3 .
16) by replacing the metric in the ( y, z ) directions by a smooth K g . In such a case, there would only be O x , y , z ) directions, but they would produce a D6-brane tadpole for branes transverse tothe ( x , y , z ) directions, as in the toroidal orientifold case above. The orientation of thesebranes is depicted in table 1. The O Before examining specific examples in more detail, let us present the type II supergravityequations of motion in a formulation that treats all the R-R field strengths democratically.This approach has been taken in [51–53]. A subsequent more complete treatment of R-Rfields in the framework of differential K-theory can be found in [54, 55]. This formulation isvery useful for checking equations of motion and Bianchi identities, especially when usingT-duality. In this paradigm, we work with a complete set of R-R field strengths, either { F , F , F , F , F , F } (4.1)for IIA, or { F , F , F , F , F } (4.2)for IIB. There are a number of places where sign conventions are rather important. Wehave described the places where sign ambiguities lurk in Appendix A, with our final choiceof sign conventions stated in ( A. ∗ F = − F , ∗ F = F , ∗ F = F , (4.3) ∗ F = − F , ∗ F = − F , ∗ F = F , (4.4) ∗ F = F , ∗ F = − F , ∗ F = − F , (4.5) ∗ F = F , ∗ F = F , (4.6)by hand at the level of the equation of motion. This puts all the R-R fields on the samefooting as F , and also makes it easy to talk about either IIA or IIB at the same time.17t also has the advantage of simplifying the R-R part of the bosonic action (when writtenin terms of field strengths), because the Chern-Simons terms are absorbed into the kineticterms.For either IIA or IIB, we have the fairly simple action: S II = 12 κ (cid:90) d x √− g (cid:40) e − (cid:18) R + 4 | d Φ | − | H | (cid:19) − (cid:88) p | F p | (cid:41) . (4.7)How do we make use of this simple form? To begin, we note that the Bianchi identity dF p = − H ∧ F p − , (4.8)implies that each field strength can be written locally in the form F p = e − B ( dC + m ) | p − form , (4.9)where C = (cid:80) p C p , and we have included the possibility of a constant m in IIA. Thisconstant corresponds to the Romans mass. Note that the potentials defined in this waywill be inconvenient for many purposes; for example, they will have ugly transformationsunder NS-NS gauge transformations. In addition, the action expressed in terms of thesepotentials will not be pretty. However, they will be related to other more convenient choiceslike ( B.
3) by field redefinitions, and this particular choice will help us derive the equationsof motion.We claim that the R-R equations of motion will be satisfied provided we satisfy theduality relations (4 . . . . . C which only appears in theaction via the term, − | dC − B ∧ dC + . . . | . (4.10)The corresponding equation of motion is simply, d ∗ F = 0 . (4.11)Now that we have an equation of motion, we can substitute in the duality relation ∗ F = F ,and the equation becomes dF = 0, which is simply the Bianchi identity for F .Next we can consider C which appears in two terms, leading to the equation of motion: d ( ∗ F − B ∧ ∗ F ) = 0 . (4.12)18sing the duality relations, this becomes0 = − d ( F + BF ) = − (( dF + HF ) + B ∧ ( dF )) , (4.13)which holds by the Bianchi identities for F and F .Let us check one more case; C leads to an equation of motion,0 = d (cid:18) ∗ F − B ∧ ∗ F + 12 B ∧ ∗ F (cid:19) = d (cid:18) F + B ∧ F + 12 B F (cid:19) = (cid:18) ( dF + H ∧ F ) + B ∧ ( dF + HF ) + 12 B ∧ ( dF ) (cid:19) . (4.14)Hopefully this is convincing evidence for the claim that the equations of motion are equiv-alent to the duality relations plus Bianchi identities.From (4.9), we also have the result that varying F p with respect to B gives us − F p − ,so the equation of motion for B becomes simply0 = d (cid:0) e − ∗ H (cid:1) + 12 (cid:88) p ≥ F p − ∧ ∗ F p . (4.15)The dilaton equation of motion is also quite simple, R + 4 ∇ Φ − ∇ Φ) − | H | = 0 . (4.16)This leaves only the Einstein equations. After subtracting a multiple of the dilatonequation, these equations become, e − (cid:18) R µν + 2 ∇ µ ∇ ν Φ − H ρσµ H νρσ (cid:19) + 18 g µν (cid:88) p | F p | − p − (cid:88) p ≥ F ρ ··· ρ p − p µ F p νρ ··· ρ p − = 0 . (4.17)There is even a further simplification; once we impose self-duality on the R-R field strengthsthen (cid:80) p | F p | = 0. For example, | F | is proportional to ∗ ( F ∧ ∗ F ) = ∗ ( F ∧ F ) = 0, and | F | + | F | is proportional to ∗ ( F ∧ ( − F ) + F ∧ F ) = 0.To summarize, the type II equations of motion are: R + 4 ∇ Φ − ∇ Φ) − | H | = 0 (dilaton) , (4.18) e − (cid:18) R µν + 2 ∇ µ ∇ ν Φ − | H | µν (cid:19) − (cid:88) p ≥ | F p | µν = 0 (Einstein) , (4.19) d (cid:0) e − ∗ H (cid:1) + 12 (cid:88) p ≥ F p − ∧ ∗ F p = 0 ( B − field) , (4.20) dF p + H ∧ F p − = 0 (Bianchi) , (4.21) ∗ F p + ( − p ( p +1) / F − p = 0 (duality) , (4.22)19here in the Einstein equations we defined: (cid:12)(cid:12) ω ( p ) (cid:12)(cid:12) µν = 1( p − ω ρ ··· ρ p − µ ω νρ ··· ρ p − . (4.23) We want to first examine some of the general features of these models. Let us againrestrict to T-duality in the ( x , y , z ) direction with the general T-dual model describedin section 2.2.5. For this general case, we find models in both conventional type IIA stringtheory and in massive IIA when the b parameter of (2 .
5) is non-zero. Performing threeT-dualities is a mirror transform in the absence of flux. Some local aspects of the mirrortransform with fluxes have been discussed in [56, 57].The original type IIB flux vacuum has a number of K¨ahler moduli, including the totalvolume of the space. We can view the total volume as the modulus corresponding to dilatingthe internal warp factor of (2 .
1) by sending w → w + λ (5.1)for some positive λ . The internal metric in string-frame for the dual theory takes the form, ds dual = e w + φB (cid:8) ( dx ) + ( dy ) + ( dz ) (cid:9) + e − w − φB (cid:8) ( dx + A x ) + ( dy + A y ) + ( dz + A z ) (cid:9) . (5.2)For the simplest case where A i = 0, the overall type IIB volume scaling contracts the( x , y , z ) T -factor while dilating the ( x , y , z ) T -factor exactly as we would expectafter applying 3 T-dualities. Rather the total volume of the space is set by the complexstructure of the type IIB model, with the complex structure typically completely fixed bythe choice of flux.The background (5 .
2) is not a large volume compactification, and we should expectlarge α (cid:48) corrections to the tree-level physics. This is precisely the same situation seen inthe torsional heterotic/type I models found by a similar procedure. Regardless of whetherthere are large corrections, the form of the solutions is rather interesting. In the torsionalsetting, those models provide examples of the topological type of compact non-K¨ahler spacesthat solve the heterotic/type I Bianchi identity. We expect something similar here for typeIIA and massive IIA.Models in type IIA with F = 0 can be supersymmetric, preserving at least N = 1 and N = 2 space-time supersymmetry. More exotic choices like N = 3 might also be possible20n this setting [58]. However, all the models we have described involve only F -flux and O F = 0, there is no H -flux or F -flux. Thereforeeach such model should be liftable to a geometric background of M-theory with a metricthat preserves the appropriate amount of supersymmetry. The CY geometries found in thelift to M-theory of models preserving N = 2 supersymmetry have been studied for specificcases in [46, 3]. If the lifts of N = 1 models are classical backgrounds then they will be G holonomy geometries that have yet to be described. However, they need not be classical!Rather, one might suspect that these backgrounds solve the gravity equations of motiononly when R corrections are included in the general case. This would be quite fascinatingto explore further.However, our main emphasis in this work are models with a Romans mass. In this case,there is an H -flux precisely correlated with F so these are honest flux backgrounds. Itwould be nice if there were a way to geometrize the F -flux data in the Romans theory, asin the M-theory lifts of type IIA vacua with only F -flux. Perhaps some extension of theattempt to geometrize massive IIA found in [59] might be useful.Solving the massive IIA equations of motion with scale separation is highly non-trivial.We know of no other examples with either Minkowski or AdS space-times with scaleseparation so it seems worthwhile to explore how these solutions work. The issue to whichwe first turn is the manner in which the D6-brane charge Bianchi identity is being satisfied. Under three T-dualities, the original type IIB D3-brane tadpole becomes a D6-brane chargetadpole for D6-branes localized in the ( x , y , z ) directions. The D3-brane tadpole is acentral ingredient in evading the usual no-go theorem forbidding supergravity flux com-pactifications. To see how this tadpole condition is satisfied in type IIA, let us examine theBianchi identity for F .Using the definitions in ( B. dF = − mH = − F H , (5.3)where F and F are the general fluxes appearing in (2 . .
3) from O O . . .
3) is δ ab ∂ a ∂ b e w = − e − φ B (cid:34) b ) + a a + a a + a a + (cid:88) i =2 ( a i ) (cid:35) , (5.4)with a, b labeling the coordinates of the ( x , y , z ) torus. For transparency, let us considera less-than-general case with only the a = a and b fluxes turned on. Then, F = − a e − φ B dx ∧ ( dz + A z ) + ∗ d (cid:0) e w (cid:1) . (5.5)The exterior derivative of this expression gives, dF = 8 a e − φ B dx ∧ dA z + d (cid:0) ∗ d (cid:0) e w (cid:1)(cid:1) (5.6)= 64 a e − φ B dx ∧ dy ∧ dz + d (cid:0) ∗ d (cid:0) e w (cid:1)(cid:1) . Combining this with the right hand side of (5 .
3) gives us back the desired specific terms inthe warp factor equation (5 . e − φ B (cid:0) a + b (cid:1) dx ∧ dy ∧ dz = − d (cid:0) ∗ d (cid:0) e w (cid:1)(cid:1) , (5.7)provided that the warp factor only depends on the coordinates of the three-torus givenby ( x , y , z ). This assumption is perfectly fine for checking that the D3-brane tadpolemaps to the D6-brane tadpole of the dual theory. For example, we could have considered anon-compact example with ( x , y , z ) ∈ R and a T -fiber with coordinates ( x , y , z ). Inthis case, the warp factor equation can be solved in supergravity with only dependence on( x , y , z ) and no exotic additional sources. If we allowed a more general choice of a i and b i coefficients, we would find equation (5 .
4) reproduced by the F Bianchi identity.We can now see why there is a precise correlation between H and F . In the originaltype IIB frame, there is a tadpole contribution proportional to ( b ) precisely from thefluxes that dualize to these terms, which is reproduced by the right hand side of (5 . a i terms in the type IIA tadpole condition is the manifestnon-closure of F . The tadpole condition is the most basic check. A much stronger constraint is satisfying theequations of motion. We will check the equations of motion are satisfied for the simplestexample of section 2.2.4. There are remarkably few explicit checks of equations of motionfor flux backgrounds. We find it gratifying that these metrics and fluxes do indeed solvethe equations of motion, modulo the inclusion of beyond supergravity sources needed forcompact solutions. 22 .2.1 type IIB starting point – the simplest case
Our starting point is IIB string theory with constant dilaton, φ B , and metric ds = e − w + φ B / η µν dx µ dx ν + e w + φ B / δ AB dx A dx B , (5.8)where µ, ν = 0 , . . . , A, B = 4 , . . . ,
9. The warp factor, w , depends on x A only. Wewill later split these coordinates further into, (cid:8) x , · · · , x (cid:9) = { x , x , y , y , z , z } , (5.9)and restrict w to dependence only on { x , y , z } , but there is no reason do that yet. Themetric (5 .
8) has determinant, √− g = e w +5 φ B / , (5.10)and Ricci tensor, R µν = 12 η µν e − w δ AB ∂ A ∂ B w, R AB = − δ AB δ CD ∂ C ∂ D w − ∂ A w∂ B w. (5.11)Thus the Ricci scalar is R = − e − w − φ B / δ AB ( ∂ A ∂ B w + 2 ∂ A w∂ B w ) = − e − w − φ B / δ AB ∂ A ∂ B (cid:0) e w (cid:1) . (5.12)We will also turn on fluxes H = − b dx ∧ dy ∧ dz , F = 8 b e − φ B dx ∧ dy ∧ dz , (5.13)and F = − (1 + ∗ ) (cid:0) d (cid:0) e − w (cid:1) ∧ vol (cid:1) . (5.14)By (4 . F = 8 b e − w vol ∧ dx ∧ dy ∧ dz . (5.15)By construction the R-R fluxes are properly self-dual, so we need only check the Bianchiidentities. For F , we just have dF = 0, which is satisfied since F is closed. For F , werequire dF + H ∧ F = 8 b d (cid:0) e − w (cid:1) ∧ vol ∧ dx ∧ dy ∧ dz (5.16)+ (8 b dx ∧ dy ∧ dz ) ∧ (cid:0) d (cid:0) e − w (cid:1) ∧ vol (cid:1) = 0 , which is true since we have been very careful with signs.23ext we turn to the Bianchi equation for F ,0 = dF + H ∧ F (5.17)= − d ∗ (cid:2) d (cid:0) e − w (cid:1) ∧ vol (cid:3) − b e − φ B dx ∧ dy ∧ dz ∧ dx ∧ dy ∧ dz . Since we can compute d ∗ (cid:2) d (cid:0) e − w (cid:1) ∧ vol (cid:3) = δ AB ∂ A ∂ B (cid:0) e w (cid:1) dx ∧ dy ∧ dz ∧ dx ∧ dy ∧ dz , (5.18)we find that this reduces to a Poisson equation for the warp factor, δ AB ∂ A ∂ B (cid:0) e w (cid:1) = − b e − φ B . (5.19)We note that this is only a local equation. On a compact space we will not be able to solveit globally unless there are extra source contributions; for example, from orientifold planes,D-branes and their supported higher derivative couplings.Let’s turn now to the NS-NS sector equations. The dilaton equation is simply0 = R − | H | = − e − w − φ B / δ AB ∂ A ∂ B (cid:0) e w (cid:1) − b e − w − φ B / , (5.20)and it is easy to see that this is equivalent to (5.19).The equation from varying the B -field becomes, d (cid:0) e − ∗ H (cid:1) + 12 F ∧ ∗ F + 12 F ∧ ∗ F = d (cid:0) e − ∗ H (cid:1) + F ∧ F (5.21)= d (cid:0) − b e − w − φ B vol ∧ dx ∧ dy ∧ dz (cid:1) − (cid:0) b e − φ B dx ∧ dy ∧ dz (cid:1) ∧ (cid:0) d (cid:0) e − w (cid:1) ∧ vol (cid:1) = 0 . Finally, we have the Einstein equations. For the µ − ν components, we find0 = e − R µν −
14 ( F ) µν −
14 ( F ) µν , (5.22)= 12 η µν e − w − φ B δ AB ∂ A ∂ B w + η µν e − w − φ B δ AB ∂ A w∂ B w + 16 η µν b e − w − φ B , which is again equivalent to (5.19). The A − B components give,0 = e − (cid:18) R AB − | H | AB (cid:19) − (cid:88) p | F p | AB . (5.23)Noting that we can rewrite − e − | H | AB − | F | AB − | F | AB = − b e − w − φ B δ AB , (5.24)24his becomes0 = e − φ B (cid:18) − δ AB δ CD ∂ C ∂ D w − ∂ A w∂ B w (cid:19) − b e − w − φ B δ AB (5.25)+ e − φ B ∂ A w∂ B w − e − φ B (cid:0) δ AB δ CD ∂ C w∂ D w − ∂ A w∂ B w (cid:1) = e − w − φ B δ AB (cid:18) − δ CD ∂ C ∂ D (cid:0) e w (cid:1) − b e − φ B (cid:19) , (5.26)which is again equivalent to our warp factor equation. That concludes our check of thetype IIB equations of motion. The democratic treatment of R-R fluxes, given in section 4,greatly simplified this analysis. After T-dualizing x , y , and z in that order, we find a dilaton φ A = − w + 14 φ B , (5.27)and a metric ds = e − w + φ B / η µν dx µ dx ν + e w + φ B / δ ab dx a dx b + e − w − φ B / δ ij dx i dx j , (5.28)where µ, ν = 0 , . . . , x a ∈ { x , y , z } and x i ∈ { x , y , z } . If the warp factor w ( x a ) is afunction of { x , y , z } then this metric has determinant, √− g = e − w + φ B , (5.29)and a Ricci tensor given by, R µν = 12 e − w η µν δ ab ( ∂ a ∂ b w − ∂ a w∂ b w ) , (5.30) R ab = − δ ab δ cd ∂ c ∂ d w + 3 ∂ a ∂ b w + 32 δ ab δ cd ∂ c w∂ d w − ∂ a w∂ b w, (5.31) R ij = 12 e − w − φ B δ ij δ ab ( ∂ a ∂ b w − ∂ a w∂ b w ) , (5.32)with Ricci scalar R = e − w − φ B / δ ab (5 ∂ a ∂ b w − ∂ a w∂ b w ) . (5.33)The H -flux is unchanged, H = − b dx ∧ dy ∧ dz , (5.34)and the R-R fields become F = − b e − φ B , F = 8 b e − w vol ∧ dx ∧ dy ∧ dz ∧ dx ∧ dy ∧ dz , = ∗ d ( e w ) , F = − vol ∧ d ( e − w ) ∧ dx ∧ dy ∧ dz , (5.35)where ∗ represents the Hodge star operator with respect to the unwarped flat T withcoordinates { x , y , z } , using (cid:15) x y z = 1.We can easily check that these R-R field strengths obey the correct duality conditions.Bianchi is of course automatic for F and F . For F , we compute dF + H F = δ ab ∂ a ∂ b (cid:0) e w (cid:1) dx ∧ dy ∧ dz + 64 b e − φ B dx ∧ dy ∧ dz = 0 , (5.36)provided that the same warp factor equation we found before is satisfied, δ ab ∂ a ∂ b (cid:0) e w (cid:1) + 64 b e − φ B = 0 , (5.37)where we are explicit about only allowing dependence on { x , y , z } . This latter restrictionwill be relaxed in the full solution with beyond supergravity sources. Finally, the Bianchiidentity for F is satisfied because dF = 0 and F = 0.Turning to the dilaton equation of motion, we need the observation that there are non-vanishing Christoffel symbols for this metric, leading to ∇ µ ∇ ν Φ = 34 e − w η µν δ ab ∂ a w∂ b w, (5.38) ∇ a ∇ b Φ = − ∂ a ∂ b w + 32 ∂ a w∂ b w − δ ab δ cd ∂ c w∂ d w, (5.39) ∇ i ∇ j Φ = 34 e − w − φ B δ ij δ ab ∂ a w∂ b w, (5.40)and ∇ Φ = 32 e − w − φ B / δ ab ( − ∂ a ∂ b w + 3 ∂ a w∂ b w ) . (5.41)Thus the dilaton equation of motion becomes, R + 4 ∇ Φ − ∇ Φ) − | H | = e − w − φ B / δ ab ( − ∂ a ∂ b w − ∂ a w∂ b w ) − b e − w − φ B / (5.42)= 0 , which is again equivalent to the warp factor equation.The B -field equation gives,0 = d (cid:0) e − ∗ H (cid:1) + 12 F ∗ F + 12 F ∗ F = d (cid:0) e w − φ B / ∗ H (cid:1) + F F d (cid:0) − b e − w − φ B vol ∧ dx ∧ dy ∧ dz (cid:1) +8 b e − φ B vol ∧ d (cid:0) e − w (cid:1) ∧ dx ∧ dy ∧ dz , (5.43)which is satisfied.We are only left with a check of the Einstein equations. We find e − ( R µν + 2 ∇ µ ∇ ν Φ) − | F | µν − | F | µν = 12 e w − φ B / η µν δ ab ∂ a ∂ b w + e w − φ B / η µν δ ab ∂ a w∂ b w + 16 b e − w − φ B / (5.44)= 0 , using the warp factor equation again. The ij component is of identical form. Lastly wehave e − (cid:18) R ab + 2 ∇ a ∇ b Φ − | H | ab (cid:19) − | F | ab − | F | ab − | F | ab = e w − φ B / (cid:18) − δ ab δ cd ∂ c ∂ d w − ∂ a w∂ b w − b e − w − φ B / δ ab (cid:19) − e w − φ B / (cid:0) δ ab δ cd ∂ c w∂ d w − ∂ a w∂ b w (cid:1) + e w − φ B / ∂ a w∂ b w + 16 b e w − φ B / = e w − φ B / δ ab (cid:20) − δ cd ∂ c ∂ d (cid:0) e w (cid:1) − b e − φ B (cid:21) , (5.45)and the warp factor equation appears one last time, so everything is consistent. Acknowledgements
We would like to thank Callum Quigley for participating in the early stages of this project.The massive IIA examples are dedicated to Alessandro Tomasiello, who requested an ex-ample of a massive IIA flux vacuum with scale separation and no smeared orientifolds. Itis our pleasure to thank Mariana Grana, Chris Hull and Michael Schulz for helpful com-ments. T. M. and S. S. are supported in part by NSF Grant No. PHY-1316960. D. R.is supported by funding from the European Research Council, ERC Grant agreement no.268088-EMERGRAV. 27
Sign Manifesto
We need to be quite careful about signs so we will keep track of the possible sign choicesby allowing α i = ± to appear in various locations. In the main text, we will use a specificchoice for α i stated below. For example, we take the convention that the epsilon tensors(not symbols) satisfy: (cid:15) ··· = α (cid:112) − g (10) , (cid:15) = α (cid:112) − g (4) . (A.1)The factors of the square root of the metric determinant ensure that (cid:15) transforms covari-antly.We define the Hodge star as follows, ∗ ( dx µ ∧ · · · ∧ dx µ p ) = 1( d − p )! (cid:15) µ ··· µ p ν ··· ν d − p dx ν ∧ · · · ∧ dx ν d − p . (A.2)In components, this reads: (cid:0) ∗ ω ( p ) (cid:1) ν ··· ν d − p = 1 p ! (cid:15) µ ··· µ p ν ··· ν d − p ω ( p ) µ ··· µ p . (A.3)Note that for d even and p odd (such as a five-form in ten dimensions), this differs by asign from the convention found in Polchinski (B.4.6) [60]. Because this might cause issues,we will also insert a sign into the duality relation for F ; namely our F will obey, ∗ F = α F , (A.4)in ten dimensions. With our choice of conventions for the T-duality rules given in Ap-28endix B, this choice implies that: ∗ F = − α F , (A.5) ∗ F = α F , (A.6) ∗ F = α F , (A.7) ∗ F = − α F , (A.8) ∗ F = − α F , (A.9) ∗ F = α F , (A.10) ∗ F = α F , (A.11) ∗ F = − α F , (A.12) ∗ F = − α F , (A.13) ∗ F = α F , (A.14) ∗ F = α F . (A.15)Let us also insert a sign in the Bianchi identity, dF p = − α H ∧ F p − , (A.16)and assume that this relation will hold with the same sign in both IIB and (massive) IIA.We use the mostly plus convention for the metric signature in Minkowski space, as do allright-hearted people.A useful relation for us follows from defining, (cid:12)(cid:12) ω ( p ) (cid:12)(cid:12) = 1 p ! ω µ ··· µ p ω µ ··· µ p , (A.17)from which we find, (cid:90) d d x √− g (cid:12)(cid:12) ω ( p ) (cid:12)(cid:12) = − α (cid:90) ω p ∧ ∗ ω p . (A.18)In passing let us note that as a result of these conventions, the volume form in flat R , is vol = √− gdx ∧ dx ∧ dx ∧ dx = α (cid:15) µ µ µ µ dx µ ∧ dx µ ∧ dx µ ∧ dx µ , (A.19)and satisfies ∗ vol = √− g(cid:15) = − α , ∗ α vol . (A.20) This can be seen as follows: for 10-dimensional flat space, a valid F in our language would be dx − α α dx . Repeated T-dualities along the (5 , . . . ,
9) directions give the stated relations. α i arbitrary, we will make a specificchoice: α i = 1 . (A.21)This also determines the various sign choices in the explicit expressions for flux appearingin the main text. B T-duality Rules
For a string background specified by a metric g , B -field and dilaton φ with isometry inthe y -direction, T-duality applied to the background gives a new background in terms ofprimed fields, g (cid:48) yy = 1 g yy , g (cid:48) µy = B µy g yy , g (cid:48) µν = g µν − g µy g νy − B µy B νy g yy , (B.1) B (cid:48) µy = g µy g yy , B (cid:48) µν = B µν − B µy g νy − g µy B νy g yy , φ (cid:48) = φ −
12 log( g yy ) . (B.2)The definition of the R-R field strengths (4 .
9) in terms of potentials given in the maintext is a particularly convenient trivialization for deriving the equations of motion. Analternate trivialization, which often appears in discussions of T-duality in the literature,defines potentials via: F ( p ) = dC ( p − + H ∧ C ( p − . (B.3)The two definitions ( B.
3) and (4 .
9) are related by a field redefinition. However, the T-duality transformations given below do not depend on the choice of trivialization. Rather,they depend only on the equations of motion and Bianchi identities.In the main text, we use either F ( p ) or F p to denote the R-R field strengths. In thisappendix, it is useful to use F ( p ) to avoid index confusion. For the most part, we will onlybe concerned with the field strengths themselves. The R-R potentials are implicitly definedby ( B. F ( p ) (cid:48) µ ··· µ p − y = F ( p − µ ··· µ p − − ( p − F ( p − µ ··· µ p − | y | g µ p − ] y g yy , (B.4) F ( p ) (cid:48) µ ··· µ p = F ( p +1) µ ··· µ p y + pF ( p − µ ··· µ p − B µ p ] y + p ( p − F ( p − µ ··· µ p − | y | B µ p − | y | g µ p ] y g yy . (B.5)The virtue of using this definition is the ability to dualize backgrounds for which the R-Rfield strengths respect the chosen isometries, while the associated R-R potentials need not.30ualizing to massive IIA always involves a situation of this type. Note that F (5) givenby ( B.
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