N=4 Gauged Supergravity from Duality-Twist Compactifications of String Theory
aa r X i v : . [ h e p - t h ] A ug N = 4 Gauged Supergravity from Duality-TwistCompactifications of String Theory R A Reid-Edwards and B Spanjaard II. Institute f¨ur Theoretische PhysikUniversit¨at HamburgDESY, Luruper Chaussee 149D-22761 Hamburg, Germany
Abstract
We investigate the lifting of half-maximal four-dimensional gauged supergravities to compact-ifications of string theory. It is shown that a class of such supergravities can arise from com-pactifications of IIA string theory on manifolds of SU (2)-structure which may be thought of as K T . Examples of these SU (2)-structure backgrounds, as smooth K H -flux, are given and we also find evidence for a class of non-geometric, Mirror-fold backgrounds. By applying the duality between IIA string theory on K T fibrewise, we argue that these SU (2)-structure backgroundsare dual to Heterotic compactifications on a class T fibrations over T . Examples of these fi-brations as twisted tori, H -flux and T-fold compactifications are given. We also construct a newset of backgrounds, particular to Heterotic string theory, which includes a previously unknownclass of Heterotic T-folds. A sigma model description of these backgrounds, from the Heteroticperspective, is presented in which we generalize the Bosonic doubled formalism to Heteroticstring theory. [email protected] [email protected] Introduction
The study of gauged supergravities has received a new impetus in recent years. This has beendue, in part, to the application of novel techniques and concepts, such as generalized geometry, inconstructing and analyzing new dimensional reduction scenarios. Generalized geometry allows usto construct internal manifolds that can be seen as generalizations of the well-known Calabi-Yauthree-folds, K SU (3)-structure manifolds explicitly break three-quarters of the supersymmetry,six-dimensional SU (2)-structure manifolds break half, while 1I-structure manifolds, of any dimen-sion, do not explicitly break any supersymmetry. The surviving supersymmetry in the effectivelower-dimensional theory may then be spontaneously broken for a given vacuum solution.Reductions on six-dimensional manifolds with SU (3)-structure have been fruitfully employed inconstructing new N = 1 and N = 2 gauged supergravities in four dimensions [5, 6, 7] which haveminimally-coupled Abelian gauge groups and scalar potentials. The topological requirement thatthe six-dimensional reduction manifold has a certain structure group does not completely specifythe internal space and, as such, there are many outstanding questions concerning the relationshipbetween these four-dimensional N = 1 and N = 2 supergravities to ten-dimensional supergravityand string theory. In particular, an issue which has yet to achieve a satisfactory resolution is whetheror not the effective four-dimensional theory correctly captures all of the low energy physics of stringor supergravity theory compactified on such a manifold.The standard procedure for producing a lower-dimensional supergravity from a higher-dimensionalone is to propose a reduction ansatz for the higher-dimensional fields, which is substituted intothe higher-dimensional action. One then integrates out all internal coordinate dependence in thehigher-dimensional action. A judicious choice of reduction ansatz can lead to a lower dimen-sional supergravity with many attractive features, such as Yang-Mills gauge symmetries and scalarpotentials. In many cases, it is possible to show that this procedure produces an action for a lower-dimensional supergravity which captures all of the low energy classical physics one would expectfrom a supergravity compactified on a compact manifold. In such cases, there is a clear relationshipbetween the four-dimensional and ten-dimensional supergravities and one can interpret the lower-dimensional supergravity as a long wavelength limit of string theory with confidence. However,for many gauged supergravities constructed in this way, it is not clear how the reduction ansatznaturally arises from considering a compactification on a conventional manifold. It is importantthen to distinguish between those cases where the ansatz can be related to a compactification on aknown manifold and the cases where the situation is not so clear. Moreover it is important to useterminology that makes this distinction clear. Below we briefly clarify the terminology pertainingto both cases as it will be used throughout this paper. Pronounced, ‘identity-structure’. compactification will refer to a decomposition of the higher-dimensional fields in terms of theharmonics of a well-defined internal manifold followed by a truncation, which sets certain highermodes in the Kaluza-Klein tower of states to zero. In order to construct an effective theory, validup to a certain energy scale, one usually chooses to truncate out modes with an effective massabove this energy scale and indeed this is what is done in conventional compactifications on toriand Calabi-Yau manifolds, where only the lowest modes are kept. So that we can be sure that allof the light modes are kept and none thrown away in the truncation, we therefore need to have agood understanding of the internal geometry.By contrast, a dimensional reduction is simply an algorithm that may be employed to obtaina lower-dimensional supergravity from a higher-dimensional one. In many cases, such as for toriand Calabi-Yau manifolds, the standard reduction algorithm is equivalent to the Kaluza-Kleincompactification and truncation described above; however, generally one may have no idea ofwhat the reduction corresponds to physically; it is merely a recipe to generate a lower-dimensionalsupergravity .One might then say that many of the N = 1 and N = 2 gauged supergravities mentionedabove are well understood at the level of a dimensional reduction - an algorithm to construct onesupergravity from another, higher-dimensional, supergravity - but are yet to be fully understoodas compactifications in the rigorous Kaluza-Klein sense. It is, in fact, quite difficult to constructexplicit examples of SU ( n )-structure manifolds of 2 n real dimensions. It is considerably easier toconstruct an SU ( n )-structure manifold of real dimension greater than 2 n . For example, one mayconstruct a (2 n + k )-dimensional SU ( n )-structure manifold, as a smooth bundle where the fibresare (2 n real-dimensional) SU ( n )-holonomy manifolds, examples of which are well known, over a k -dimensional base. Examples of seven-dimensional SU (3)-structure manifolds of this kind, con-structed as Calabi-Yau bundles over S , were given in [8]. Further examples, giving six-dimensionalmanifolds of SU (2)-structure, will be given in this paper (see also [9] for a related class of SU (3)-structure backgrounds constructed in a similar manner).Whilst a central motivation of this growing field of generalized reductions has been to addressthe urgent need to construct credible moduli stabilization and supersymmetry-breaking scenariosfrom flux compactifications, these studies have also led to additional insights into the fundamentalstructure of string theory. In particular, there is mounting evidence that many gauged supergrav-ities can not arise from a compactification of a higher dimensional supergravity on a conventionalmanifold, but can be lifted to string theory on a non-geometric background. Such backgroundshave no analogue in field theories such as General Relativity and shed light on possible string-or M-theoretic generalizations of spacetime [10, 11, 12]. The need for an explicit constructionsof the internal background is perhaps of greater importance in the non-geometric case, where ourexperience is more limited. Many non-geometric 1I-structure backgrounds can be thought of asconventional manifolds locally, with transition functions between coordinate patches that includeT-dualities, making the background globally non-geometric. The viability of such backgrounds as A consistent reduction is a reduction in which solutions of the reduced theory lift to solutions of the full, higher-dimensional, theory. Similarly one may speak of a consistent truncation as a truncation such that the survivingmodes solve the higher-dimensional equations of motion. SU (3)-and SU (3) × SU (3)-structure backgrounds.Ultimately, one would like to have not only an understanding of which gauged supergravities liftto string theory and how, but also a description of the vacuum solutions of these gauged supergrav-ities from the worldsheet perspective. Whilst the reduced structure techniques pioneered in [3, 4]have found very general application to the construction of N = 1 and N = 2 gauged supergravitiesand their vacuum solutions, the construction of a worldsheet description remains a challenge in allbut the simplest cases. By contrast, compactifications which give rise to four-dimensional max-imally supersymmetric supergravities have been usefully described at the worldsheet level usingthe doubled formalism of [12]. These maximal gauged supergravities arise from compactificationsof Type II supergravity on six-dimensional 1I-structure backgrounds. Such backgrounds are par-allelizable and do not break any of the 32 supersymmetries explicitly, resulting in N = 8 gaugedsupergravities in four dimensions. The general form for the Lagrangian of such maximal super-gravities and the conditions that the gauging does not explicitly break supersymmetry were givenin [13].In contrast to the well-studied N = 1, 2, and 8 cases discussed above, our interest in this paperis to study the half-maximal case of N = 4 gauged supergravity in four dimensions. The halfmaximal case is of particular interest as it represents a situation in which the techniques used incompactifications on six-dimensional manifolds of SU (2)-structure and 1I-structure make contact.We will show that many of these massive four-dimensional N = 4 supergravities can be realizedeither as a compactification of Type IIA string theory on a manifold of SU (2)-structure or as amaximally supersymmetric compactification of Heterotic string theory on a manifold of 1I-structureand present explicit constructions of these backgrounds. Overview and Results
There are, in principle, at least three routes to constructing N = 4 gauged supergravities bydimensional reduction from ten dimensions; we may reduce the Type I, Spin (32) / Z or E × E Heterotic string theories on manifolds with 1I-structure group, reduce Type IIA string theory (inwhich we include flux compactifications of M-Theory [14]) on SU (2)-structure manifolds, or reduceType IIB (in which we include compactifications of F-Theory [15]) on SU (2)-structure manifolds.We will only consider reductions of IIA and Heterotic string theory on SU (2)- and 1I-structurebackgrounds respectively here and we shall be particularly interested in realizing these reductionsas compactifications on internal spaces which we shall identify.The first step of the reduction of the Heterotic supergravity consists of a conventional Kaluza-Klein compactification on T (with coordinates z m , m, n = 1 , , , N = 2 theory with rigid O (4 ,
20) symmetry, which lifts to a O (4 , Z ) T-duality symmetry ofthe string theory. We then compactify on a further T , twisting by elements of this T-duality Of course, for a given solution, some degree of supersymmetry may be spontaneously broken. T (with coordinates y i , i = 5 ,
6) to give a four-dimensional N = 4gauged supergravity. The six-dimensional internal space can then be thought of as a T fibrationover T with monodromies taking values in O (4 , Z ). For example, twisting by a geometric SL (4; Z ) ⊂ O (4 , Z ) results in a six-dimensional internal manifold which can be thought of as asmooth T bundle over T . For a general duality-twist reduction of this kind, the resulting N = 4theory has non-Abelian gauge symmetry generated by the algebra[ Z i , Z m ] = f imn Z n + M ima Y a + K imn X n [ Z m , Z n ] = K imn X i [ X m , Z n ] = f imn X i [ X m , X n ] = Q imn X i [ Z i , X m ] = f inm X n + W ima Y a + Q imn Z n [ Z i , Y a ] = − δ ab W imb Z m + M ima X m + S iab Y b [ Z m , Y a ] = M ima X i [ Y a , Y b ] = S iab X i where the generators ( Z m , Z i ) are related to diffeomorphisms of the six-dimensional internal space,( X m , X i ) are related to B -field antisymmetric tensor transformations and the Y a ( a, b = 1 , , .. U (1) internal gauge symmetry.The gauge algebra contains a lot of information about the ten-dimensional lift of the four-dimensional supergravity. A particular challenge to realizing lower-dimensional gauged supergrav-ities as compactifications of string theory is to give a string theoretic interpretation to the param-eters, such as the structure constants of this gauge algebra. In particular, the structure constants f imn encode information about the local geometry of the compactification manifold and the struc-ture constants K mnp and M mna contain information about the fluxes in the internal geometry.It has become commonplace to refer to the higher-dimensional description of all such structureconstants, regardless of their physical interpretation, as fluxes [16, 17] and we shall adopt thisnomenclature here. In addition to the geometric, K - and M -fluxes, which are related to thosedescribed in [18], we also find new Q -, W - and S -fluxes. It will be argued that the Q -flux is akinto that found for the T-folds studied in [12, 19, 20, 21] where the monodromy around the T cyclesinclude a strict T-duality and mixes metric and B -field degrees of freedom. The W -flux will beshown to be indicative of a T-fold background, but of a kind specific to Heterotic string theoryand not previously studied. The S -flux will be understood as a non-trivial fibering of the internal U (1) gauge bundle over the T . Remarkably, we shall see that all of these backgrounds can beunderstood in terms of a worldsheet sigma model of the form pioneered in [12] and [19].The IIA theory can be similarly compactified on a K O (4 ,
20) rigid symmetry, dual to the Heterotic theory compactified on T discussed above.Performing a duality-twist reduction, as above, over the cycles of a further T gives an N = 4gauged supergravity with non-Abelian gauge symmetry generated by the Lie algebra[ Z i , J ] = K iA T A [ Z i , ˜ J ] = Q iA T A [ Z i , T A ] = D iAB T B − K iA ˜ J − Q iA J [ T A , T B ] = D ABi X i [ J, T A ] = K iA X i [ ˜ J , T A ] = Q iA X i where the generators Z i and X i are as in the Heterotic case above. J , ˜ J and T A generate gaugetransformations of the Ramond-Ramond fields where the indices A, B = 1 , , ..
22 run over thetwenty-two harmonic two-cycles of the K (3 , Z ) mapping class group of the K
3, the internal background is a smooth K D iAB - a geometric flux for SU (2)-structure reductions. It will be shown that the K -fluxes correspondto H -flux compactifications and the Q -flux corresponds to a new non-geometric flux. We willargue that this non-geometric background can be thought of as a K T in which themonodromy includes a Mirror Symmetry in the K Mirror-fold [22].The content of this paper is organi s ed as follows. In the next section we review the salientfeatures of N = 4 gauged supergravity in four dimensions. The ungauged, massless, theory has arigid SU (2) × O (6 ,
22) symmetry. The general form of the Lagrangian for the gauged supergravitywas found in [24] where the gauge symmetry includes a non-Abelian subgroup of the rigid SU (2) × O (6 ,
22) and we shall only consider electric gaugings of the O (6 , ⊂ SL (2) × O (6 , SU (2)-structure manifolds. In section four webriefly review evidence, at the level of the massless supergravity, for the conjectured duality betweenthe compactification of IIA supergravity on K T . In section fivewe consider a further duality-twist reduction, down to four dimensions, on T . In section six, theinterpretation of the duality twist reductions as compactifications of ten-dimensional theories onexplicitly constructed 1I- and SU (2)-structure backgrounds is discussed. Section seven considersa worldsheet construction of the Heterotic backgrounds considered in section six along the linesof the doubled formalisms introduced in [12] and [25] and section eight concludes and discussespossible avenues for future research. To streamline the arguments, the details of most calculationsare included in the Appendices. N = 4 Gauged Supergravity in Four Dimensions
In [24] the general form of the Lagrangian for N = 4 gauged supergravity in four dimensions waspresented. The massless Abelian gauge theory has a rigid SL (2) × O (6 , n ) symmetry [23], certainsubgroups of which can be promoted to a non-Abelian local symmetry in a manner consistentwith N = 4 supersymmetry. The non-Abelian gauging breaks the rigid symmetry to a subgroup,however there is still a natural action of SL (2) × O (6 , n ) on the fields of the gauged theory. In [24],following the work of [13, 26, 27, 28] (see also [29] for an excellent review), this fact was exploitedin order to write all possible N = 4, four dimensional, gauged supergravities in terms of a single,universal, Lagrangian. The action of SL (2) × O (6 , n ) does not preserve the gauging, but generallymaps one gauged supergravity into another. The gauging introduces the constant deformationparameters f αMNP , ξ αM , (2.1)where α = ± are indices for a vector representation of SL (2) and M, N = 1 , , .. n are indicesfor a vector representation of O (6 , n ). These deformation parameters determine the gauging andmust satisfy various constraints in order to be consistent with N = 4 supersymmetry. The task of The group of large diffeomorphisms. N = 4 gauged supergravities in four dimensions reduces to one of finding the constants(2.1) compatible with these constraints. We focus on the case n = 22 as this is of most relevanceto the string theory compactifications we shall be considering. The parameters (2.1) transform astensors under the rigid SL (2) × O (6 , G = SL (2) × O (6 , H = SO (2) × O (6) × O (22) ⊂ G .The field content of the theory is made up of an N = 4 gravity multiplet which consists ofa vielbein e µa , four gravitini ψ µi , six spin-one gravi-photons A µ , four spin-half fermions χ i anda complex scalar τ respectively. To this are coupled twenty-two N = 4 vector multiplets, eachconsisting of a vector A µ , four spin-half gauginos λ ai and six real scalars. It is useful to combinethe vectors from the vector and gravity multiplets and write them as A M (1) and the 132 scalars ofthe 22 vector multiplets into an array M MN . The scalars τ and M MN take values in the productof cosets SL (2) SO (2) × O (6 , O (6) × O (22) . The SL (2) has a fractional-linear action on τ as τ → ( aτ + b ) / ( cτ + d ). This can be realized as alinear action on the array M αβ ( τ ) given by M αβ = 1 ℑ ( τ ) | τ | ℜ ( τ ) ℜ ( τ ) 1 ! , where ℜ ( τ ) and ℑ ( τ ) denote the real and imaginary parts of τ .Here, we shall consider electric gaugings of the O (6 , f + MNP ≡ t MNP which plays the role of structure constants in the gauge algebra. Theconstants t MNP are antisymmetric in all indices. The four-dimensional gauged Lagrangian for thebosonic sector is then L = R ∗ D M MN ∧ ∗ D M MN + 12 dM αβ ∧ ∗ dM αβ − ℑ ( τ ) M MN F M (2) ∧ ∗F M (2) − ℜ ( τ ) L MN F M (2) ∧ F M (2) − g V ∗ , where the scalar potential is V = 148 ℑ ( τ ) t MNP t QRS (cid:0) M MQ M NR M P S − M MQ L NR L P S (cid:1) + 124 ℑ ( τ ) t MNP t MNP (2.2)and F M (2) is the gauge covariant field strength for A M (1) . The only constraint on this gauging isthe Jacobi identity t [ MN | R t | P Q ] R = 0 so that, a priori, any Lie subgroup of O (6 ,
22) should give aconsistent gauging. The invariant of O (6 ,
22) is L MN , which can be used to raise and lower indiceson the constants t MNP . In particular t MNP = L MQ t NP Q
Our goal will be to understand how some of these gauged supergravities lift to compactificationsof string theory. In [24] a basis was chosen in which L MN =diag( − , ), the basis choice here is given by (5.7). Six-dimensional manifolds with SU (2) -structure One way in which a four-dimensional, half-maximal, gauged supergravity can be realized as astring theory reduction is as IIA supergravity, reduced on a manifold with SU (2)-structure. In thissection, we will recall some of the salient features of manifolds with reduced structure, and thendiscuss the specific case of a six-dimensional manifold with SU (2)-structure in detail. Much of whatis discussed here is published elsewhere (in particular, see [2, 30, 31] and references therein), butthe SO (3)-symmetry of j and ω , and the discussion about the forms we expand the ten-dimensionalfields in, are original.One of the main questions in the reduction of a supersymmetric theory is how much supersym-metry of the higher-dimensional theory is explicitly broken by the dimensional reduction. In thecase of a ten-dimensional supergravity theory the reduction ansatz of the ten-dimensional spacetime M , is M , = M , × Y, where Y is a compact six-dimensional manifold. The ten-dimensional Lorentz group decomposesas SO (1 , → SO (1 , × SO (6) , and the 16-dimensional Majorana-Weyl spinor representation therefore decomposes as → ( , ) ⊕ ( ¯2 , ¯4 ) . A ten-dimensional supersymmetry parameter ε can be written as ε = ξ + ι ⊗ η ι + + ξ − ι ⊗ η ι − , (3.1)with ξ ι four-dimensional spinors and η ι the available six-dimensional spinors where the index ι runsover 1 and 2 and the subscripts ± denote the chirality of the spinor. Clearly, the amount of super-symmetry of the four-dimensional theory depends on the number of supersymmetry parameters ε of the original theory and on the number of six-dimensional spinors η ι .The spinors η ι in equation (3.1) need to be globally well-defined on Y . In other words, theyneed to be singlets under the structure group G ⊆ SO (6) of Y . For example, on a manifold with SU (3)-structure, the spinor representation decomposes as SO (6) ∼ = SU (4) → SU (3) : → ⊕ , (3.2)so there is one spinor that transforms as a singlet under the structure group. This is the one spinorwe can use in (3.1).The spinors η ι need not be covariantly constant, but there does exist a unique metric-compatibleconnection ∇ T such that ∇ T η = 0 . The structure group is the group the transition functions of the tangent bundle
T Y take values in. ∇ T is not the Levi-Civita connection, it must be torsional. This torsion can be described by fivetorsion classes W , . . . , W [2] and so manifolds with reduced structures can be characterized bytheir torsion classes.Equivalently, a manifold of SU ( n )-structure in n complex dimensions can be defined in termsof a two-form J and an n -form Ω. These forms obey the relations J ∧ Ω = 0 , Ω ∧ ¯Ω = i n ( n +2) n n J n . Furthermore, J with one index raised is an almost complex structure I , I wv ≡ J vx g xw I = − , and with respect to this almost complex structure, J is a (1 , n, v, w = 1 , . . . n the indices on the real coordinates). The exterior derivatives of J and Ω aredetermined by the torsion classes as follows: dJ ∈ W ⊕ W ⊕ W , d Ω ∈ W ⊕ W ⊕ W . In three complex dimensions, the equivalence between the definition of SU ( n )-structure in termsof the spinors η ± and that given in terms of the forms ( J, Ω) is given by formulas that expressthe forms in terms of the spinors. A manifold with SU (3)-structure has one globally well-definedspinor η , and in terms of this spinor the forms J and Ω can be written as J vw = iη †− γ vw η − , Ω vwx = iη †− γ vwx η + , v, w = 1 , . . . , . Using Fierz identities, it can be shown that the forms, so defined, indeed satisfy the constraints ofan SU (3)-structure.Let us now specialise to the case of a six real-dimensional manifold with SU (2)-structure.By the reasoning presented before, such a manifold has two globally well-defined spinors η ι ( ι =1 , SU (2)-structure would give a four-dimensionalsupergravity with 16 supercharges, or N = 4.For manifolds with SU (2)-structure one can define a pair of SU (3)-structures: a pair of 2-forms J ι and a pair of 3-forms Ω ι via J ιvw = iη ι †− γ vw η ι − , Ω ιvwx = iη ι †− γ vwx η ι + , v, w = 1 , . . . , . By raising an index with the metric one obtains two almost complex structures I ι wv ≡ J ιvx g xw , ( I ι ) = − , which generically are not integrable since the Nijenhuis-tensor is not necessarily vanishing. Withrespect to I ι the two-forms J ι are (1 , ι are (3 , SU (2)-structure, the two almost complex structures commute, [ I , I ] =0, and define an almost product structure π via π wv ≡ I xv I wx with π = 1I . One can check that π has four negative and two positive eigenvalues which in turn implies that thetangent space splits into a four-dimensional and a two-dimensional component. It follows, then,that this split also holds for all tensor products of tangent and cotangent spaces. A form χ on thesix-dimensional manifold Y can therefore always be written as a wedge product χ = χ (2) ∧ χ (4) of a form χ (2) with its legs in the two-dimensional directions and a form χ (4) with its legs in thefour-dimensional directions. Furthermore, it can be shown that the almost product structure π isintegrable, which means that it is possible, on a chart U , to choose coordinates { y i , z m } for i = 1 , m = 1 , . . . , (cid:26) ∂∂y i , ∂∂z m (cid:27) spans the tangent space to U . This means that on U , the metric can be written in the block-diagonalform ds = g mn ( y, z ) dz m dz n + g ij ( y, z ) dy i dy j . In other words, locally the six-manifold Y is a product of the form Y ≃ Y (2) × Y (4) where Y (4) is areal four-dimensional manifold while Y (2) is a real two-dimensional manifold. The almost productstructure becomes more apparent when we look at the forms that can be defined using the spinors.Since the spinors are never parallel, a globally defined complex one-form σ v ≡ σ v − iσ v ≡ η † + γ v η − exists. With this information, the four tensors J ι and Ω ι can be expressed in terms of the oneforms σ i , a (1 , j and a (2 , ω via [2, 30] J , = j ± σ ∧ σ , Ω , = ω ∧ ( σ ± iσ ) , or equivalently [31] j = 12 ( J − J ) , ω vw = iη †− γ vw η − . As shown in [2, 30], the σ i can be viewed as one-forms on the two-dimensional component Y (2) while j and ω define an SU (2)-structure on the four-dimensional component Y (4) .The simplest example of a six-dimensional manifold with SU (2)-structure is truly a productstructure: it is T × K
3. The forms σ i are then simply dy i , and the forms j and ω that determinethe structure of the K K SO (3)-rotation of the vector j Re ω Im ω K
3. In Appendix B we show that a similar symmetry existsfor a general manifold with SU (2)-structure: the SU (2)-structure is defined up to an SU (2)-rotation of the spinors η i . This rotation leaves σ invariant, but translates into an SO (3)-rotationon ( j, Re ω, Im ω ).In order to perform a dimensional reduction of a supergravity, we need to know what forms toexpand the ten-dimensional fields in. Since six-dimensional manifolds with SU (2)-structure seemto be, in many aspects, a generalization of T × K
3, it is reasonable to base the available formson T × K σ i andan arbitrary number n of two-forms e Ω A . These two-forms contain the (2 , ⊕ (0 , ω , the(1 , j and n − , η AB as η AB ǫ ij ≡ Z Y σ i ∧ σ j ∧ e Ω A ∧ e Ω B . In contrast to the forms ( dy i , Ω A ) on T × K
3, however, the forms ( σ i , e Ω A ) of the SU (2)-structure manifold need not be closed. This means the most general consistent formulas for theexterior derivatives of the forms are dσ i = 12 D jki σ j ∧ σ k + D iA e Ω A , d e Ω A = D iBA σ i ∧ e Ω B . (3.3)and we shall assume that D ijk , D Ai and D iAB are all constant. Furthermore, there are someadditional constraints on the matrices D . These come from requiring d = 0 (integrability of σ i and e Ω A ) and Stokes’ theorem to hold. Let us look at what that implies for D iA = 0. Requiring d = 0 then means D iBC D jAB − D jBC D iAB = D ijk D kAC , (3.4)but it does not give a similar constraint on the D kij since a triple wedge product of σ i ’s is zeroregardless of the coefficient. Instead, we can obtain the constraint D ilk D jml − D jlk D iml = D ij k D lml (3.5)by explicitly writing out the indices i, j . Finally, Stokes’ theorem yields the constraint − η AB D ikk = η AC D iC B + η BC D iC A . (3.6)We shall be particularly interested in a class of simple examples, where D ij k = D iA = 0 and A = 1 , . . .
22, so that dσ i = 0 , d e Ω A = D iBA σ i ∧ e Ω B . These Bianchi identities may be solved by σ i = dy i , e Ω A = exp ( D iBA y i )Ω B where we have introduced the harmonic two-form Ω A ( z ) ∈ H ( Y (4) ), so that d Ω A = 0. We maythen identify Y (4) ≃ K Y is a deformation of the SU (2)-holonomy manifold10 × K K ֒ → Y ↓ T (3.7)where Y (4) ≃ K Y (2) ≃ T and Y , therefore, is a non-trivial K T . The monodromy of the fibrations over the cycle with coordinate y i ∼ y i + ξ i is givenby e D iAB ξ i and we require that this monodromy takes values in the mapping class group of the K Y to be a smooth bundle.We shall see, in the sections that follow, that many compactifications on manifolds of this kindmay be realized by the duality-twist reductions of [32] which we review in section five. N = 2 Supergravity in Six Dimensions
Our goal is to realize a certain class of four-dimensional, half-maximal, supergravities in whicha non-Abelian subgroup of O (6 ,
22) is gauged, as compactifications of ten-dimensional string andsupergravity theories. As stated in the introduction, we shall first consider standard Kaluza-Klein compactifications of ten-dimensional IIA and Heterotic supergravities to six dimensions. Thecompactifications of the Heterotic and IIA theories on T and K N = 2 theories in six dimensions. The conjectured duality between Heterotic and IIAstring theories in six dimensions is then used to identify these theories as a precursor to furtherdimensional reduction to four dimensions, which will be performed in the next section. The detailsof the compactification to six dimensions will be of importance in understanding the lift of the fourdimensional gauged supergravities to string theory as will be discussed in section six and seven. T The bosonic sector of the Heterotic supergravity consists of a scalar dilaton Φ, a two-form potential B (2) with associated three form field strength H (3) = d B (2) + ... , and gauge bosons A a (1) with fieldstrength F a (2) , taking values in the adjoint representation of either E × E or Spin (32) / Z [33, 34,35]. It will be assumed that the internal E × E or Spin (32) / Z gauge group is broken to theCartan subgroup U (1) by some mechanism, such as Wilson lines in the toroidal compactification.The internal gauge fields A a then take values in the Lie algebra of U (1) where a, b = 1 , , ... E × E or Spin (32) / Z can be identified as isometriesof the Cartan torus T , with the remaining 480 generators related to massless solitons [33]. We The
Spin (32) / Z Heterotic string theory is more usually referred to as the SO (32) Heterotic string. Theworldsheet theory does indeed have a rigid SO (32) symmetry; however, the worldsheet fermions λ a , taking valuesin Spin (32) - the cover of SO (32) - are subject to a GSO-type projection, which means that only two of the fourconjugacy classes of Spin (32) play a role in the spectrum of the theory [33]. This distinction will not play an importantrole here but, in keeping with our aim to describe the supergravities in terms of string theory, we shall adopt themore accurate description of the ten-dimensional gauge group, as
Spin (32) / Z throughout. U (1) directly to the geometric symmetries of the Cartan torus. Thisperspective will be important in section seven where we consider the worldsheet description of thebackgrounds discussed in the following sections.The supergravity may be thought of as a low energy, weak coupling, effective field theory forthe Heterotic string. The low energy description is truncated to first order in α ′ , where the internalgauge fields first appear. A one-loop calculation in the string coupling gives further corrections,involving the spin connection, at first order in α ′ . Such corrections are important for issues such asanomaly cancelation in the theory and any full treatment should take such corrections into account[36] . Here, we shall neglect such string loop effects and consider only the tree level contributionsto the effective action, leaving a more complete analysis to be considered at a later date. Thebosonic sector of the effective ten-dimensional theory we shall consider is given, in string frame, bythe Lagrangian L Het = e − Φ (cid:18) R ∗ ∗ d Φ ∧ d Φ − H (3) ∧ ∗ H (3) − δ ab F a (2) ∧ ∗ F b (2) (cid:19) , (4.1)where we have set α ′ = 1 and the field strengths are given by F a (2) = d A a (1) , H (3) = d B (2) − δ ab A a (1) ∧ F a (2) . The particular case of interest is that in which four of the space coordinates are compactified intoa T with internal coordinates z m , where m, n = 6 , , ,
9. The standard Kaluza-Klein reductionansatz is used, for which the reduction ansatz for the fields are the zero modes of harmonic ex-pansions on the internal space, and as such the ansatz does not depend on the coordinates z m .The details of this reduction are given in the Appendix C. Inserting the reduction ansatz givenin (C.2) into the Lagrangian (4.1) and integrating over the T gives the effective N = 2 theoryin six dimensions. This six-dimensional theory has a rigid O (4 ,
20) symmetry [38] and the fieldscan be combined into multiplets of this rigid symmetry so that the Lagrangian can be written in amanifestly O (4 ,
20) invariant way [38] L Het = e − b φ (cid:18) b R ∗ ∗ d b φ ∧ d b φ + 14 d c M IJ ∧ ∗ d c M IJ − b H (3) ∧ ∗ b H (3) − c M IJ b F I (2) ∧ ∗ b F J (2) (cid:19) , where the indices I, J run from 1 to 24. The two- and three-form O (4 , b H (3) = d b C (2) − L IJ b A I (1) ∧ d b A J (1) b F I (2) = d b A I (1) (4.2)where the potential b C (2) is related to the B -field B .The vector fields coming from the reduction of the gauge fields A a , the off-diagonal parts ofthe metric, and the B -field components with one leg on the T and the other in the six-dimensionalspacetime, combine into the O (4 ,
20) vector b A I (1) . The array of scalars c M IJ is given by c M IJ = b g mn + b A ma b A na − b C mp b g pq b C qn b C mn b g np b A ma − b C mn b g np b A pa − b g mp b C pn b g mn − b g mn b A na b A ma + b A pa b g pn b C nm − b A na b g nm δ ab + b A ma b g mn b A nb , (4.3) See also Chapter 13 of [37] for further discussion. b g mn are the metric moduli for the T , b C mn is related to the B -field with both legs on the T and b A ma are the components of the gauge field A a along the T . The scalars c M IJ take values inthe Grasmannian O (4 , / ( O (4) × O (20)) and the inverse of c M IJ is given by c M IJ = L IK c M KL L LJ where L IJ = is the invariant of O (4 , U (1) gauge symmetry, with gauge bosons b A I (1) ,where U (1) ⊂ U (1) arises from the internal gauge symmetry inherited directly from the ten-dimensional theory. In addition, a U (1) comes from diffeomorphisms z m → z m + ω m of theinternal T and a further U (1) arises from the antisymmetric tensor transformations of the ten-dimensional B -field where one leg of the B -field lies along a cycle in the T and the other along thesix-dimensional non-compact spacetime. Let us denote the sixteen generators of the ten-dimensional(internal) gauge transformations by Y a , the four generators of diffeomorphisms along the cycles ofthe torus as Z m and the generators of the antisymmetric tensor transformations as X m . Thegenerators may be arranged into an O (4 ,
20) vector T I = Z m X m Y a , (4.4)which generates the full U (1) with Abelian gauge algebra [ T I , T J ] = 0. K The Lagrangian of the bosonic sector of Type IIA supergravity in ten dimensions is L IIA = e − Φ (cid:18) R ∗ d Φ ∧ ∗ d Φ − d B (2) ∧ ∗ d B (2) − d A (1) ∧ ∗ d A (1) −
12 ( d C (3) − A (1) ∧ d B (2) ) ∧ ∗ ( d C (3) − A (1) ∧ d B (2) ) − B (2) ∧ d C (3) ∧ d C (3) (cid:19) , where Φ is the dilaton, B (2) is the Kalb-Ramond field and A (1) and C (3) are Ramond-Ramondfields. The O (4 , N = 2 theory above can also be obtained from aKaluza-Klein reduction of IIA supergravity on K K A , where A, B =1 , , ...
22. The Hodge dual of Ω A is also a two-form and can be expanded in the basis { Ω A } as ∗ Ω A = H AB Ω B . The array of coefficients H AB take values in the Grassmannian H AB ∈ SO (3 , SO (3) × SO (19)and encode fifty-seven of the fifty-eight metric moduli of the K H AB satisfies H AC H C B = δ AB and13 [ A | C H C | B ] = 0, where η AB is the intersection matrix for K η AB ≡ Z K Ω A ∧ Ω B , and η AB is defined as its inverse. A final metric modulus, e − ρ , controls the overall volume of the K
3. When compactifying on a K
3, the moduli H AB and ρ , may depend on the six non-compactdirections. We will later give the explicit dependence of these moduli fields on the two torusdirections, and the fields with this specific form of dependence will be denoted by e H AB and e ρ andwe shall henceforth use H AB and ρ to denote moduli that do not depend on the T coordinates y i .The six-dimensional supergravity Lagrangian is given by L IIA = e − e φ (cid:18) e R ∗ ∗ d e φ ∧ d e φ + 14 d f M IJ ∧ ∗ d f M IJ − e H (3) ∧ ∗ e H (3) − f M IJ e F I (2) ∧ ∗ e F J (2) (cid:19) − L IJ e B (2) ∧ e F I (2) ∧ e F J (2) , (4.5)where the field strengths are e H (3) = d e B (2) , e F I (2) = d e A I (1) . The relationship between e B and the ten-dimensional B -field is given in Appendix D, as are otherdetails of this reduction. The Ramond potential A (1) and the part of the Ramond field C (3) whichwraps the twenty-two harmonic cycles Ω A combine with the six-dimensional Hodge-dual of thethree-form Ramond field to give the O (4 ,
20) vector e A I (1) .The matrix f M IJ takes values in the Grassmannian O (4 , / ( O (4) × O (20)) and is given by f M IJ = e − e ρ + e H AB e b A e b B + e e ρ e C e e ρ e C − e H CB e b C − e e ρ e b B e Ce e ρ e C e e ρ − e e ρ e b B − e H BA e b B − e e ρ e b A e C − e e ρ e b A η AC e H C B + e e ρ e b A e b B , (4.6)where e b A are the twenty-two components of the B -field which wrap the harmonic cycles Ω A , e C = η AB e b A e b B and the indices on e H AB are raised and lowered using η AB and its inverse η AB respectively.The symmetric matrix of scalars f M IJ satisfy f M IK L KL f M LJ = L IJ with L IJ , the invariant of O (4 , L IJ = − − η AB . The gauge algebra of this theory is U (1) , where U (1) ⊂ U (1) is generated by antisymmetrictensor transformation of the Ramond-Ramond fields with parameters Λ A associated to each of theharmonic two cycles of the K
3. A further U (1) is inherited directly from ten dimensions as theAbelian gauge transformation of the A (1) Ramond field δ A (1) = d Λ. We denote the generator ofthis transformation by J . In six dimensions the three form part of the Ramond field C (3) is dualto a one form ˜ C (1) and so a final U (1) comes from the Abelian gauge transformations of this field δ e C (1) = d e λ (0) , generated by ˜ J . These generators can be written as a O (4 ,
20) vector T I T I = J ˜ JT A . (4.7)14hich generate the Abelian gauge algebra [ T I , T J ] = 0. At the level of the supergravity, one can show that the Heterotic and IIA compactifications consid-ered above give equivalent six-dimensional Lagrangians. It has been conjectured [39, 40] that thefull quantum string theories are in fact dual, a conjecture which we shall assume to be true in whatfollows. Evidence for the duality can, for example, be found in the six-dimensional supergravitiesand a study of the BPS sectors as discussed in [41, 42, 43, 44] and references therein. We reviewthe basic supergravity argument below. An excellent introduction to this conjectured duality andits consequences may be found in [45].The IIA Lagrangian L IIA can be brought to the form of the Heterotic Lagrangian L Het byfirst identifying f M IJ → c M IJ , e A I (1) → b A I (1) . The next step is to dualise the three form e H (3) . We introduce the term d b C (2) ∧ e H (3) into the Lagrangian (4.5). The two-form b C (2) is a Lagrange multiplier which imposes the Bianchiidentity d e H (3) = 0, the variation of the Lagrangian with respect to the field strength e H (3) gives b H (3) = d b C (2) − L IJ b A I (1) ∧ d b A J (1) = e − b φ e H (3) , which we recognize as the three-form field strength in the six-dimensional Heterotic theory (4.2).Substituting this back into the six-dimensional IIA Lagrangian, and changing the sign of the dilaton e φ → − b φ , gives the Lagrangian of the six-dimensional Heterotic theory. We see then that, at thelevel of classical supergravities, the theories are indeed equivalent descriptions of the same physics. T We have seen that the Heterotic theory compactified on T and the IIA theory compactified on K L = e − b φ (cid:18) b R ∗ ∗ d b φ ∧ d b φ + 14 d c M IJ ∧ ∗ d c M IJ − b H (3) ∧ ∗ b H (3) − c M IJ b F I (2) ∧ ∗ b F J (2) (cid:19) . (5.1)As noted in [38], the theory has O (4 ,
20) rigid symmetry, a discrete subgroup of which lifts to aU-duality symmetry of the full string theory [46]. In this section we consider a further reduction on T , twisting by two commuting elements of the discrete U -duality subgroup O (4 , Z ) ⊂ O (4 , T , to give an effective theory in four dimensions. Let y i , i = 1 , T coordinates. The reduction ansatz for the duality twist compactifications introduces a Up to total derivative terms which vanish in the action. i -dependence in the fields according to their transformation properties under O (4 ,
20) [20, 32] sothat, for the scalar and vector fields, c M IJ and b A I (1) , the reduction ansatz is given by c M IJ ( x, y ) = O I K ( y ) M KL ( x ) O LJ ( y ) , b A I ( x, y ) = O I J ( y ) A J ( x ) , (5.2)where O I J ( y ) = exp( N iJ I y i ), with N I J taking values in the Lie algebra of SO (4 , O I J as the twist matrix and N iI J as the mass matrix. Thenotation we adopt is that the fields with hats or tildes are y -dependent.The structure constants N iJ I encode the monodromy around the i = 1 , N iJ I = (cid:16) α J I , β
J I , (cid:17) (5.3)where e α is the SO (4 ,
20) monodromy around the y ∼ y + 1 direction and e β is that around the y ∼ y + 1 direction where [ α, β ] = 0. The condition that the two twists commute is [ α, β ] I J = α I K β K J − β I K α K J = 2 N I [ i | K N | j ] KJ = 0 which is equivalent to a Bianchi identity for the twistmatrix d ( e N · y ) I J = ( e N · y ) I L N LiK N jK J dy i ∧ dy j = 0 . (5.4)Putting the reduction ansatz (5.2) into the six-dimensional Lagrangian (5.1) and integratingover the y i -coordinates gives the Lagrangian for an N = 4, four-dimensional gauged supergravity.This Lagrangian may be written in an O (6 ,
22) covariant way L = e − φ (cid:18) R ∗ ∗ dφ ∧ dφ + 12 ∗ H (3) ∧ H (3) + 14 ∗ D M MN ∧ D M MN − M MN ∗ F M (2) ∧ F N (2) (cid:19) + V ∗ . (5.5)The scalar potential is given by V = e − φ (cid:18) − M MQ M NT M P S t MNP t QT S + 14 M MQ L NT L P S t MNP t QT S (cid:19) , (5.6)where L MQ t NP Q = t MNP are the structure constants for the gauged supergravity. At first glance,there seems to be a discrepancy between this scalar potential and the general one given by (2.2).The expressions are reconciled if we note that t MNP t QT S L MQ L NT L P S = N iIJ N jKL L ij L IK L JL = 0since L ij = 0. The scalar fields M MN take values in the coset O (6 , / ( O (6) × O (22)) where M MN = g ij + M IJ A Ii A Jj + g kl C ik C jl g ik C jk g jk C ij L IK A Kk + M IK A Ki g ik C jk g ij g ij L IK A Kj g jk C ij L JK A Kk + M JK A Ki g ij L IK A Kj M IJ + g ij L IK L JL A Ki A Lj . The O (6 ,
22) invariant is L MN = L IJ . (5.7) One can consider these directions as describing the canonical α and β cycles of the torus.
16n four dimensions, we may write this in the Einstein frame using the four-dimensional Weylrescaling g µν ( x ) → e φ ( x ) g µν ( x ) (5.8)and then dualizing the H (3) field to a scalar χ as described in Appendix E.3. Introducing theaxio-dilaton τ = χ + ie − φ and defining M αβ = e φ χ + e − φ χχ ! , (5.9)the Lagrangian may then be written as L = R ∗ dM αβ ∗ dM αβ + 14 D M MN ∧ ∗ D M MN − ℑ ( τ ) M MN ∗ F M (2) ∧ F N (2) + 12 ℜ ( τ ) L MN F M (2) ∧ F N (2) + V ∗ . (5.10)Written in this form, it is easier to see that the scalars M αβ and M IJ parameterize the space SL (2) U (1) × O (6 , O (6) × O (22) . (5.11) Introducing generators for the diffeomorphisms y i → y i + ω i of the T as Z i and the antisymmetrictensor transformations of the B -field, with one leg along the T and the other in the non-compactfour-dimensional spacetime, as X i , the gauge algebra of this four-dimensional gauged supergravityis [ Z i , T I ] = N IiJ T J , [ T I , T J ] = N IJi X i , (5.12)with all other commutators vanishing. We have defined N IJi = − N JIi = L IK N JiK . (5.13)The gauge generators can be combined into a O (6 ,
22) vector T M as T M = Z i X i T I , (5.14)the algebra (5.12) may be written as [ T M , T N ] = t MN P T P where the structure constants of thegauge group are t iI J = N iI J , N IJi = L JK f iI K , (5.15)and all other structure constants are zero. The derivation of this gauge algebra is given in AppendixE.2. 17 Lifting to String Theory
In this section we elucidate the structure of the internal space given by the duality-twist constructiondescribed above. Our main goal will be to understand the lift of the four-dimensional gaugedsupergravity to a ten-dimensional string theory background. We shall see that, whilst many of theduality-twists give supergravities which can be lifted to a compactification of string theory on aconventional manifold, the generic situation cannot be understood in this way. Instead, we canunderstand the lift of the supergravity as a string theory on a non-geometric background. Suchbackgrounds do not have a realization as a compactification of ten-dimensional supergravity, butnonetheless are good candidates for consistent string theory backgrounds.The string theory sigma model describes the embedding of the two-dimensional worldsheetinto the ten-dimensional target spaces described above where the O (4 , Z ) ⊂ O (4 ,
20) acts as aperturbative duality symmetry of the string theory. Therefore, in order for the four-dimensionalsupergravity to have a physical ten-dimensional interpretation, in which the string theory is globallydefined, the monodromies around the T cycles must take values in the discrete O (4 , Z ) subgroupof O (4 , The duality-twist reduction leads to a natural interpretation of the four-dimensional supergravitiesas arising from a compactification of Heterotic theory on a six-dimensional background. In partic-ular, we can think of the six-dimensional space as a T fibration with monodromies along the twocycles of the T base. For there to be a well-defined string theory background, the monodromies e α and e β are required to commute and take values in the discrete T-duality group O (4 , Z ).Not all of these backgrounds will be geometries in the conventional sense and there are manyexamples of duality-twist reductions lifting to non-geometric compactifications of string theory[12, 20, 21]. As a warm-up, let us recall the example of a compactification of bosonic string theoryon T d . The resulting theory will have an O ( d, d ) rigid symmetry which can be used to perform aduality-twist reduction, over a circle S x with coordinate x ∼ x + 1, of the kind described in theprevious section and in [20, 21, 47]. The monodromy e N takes values in O ( d, d ; Z ) ⊂ O ( d, d ), thediscrete T-duality group of the string theory in the fibres. It is useful to decompose the 2 d × d twist matrix N I J into SL ( d ; Z ) blocks [47] N I J = f mn K mn Q mn − f nm , ! (6.1)where exp( f mn ) ∈ SL ( d ; Z ). The resulting gauge algebra is [47][ Z x , Z m ] = f mn Z n + K mn X n , [ Z m , Z n ] = K mn X x , [ X m , Z n ] = f mn X x , [ X m , X n ] = Q mn X x , [ Z x , X m ] = f nm X n . Z x and Z m generate diffeomorphisms of the S x and T d cycles respectively and X x and X m generate antisymmetric tensor transformations of the B -field. As discussed in [47], a compactifi-cation in which Q mn = 0, can be understood as a reduction on a conventional torus bundle, inwhich the fibres are patched together by a large diffeomorphism, and K mn gives a non-trivial flux H = K mn dx ∧ dz m ∧ dz n for the H -field strength. Compactifications in which Q mn = 0 have mon-odromies for which the theory in the T fibres must be patched together by a more general elementof O ( d, d ; Z ) involving T-dualities. Such backgrounds, called T-folds, are conventional geometriesin a contractible patch (or, for example, on the cover S x → R x ), but will not be a conventionalgeometry globally. There is evidence that such backgrounds, with only one type of structure con-stant turned on, are related by T-duality [10, 16, 47, 48], a relationship which may be summarizeddiagrammatically as K mn → f mn → Q mn , (6.2)where the arrows denote a T-duality along a z m -direction of the torus fibre.We now return to the Heterotic theory and consider an open contractible patch U α on thebase T . The sigma model, describing the embedding of the Heterotic worldsheet into the region U α × T , has the discrete T-duality symmetry O (4 , Z ). This sigma model determines the localphysics of the string theory, a global description requires one to specify the transition functionsbetween coordinates in different patches, U α and U β say, on the overlap U α ∩ U β . From the dualitytwist construction we know that the transition functions O αβ will be elements of O (4 , Z ) andso in order to understand the background we need to understand the action of O (4 , Z ) on theconformal field theory in the fibres. The monodromy around a cycle with coordinate y i ∼ y i + ξ i is given by exp( ξ i N iI J ). Following the description of the Bosonic theory above, we decompose thetwist matrix as N iI J = f imn K imn M imb Q imn − f inm W ibm − W ian − M ina S iab , (6.3)where M ina = δ ab M bin , K mni = − K nmi , Q imn = − Q inm and W iam = δ ab W ibm . This is the mostgeneral form for which N iIJ = N iI K L KJ = − N iJI , where N iIJ = K imn f imn M ima − f inm Q imn W iam − M inb − W ibn S iab , (6.4)where S iab = δ ac S ibc = − S iba . There are further quadratic constraints on the elements of (6.3)coming from the Jacobi identity N [ i | I K N j ] K J = 0. Decomposing the generators T M using (5.14)and (4.4), the four-dimensional gauge algebra (5.12) can then be written as[ Z i , Z m ] = f imn Z n + M ima Y a + K imn X n , [ Z m , Z n ] = K imn X i , [ X m , Z n ] = f imn X i , [ X m , X n ] = Q imn X i , [ Z i , X m ] = f inm X n + W ima Y a + Q imn Z n , [ Z i , Y a ] = − δ ab W imb Z m + M ima X m + S iab Y b , [ Z m , Y a ] = M ima X i , [ Y a , Y b ] = S iab X i .
19s mentioned in the previous section, the generators Z i and Z m are related to the diffeomorphisms y i → y i + ω i and z m → z m + ω m . The X i and X m are related to antisymmetric tensor trans-formations of the B -field with one leg in the y i and z m directions respectively. The Y a generatethe U (1) internal gauge symmetry. Our task now is to understand what it means physically to‘switch on’ the structure constants (or ‘fluxes’) in the above gauge algebra.Upon circumnavigating the cycles of the base T the effect of the twist matrix O I J = exp( N iI J y i )on the fields on the T is c M = O T MO where c M IJ is given by (4.3). To understand the effectof the duality twist on the T fields, it is considerably easier to work in terms of the vielbein V : O (4 , → O (4) × O (20) given by c M = b V T b V where c M is given by (4.3) and b V Λ I = b e mα − b e mα b C mn − b e mα b A am b e mα b A am δ ab (6.5)where e : GL (4) → O (4) is the vielbein for the T with metric b g mn = δ αβ b e αm b e βn and b C mn denotes b C mn = b B (0) mn + 12 δ ab b A a (0) m b A b (0) n . (6.6)The duality twist reduction ansatz is then simply b V ( y ) = V · O ( y ). f imn Consider first the case where the only non-zero structure constant in 6.3 is N imn = f imn . Theduality-twist matrix is O I J ( y, f ) = exp( y i f imn ) 0 00 exp( − y i f inm ) 00 0 δ ab , (6.7)and it is equivalent to multiplying, or twisting, every z m fibre index in the reduction ansatz byexp( y i f imn ). For example, the metric of the T becomes b g mn ( y ) = ( e y i f i ) mp g pq ( e y j f j ) qn , with line element ds = b g mn ( y ) dz m ⊗ dz n on the T fibres. Equivalently, this twist may be implemented by twisting the harmonic forms onthe T fibres dz m → σ m = ( e − y i f i ) mn dz n , dy i → σ i = dy i , so that they are no longer harmonic but satisfy the Maurer-Cartan equations dσ m + f inm σ i ∧ σ n = 0 , dσ i = 0 . The reduction ansatz for the fields is then given by replacing ( dz m , dy i ) in the standard Kaluza-Klein reduction with ( σ m , σ i ). For example the reduction ansatz for the metric becomes ds = g mn σ m ⊗ σ n . σ i , σ m ) are the natural left-invariant one forms on the group manifold G withalgebra generated by the vector fields K m = ( e y i f i ) mn ∂ n and K i = ∂ i , dual to the forms ( σ i , σ m ),which satisfy the commutator relations[ K i , K m ] = f imn K n , [ K m , K n ] = 0 . The reduction can then be thought of as a standard compactification on the six-dimensionalgroup manifold G . In fact, one need only require that the manifold is locally of the form G .Globally the internal manifold can be the twisted torus Γ \ G where Γ ⊂ G is a discrete subgroupof G , acting from the left, such that Γ \ G is compact [49]. Such discrete groups, which ensure thatΓ \ G is compact, are called cocompact groups. Examples of flux such compactifications on twistedtori were studied at length in [14, 15, 18, 49]. Generally the group G will not be compact and soa duality-twist reduction with such a monodromy can only be realized as a compactification whensuch a cocompact subgroup can be found.We see then that such a N = 4 gauged supergravity lifts to a compactification of Heteroticstring theory on a six-dimensional twisted torus Γ \ G which may be thought of as a topologicallytwisted T bundle over T . H -flux We now consider the case where the only non-zero structure constant in (6.3) is N imn = K imn . Forthis choice of structure constant, the calculation of the twist matrix is simplified greatly by the factthat N iI J N jJ K = 0, so that O I J ( y, K ) = δ I J + y i N iI J , or O I J ( y, K ) = δ mn y i K imn δ mn
00 0 δ ab . The duality twist, with this parameter, reproduces the result of the following reduction ansatz forthe vielbein b V ( y ) = VO ( y ): b V Λ I ( y ) = e mα − e mα (cid:0) C mn − K mni y i (cid:1) − e mα A am e mα A am δ ab , (6.8)which may be summarized as the reduction ansatz b g mn ( x, y ) = g mn ( x ) , b B (0) mn ( x, y ) = B (0) mn ( x ) − K mni y i , b A am ( x, y ) = b A am ( x ) . thus, the effect of this duality-twist reduction is equivalent to introducing a constant H -flux b H = H + 12 K mni dz m ∧ dz n ∧ dy i . and the gauged supergravity lifts to a Kaluza-Klein compactification on T with constant H -fluxon three cycles of the six-torus. 21 .1.3 Compactification with F -Flux A slightly less trivial example is the monodromy corresponding to N ima = M ima . It is not hard toshow that the twist matrix in this case is given by O I J ( y, M ) = δ I J + y i N iI J + y i y j N iI K N jKJ sothat O ( y, M ) = δ mn − y i y j M ima M jna y i M ima δ mn − y i M imb δ ab . The twist matrix is of the same form as the vielbein V , so that the monodromy will preserve theform of V and not mix the components. This is a clue that the monodromy has a geometric actionon the internal fields. The vielbein reduction ansatz b V Λ I ( y ) = e mα − e mα (cid:0) B mn + δ ab ( A ma − M mia y i )( A nb − M nib y i ) (cid:1) − e mα (cid:0) A ma − M mia y i (cid:1) e mα A ma − M mia y i δ ab reproduces the results of this monodromy, and this duality-twist ansatz is equivalent to the reduc-tion ansatz b g mn ( x, y ) = g mn ( x ) , b B (0) mn ( x, y ) = B (0) mn ( x ) , b A am ( x, y ) = A am ( x ) − M mni y i . This duality twist reduction therefore can be thought of as adding a constant flux to the U (1) gauge bosons b F a (2) = F a (2) − M ima dy i ∧ dz m . so that the half-maximal gauged supergravity with structure constants M ima can be lifted toa compactification of Heterotic string theory on T with constant flux on the F -field strengthswrapping T cycles inside the T . Compactifications on twisted tori with such fluxes on the F -fieldand also on the H -field as discussed above were studied in detail in [18] for more general geometricbackgrounds than the torus fibrations considered here. Q imn Let us now consider the duality-twist arising from setting all structure constants in (6.3) to zero,with the exception of N imn = Q imn = ( α mn , β mn ). The only requirement is that the monodromiesaround the cycles of the T take values in the T-duality group O (4 , Z ) ⊂ O (4 , Z ). Such aduality-twist reduction of the supergravity lifts to string theory compactified on a T-fold backgroundof the kind discussed in [12, 19, 47]. Let us consider this in more detail. The twist matrix is O I J ( y, Q ) = δ mn y i Q imn δ mn
00 0 δ ab . (6.9)The twist matrix is not of the same form as the vielbein and so will not generally preserve the uppertriangular form of the vielbein (6.5). This is an indication that the background corresponding tothis gauging is non-geometric. 22 IJ is the invariant of O (4 ,
20) given in (4.4) and acts as a T-duality along all directions inthe T [50]. The easiest way to see that this reduction is equivalent to a T fibration over T ,the monodromies of which includes a T-dualities, is to note that this twist matrix can be realizedas the H -flux twist matrix O ( K ), conjugated by the action of T-duality along all of the T fibrecoordinates O I J ( Q ) = L I K O K L ( K ) L LJ , (6.10)where Q imn = δ mp δ nq K ipq . Let the periodicities of the α and β cycle coordinates on the T be y i ∼ y i + ξ i , then the monodromy in M IJ , as we circumnavigate the cycles of the T , is given by M IJ ∼ O I K ( Q ) M KL O LJ ( Q ) | y i = ξ i , where the twist matrices are evaluated at y i = ξ i . Locally, overa simply connected region of the base T , the theory in the fibres is described by a free HeteroticCFT on T . As we circumnavigate the base the theory is identified by this monodromy, which isan action of O (4 , Z ) which includes a T-duality. Since the action of T-duality is a symmetry ofthe T CFT the background is smooth from the point of view of the string theory, even though itis not a conventional geometry and does not admit a ten-dimensional supergravity description. W ima The flux W ima plays a similar role to the flux Q imn . It is not hard to show that O ( y, W ) = δ mn − y i y j W ima W jan δ mn y i W ima − y i W ian δ ab , (6.11)which is equivalent to the F -flux twist matrix O ( M ) considered above, conjugated by a T-duality O ( y, W ) I J = L JK O K L ( M ) L IL . (6.12)The background is a T-fold in the spirit of the previous example, in that the monodromy includes aT-duality and can be thought of as a conventional flux compactification, seen through the distortinglens of a series of T-dualities along all fibre coordinates. However, this Heterotic T-fold backgroundhas no analogue in the Type II or bosonic string theories due to the important role the internalgauge fields A a play which are only present in the Heterotic and Type I string theories. It isinteresting to note that the quadratic part of the twist, − y i y j W ima W jan , plays a similar role tothe non-geometric flux y i Q imn in a conventional T-fold, except there is a symmetry as opposed toan antisymmetry of the un-contracted indices. If we treat the Cartan torus of the E × E or Spin (32) / Z as a bona-fide geometry, then S iab canbe thought of as a geometric flux which describes a non-trivial fibration of the Cartan torus overthe base T O I J ( y, S ) = δ mn δ mn
00 0 exp( S iab y i ) . (6.13)23e can then think of this gauged supergravity, where all structure constants except S iab vanish,as arising from a standard geometric compactification on T in which the U (1) fibration over theinternal T is not trivial but has a topological twist with monodromy exp( S iab ξ i ) as we circum-navigate the cycles of the T . The monodromy must take values in O (16; Z ) in order for the gaugebundle to be smooth.It is worth noting that this interpretation of the different components has only limited viability.While we have given here an interpretation of every single component, this cannot be used to inter-pret every combination of components. Some combinations, such as a combination of a geometricflux f - and an H -flux, can be interpreted in this way (as an H -flux compactification on a twistedtorus), but others, for example a combination of H - and Q -flux may not have so straightforwardan interpretation. We now consider lifting the gauged supergravities (5.10) to IIA string theory. Although the proce-dure we shall follow, of analyzing the effect of specific classes of monodromies on the fields in thefibre, is similar, there are important qualitative differences to the Heterotic case discussed above.For the Heterotic case, the metric moduli of the T fibres are in one-to-one correspondence withthe components of the T metric. We may therefore consider the monodromy to act directly onthe metric of the T . Indeed, in the previous section, we saw that it was possible to show howthe SL (4; Z ) monodromy acts on the coordinates of the T fibre. The fact that we can describehow the coordinates z m transform under the monodromy will allow us to construct a worldsheetdescription of these backgrounds in section seven. By contrast, the moduli of the K K K A and b A and the volume modulus ρ . Thisis enough to give an accurate description of how the K T base.As before, the possible mass matrices N iI J take values in the Lie algebra of SO (4 ,
20) such thatthe monodromies around the T cycles are in the U-duality group O (4 , Z ) ⊂ O (4 , K O (3 , Z ). The generatorsof O (4 , Z ) can be decomposed in terms of the generators of O (3 , Z ). Similar to the Heteroticdecomposition of N iI J according to SL (4; Z ) discussed above, it can be shown that a general twistelement taking values in the Lie algebra of the continuous group SO (4 , N iI J = − Λ i K iB i Q iB −Q iA −K iA D iAB , N iIJ = − Λ i K iA Λ i Q iA −K iA −Q iA D iAB . (6.14)As we will now show, the requirement that the monodromy is in O (4 , Z ) restricts the form,or texture , of the twist matrix further. Let us consider first the monodromy with mass matrix The term texture is often used to describe the qualitative features of quark and neutrino mass matrices. Theterm seems well-suited to also describe the qualitative features of the mass and twist matrices here. i = Λ i . The twist matrix is O I J ( y, Λ) = e − Λ i y i e Λ i y i
00 0 δ AB (6.15)which, upon circumnavigating the cycles of the T , simply has the effect of re-scaling the volumeof the K e H AB ( x, y ) = H AB ( x ) , e ρ ( x, y ) = ρ ( x ) + 2Λ i y i , e b A ( x, y ) = b A ( x ) . (6.16)A priori, one might expect this to provide a reasonable reduction ansatz, however the constraintthat the monodromy is in O (4 , Z ) means that e Λ and e − Λ must be integral. This is only thecase for Λ = 0, and so, only in the trivial case, can such a reduction ansatz be realized as acompactification of string theory.Thus, the most general mass matrix N iI J , such that exp( N iI J ξ i ) ∈ O (4 , Z ) is then N iI J = K iB Q iB −Q iA −K iA D iAB , (6.17)where K iA ξ i = η AB K iB ξ i ∈ Z , , Q iA ξ i = η AB Q iB ξ i ∈ Z , and e D iAB ξ i ∈ O (3 , Z ) and wehave taken the identifications of the T coordinates to be y i ∼ y i + ξ i . The gauge algebra of thegeneral gauged supergravity, arising from a reduction of this kind, is then[ Z i , J ] = K iA T A , [ Z i , ˜ J ] = Q iA T A , [ Z i , T A ] = D iAB T B − K iA ˜ J − Q iA J, [ T A , T B ] = D ABi X i , [ J, T A ] = K iA X i , [ ˜ J , T A ] = Q iA X i , where the physical significance of the generators was discussed in sections two and three. Notethat the requirement that the monodromy be an element of O (4 , Z ) means that the commutator[ J, ˜ J ] = Λ i X i = 0, a result which would not have been possible from a direct consideration of thefour-dimensional Lagrangian.As for the Heterotic case, it is useful to define a vielbein V : O (4 , → O (4) × O (20) such that M = V · V T where M is given by (4.6). The y -dependent array f M is given in terms of the twistedvielbein e V ( y ) = O ( y ) · V . The vielbein can be given by V Λ I = e − ρ e ρ C − b A ν Aa e ρ − e − ρ b A ν Aa , (6.18)where ν : SO (3 , → SO (3) × SO (19) is a vielbein such that H AB = ν Aa ( ν T ) aB where ν Aa = η AB ν Ba . 25 .2.1 Geometric O (3 , Z ) Twisted Reduction
The gauged supergravity with structure constants N iAB = D iAB corresponds to duality twistcompactifications with twist matrix O I J ( y, D ) = e D iAB y i (6.19)and can be realized as a smooth, geometric, K T for exp( D iAB ) ∈ O (3 , Z ), for i = 1 ,
2. The action of this monodromy on the K e H AB ( x, y ) = ( e D i y i ) AC H C D ( x )( e D i y i ) DB , e ρ ( x, y ) = ρ ( x ) , e b A ( x, y ) = ( e D Ti y i ) AB b B ( x ) . (6.20)To see what this means, recall the definition of H AB in terms of the holomorphic two-forms definedon the K ∗ Ω A = H AB Ω B . In order for this relation on the four-dimensional K SU (2)-structure internal space, this equation must be replaced by ∗ e Ω A = e H AB e Ω B where the e Ω A = ( e D i y i ) AB Ω B . Thus the duality-twist reduction with mass matrix D iAB is equivalent to an SU (2)-structure compactification on a six-dimensional manifold where theharmonic two-forms Ω A of the the K e Ω A ( y ) which are notclosed but satisfy d e Ω A − D iBA e Ω B ∧ dy i = 0 . (6.21)We encountered this SU (2)-structure manifold earlier in section three. To see why the struc-ture constants D iBA = ( α AB , β AB ) must take values in the generators of the discrete subgroup SO (3 , Z ), recall that although K
3, like a Calabi-Yau three-fold, does not have any continuousisometries, it does have discrete isometric symmetries. These large diffeomorphisms preserve thelattice Z , and generate the discrete group SO (3 , Z ). Let the T coordinates be given by y ∼ y + 1 and y ∼ y + 1. Upon circumnavigating the cycles of the T base, the K α AB ) along the y cycle and exp( β AB ) around the y cycle.The background will only be smooth if α AB and β AB are elements of the mapping class group SO (3 , Z ) of the K . H -flux Let us now consider the gauged supergravity with non-zero structure constants K iA for i = 1 , O I J ( K , y ) = − K iA K j A y i y j K iA y i −K iB y i δ AB (6.22)which is equivalent to the reduction ansatz e H AB ( x, y ) = H AB ( x ) , e ρ ( x, y ) = ρ ( x ) , e b A ( x, y ) = b A ( x ) + K Ai y i , (6.23) This is analogous to the simpler three-dimensional nilmanifold construction in a T is fibred over a circle withmonodromy in SL (2; Z ) - the mapping class group of the T fibre [47]. K × T with constant H -flux e H = H + K iA dy i ∧ Ω A . (6.24)The geometric reductions discussed above where D iAB , K iA = 0 only make use of the subgroup O (3 , Z ) × Z , ⊂ O (4 , Z ) as topological twists in the K T . We shall referto this as the geometric monodromy subgroup. This is in contrast with the other monodromies in O (4 , Z ) which produce backgrounds which cannot be thought of geometric compactifications inthe standard sense. It is to these non-geometric compactifications to which we turn next. Q iA Flux
The description of these backgrounds as K T suggests that there should be afamily of IIA reductions constructed as K T with monodromy taking values,not just in the geometric subgroup O (3 , Z ) × Z , ⊂ O (4 , Z ), but in the full duality group O (4 , Z ). In particular, we are interested in duality-twist reductions where the twist matrix takesthe form O I J ( y, Q ) = − y i y j Q iA Q j A y i Q iA − y i Q iB δ AB , (6.25)which produces a gauged supergravity with structure constants Q iA . The first thing we notice isthat this twist matrix will not preserve the form of the vielbein (6.18), i.e. the zero entries in thematrix V will generally not be preserved in the twisted vielbein e V . An O (4) × O (20) transformationcan be used to restore e V to the form (6.18) but, as for the Heterotic T-fold example consideredin the last section, this mixing of ρ , b A and H AB in the reduction ansatz, is indicative of a twistgiving rise to a non-geometric background. Another similarity between the T-fold monodromy andthe case we are considering here is that the twist matrix may be given in terms of the H -flux twistmatrix O ( y, K ), conjugated by L IJ O I J ( y, Q ) = L JK O K L ( y, K ) L LI , Q iA = K iB η BA , which is reminiscent of the relationship between O ( y, K ) and O ( y, Q ) in (6.10). In (6.10), theaction of the O (4 ,
20) invariant L IJ was identified as a T-duality along all cycles of the T fibre.Here, although the interpretation is not as clear, we will argue that gauged supergravities withstructure constants N iA = Q iA lift to a compactification on a non-geometric background whichexhibits many characteristics similar to that of the T-fold.It was shown in [40] that the moduli space of N = (4 ,
4) conformal field theories, describingembeddings of the worldsheet into K B -field, is given by O (4 , Z ) \ O (4 , / ( O (4) × O (20))where the discrete U-duality group O (4 , Z ) is generated by the following symmetry transforma-tions [51, 52]: 27 apping Class Group : O (3 , Z ) is the mapping class group of a K D iAB has been shown above to generate all suchfibrations with monodromies in O (3 , Z ). Integral B -Field Shifts : The H -flux background, given by the duality-twist reduction withmass matrix K iA gives a monodromy to the B -field b A ∼ b A + K iA ξ i , where K iA ξ i takes valuesin the discrete lattice Z , . All elements of O (4 , Z ) corresponding to integral B -field shiftsare therefore accounted for by the mass matrix K iA . Mirror Symmetry : The mass matrices D iAB and K iA together account for the geometricsubgroup O (3 , Z ) × Z , ⊂ O (4 , Z ). In addition, there is a version of Mirror Symmetryfor K Z contribution to the duality group O (4 , Z ). In fact, ithas been shown that the geometric symmetries O (3 , Z ) × Z , and this Z Mirror mapgenerate the full O (4 , Z ) duality group.The familiar Mirror symmetry relation between Calabi-Yau three-folds involves the exchangeof complex and complexified K¨ahler structures on mirror pairs of manifolds . For K
3, one maychoose any one of the real two-forms j , Re ( ω ) and Im ( ω ) to be paired with the B -field to give acomplexified K¨ahler structure, the other two give a complex structure. One may think of MirrorSymmetry as exchanging these complex and complexified K¨ahler structures, as is the case withMirror Symmetry for Calabi-Yau three-folds. However, this split of the hyperk¨ahler structureinto complex and K¨ahler structures is not unambiguous and the SO (3) relates different choices ofcomplex structure to each other. A clear definition of Mirror symmetry for N = (4 ,
4) string theoryon K L IJ isrelated to this Z mirror symmetry. Inspired by the T-fold construction, we conjecture that thisbackground, with a duality-twist given by the mass matrix Q iA , can be interpreted as a Mirror-fold.The idea of a Mirror-fold is not new (see, for example, [20]) and in [22] a CFT for the interpo-lating orbifold corresponding to a Mirror-fold was explicitly constructed. The Mirror-fold proposedhere is a smooth, non-geometric, string theory background, given by a K T in whichthe K K K T fibration over C P ≃ S , where the complex structure of the fibres is a holomorphic functionof the base coordinates. Mirror symmetry for the K T fibres [53]. We may consider performing the T-duality fibre-wise, despitethe absence of continuous isometries in the K
3. In [12], a three-dimensional T-fold was constructedas a T fibration over S , where the theory in the T fibres is patched, upon circumnavigating thebase, by a double T-duality along the fibres of the T . If we now think of this T as correspondingto the T fibres of an elliptically fibred K
3, then we can think of this five-dimensional Mirror-fold See [52] and references contained therein. The K¨ahler form J is complexified by including the B -field to give the complexified K¨ahler form J + ib .
28s a particular example of a three-dimensional T-fold, fibred over C P base. The generalization toa six-dimensional Mirror-fold, by including an extra S is straightforward.Similar to the heterotic case, the interpretation of individual components of the mass matrixdoes not have a straightforward generalization to multiple non-zero components. It is possibleto reproduce a dimensional reduction with both D iAB and K iA non-zero as a reduction of IIAsupergravity on a K H -flux, but an interpretation of a dimensional reductionwith, for example, K iA and Q iA both switched on is not so readily given. In the previous two subsections we considered the lifting different gauged supergravities to com-pactifications, possibly with fluxes, of string theory. Of the reductions considered, it was useful tosplit the mass matrices into different classes and we shall be concerned here with understanding theway in which these different classes of compactifications may be related to each other. In particular,we shall discuss to what extent the action of the group O (4 , Z ) on the space of all N = 4 gaugedsupergravities can be thought of as a duality symmetry of the string theory.The massless N = 4 gauged supergravity has a rigid SL (2) × O (6 ,
22) symmetry which actson the bosonic degrees of freedom (the fermions transform under the maximal compact subgroup)where the discrete subgroup SL (2; Z ) × O (4 , Z ) is conjectured to lift to a U-duality symmetryof the ten dimensional string theory [46]. As shown in [24], the gauged theory can be written in an SL (2) × O (6 , SL (2) × O (6 ,
22) does not preserve the gauging and the structure constants ofthe gauge group transform covariantly under SL (2) × O (6 ,
22) as t αMN P → U αβ U M Q U N T t βQT S ( U − ) SP ξ αM → U αβ U M N ξ βN where U αβ ∈ SL (2) and U M N ∈ O (6 , SL (2) × O (6 ,
22) maps one gauged supergravity into another,inequivalent gauged supergravity; however, one might conjecture that the action of the discretesubgroup SL (2; Z ) × O (6 , Z ) still lifts to a symmetry of the string theory. Another possibility isthat only a subgroup of SL (2; Z ) × O (6 , Z ) survives as a duality symmetry of the string theory.The gaugings arising from the O (4 , Z ) duality-twist reductions considered in this paper giverise to N = 4 gauged supergravities, the gauge groups for which are characterized by the structureconstants t MN P = δ M i δ N I δ P J N iI J where M = 1 , , .. i = 1 , I, J = 5 , , ...
28. Here, we shall only consider the action of O (4 , Z ) ⊂ O (6 , Z ) on these structure constants and leave a discussion of the action of thefull O (6 , Z ) for section eight. Under this action of O (4 , Z ) the structure constants transformas t iI J → U I K t iK L ( U − ) LJ U I J ∈ O (4 , Z ) (recall that for the gaugings considered here ξ αM = 0). This is equivalentto the action on the mass matrix N iI J → U I K N iK L ( U − ) LJ .Within O (4 , Z ) are sets of discrete Z duality symmetries which one might call strict dual-ities . These include the Z T-duality symmetries discovered by Buscher in [54] or the K O ( d, d ; Z ) discrete symmetry ofthe Bosonic string theory compactified on a d dimensional torus background may be decomposedinto the action of SL ( d ; Z ) (the mapping class group of T d ), discrete (constant) B -field shifts and a set of d Z strict T-duality symmetries which exchange momentum and winding modes. Weconsider now how the action of such strict Z dualities in the O (4 , Z )-action of U I J are expectedto relate the backgrounds considered in the previous sub-sections.
For the Heterotic string compactified on T , the O (4 , Z ) is a T-duality symmetry of the theoryand may be decomposed into the SL (4; Z ) mapping class group of the T , discrete shifts in the B -field and gauge potential A a and a set of strict Z T-duality symmetries. The action of the strictT-dualities relates different gaugings and is summarized in the following diagram K imn → f imn → Q imn ↓ ↓ M ima → W ima ↓ S iab (6.26)where the horizontal arrows denote the action of a Z ⊂ O (4 , Z ) ⊂ O (4 , Z ) strict dualitywhich is common to the Heterotic and Bosonic string theories. The vertical arrows denote theaction of those strict dualities that do not have a counterpart in the Bosonic theory, i.e. thoseT-dualities which directly involve the sixteen gauge fields A a . One may think of this as a Heteroticgeneralization of the bosonic T-duality sequence (6.2). There is some evidence [19] that, for certainchoices of structure constants, some of these backgrounds are indeed T-dual to each other. Inparticular, it was shown in [48] that certain H -flux backgrounds, characterized by the structureconstants K imn , are physically equivalent as string backgrounds, to the twisted tori, characterizedby the structure constants f imn . The IIA case is much simpler as there is only one strict duality - the K K iA → Q iA If we treat the Mirror Symmetry as two T-dualities along the fibres of an elliptically fibred K which preserve the partition function Heterotic Worldsheet Theory
A principal concern of this investigation has been to realize a class of half-maximal gauged super-gravities within the framework of Kaluza-Klein theory and thus give a higher-dimensional inter-pretation to the these four-dimensional effective theories. The ultimate goal of this programme isto obtain such gauged supergravities, not just from higher-dimensional supergravities, but also asfull string theory scenarios. We have seen that many of the half-maximal gauged supergravities donot lift to compactifications of field theory on a smooth manifold, but are possible candidates fornon-geometric string theory backgrounds.As a first step in the study of such backgrounds from the string theory perspective, we presentin this section a sigma model description in which the backgrounds studied in previous sectionscan be described. Due to the difficulties of an explicit construction of sigma models on K K T coordinates z m as in [12] by introducing dual coordinates˜ z m for a dual torus e T , conjugate to the winding modes wrapping the cycles of T . A self-dualityconstraint is then imposed to ensure the doubled theory describes the correct numbers of degreesof freedom.We shall see that the O ( p, q ; Z )-covariance of the heterotic theory, where p = q , introduces newissues not found in the O ( p, p ; Z )-covariant models studied in [12, 19]. In particular, we have toconsider the doubling of the left-moving scalars taking values in U (1) . These scalars χ La can bethought of as describing embeddings of the left-moving string modes into the Cartan torus, T c ,of E × E or Spin (32) / Z . A priori, it is not clear how to treat these chiral fields in a doubledformalism. The solution we propose here is to double these degrees of freedom by introducing right-moving modes χ Ra , also describing embedding into the Cartan torus. The self duality constraintbecomes a chirality constraint on the fields χ a which ensures that the theory has the correct numberof chiral degrees of freedom. The coordinates on the doubled fibres are X I = ( z m , ˜ z m , χ a ) wherethe ‘internal’ bosonic coordinates χ a take values in the Cartan torus of E × E or Spin (32) / Z .Initially we impose no chirality constraint on the χ a so we may think of the internal coordinatesas having been ‘doubled’. In this way, we may now treat the Cartan torus T c of E × E or Spin (32) / Z as we would a conventional target space geometry and consider embeddings of theworldsheet Σ into this T ≃ T × e T × T c . 31 .1 Heterotic Doubled Geometry Consider an O (4 , Z )-twisted T fibration over T . We are interested in constructing a doubledformalism which encodes the data of this background, including the gauge fields, geometrically. Wedenote the 26-dimensional T fibration over T by T , where T ≃ T × e T × T c ֒ → T↓ T (7.1)The coordinates on T are given by ( y i , X I ), where y ∼ y + 1 and y ∼ y + 1 are coordinateson the T base and X I (the index I runs from 1 to 24) are coordinates on the doubled fibres. For abundle with a monodromy taking values in O (4 , Z ), the doubled coordinates are subject to theidentifications (cid:0) y i , X I (cid:1) ∼ (cid:16) y i + ξ i , ( e − N · ξ ) I J X J (cid:17) , (cid:0) y i , X I (cid:1) ∼ (cid:0) y i , X I + α I (cid:1) , where ξ i = (1 ,
1) and the cycles of the doubled torus fibre can be normalized such that all entriesin α I are also unity.In order to recover a conventional description of the background a polarization Π must be chosen- a projection that selects which of the twenty-four X I are to be identified as the four spacetimecoordinates. This may be written as z m = Π mI X I . It is generally not possible to define the polarization globally. There are two cases in particular inwhich a global description of the spacetime cannot be given. The first, where we consider the onlynon-zero entry in the twist matrix to be N imn = Q imn , then the coordinates on T are subject tothe identifications (cid:0) y i , z m , ˜ z m , χ a (cid:1) ∼ (cid:0) y i + ξ i , z m + ξ i Q imn ˜ z n , ˜ z m , χ a (cid:1) , and we see that the monodromy mixes the z m and ˜ z m coordinates together and so the polarizationis not well-defined under this monodromy. This phenomenon has been discussed for the bosonicstring in [20, 57]. A second example, which does not arise for the bosonic string, occurs when theonly non-zero entry in the twist matrix is N iam = W iam so that the coordinates on T are thensubject to the identifications (cid:0) y i , z m , ˜ z m , χ a (cid:1) ∼ (cid:18) y i + ξ i , z m − ξ i ξ j W ima W jan ˜ z n − ξ i W iam χ a , ˜ z m , χ a + ξ i W ima ˜ z m (cid:19) , and we see explicitly that the monodromy mixes the z m , ˜ z m and χ a coordinates and therefore doesnot preserve the polarization.Once a polarization is chosen, the metric b g mn ( y ), B -field b B mn ( y ), and gauge fields b A ma ( y )on the T fibres define a y -dependent metric c M IJ ( y ) = ( e − N · y ) I K M KL ( e − N · y ) LJ where the y -32ndependent M can be written, in this polarization, as M IJ = g mn − C mp g pq C qn + A ma A na C mn g np A ma − C mn g np A pa − g mp C pn g mn − g mn A na A ma + A pa g pn C nm − A na g nm δ ab + A ma g mn A nb , (7.2)where C mn = B mn + A ma A na . The T metric, B -field and gauge fields, each with one leg on the T , define a connection of the T bundle b A I ( y ) = ( e − N · y ) I J A J i dy i , where A I i = A mi B mi + B mn A ni A ai . Following the strategy pioneered in [12] (see also [58, 59]), a sigma model, describing the embeddingof a worldsheet Σ into the doubled target space T , is given by S [ y i , X I ] = 14 I Σ c M ( y ) IJ d X I ∧ ∗ d X J + 14 I Σ Θ IJ d X I ∧ d X J + 12 I Σ d X I ∧ ∗ b J I ( y )+ 12 I Σ G ij dy i ∧ ∗ dy j + 12 I Σ B ij dy i ∧ dy j , (7.3)where b J I ( y ) = c M ( y ) IJ b A J − ∗ L IJ b A J ( y ) , G ij = g ij + 12 M IJ A I i A J j , and g ij is the metric on the T base. b A I now denotes the pull-back of the connection form to theworldsheet, so that b A I = b A I i ∂ α y i dσ α where σ α = ( τ, σ ) are coordinates on Σ and d = dσ α ∂ α isthe worldsheet exterior derivative. We choose to omit contributions to the sigma model comingfrom embedding Σ into the four-dimensional noncompact space. These contributions are importantin finding exact solutions based on these fibrations that can be included without difficulty. Theiromission here is simply for clarity of exposition. We also defineΘ IJ = − , L IJ = . (7.4)It is useful to define the one forms on T P i = dy i , P I = (cid:16) e N i y i (cid:17) I J d X J . These forms are globally defined on T and satisfy the worldsheet Bianchi identities dP i = 0 , d P I − N iJ I P i ∧ P J = 0 . (7.5)The sigma model may be conveniently written in terms of these one-forms as S [ y i , X I ] = 14 I Σ M IJ P I ∧ ∗P J + 14 I Σ Θ IJ d X I ∧ d X J + 12 I Σ P I ∧ ∗ J I + 12 I Σ G ij dy i ∧ ∗ dy j + 12 I Σ B ij dy i ∧ dy j , (7.6)33here J I = M IJ A J − ∗ L IJ A J . The correct number of physical degrees of freedom are ensured by the imposition of the self-dualityconstraint [12] P I = L IJ (cid:0) M JK ∗ P K + ∗ J J (cid:1) . (7.7)This constraint is compatible with the equations of motion of the action (7.6) and Bianchi identities(7.5). In the classical theory, once a polarization is chosen, the self-duality constraint may be usedto eliminate the auxiliary degrees of freedom ˜ z m and χ Ra in the equations of motion. One thenfinds a conventional description of the system described by a set of equations of motion written interms of the physical fields z m and χ La . We now turn to the more involved issue of imposing thisconstraint in the quantum theory. There is a U (1) isometry symmetry of the target space X I → X I + ǫ I which is manifest in thesigma model as a rigid symmetry of the two-dimensional field theory. For the bosonic string, it wasshown in [19], that a self-duality constraint of the form (7.7) can be imposed in the quantum theoryby gauging a maximally isotropic subgroup of the rigid U (1) symmetry. The rigid symmetry wewish to gauge is a U (1) which acts on the coordinates as δz m = 0 , δ ˜ z m = ǫ m , δχ a = ǫ a . This may at first appear to produce a gauging of U (1) ; however, we shall require that dǫ a = ∗ dǫ a so that ǫ a only describes eight independent parameters and therefore only gauges a U (1) . Includingalso the four parameters ǫ m , there are then twelve independent parameters, corresponding to thegauge group U (1) . The effect of this chirality condition on the gauge parameter will be thatonly the right-moving part of χ a will be gauged. Following [19], the gauging proceeds by minimalcoupling in the kinetic term of the sigma model d X I → D X I = d X I + C I , where the gauge worldsheet one-forms are C I = (0 , C m , C a ) which transform as δC m = − dǫ m , δC a = − dǫ a , and C a = ∗ C a is chiral. One must also introduce the Wess-Zumino term12 L IJ P I ∧ C J . where we have introduced the duality-twisted one-forms C I = (cid:0) e N · y (cid:1) I J C J . We shall choose to setthe background one-forms A I to zero. This is done to simplify the exposition and the general case,in which A I = 0, follows in a straight-forward manner.34he gauged sigma model is S [ y i , X I , C m , C a ] = 14 I Σ M IJ P I ∧ ∗P J + 14 I Σ Θ IJ d X I ∧ d X J + 12 I Σ P I ∧ ∗J I + 12 L IJ P I ∧ C J + 14 M IJ C I ∧ ∗C J + 12 I Σ G ij dy i ∧ ∗ dy j + 12 I Σ B ij dy i ∧ dy j , (7.8)where J I = M IJ P J − L IJ ∗ P J . Following [19], one can show that, by completing the square in C m that the action splits into twoparts S [ y i , X I , C m , C a ] = S [ y i , z m , C a ] + S [Λ m ] , where Λ m = (cid:0) e N · y (cid:1) m n d ˜ z n + C m + ... appear quadratically in the action. In the path integral oneperforms a change of variables in the integration measure from a functional integration over C m toone over Λ m . We may then integrate out the Λ m . The determinant coming from integrating outthese fields, in addition to the Jacobian factor arising from the change of variables in the integrationmeasure, will contribute to a shift in the dilaton as described in [19]. One then finds that theremaining chiral gauge field C a acts as a Lagrange multiplier in the path integral, constrainingthe right-moving part of χ a to vanish and giving a sigma model for the physical fields ( y i , z m , χ aL ),where χ aL is the left-moving part of χ a . Since χ a is chiral, it is often easier to write these degreesof freedom in terms of a chiral fermion λ a . Let us consider explicitly the procedure of recovering the standard formulation from the doubledgeometry in the following example of a trivial fibration in which the mass matrix N iI J is zero, i.e.the internal space is simply T ≃ T × T . Imposing the constraint by gauging is subtle in the casewhere N iI J = 0 and the U (1) isometry group is not always well-defined on the doubled geometry T . In particular, the target space vector fields which generate the U (1) isometries are defined ineach T fibre, but may not be well defined on the full bundle T . Such issues where not addressed in[19] and we shall not elaborate on them further here, except to point out that a detailed discussionof these issues will be presented in [60]. For the simple example presented here, where N iI J = 0,no such global issues arise and we can follow the procedure outlined in [19]. We first consider thecase where the background is given by the spacetime fields M IJ and A I i , all ( y i , X I )-independent.It will aid the clarity of the exposition to assume also that A I i = 0, although this condition maybe relaxed. The constraint (7.7) is imposed by gauging a null subgroup of the algebra [ T I , T J ] = 0.The action for the gauged theory is S [ y i , X I , C I ] = S [ y i ] + S [ X I , C I ] where the theory on the baseis given by the action S [ y i ] = 12 I Σ g ij dy i ∧ ∗ dy j + 12 I Σ B ij dy i ∧ dy j , and the theory in the T × e T × T c fibres is given by S [ X I , C I ] = 14 I Σ M IJ D X I ∧ ∗D X J + 14 I Σ Θ IJ d X I ∧ d X J + 12 I Σ L IJ d X I ∧ C J , D X I = d X I + C I can be written as D X I = (cid:16) dz m d ˜ z m + C m dχ a + C a (cid:17) . For the trivial T fibration over T , which we consider in this example, the theory on thebase will not play an important role and we shall focus on the theory in the fibres given by theLagrangian L ( X I , C I ). Expanding this Lagrangian out using (7.2) and (7.4) gives L ( X I , C I ) = 14 ( g mn + A ma A na − C mp g pq C qn ) dz m ∧ ∗ dz n + 14 g mn D ˜ z m ∧ ∗D ˜ z n + 14 ( δ ab + A ma g mn A nb ) D χ a ∧ ∗D χ b − g mp C pm D ˜ z m ∧ ∗ dz n A na g nm D χ a ∧ ∗D ˜ z m + 12 ( A ma + A pa g pn C nm ) D χ a ∧ ∗ dz m + 12 dz m ∧ d ˜ z m + 12 dz m ∧ C m + 12 dχ a ∧ C a . where D ˜ z m = d ˜ z m + C m and D χ a = dχ a + C a . Completing the square in C m we have L = 12 G mn dz m ∧ ∗ dz n + 12 B mn dz m ∧ dz n + 14 D χ a ∧ ∗D χ a + 12 A ma D χ a ∧ ( ∗ dz m − dz m ) + 12 dχ a ∧ C a + 14 g mn Λ m ∧ ∗ Λ n , where Λ m = C m + d ˜ z m − g mn ∗ dz n − C mn dz n − A ma D χ a gives a ln(det( g mn )) contribution to the dilaton when we integrate over the gauge fields C m inthe path integral. The metric G mn of the sigma model is given by G mn = g mn + 12 A ma A na and is the form of the effective metric required in order for Green-Schwarz anomaly cancellation tobe consistent with supersymmetry, as found in [61]. It is helpful to define the light-cone worldsheetcoordinates ζ ± = 12 ( τ ± σ ) , ∂ ± = ∂∂ζ ± = ∂ τ ± ∂ σ . Recalling that C a = ∗ C a , we see that C − a = 0 and the Lagrangian then becomes L = − G mn ∂ + z m ∂ − z n − B mn ∂ + z m ∂ − z n − ∂ + χ a ∂ − χ a − A ma ∂ + χ a ∂ − z m − ( ∂ − χ a + A ma ∂ − z m ) C + a . We see that, as expected, the chiral one-form C + a acts as a Lagrange multiplier to impose theconstraint ∂ − χ a + A ma ∂ − z m = 0 . (7.9)In the A ma = 0 background this is just the familiar constraint that χ a is chiral, i.e. a left-mover.The condition (7.9) is a covariantization of this chirality constraint. Imposing the constraint givesthe Lagrangian L = − G mn ∂ + z m ∂ − z n − B mn ∂ + z m ∂ − z n + 12 A ma ∂ + χ a ∂ − z m .
36e encounter the familiar problem that a chiral scalar does not have a Lorentz-covariant kineticterm. In the next sub-section we consider a solution to this problem following Tseytlin [25] andpropose a Lagrangian in which manifest Lorentz invariance is lost. Alternatively, one may writethe scalars χ a as chiral fermions λ a where ∂ + χ a = 2¯ λρ + T a λ.T a is a generator in the Cartan subalgebra of E × E or Spin (32) / Z and ρ ± = ρ ± ρ + areworldsheet gamma matrices in light-cone coordinates. Substituting for ∂ + χ a in the Lagrangianabove and introducing a kinetic term for the chiral fermions, the Lagrangian becomes L = − ( g ij + B ij ) ∂ + y i ∂ − y j − ( G mn + B mn ) ∂ + z m ∂ − z n + i ¯ λρ + ∂ + λ + ¯ λρ + T a λA ma ∂ − z m , where the contribution from the theory on the T base, with coordinates y i , has been included.This is the conventional form of the sigma model for a Heterotic string on a flat background withconstant gauge and B -field [62]. In the (worldsheet) Lorentz-convariant sigma model considered above, the chirality constraint onthe bosons χ a must be imposed at the level of the equations of motion. There is a manifestlyduality-symmetric formalism proposed by Schwarz and Sen [63, 64], building on the earlier workof Tseytlin [25] in which the chirality constraint is imposed at the level of the Lagrangian. Thedrawback of this formalism is that general covariance of the worldsheet is not manifest. In [63, 64]the duality-covariant Lagrangian for a reduction of Heterotic String theory on T d was proposed L = 12 g ij η αβ ∂ α i i ∂ β y j − L IJ D τ X I D σ X J − M IJ D σ X I D σ X J + 12 ǫ αβ (cid:0) B ij ∂ α y i ∂ β y j − L IJ A I i ∂ α y i D β X J (cid:1) (7.10)where D α X I = ∂ α X I + A I i ∂ α y i (7.11)In order to describe the O (4 , Z )-twisted reduction of the kind we have been examining, it is asimple matter to twist the embedding fields ∂ α X I according to the duality-twist reduction ansatz.The non-Lorentz-covariant version of the action (7.8) is then given by L = 12 g ij η αβ ∂ α y i ∂ β y j − L IJ b P τ I b P σJ − M IJ b P σI b P σJ + 12 ǫ αβ (cid:16) B ij ∂ α y i ∂ β y j − L IJ A I i ∂ α y i b P βJ (cid:17) where we have defined b P αI = (cid:0) e N · y (cid:1) I J D α X J . Duality-twist models of this kind, for the Bosonic String, have recently been studied in [65].37
Conclusion, Discussion and Future Directions
In this article, we have shown that certain four-dimensional gauged N = 4 theories can be obtainedfrom dimensional reduction of either IIA or Heterotic supergravity. Our starting point has been theungauged N = 2 supergravity in six dimensions obtained by either compactifying IIA supergravityon a K T . We have used the O (4 , T , leading to a gauged N = 4supergravity in four dimensions.We have analyzed the possible gaugings of the four-dimensional theory and interpreted thecomponents of the structure constants in terms of both IIA and Heterotic theories. A twist matrixcan be divided into different classes for a IIA origin and a Heterotic origin. For IIA reductions, wewere able to identify the three different classes of mass matrix: D iAB , K iA and Q iA , as defining amanifold of SU (2)-structure, an H -flux and what might be called a Mirror-flux respectively. ForHeterotic reductions, we found a geometric flux, H and F -fluxes, two new classes of T-fold and aflux corresponding to a topological twisting of the Cartan torus of the the U (1) internal gaugegroup over spacetime.It should be stressed that the interpretation we gave to each class of twist holds individuallyfor each case where the only non-zero elements in the mass matrix N iI J take values in any oneof the classes D iAB , K iA or Q iA . However, for more general cases, where the mass matrix N iI J is composed of non-zero elements from more than one of the classes, the interpretation may bemore subtle. For example, if a mass matrix has components of the geometric class D iAB andcomponents of the flux K iA switched on, then the interpretation of the duality-twist reduction as acompactification is simple - we can think of it as a compactification on a K H -flux. More generally, we might consider a reduction in which elements of the class D iAB and,say, Q iA are switched on. In this case the interpretation of the reduction is not straightforward.It is not immediately clear how to make sense of such a K K T are con-jectured to be dual descriptions of the same physics. Whether or not this implies that IIA stringtheory on the six-dimensional SU (2)-structure manifolds and Heterotic string theory on the six-dimensional 1I-structure manifolds we described should also be conjectured to be dual remainsunclear. One might argue that since the duality is clearly true for the theory in a single T or K T d bundle with base S of the kind constructed in [47]. There, one can demonstratethat two descriptions are T-dual by gauging an isometry of the T d background as described by[54]. The proof of T-duality requires the existence of a well-defined isometry which generates aninvariance of the string background. One can show [60] that if one gauges an isometry that iswell-defined in the T d fibres, yet is not well-defined on the ( d + 1)-dimensional bundle, then theresult is a globally non-geometric background, such as a T-fold. One point of view is that theT-duality is still possible and indeed, there is evidence from Mirror symmetry [53] that T-dualitydoes not require the existence of globally well-defined isometries. Furthermore, there is evidencethat the existence of an isometry, even locally, is not a requirement for T-duality to be possible[21]. Another, more conservative, perspective is that, in some cases, the monodromy acts as anobstruction to the existence of the structure - a well-defined isometry - that allows T-duality to bepossible.Given this note of caution from T-duality, one might hesitate to proclaim that the IIA andHeterotic compactifications considered in this paper are dual descriptions of the same Physics. Atthe very least, one may worry that a certain class of monodromies may obstruct the duality. Theadded difficulty here is that the strong/weak duality cannot be derived from any known worldsheetconstruction, such as the gauging of isometries in T-duality, and so it is unclear what structure(playing the role of the global isometry in T-duality) one might wish to preserve in order to ensurethe duality holds. For the T-duality case discussed above, we know that if the monodromy doesnot obstruct the existence of a global isometry, then T-duality still holds. If the monodromy doesprovide an obstruction, then we are on uncertain ground. By contrast, it is difficult to say anythingabout the validity of the IIA/Heterotic duality in any of the cases we have been considering here.One way in which the proposal for such a duality might be checked would to be to directly calculateand compare non-perturbative effects such as BPS-states for the generalized case, akin to the teststhat were done for the original IIA/heterotic duality.If the Heterotic/IIA duality holds in the generalized case treated in this article, it would havesome interesting implications for string theory compactifications on generalized manifolds. Forexample, some compactifications on the IIA side that are geometric would be dual to non-geometriccompactifications of the Heterotic theory and vice versa. This would provide strong evidence thatnon-geometric backgrounds play a central role in string and M-theory.In section four we reviewed some of the arguments, coming from supergravity, for a strong/weakcoupling duality between the Heterotic string on T and the Type IIA string on K
3. By contrast, theIIB theory, compactified on K
3, leads to a six-dimensional chiral theory which cannot be directlyrelated to the the Heterotic and IIA string. However, there is evidence that a further Kaluza-Klein compactification on T , down to four dimensions, allows for a link to the IIB theory to beestablished and the duality between the Heterotic and IIA theories in four-dimensions is extended toa triality between the IIA, IIB and Heterotic theories . This triality has many remarkable featuresand was studied at length in [66]. In particular, there is a ( SL (2; Z )) ⊂ SL (2; Z ) × SO (6 , Z )symmetry for which the roles of the three SL (2; Z )’s are permuted under the triality. For example,in the Heterotic compactification, there is a SL (2; Z ) τ × SL (2; Z ) ω ⊂ SO (6 , Z ) which acts on The two Heterotic strings are T-dual to each other. τ and ω respectively) of the T base and arisesfrom the ( SL (2; Z ) τ × SL (2; Z ) ω ) / Z ≃ O (2 , Z )T-duality symmetry of the metric and B -field on the T . The third SL (2; Z ) is the manifest SL (2; Z ) symmetry in the reduced action which acts linearly on the Heterotic axio-dilaton M αβ .A natural question is whether or not this triality occurs in the compactifications we havebeen considering here, where the fibration of K T over the T base is not trivial, but hasmonodromy in O (4 , Z ). This question may be phrased in a slightly different way: can we liftthe gauged supergravities considered here to compactifications of IIB string theory? One couldconsider performing a T-duality along one of the circles of the T to obtain a description in termsof IIB string theory from the IIA theory. However, a monodromy around a cycle of the T will bean obstruction to the existence of an Abelian isometry along that direction. In the absence of anisometric direction in which to perform the duality, it is not clear that a T-dual description evenexists . If we twist around only one of the T cycles, i.e. we set β I J = 0, then there is a non-trivialgauging with structure constant t I J = α I J . Then, the circle with coordinate y ∼ y + 1 hasmonodromy exp( α I J ) and the circle with coordinate y has an Abelian isometry. We may thenperform a (possibly fibre-wise) T-duality along y to give a dual IIB description.More generally, one could consider dualizing along a cycle with non-trivial monodromy. Thereis some evidence that such a non-isometric duality can be performed [21] and, in this case, wouldgive rise to Heterotic compactifications with what has come to be known as ‘ R -flux’. For example,if we consider a generic double duality-twist compactification of the Heterotic string and dualizealong both directions of the T base we expect to find a theory with gauge algebra[ X i , Z m ] = Q min Z n + W mia Y a + f mni X n [ Z m , Z n ] = f mni Z i [ X m , Z n ] = Q mni Z i [ X m , X n ] = R mni Z i [ X i , X m ] = Q nmi X n + V ima Y a + R imn Z n [ X i , Y a ] = − δ ab V imb Z m + W mai X m + G aib Y b [ Z m , Y a ] = W mai Z i [ Y a , Y b ] = G abi Z i (8.1)The diagram (6.26) is then extended to K imn → f imn → Q imn → R imn ↓ ↓ ↓ M ima → W ima → V ima ↓ ↓ S abi → G abi ↓ h abc A potential difficulty, even for isometric duality, arises from the suggestion in [67] that the existence of an isometryis not enough to allow for the application of the Buscher rules in certain circumstances. In particular, some evidencewas presented which suggested the expected application of T-duality along the T of even a T × K y i directions and we have includedthe obvious generalization to accommodate structure constants of the form h abc .One might wonder how such structure constants h abc may be realized in compactifications. Onepossibility is that at special points in the moduli space of the Heterotic and Type II compactifi-cations, the symmetry group is enhanced [39]. For example, the U (1) internal symmetry of theHeterotic theories may be enhanced to non-Abelian E × E or Spin (32) / Z . At these enhancedpoints, the algebra includes the commutator [ Y a , Y b ] = h abc Y c + ... where the index a now runs from1 to n ≤ h abc are structure constants for E × E or Spin (32) / Z or some sub-group. Compactifications with non-zero f imn and h abc wereconsidered in [68] and are discussed briefly in Appendix A.2. It would be interesting to study thissymmetry enhancement, in a duality-covariant manner using the doubled sigma model introducedin section seven.One may think of backgrounds for which all structure constants except R imn vanish as a T fibration over the coordinates ˜ y i of the dual torus e T , conjugate to the winding modes of theHeterotic string around the T [21]. Similarly, one can view a reduction which gives the structureconstant V ima as a Heterotic generalization of R -flux backgrounds in which the gauge fields ofthe U (1) internal symmetry play an important role. The structure constants G abi indicate abackground in which the internal U (1) fibration is topologically twisted over the cycles of thedual e T .From the Type II perspective, we can think of such R -flux backgrounds as K y i , conjugate to the winding modes of the IIA string around the T . We conjecturethat N = 4 gauged supergravities with such structure constants lift to compactifications of stringtheory on such locally non-geometric backgrounds. The corresponding gauge algebra for the lower-dimensional supergravities is then[ X i , J ] = W iA η AB T B , [ X i , ˜ J ] = S iA T A , [ X i , T A ] = R iAB η BC T C − W iA ˜ J − η AB S iB J, [ T A , T B ] = R iAB Z i , [ J, T A ] = W iA Z i , [ ˜ J , T A ] = η AB S iB Z i , We can think of the W - and S -fluxes as being similar to the K - and Q -fluxes, but involving thedual torus e T . For example, the reduction ansatz for the B -field in the W -flux background isˆ b A = b A + W Ai ˜ y i , the correct interpretation of which is not clear. The R -flux denotes a smooth K e T . It remains to be seen if such constructions canprovide a basis for good Type II string backgrounds.Unlike the duality-twist backgrounds constructed in section five, which can be understood asa conventional background in a contractible patch, the only picture we have of the backgroundsconstructed as a fibration over e T is through the doubled formalism. Following the construction ofthe R -flux doubled geometry in [47], the doubled geometry for such Heterotic backgrounds wouldbe given by X = Γ \G where G is the (non-compact) group manifold for the gauge algebra (8.1)and Γ ⊂ G is a discrete (cocompact) subgroup, acting from the left, such that X is compact. Thegauge algebra could then be written as[ X i , T I ] = N I iJ T J [ T I , T J ] = N IJ i Z i X are Q i = d ˜ y i + 12 N iIJ X I d X J P i = dy i P I = (cid:0) e N · ˜ y (cid:1) I J d X J where the coordinates on the 28-dimensional doubled space X are ( y i , ˜ y i , X I ). For example, in theHeterotic case we have seen that X I = ( z m , ˜ z m , χ a ) as in the previous section. We shall return toa supersymmetric sigma model description of the doubled geometry X elsewhere. Acknowledgments
We would like to thank Jan Louis, Dan Waldram and Stefan Groot Nibbelink for helpful discussionsand Andrei Micu for helpful comments and for pointing out a number of typos in an earlier versionof the manuscript. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) inthe SFB 676 “Particles, Strings and the Early Universe”.
A Generalized Reductions and Compactifications
In this Appendix we briefly review various features of the most prominent generalized reductiontechniques.
A.1 SU (3) - and SU (3) × SU (3) -Structure Reductions As an example of the methodology and limitations of constructing lower-dimensional supergravitiesby reduction algorithms, we recall here SU (3)- and SU (3) × SU (3)-structure reductions. TheCanonical example is of a compactification on a three-dimensional compact Calabi-Yau manifold.One can find a (symplectic) basis of harmonic three-forms ( α (0) I , β (0) I ) where I = 0 , , ..h , + 1 anda basis of two-forms ω (0) A and, Hodge dual, four-forms ˜ ω (0) A , where A = 0 , , ..h , +1. The effective,massless, four-dimensional supergravity is obtained by expanding the ten-dimensional form-fieldsand fluctuations of the ten-dimensional metric in terms of this basis of forms. This constructionhas been well-studied and is equivalent to a compactification on a Calabi-Yau manifold followed bya truncation to the zero modes of the Kaluza-Klein spectrum.Massive supergravities can be constructed by adding constant fluxes to Ramond-Ramond and H -field strengths, wrapping harmonic cycles of the Calabi-Yau. For example, one can add constantelectric ( e I ) and magnetic ( m I ) fluxes to the H -field strength H = e I β (0) I − m I α (0) I + ... . Suchreductions have a clear interpretation as a flux-compactifications on compact Calabi-Yau manifolds.Following [5], it is then natural to consider expanding the reduction ansatz in terms of a non-harmonic, twisted basis of forms ω A = e B ω (0) A ˜ ω A = e B ˜ ω (0) A α I = e B α (0) I β I = e B β (0) I where dB = e I β I − m I α I is the flux on the H -field strength. Up to terms which vanish under the42ukai pairing these twisted forms of mixed degree satisfy dα I ∼ e I ˜ ω dβ I ∼ m I ˜ ω dω = m I α I − e I β I (A.1)where all other forms are harmonic. In particular, if m I = 0 so that the flux is purely electric then,under the conjectured mirror symmetry which involves the exchange of mixed-odd and mixed-evendegree forms, this background is dual to a particular class of SU (3)-structure manifolds, called half-flat manifolds as discussed in [5]. On the assumption that such a generalization of Mirror Symmetrydoes hold in this case, the fact that we can construct a rigorous Kaluza-Klein compactification fora flux compactification on a Calabi-Yau lends credence to the idea that the half-flat reduction doesindeed lift to a genuine Kaluza-Klein compactification.More generally, one can consider reductions on SU (3) × SU (3)-structure backgrounds for whichthe reduction ansatz is expanded in terms of the forms α I , β I , ω A and ˜ ω A where dα I ∼ p I A ω A + e IA ˜ ω A dβ I ∼ q IA ω A + m AI ˜ ω A dω A ∼ m AI α I − e IA β I d ˜ ω A ∼ − q IA α I + p I A β I where the notion of the exterior derivative on the internal space d : V p → V p +1 is generalized toone in which d : V p → V p ± . Although it is clear that such a reduction ansatz, in which the degreesof forms are not preserved, cannot be realized as a compactification on a conventional manifold,there is little understanding of precisely what the internal background is. There is some evidence[7, 69] that such non-geometric backgrounds should share many of the qualitative characteristicsof non-geometric 1I-structure backgrounds constructed in [10, 20, 21] and discussed briefly below. A.2 1I-Structure Flux Compactifications and Duality-Twist Reductions N = 4 gauged supergrav-ities. Of particular relevance to the 1I-structure reductions studied here, are those first presentedin the seminal work of Scherk and Schwarz [32]. There, two types of dimensional reduction wereintroduced; the first, which has become known as a compactification on a twisted torus, involved atwisting of the frame bundle over the internal manifold in the reduction ansatz so that the ansatzis written in terms of the forms σ m = σ mi ( y ) dy i , where m, n = 1 , , ..d are frame indices, y i ( i, j = 1 , , ..d ) are coordinates on the d -dimensional internal manifold. In order for the reductionto be consistent (in the sense that solutions to the lower dimensional equations of motion lift tosolutions of the full, higher-dimensional theory), the y -dependent vielbeins σ mi must be such thatthe one forms σ m satisfy the structure equation dσ m + 12 f npm σ n ∧ σ p = 0 The O (6 , h , i gives Z CY h α I , β J i = δ IJ Z CY h ω A , ˜ ω B i = δ AB f npm are structure constants for some, possibly non-compact, d -dimensional group G . Wesee then that the condition for consistency is a parallelizability condition and the twisted torus isan 1I-structure manifold. So, for example, the reduction ansatz for the metric takes the form g ( y ) = g mn σ m ⊗ σ n Because of the T-duality symmetric role H -fluxes and the structure constants f mnp play in suchreductions, the structure constants are often referred to a ‘geometric fluxes’. It was shown in [49]that such reductions could be understood as compactifications on a twisted torus , Γ \ G , where Γ ⊂ G is a discrete group, acting on the left, such that Γ \ G is compact. This identification of the twistedtorus as the compactification manifold for Scherk-Schwarz reductions elevates the Scherk-Schwarzreduction algorithm to the status of a Kaluza-Klein compactification scenario.In [18], compactifications of Heterotic supergravity on d -dimensional twisted tori were consid-ered with fluxes for the H -field and internal gauge field strength switched on so that H = 16 K mnp σ m ∧ σ n ∧ σ p + ... F a = 12 M mna σ m ∧ σ n + ... where K mnp and M mna are constants. The case of interest here is d = 6. In [18], it was assumedthat Wilson lines break the Heterotic gauge groups from E × E or Spin (32) / Z to U (1) (andwe have assumed the same is true in the Heterotic reductions to be considered in this paper). Itwas shown that the resulting four-dimensional, N = 4, gauged supergravity could be written in amanifestly O (6 , [ Z m , Z n ] = f mnp Z p + K mnp X p + M amn Y a [ X m , Z n ] = f npm X p [ Y a , Z m ] = δ ab M mnb X n (A.2)where the Z m ( m, n = 1 , , ...
6) are related to diffeomorphisms of the twisted torus, the X m arerelated to B -field antisymmetric tensor transformations and the Y a ( a, b = 1 , , ..
16) generate the U (1) internal gauge symmetry. All other commutators vanish. One can see that the gauge algebracontains a lot of information about the ten-dimensional lift of the four-dimensional supergravity.In particular, the structure constants f mnp tell us what the local geometry of the compactificationmanifold is and the structure constants K mnp and M mna contain information about the fluxes inthe internal geometry.A complementary situation, in which not all of the internal gauge group is broken (but nofluxes were turned on) was considered in [68]. The resulting N = 4 gauged supergravity has ascalar potential and a non-Abelian gauge symmetry generated by the Lie algebra[ Z m , Z n ] = f mnp Z p [ X m , Z n ] = f npm X p [ Y a , Y b ] = h abc Y c (A.3)where h abc are the structure constants for the non-Abelian internal gauge symmetry group H wherenow the indices a, b, c run from one to the dimension of H . Again we see that the structure constants f mnp encode the local structure of the compactification manifold and h abc tells us the subgroup H It was shown in [14] that the local symmetry of the supergravity was in fact generated by a Lie-algebroid, whichcontains this Lie algebra. E × E or Spin (32) / Z which is preserved by the compactification. From such reasoning, it ispossible to deduce a string theory origin of many gauged supergravities.Duality-twist dimensional reduction was introduced in [32] and was reviewed in section five.Excellent discussions of this procedure may be found in [20, 71, 72]. The general idea is to performthe compactification of the higher-dimensional supergravity on two stages: the first consists of aconventional Kaluza-Klein compactification from D + d + 1 dimensions down to D + 1 dimensionson some d -dimensional manifold, followed by a truncation to the zero modes of the Kaluza-Kleinspectrum. The resulting ( D + 1)-dimensional supergravity has a rigid symmetry symmetry group K which will generally lift to a discrete U-duality symmetry group K ( Z ) ⊂ K of the string theory[46]. The second step of the reduction involves compactification down to D dimensions on a circle,where we introduce a topological twist such that the monodromy around the circle is an element of K ( Z ). Since K ( Z ) is a symmetry of the string theory, the string theory is well defined on the ( d +1)-dimensional internal background. It can be shown that the twist by K ( Z ) does not explicitly breakany supersymmetry so that, if the d -dimensional manifold has, for example, SU ( n )-holonomy, the( d +1)-dimensional background will have SU ( n )-structure. Moreover, if the d -dimensional manifoldis of 1I-holonomy, then the ( d + 1)-dimensional background is of 1I-structure and admits a consistenttruncation of the Kaluza-Klein spectrum. Such reductions have become known as duality-twist reductions and, interpreting such duality-twist reductions in terms of compactifications of stringtheory is not always as simple as for the twisted torus compactifications reviewed above.Generally the action of K ( Z ) on the d -dimensional manifold will not be geometric and the ( d +1)-dimensional internal background will not be a Riemannian manifold, but may be non-geometric.If however, K ( Z ) does act geometrically then, it was shown in [49] that the ( d + 1)-dimensionalbackground is a twisted torus of the kind described above. In [70] duality-twist reductions ofHeterotic supergravity, where the theory was reduced to five dimensions and then twisted over acircle with monodromy in the geometric SL (5; Z ), were considered . Such reductions give rise togauged N = 4 supergravities as described above and may be thought of as a compactification on asix-dimensional twisted torus [14]. B SO (3) -symmetry of six-dimensional manifolds with SU (2) -structure Here, we want to show that the definition of j and ω in terms of spinors has an inherent SO (3)-symmetry which is analogous to the hyperk¨ahler-structure of K
3. In fact, we will show the existenceof a symmetry operator g acting on the spinors, whose action translates into an SO (3)-rotation onthe vector j Re ω Im ω , and that leaves the one-form σ invariant. Consistent truncations to N = 1 models, characterised by a positive semi-definite potential, were also studied in[70].
45e start by noting that there is an arbitrariness in our choice of spinors. Any linear combinationof the two globally well-defined spinors η , η , as long as it leaves the lengths invariant, would yieldthe two-forms j, Re ω, Im ω and a one-form σ . Let us consider a complex 2 × g acting on η − ≡ ( η − , η − ) such that η †− η − → η †− g † gη − = η †− η − . This means g ∈ SU (2), so we can write it as g = a b − b ∗ a ∗ ! , with a, b ∈ C satisfying | a | + | b | = 1. While the negative chirality spinors transform as η − → gη − ,η †− → η †− g † , the positive chirality-spinors, defined as η † + = η T − C, with C the charge conjugation matrix, transform as η + → g ∗ η + ,η † + → η † + g T . This SU (2)-rotation of the spinors corresponds to an SO (3)-rotation of the three two-forms j, Re ω, Im ω . Let G ( g ) be the action of g on any spinor bi-linear given by the SU (2)-rotation g ofthe spinors, written as a matrix multiplying the vector j Re ω Im ω . We want to consider the action of g on this vector. In terms of spinors, we have j vw = i (cid:16) η †− γ vw η − − η †− γ vw η − (cid:17) , Re ω vw = i (cid:16) η †− γ vw η − + η †− γ vw η − (cid:17) , Im ω vw = 12 (cid:16) η †− γ vw η − − η †− γ vw η − (cid:17) , and writing a = t + ix , b = y + iz we find G ( g ) = t + x − y − z ty + 2 xz xy − tz xz − ty t − x − y + z tx + 2 yz xy + 2 tz yz − tx t − x + y − z . For g ∈ SU (2) we have t + x + y + z = 1, and using this we can calculate that GG T = 1. Thismeans G ∈ O (3), and since det G = 1 for t = 1 , x = y = z = 0, we conclude G ∈ SO (3).46n the other hand, the one-form σ is left invariant under this rotation. To show this we needthat σ v ≡ η †− γ v η = − η †− γ v η . This can be shown as follows: η †− γ v η = ( η ) T Cγ v η = − ( η ) T γ Tv Cη = − ( η ) T γ Tv ( η †− ) T = − η †− γ v η . We calculate G ( g ) σ v = ( − bη †− + aη †− ) γ v ( a ∗ η + b ∗ η ) = ( | a | + | b | ) σ v = σ v . C Kaluza-Klein Compactification of the Heterotic Theory on T The Heterotic theory in ten dimensions is given, in string frame, by the Lagrangian L = e − Φ (cid:18) R ∗ ∗ d Φ ∧ d Φ − H (3) ∧ ∗ H (3) − δ ab F a (2) ∧ ∗ F b (2) (cid:19) , (C.1)where F a (2) = d A a (1) , H (3) = d B (2) − δ ab A a (1) ∧ F a (2) . We compactify the spacetime of the theory on T using the standard Kaluza-Klein reduction ansatz ds = ds + g mn ν m ⊗ ν n , B (2) = b B (2) + b B (1) m ∧ ν m + 12 B mn ν m ∧ ν n , A a (1) = b A a (1) + b A am ν m , Φ = b φ + 12 ln( g ) , (C.2)where ν m = dz m − b A m (1) , b F m (2) = d b A m (1) . The graviphoton of the reduction is b A m (1) and z m ( m = 6 , , ,
9) are coordinates on T . The notationis that a field Ψ ( p ) is of degree p in ten dimensions and a field b ψ ( p ) is of degree p in six dimensions,where the subscript for scalars is suppressed.Inserting the reduction ansatz (C.2) into the Lagrangian (C.1) gives the effective theory in sixdimensions L = e − b φ (cid:18) b R ∗ ∗ d b φ ∧ d b φ − dg mn ∧ ∗ dg mn − g mn ∗ b F m (2) ∧ b F n (2) − b H (3) ∧ ∗ b H (3) − g mn b H (2) m ∧ ∗ b H (2) n − g mn g pq b H (1) mp ∧ ∗ b H (1) nq − δ ab b F a (2) ∧ ∗ b F b (2) − δ ab g mn b F a (1) m ∧ ∗ b F b (1) n (cid:19) , b H (3) = d b B (2) + b B (1) m ∧ b F m (2) − δ ab b A a (1) ∧ d b A b (1) − δ ab b A am b A b (1) ∧ b F m (2) , b H (2) m = d b B (1) m + B mn b F n (2) − δ ab b A a (1) ∧ d b A bm − δ ab b A am d b A b (1) − δ ab b A am b A bn b F m (2) , b H (1) mn = dB mn − δ ab b A am d b A bn , b F a (2) = d b A a (1) − b A am b F m (2) , b F a (1) m = d b A am . (C.3)Using the field redefinitions b C (2) = b B (2) − B (1) m ∧ b A m (1) , b C (1) m = b B (1) m − δ ab b A am b A b (1) , b C mn = B mn + 12 δ ab b A ma b A nb , we see that the reduced theory has gauge group U (1) and may be written in a manifestly O (4 , L = e − b φ (cid:18) b R ∗ ∗ d b φ ∧ d b φ + 14 T r (cid:16) d c M ∧ ∗ d c M − (cid:17) − b H (3) ∧ ∗ b H (3) − c M IJ b F I (2) ∧ ∗ b F J (2) (cid:19) (C.4)where b H (3) = d b C (2) − L IJ b A I (1) ∧ d b A J (1) , b F I (2) = d b A I (1) . Also
I, J = 1 , , ..
24 and c M IJ = g mn − B np g pm − g mn b A na − B mp g np g mn + g pq B mp B nq + δ ab b A ma b A nb b A ma + B mp g pn b A na − b A na g mn b A ma + b A na g np B mp δ ab + b A ma g mn b A nb , where the O (4 ,
20) vector b A I and corresponding field strength b F I are b A I = b A (1) m b B (1) m b A (1) a , b F I = b F m (2) b H (2) m − B mn b F n (2) − δ ab b A a (1) ∧ d b A b (1) b F a (2) , and the O (4 ,
20) invariant is L IJ = . D Kaluza-Klein Compactification of IIA Supergravity on K The O (4 , K
3. We define ∗ Ω A ≡ H AB Ω B , and it is not48ard to show that it satisfies H AC H C B = δ AB and η [ A | C H C | B ] = 0, so that H AB ∈ SO (3 , R ) SO (3 , R ) × SO (19 , R ) .H AB therefore parametrizes 57 of the metric moduli coming from the two-forms, the last one isgiven by an overall size modulus. The moduli H AB and ρ , as defined, do not depend on the sixnon-compact directions. It is useful to introduce e H AB and e ρ which, as we shall see in the nextsection, do depend explicitly on the remaining six directions. D.1 Dimensional Reduction
The Lagrangian of Type IIA supergravity is L IIA = e − Φ (cid:18) R ∗ d Φ ∧ ∗ d Φ + 12 d B (2) ∧ ∗ d B (2) + 12 d A (1) ∧ ∗ d A (1) + 12 ( d C (3) − A (1) ∧ d B (2) ) ∧ ∗ ( d C (3) − A (1) ∧ d B (2) ) − B (2) ∧ d C (3) ∧ d C (3) (cid:19) , where Φ is the dilaton, B is the Kalb-Ramond field and A (1) and C (3) are Ramond-Ramond fields.We consider a Kaluza-Klein reduction of IIA supergravity on K
3. The Kaluza-Klein reductionansatz is A (1) = e A (1) , B (2) = e B (2) + e b A Ω A , C (3) = e C (3) + e C (1) A ∧ Ω A . The resulting six-dimensional theory is best written in terms of an O (4 ,
20) matrix f M IJ whichtakes values in the coset O (4 , / ( O (4) × O (20)) and is given by f M IJ = e − e ρ + e H AB e b A e b B + e e ρ e C e e ρ e C − e H CB e b C − e e ρ e b B e Ce ρ e C e e ρ − e e ρ e b B − e H BA e b B − e e ρ e b A e C − e e ρ e b A η AC e H C B + e e ρ e b A e b B , where e C = η AB e b A e b B and η AB is the intersection matrix for K
3. The symmetric matrix of scalarssatisfy f M IK L KL f M LJ = L IJ with L IJ , the invariant of O (4 , L IJ = − − η AB . In the theory are also a metric, a dilaton e φ and a two-form field e B , and 24 gauge fields. Ofthese gauge fields, one comes from the ten-dimensional gauge field, 22 from the expansion of thethree-form field in the two-forms of K
3, and the last one is the dual of the three-form field in sixdimensions. The six-dimensional supergravity Lagrangian (this can be found, for example, in [ ? ])49s given by L IIA = e − e φ (cid:18) e R ∗ ∗ d e φ ∧ d e φ + 14 d f M IJ ∧ ∗ d f M IJ − e H (3) ∧ ∗ e H (3) − f M IJ e F I (2) ∧ ∗ e F J (2) (cid:19) − L IJ e B (2) ∧ e F I (2) ∧ e F J (2) , where the field strengths are e H (3) = d e B (2) , e F I (2) = d e A I (1) . D.2 Gauge Algebra
The gauge algebra of this theory is U (1) , where U (1) ⊂ U (1) is generated by antisymmetrictensor transformation with parameters λ A associated to each of the harmonic two cycles of the K U (1) is inherited directly from ten dimensions as the Abelian gauge transformation ofthe Ramond field A . We denote the generator of this transformation by J . In six dimensions thethree form part of the Ramond field C (3) is dual to a one form ˜ C (1) . A final U (1) comes from theAbelian gauge transformations of this field, generated by ˜ J . These generators can be written as an O (4 ,
20) vector T I , with algebra [ T I , T J ] = 0 where T I = J ˜ JT A . E Duality-Twist Reductions Over T We have seen that the Heterotic theory compactified in T and the IIA theory compactified on K L = e − b φ (cid:18) b R ∗ ∗ d b φ ∧ d b φ + 14 d c M IJ ∧ ∗ d c M IJ − b H (3) ∧ ∗ b H (3) − c M IJ b F I (2) ∧ ∗ b F J (2) (cid:19) . As noted in [38], the theory has SL (2) × O (4 ,
20) rigid symmetry, a discrete subgroup of which liftsto a duality symmetry of the full string theory [46]. In this Appendix we present we consider afurther reduction on T , twisting with two commuting elements of O (4 ,
20) over the two cycles ofthe T , to give an effective theory in four dimensions. E.1 Dimensional Reduction
Let y i , i = 1 , T coordinates. The reduction ans¨atze are ds = ds + g ij ν i ⊗ ν j , b A I (1) ( x, y ) = ( e N · y ) I J (cid:16) A J (1) ( x ) + A Jj ( x ) ν j (cid:17) , c M IJ ( x, y ) = ( e N · y ) I K M KL ( x )( e N T · y ) I L , b C (2) ( x, y ) = B (2) ( x ) + B (1) i ( x ) ∧ ν i + 12 B ij ( x ) ν i ∧ ν j , ν i = dy i − V i (1) , and the twist matrix is ( e N · y ) I J = exp( N iJ I y i ) . The structure constants N iJ I encode the monodromy around the i = 1 , N iJ I = (cid:16) α J I β J I (cid:17) , where e α is the SO (4 ,
20) monodromy around the y ∼ y + 1 direction and e β is that around the y ∼ y + 1 direction where [ α, β ] = 0. The condition that the two twists commute is [ α, β ] I J = α I K β K J − β I K α K J = 2 N I [ i | K N | j ] KJ = 0. This is equivalent to the Bianchi identity d ( e N · y ) I J = ( e N · y ) I L N LiK N jKJ dy i ∧ dy j = 0 (E.1)which states that the second cocycle is trivial.The field strength reductions are b F I (2) ( x, y ) = (cid:0) e N · y (cid:1) I J n(cid:16) f J (2) + A Ji G i (2) (cid:17) + f J (1) i ∧ ν i o , where f I (2) = d A I (1) − N iJ I V i (1) ∧ A J (1) ,f I (1) i = d A Ii − N jJ I V j (1) A Ji − N iJ I A J (1) ,G i (2) = dV i (1) . It is useful to make the following field redefinitions: C (1) i = dB (1) i − L IJ A Ii A J (1) , C ij = B ij + 12 L IJ A Ii A Jj . These potentials and field strengths can be combined into the O (6 ,
22) multiplets C = V i (1) C (1) i A I (1) , F M = G i H i f I , where H (2) i = dC (1) i + 12 N iIJ A I (1) ∧ A J (1) , and N iIJ = − N iJI = L IK N iJ K . The reduced Lagrangian may be written as L = e − φ (cid:18) R ∗ ∗ dφ ∧ dφ + 12 ∗ H (3) ∧ H (3) + 14 ∗ D M MN ∧ D M MN − M MN ∗ F M (2) ∧ F N (2) + V ∗ (cid:19) , where L MQ t NP Q = t MNP and the scalar potential is given by V = − M MQ M NT M P S t MNP t QT S + 14 M MQ L NT L P S t MNP t QT S . M MN span the coset O (6 , /O (6) × O (22) where M MN = g ij + M IJ A Ii A Jj + g kl C ik C jl g ik C jk g jk C ij L IK A Kk + M IK A Ki g ik C jk g ij g ij L IK A Kj g jk C ij L JK A Kk + M JK A Ki g ij L IK A Kj M IJ + g ij L IK L JL A Ki A Lj . The O (6 ,
22) invariant is L MN = L IJ . E.2 Gauge Symmetry
In ten dimensions the theory has the antisymmetric tensor transformation symmetry B → B + d Λ (1) . The reduction ansatz for the parameter Λ (1) on T is Λ (1) = b λ (1) + b λ m ν m . The remainder of the U (1) gauge symmetry comes from the four U (1) isometries of the T , z m → z m + ω m , underwhich δ b A m (1) = dω m and the U (1) gauge transformations δ b A a (1) = dǫ a . In six dimensions this U (1) gauge symmetry acts on the fields as δ T b A I (1) = d b λ I , δ T b C (2) = d b λ (1) + 12 L IJ b λ I b F J (2) . where we have defined b λ I = ω m λ m ǫ a . Antisymmetric tensor transformations
The duality twist reduction ansatz for the six-dimensional gauge parameters b λ (1) and b λ A is b λ I = (cid:0) e N · y (cid:1) I J λ J , b λ (1) = λ (1) + λ i ν i . We denote the infinitesimal variation of the fields under this transformation by δ T . It is easy toshow, by calculating d b λ I , that the four-dimensional fields transform as δ T A I (1) = dλ I + N JiI λ J V i (1) ,δ T A I = N JiI λ J ,δ T B (2) = dλ (1) + λ i F i (2) − L IJ λ I (cid:16) F J (2) − A Ii G i (2) (cid:17) ,δ T B (1) i = dλ i − L IJ λ I F J (2) . (E.2)Using the field redefinition C (1) i = B (1) i − L IJ A Ii A J (1)
52e see that C (1) i transforms as a connection: δ T C (1) i = dλ i + L JK N IiK λ I A J (1) . (E.3) T Diffeomorphisms
The theory must be invariant under reparametrizations of the circle coordinate y i → y i + ω i . The matrix e N · y changes as (cid:0) e N · y (cid:1) I J → (cid:0) e N · y (cid:1) I K (cid:0) e N · ω (cid:1) K J = (cid:0) e N · y (cid:1) I K (cid:0) δ K J + N JiK ω i + ... (cid:1) .From this is it easy to see how the four dimensional fields must transform in order for the six-dimensional ansatz to be invariant δ Z A I = − N JiI A J ω i , δ Z A I (1) = − N JiI A J (1) ω i , δ Z V i (1) = dω i . (E.4) Symmetry Algebra
We define δ Z = ω i Z i , δ T = λ I T I , δ X = λ i X i (E.5)where Z i , X i and T I are generators of gauge transformations with parameters ω i , λ i and λ I respec-tively. It is not hard to show from (E.2), (E.3) and (E.4) above that the Lie algebra of the gaugegroup is [ Z i , T I ] = N IiJ T J , [ T I , T J ] = N IJi X i , with all other commutators vanishing, where we have defined N IJi = − N JIi = L IK N JiK . Note that N ( IJ ) i = 0 as we are gauging a subgroup of O (6 , E.3 Rewriting the Four-Dimensional Lagrangian
The gauged theory we have found by dimensional reduction takes the general form L = e − φ (cid:18) R ∗ dφ ∧ ∗ dφ − H (3) ∧ ∗H (3) + 14 ∗ D M MN ∧ D M MN − M MN ∗ F M ∧ F N (cid:19) + V ∗ . (E.6)In four dimensions, we may write this in the Einstein frame using the four-dimensional Weylrescaling g µν ( x ) → e φ ( x ) g µν ( x ) . The individual terms in the string frame action rescale as e − φ √− gR → √− g (cid:18) R + 32 ( ∂φ ) (cid:19) ,e − φ √− gg µλ g νσ g ρτ H µνρ H λστ → e − φ √− g H .
53n the Einstein frame the action is written as L = R ∗ dφ ∧ ∗ dφ − e − φ H (3) ∧ ∗H (3) + 14 ∗ D M MN ∧ D M MN − e − φ M MN ∗ F M (2) ∧ F N (2) + V ∗ . We now consider the dualization of C (2) to a scalar. Let G (3) = dC (2) = H (3) − Ω (3) , where Ω (3) is a Chern-Simons term such that d Ω (3) = 12 L MN F M (2) ∧ F N (2) The Bianchi identity dG (3) = 0 is imposed by the Lagrange multiplier χ L = R ∗ dφ ∧ ∗ dφ − e − φ H (3) ∧ ∗H (3) + 14 ∗ D M MN ∧ D M MN − e − φ M AB ∗ F A (2) ∧ F B (2) + V ∗ dχ ∧ G (3) . The G (3) equation of motion is G (3) = e − φ ∗ dχ . Substituting this into the Lagrangian gives thedual formulation L D = R ∗ dφ ∧ ∗ dφ + 12 e φ ∗ dχ ∧ dχ + 14 ∗ D M MN ∧ D M MN − e − φ M MN ∗ F M (2) ∧ F N (2) + V ∗ − dχ ∧ Ω (3) . 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