hepth/0102110
Massive Type IIA Theory on K3 ^{1}^{1}1Work supported by: AvH – the Alexander von Humboldt foundation, DFG – the German Science Foundation, GIF – the German–Israeli Foundation for Scientific Research, DAAD – the German Academic Exchange Service.
Michael Haack, Jan Louis and Harvendra Singh ^{2}^{2}2email: haack, j.louis,
Fachbereich Physik, MartinLutherUniversität HalleWittenberg,
FriedemannBachPlatz 6, D06099 Halle, Germany
ABSTRACT
In this paper we study compactification of tendimensional massive type IIA theory with all possible RamondRamond background fluxes turned on. The resulting sixdimensional theory is a new massive (gauged) supergravity with an action that is manifestly invariant under an duality symmetry. We discover that this sixdimensional theory interpolates between vacua of tendimensional massive IIA supergravity and vacua of massless IIA supergravity with appropriate background fluxes turned on. This in turn suggests a new 11dimensional interpretation for the massive type IIA theory.
1 Introduction
Recently there has been renewed interest in the study of gauged/massive supergravity theories due to the AdS/CFT correspondence [1]. In gauged supergravity theories either a subgroup of the symmetry group (the automorphism group of the supersymmetry algebra) or isometries of the scalar manifold are gauged by some of the vector fields in the spectrum [2]. Such theories can be constructed from their ungauged ‘cousins’ by adding appropriate terms to the action or field equations and changing the supersymmetry transformation laws accordingly. This procedure does not change the spectrum and the number of supercharges but generically does change the properties of the ground state. For example, a Minkowskian spacetime which always is a solution of the ungauged theory ceases to be a ground state of the gauged supergravity in most cases. Instead the supersymmetric ground states are often of the antideSitter (AdS) type or domainwall solutions.
A particular subclass of gauged supergravities are the massive supergravities. In these theories some of the vector (or tensor) fields become massive through a generalized Higgs mechanism. Such theories occur for a specific gauging which allows the possibility of a Higgstype mechanism. A prominent example is the massive type IIA supergravity in constructed by Romans [3]. In string theory massive supergravities typically arise in lower dimensions through generalized ScherkSchwarz reduction [4] provided that some field strength of the form tensor field is given a nontrivial background value (flux) along the compact directions [5, 6, 7]. Such background fluxes can be turned on consistently if the action or the field equations depend on only through its field strength .
Gauged supergravities have also been studied in the context of string dualities and branes. For example, the massive type IIA supergravity has a domain wall solution which preserves of the 32 supercharges of type IIA [5] and can be given an interpretation of a type IIA D8brane [8]. This observation has led to the search for possible duality connections involving massive supergravity theories analogous to the existing Uduality relations in the massless cases. This required the construction of other massive supergravities in lower dimensions through generalized dimensional reduction [5, 6, 11, 13, 14, 7, 9, 10, 12, 16, 17, 15]. However, it has remained an interesting and open question to what extent these generalized compactifications respect the duality properties of the massless cases. Along this line, Kaloper and Myers [16] showed that a generalized ScherkSchwarz reduction of the heterotic string on a torus can still be written in a manifestly invariant form provided Tduality transformations also act on the background fluxes.
In a parallel line of developments CalabiYau compactifications with background fluxes have also been studied because of their phenomenological properties [19, 18, 20, 22, 23, 24, 28, 29, 30, 31, 33, 34, 35, 36, 38, 21, 26, 27, 25, 32, 37]. One finds that background fluxes generically generate a potential for some of the moduli fields of the theory without fluxes and as a consequence the moduli space – and hence the arbitrariness of the theory – is reduced. In addition the resulting ground states can break supersymmetry spontaneously.
In this paper we study a generalized reduction of tendimensional massive typeIIA theory with all possible background fluxes turned on. Our goal is to investigate the fate of the perturbative duality symmetry and the nonperturbative Sduality with the heterotic string compactified on . We also discuss the relation of massive type IIA theory with Mtheory. The paper is organized as follows. In section 2 we recall the massive type II theory and derive its compactification with all RR background fluxes turned on. We find that the resulting sixdimensional theory is a new massive (gauged) supergravity with manifest symmetry. In section 3 we study domain wall solutions of this massive supergravity. We show that the D8 brane of massive type IIA when wrapped on is Tdual to a solution of massless type IIA with an appropriate fourform flux turned on. Thus Tduality interpolates between vacua of massive and massless type IIA theories. This property also allows to further relate massive type II vacua to eleven dimensional solutions. This is discussed in section 4 where massive IIA theory is given a concrete elevendimensional interpretation. Finally we conclude in the section 5.
2 Compactification of Massive Type IIA Supergravity
The type IIA supergravity in ten dimensions, which describes the low energy limit of type IIA superstrings, contains in the massless spectrum the graviton , the dilaton , an NSNS twoform , a RR oneform and a RR threeform . The fermionic fields consist of two gravitini and two Majorana spinors. It was shown by Romans [3] that this supergravity can be generalized to include a mass term for the field without disturbing the supersymmetry. The field becomes massive through a Higgs type mechanism in which it eats the vector field . The supersymmetric action for massive IIA theory in the string frame is given by [3]
(1)  
where we have adopted the notation that every product of forms is understood as a wedge product. The signature of the metric is and for a form we use the convention
(2) 
while the Poincare dual is given by
(3) 
and . is the mass parameter and the various field strengths in the Lagrangian (1) are defined as
(4) 
and only appear through their derivatives in the Lagrangian (1) and thus obey the standard form gauge invariance . The twoform on the other hand also appears without derivatives but nevertheless the ‘Stueckelberg’ gauge transformation
(5) 
leave the Lagrangian invariant. Finally, the conventional massless type IIA theory is recovered from the action (1) in the limit .
Before we turn to the compactification let us first recall some facts about manifolds. is a compact, Ricci flat complex manifold with Betti numbers . Thus there exist 22 harmonic twoforms and their intersection matrix
(6) 
is a Lorentzian metric with signature . We choose conventions for the twoforms such that is given by
(7) 
and represents the dimensional identity matrix. Since is fourdimensional the Hodgedual of the harmonic 2forms can again be expanded in terms of twoforms. More precisely one has
(8) 
where the matrix depends on the moduli parameterizing the deformations of the metric of constant volume on . For holds which implies
(9) 
Let us now turn to the compactification of the massive type IIA theory on . The standard KaluzaKlein reduction considers the theory in a spacetime background , where is a noncompact dimensional manifold with Lorentzian signature while is a dimensional compact manifold. This ansatz is consistent whenever the spacetime background satisfies the dimensional field equations. However, for massive type IIA there are no direct product solutions; instead the ground states are domainwall solutions [5]. A similar situation occurs in the compactification of massive type IIA discussed in [5]. However, as argued there one can also expand around a solution which strictly speaking is not a direct product , but rather the compact manifold is allowed to vary over the spacetime manifold .^{3}^{3}3Or in other words the moduli of are not constant in the background but vary over . For the case at hand such a solution exists and is given by the D8brane solution of massive type IIA theory wrapped on a [5, 42]. We discuss this solution in the next section and will find that it can be interpreted as the product of a dimensional domain wall with a warped whose volume varies over the transverse direction. This spacetime dependence of the volume ensures that the equations of motion of massive type IIA theory are fulfilled.
Thus for the 10dimensional metric we take the standard ansatz
(10) 
where is the metric and are the coordinates of . is a fixed background metric on (with coordinates ) and denotes the allowed deformations of the metric. These deformations are parameterized by moduli where the extra modulus corresponds to the overall volume of .^{4}^{4}4The metric on the moduli space of sigma models on does not change with respect to the standard KaluzaKlein reduction of massless type IIA theory and can therefore be taken from that case [41, 39, 40].
For the dilaton and the twoform we take the standard ansatz precisely as in massless type IIA theories
(11) 
where is a twoform in and the are 22 additional scalar fields. (Thus the total number of scalars is .)
For the oneform and the threeform we take a generalized KaluzaKlein ansatz where background values (fluxes) of the corresponding field strengths are included
(12) 
This ansatz introduces 23 new mass parameters and parameterizing fluxes along the 22 twocycles and the fourcycle on . This generalization is possible since appear in the action only with derivative couplings (i.e. via their field strength) and an appropriate background value can be consistently turned on [7].
Altogether, the bosonic spectrum of the reduced sixdimensional theory consists of the graviton , the two form , 22+1 oneform gauge fields , a threeform and 81 scalar fields . From their definition it is clear that and both transform in the vector representation of . In a threeform is Poincare dual to a oneform and thus the above spectrum combines into a gravitational multiplet consisting of the graviton, the twoform, four vector fields and the dilaton and 20 vector multiplets each containing a oneform and four scalars.
In order to obtain the action of the massless modes for this theory we substitute the ansatz (10)(12) into the action (1). The resulting sixdimensional bosonic action reads
(13)  
where
(14) 
The index takes values and we have defined
(15) 
is the Poincare dual of the 4form defined as
(16) 
where the indices are contracted with the metric and is the modulus associated with the overall volume of the
(17) 
From eq. (16) we learn that is an invariant 2form field strength. Finally, the scalar matrix which appears in the action (13) depends on the moduli of and the 22 in the following way
(18) 
where . The submatrix depends only on the moduli of (without the volume) and determines the inverse of the matrix introduced in (8)
(19) 
The matrices and satisfy
(20) 
where is the metric
(21) 
The action (13) is invariant under global transformations acting according to
(22) 
where . These symmetry transformations except the transformations of the mass parameters are precisely the Tduality transformations of the standard (massless) type IIA theory on . Indeed, in the limit the action (13) reduces to the action of massless type IIA on [41]. In the massive case the symmetry can be maintained if the 24 mass parameters transform in the vector representation of . This transformation of masses means that under the action of the duality group a massive IIA compactified on with fluxes transforms into another massive IIA with a different set of background fluxes as determined by the symmetry. We emphasize that the action (13) represents a unification of a wide class of massive supergravities related to each other via the action of symmetry. Any particular choice of the mass vector represents a different massive theory. For example, if we set in the action (13) the theory represents a pure massive IIA compactified on without any fluxes. Similarly a different choice in the above action represents a sixdimensional massive theory obtained through generalized reduction of type IIA on with 4form flux. We will show explicitely in the next sections that these two theories with single mass parameters are related via an element of the duality symmetry (22).
Apart from ordinary gauge invariance the field strengths given in (14), the Bianchi identities as well as the action (13) exhibit a Stueckelberg type gauge symmetry of the form
(23) 
where is a oneform. This gauge invariance can be used to absorb one of the 24 gauge fields into and render it massive. Such a gauge fixed version of the theory breaks the symmetry spontaneously since in a given vacuum only one of the fields can be absorbed. Nevertheless, at the level of the action the symmetry is manifest and will play an important role in determining new vacuum configurations.
3 Domain Wall solutions
Generally, massive supergravities admit domainwall solutions which preserve half of the supersymmetries. So we also expect this to be the case for the sixdimensional massive theory in (13). It is known that the tendimensional massive IIA theory has a D8brane (domainwall) solution which preserves 16 supercharges [5]. In the string frame metric it is given by
(24) 
where is a harmonic function of the transverse coordinate and all other fields have vanishing backgrounds. This solution can be compactified by wrapping four of the worldvolume directions on . In other words one can also write a D8brane solution with four of its brane directions being along
(25) 
That this is indeed a solution of the equations of motion has been shown in [42], where it is argued that one can replace the spatial part of the D8brane’s worldvolume by any manifold of the form with a Ricciflat . It is further shown that for the solution preserves 8 supercharges.^{5}^{5}5As anticipated in section 2 the solution (3) is a warped product of a sixdimensional domain wall and a . The corresponding sixdimensional domainwall solution can be written as
(26) 
while all other sixdimensional background values vanish. The breathing mode defined in (17) is given by in this example. Furthermore in (3) we have chosen a special point in the moduli space of where (8) reads
(27) 
The solution (3) still has unbroken supersymmetries.
Now, by applying an duality transformation (22) on the background in (3) new solutions with nontrivial RR fluxes can be generated. Let us first consider the special case where the matrix is taken to be
(28) 
Inserting and the configuration (3) in (22) we get and
(29) 
while the sixdimensional metric and the dilaton remain the same. The transformed mass vector implies that the new configuration is a solution of a massless IIA compactified on with an equivalent amount of fourform flux turned on along . When (29) is lifted to ten dimensions we get the following new configuration
(30) 
Since this solution is obtained by the duality transformation (22) the number of preserved supercharges is unchanged. It can also be checked explicitly that (3) leaves supercharges unbroken. Since under the duality transformation (29) , which amounts to making Tduality along all the four directions of , the background (3) represents a stack of D4branes filling the with nontrivial 4form flux along . This configuration can be further lifted to eleven dimensions as we will see in the next section.
By applying an Tduality transformation we transformed a solution of massive type IIA to a solution of massless type IIA with nontrivial fourform flux. Thus, the duality interpolates between vacua of massive IIA and massless IIA. In spirit this is similar to the situation encountered in the case of massive type II duality in [5].
Further solutions in can be generated by using other elements of the duality group which mix the mass with the fluxes of the 2cycles. Let us consider the case
(31) 
which mixes and . Note that the duality element in (31) preserves the metric as can be checked using the explicit form for given in (7). When we apply (31) on the configuration (3), using (22) we find and
(32) 
while all other sixdimensional fields remaining unchanged. Again from the analysis of the new massvector it can be seen that this new sixdimensional configuration corresponds to massless IIA compactified on but now with a twoform flux along a 2cycle of . When lifted to ten dimensions we have a new solution of massless IIA theory as
(33) 
where is the deformed metric on such that the deformation corresponds to the nontrivial moduli matrix in (32). is a 2cycle on . This background configuration (3) preserves the same amount of supersymmetries as the one in (3).
One can go through a similar analysis for solutions which depend on more than one mass or flux parameter. Let us consider the following solution of the massive IIA theory in (1)
(34) 
For the special case when it reduces to
(35) 
which preserves supercharges as can be checked explicitly. This can be compactified to six dimensions and using the matrix of (31) it can be transformed as
(36) 
Lifting this solution back to ten dimensions we get
(37) 
is selfdual since . This is the solution obtained in [7] for massless type IIA.
4 A Lift to Mtheory
It is conjectured that the strong coupling limit of type IIA is governed by Mtheory [43] whose low energy limit is believed to be 11dimensional supergravity [44]. In fact one can obtain massless type IIA supergravity as an compactification of 11dimensional supergravity and in this way all brane solutions of massless type IIA can be lifted to eleven dimensions. However, the 8brane solution of massive IIA given in (24) cannot be lifted easily to 11dimensions since that would require a massive version of 11dimensional supergravity [11, 13] which does not exist [45, 46]. In ref. [15] a general relation between massive IIA theory and M and Ftheory has been proposed. We will see here that taking the detour of compactifying massive IIA on provides another possibility to relate the 8brane solution to Mtheory.^{6}^{6}6It would be interesting to understand the connection with ref. [15] in more detail. In order to show this in slightly more detail let us write down the map among the fields of massive and massless type IIA on . As discussed above, the sixdimensional theory in (13) with the mass vector represents an ordinary type IIA on with RR4flux. On the other hand the theory with mass vector represents massive IIA on without any RRflux. These two massive sixdimensional theories are related by the duality element (28). In section 3, eqs. (3)–(3) we displayed the Tduality between a domain wall solution of massive IIA and a stack of solitonic D4branes of massless type IIA. In fact one can not only map the solutions onto each other but the entire actions. Under the transformation given in (28) massive IIA with no fluxes transforms into massless IIA with 4form flux as
(38) 
Under this map the mass parameter of massive type IIA is mapped to the fourform flux on of massless IIA and vice versa.
Since massless type IIA on is equivalent to Mtheory on , all solutions of massive IIA can be lifted to eleven dimensions by first mapping them to solutions of the massless theory using the map (38). The solution given in (3) corresponds to the following eleven dimensional solution
(39) 
where is the 4form field strength of elevendimensional supergravity. The solution (4) is a stack of M5branes which couple magnetically to the 4form field strength.^{7}^{7}7 It is possible to establish a similar map as in (38) between the backgrounds of massive IIA compactified on (with ) and the backgrounds of massless IIA on (with ). However, it is not clear whether those can be lifted to 11 dimensions since this would require a globally defined .
Thus it seems that the conjectured duality between type IIA compactified on and Mtheory on extends to the massive case. It is shown in ref. [10] that for Mtheory on the compactifications on and commute even in the presence of 4form flux on . Here we have seen that massless type IIA compactified on with ‘4form flux along ’ is Tdual to massive IIA compactified on without flux. Thus we are led to conjecture that Mtheory on with ‘4form flux on ’ is dual to massive type IIA compactified on a without flux. In this duality the mass of Romans theory is mapped to the 4form flux of Mtheory along .
Mtheory on 4Flux  
massive IIA on  massless IIA on 4Flux 
5 Conclusion
In this paper we derived the action for compactifications of tendimensional massive type IIA theory with all RamondRamond background fluxes turned on. We found that the resulting sixdimensional theory is a new massive (gauged) supergravity with an action having manifest duality symmetry provided the mass (flux) parameters transform accordingly. ¿From this we learn that the perturbative Tduality survives even at the massive level when appropriate masses and fluxes are switched on.
We have seen in this paper that the massive sixdimensional theory of (13) interpolates between tendimensional massive and massless type IIA theories. The wrapped D8brane solution of massive type IIA turns out to be Tdual to a supersymmetric solution of massless IIA theory on with fourform flux. The relationship between massless and massive IIA on also suggests a new 11dimensional interpretation of massive IIA theory.
As it is known, in addition to the perturbative Tduality there also is a nonperturbative Sduality between massless type IIA on and massless heterotic string on [39, 40, 41]. Both theories have the same perturbative Tduality group and their respective dilatons are related by . Let us recall that there also exists an symmetric massive compactification of the heterotic string on [16]. However, the nonperturbative Sduality seems no longer to be valid in the massive theories. The major difference is that in the type IIA theory (13) it is the tensor field B which becomes massive after eating the vector fields while in the heterotic theory vector fields become massive after eating some scalars [16]. This is similar to the situation encountered in the duality between massive Mtheory on and heterotic theory on [7]. Furthermore there is a runaway dilaton potential in both theories driving the dilaton to weak coupling. As a consequence strongweak Sduality can no longer hold.
Finally, it is interesting to consider a further reduction of the massive sixdimensional theory (13) on and explore some duality relationship with type IIB on with fluxes. It is clear from the action (13) that there are no axions in this theory so the further reduction on will not produce any new mass parameter. On the other hand the standard reduction of type IIB on produces exactly 24 axions in six dimensions which upon further generalized compactification on will generate 24 mass parameters in five dimensions. Through a duality relation one should be able to relate the 24 mass parameters of massive IIA on to those of type IIB on . This should be related to the duality of massive type IIA and type IIB in [5].
ACKNOWLEDGMENTS
The work of M.H. is supported by the DFG (the German Science Foundation) and the DAAD (the German Academic Exchange Service). The work of J.L. is supported by GIF (the German–Israeli Foundation for Scientific Research), the DFG and the DAAD. The work of H.S. is supported by AvH (the Alexander von Humboldt foundation).
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