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Massive IIA supergravities

We perform a systematic search for all possible massive deformations of IIA supergravity in ten dimensions. We show that there exist exactly two possibilities: Romans supergravity and Howe-Lambert-West supergravity. Along the way we give the full details of the ten-dimensional superspace formulation of the latter. The scalar superfield at canonical mass dimension zero (whose lowest component is the dilaton), present in both Romans and massless IIA supergravities, is not introduced from the outset but its existence follows from a certain integrability condition implied by the Bianchi identities. This fact leads to the possibility for a certain topological modification of massless IIA, reflecting an analogous situation in eleven dimensions.


Introduction
Romans massive supergravity [1] has attracted a lot of interest following the observation that its mass parameter (cosmological constant) may be thought of as sourced by the D8 brane of type IIA [2]. If string theory is to be understood as embedded in M-theory, it would be desirable to have an eleven-dimensional understanding of Romans supergravity. The latter, however, has no covariant eleven-dimensional lift 1 , owing to certain no-go theorems forbidding any straightforward introduction of mass in eleven-dimensions [6].
There does exist however a topological modification of eleven-dimensional supergravity (subsequently dubbed 'MM-theory'), as pointed out by Howe in [7], which allows the introduction of a mass parameter upon compactification to ten dimensions. In this way one obtains the Howe-Lambert-West supergravity of reference [8]. The latter contains a lot of interesting physics [9,10,11]; it is nevertheless much less studied despite that, contrary to Romans supergravity, it has a well-understood covariant eleven-dimensional origin and it has been shown to admit a de Sitter vacuum 2 . HLW supergravity can also be obtained by a generalized Scherk-Schwarz reduction of the equations-of-motion of ordinary eleven-dimensional supergravity [12].
It would be desirable to have an understanding of the relation between Romans and HLW supergravity from a purely ten-dimensional perspective. We would also like to know how unique these supergravities are and whether there exist or not other massive deformations of type IIA.
In this paper we shall address these questions by working in ten-dimensional IIA superspace. The starting point of our search is the supertorsion Bianchi identities (BI) -which every supersymmetric system should satisfy. In this purely-geometric approach no form superfields are introduced by hand, unlike in the usual superspace formulation of IIA supergravity [13]; the field-strengths of the various supergravity forms 'sit' inside the components of the torsion. We do not make any further assumptions other than that any deviation from massless IIA should appear at canonical mass dimension one or higher. I.e. we demand that up to dimension one-half, the supertorsion components are (equivalent to) those of massless IIA supergravity. As there are formulations of the latter (see for example [14]) in which the scalar superfield does not appear explicitly in the torsion components of dimension zero or one-half, we shall assume that the field content at dimension zero consists of at most one scalar superfield, while the field content at dimension one-half consists of one chiral and one antichiral Majorana spinor superfields.
In fact, an interesting feature of the formulation presented here is that the scalar superfield at dimension zero (whose lowest component can be identified with the dilaton), present in both Romans and massless IIA supergravities, is not introduced from the outset but it arises as a 'potential' for the spinor superfields at dimension one-half: its existence follows from a certain integrability condition implied by the BI's. In the case of HLW supergravity the aforementioned integrability condition fails and such a 'potential' does not exist. More specifically: as we show at the end of section 3.4, both in the case of massless IIA and in Romans supergravity, one can construct a closed one-form superfield whose lowest component is identified with the spinor superfields at dimension one-half. In a topologically nontrivial spacetime this one-form may not be exact, a fact which leads to a topological modification of massless IIA. This is the ten-dimensional version of the possibility to modify ordinary eleven-dimensional supergravity to MM-theory.
We generally expect that taking the supertorsion Bianchi identities as the starting point, should allow for more freedom than starting with superforms. Evidence for this was recently provided in [15], where the supertorsion BI's of eleven-dimensional supergravity where solved at first order in a deformation parameter related to the Planck length 3 . Although it has been shown that the four-form formulation 2 Presumably this apparent neglect should be attributed to certain unconventional features of HLW supergravity. For example, its equations-of-motion cannot be integrated to a local Lagrangian. More importantly, it is not clear at present if the theory can be made quantum-mechanically self-consistent. 3 The first supersymmetric deformation occurs at order l 3 and it is of topological nature [16]. The next deformation, of 11d supergravity implies the supertorsion formulation [17] 4 , it was realized in [15] that the converse may not be true.
The main result of this paper can be stated as follows. Depending on the values of two scalar superfields (L, L ′ of equation (4.7) below) arising at canonical mass dimension one, there exist exactly two massive deformations of IIA supergravity: Romans supergravity and HLW supergravity.
In the following section we introduce IIA superspace and establish our notation and conventions. We also examine the possible field redefinitions, in preparation for the analysis of the BI's in section 3. The reader who is not interested in the derivation of the final result, may skip directly to section 4 where the outcome of the analysis of the BI's is summarized. We conclude in section 5 with some possible future directions. The appendix contains our conventions on certain gamma-traceless projections used in section 3.

Type IIA superspace
Let us begin by introducing the usual superspace machinery of vielbein (E A ), connection (Ω A B ), torsion (T A ) and curvature (R A B ), via A flat superindex is denoted by a capital Latin letter from the beginning of the alphabet and stands for both bosonic (a) and fermionic (α) indices. Underlined Greek indices from the beginning of the alphabet stand for flat fermionic indices of both chiralities. For example: where S α is chiral and S α is antichiral. Note that we never raise/lower chiral fermionic indices, so that the position of the index denotes a definite chirality. In IIA superspace the spinor part of the vielbein contains both chiralities: The torsion and curvature satisfy the Bianchi identities If the connection is Lorentzian, as we assume to be the case in the present paper, the second BI follows from the first [20]. Hence we need only analyze the first of equations (2.3), i.e. the supertorsion BI.
related to the R 4 superinvariant, is expected to occur at order l 6 .

Field redefinitions
Before coming to the analysis of the BI's, let us examine what the possible field redefinitions are. We have at our disposal vielbein and connection redefinitions which can be used to gauge-fix some of the torsion components. Specifically, let Let us analyze the possible field redefinitions in the order of increasing canonical mass dimension.
• Dimension 0 We search for supergravities which, in the massless limit, reduce to massless type IIA. Therefore, as explained in the introduction, we shall assume that the field content at dimension zero consists of at most a scalar superfield (φ). Hence, the most general form of the dimension-zero torsion components is If a dimension-zero scalar superfield φ exists, c 1 , c 2 can be arbitrary functions of φ. From (2.4) we see that we can use two independent linear combinations of h α β , h α β , h a b to set T αβ c = −i(γ c ) αβ , Note that in the case where there exists no scalar superfield φ at dimension zero, the remaining linear h-combination is a constant. Therefore it cannot be used as a redefinition at dimension one-half or higher, as Dh vanishes.
• Dimension 1 2 At canonical mass dimension one-half the field content consists of one right-handed and one left-handed Majorana spinor superfield µ α , λ α , respectively 5 . In the massless IIA limit we have λ α = D α φ, µ α = D α φ. As we have already remarked, we shall not assume this to be the case a priori. The most general form of the vielbein and connection redefinitions is and respectively. Note that ∆ αβ γ is not independent, but is related to ∆ αb c via the Lorentz condition. The most general form of the torsion component T αb c reads and similarly for T α b c . Hence we can use the field redefinitions (2.6, 2.7) to set T αb In the case where there exists no scalar superfield at zero dimension, this is all we can do in the way of gauge-fixing. However, when φ exists T αβ γ can also be partially gauge-fixed as follows. The most general form of T αβ γ is As we can see from (2.4) and the fact that D α φ = λ α , the remaining independent linear combination of h α β , h α β , h a b can be used to gauge-fix the coefficient g 1 (φ) in (2.9).
• Dimension 1 As usual, ∆ ab c can be used to set T ab c = 0.
To summarize the results of this subsection: the field redefinitions can be used to set If, in addition, there is a scalar superfield φ such that D α φ = λ α , then T αβ γ can also be partially gauge-fixed, as explained below (2.9).

Analysis of the torsion Bianchi identities
We are now ready to come to the solution of the torsion Bianchi identities. We shall proceed systematically, in increasing order of canonical mass dimension. The readers who are not interested in the details of this analysis, may skip directly to section 4 where the final result is summarized.

Dimension-1 2 BI
Taking the discussion of section 2.2 into account and imposing the Lorentz condition, the dimension one-half torsion Bianchi identity reads We distinguish the following cases.

Dimension-1 BI
For the gamma-matrix manipulations of this and the remaining subsections, we have found [21] extremely useful. We have also made use of [22] in evaluating tensor products of representations.
Let us expand the spinor derivatives of the dimension-1 2 spinor superfields as follows and We also expand the dimension-one torsion as follows: The superfields appearing on the right-hand-side of (3.12) are all forms. We can see that there can be no hooks in the above expansions, for the following reason. Assuming there is a hook superfield (U ) at dimension one, we can expand U = mU (0) + U (1) , where m is a mass parameter, so that U (0) is of canonical mass dimension zero, U (1) is of canonical mass dimension one and does not depend on m.
Taking the massless m → 0 limit we see that U (1) has to vanish, as no hook superfields can appear in the torsion components of massless IIA. Also U (0) has to vanish, since at dimension zero there can exist at most a scalar superfield. Using the same argument we can see that there can be no five-forms in the expansions (3.10-3.12).
Taking the Lorentz condition into account, the Bianchi identities at dimension one read and (3.14) Let us analyze (3.13) first. We distinguish the following cases.
• Case 1: (α, β) = (α, β) Demanding that the right-hand-side of (3.13) be antisymmetric in c, d implies (3.15) and the curvature is given by Similarly to the previous case we get and Demanding that the right-hand-side of (3.13) be antisymmetric in c, d in this case implies and the curvature is given by Let us now come to (3.14). We distinguish the following cases.
• Case 1.1: (α, β, γ, δ) = (α, β, γ, δ) We shall work out this case in some detail in order to illustrate the general procedure. Taking the Lorentz condition into account, we obtain where we have allowed for the possibility that λ α = D α φ and we have setġ 1 := d dφ g 1 . Note that the vector drops out of the BI due to the identity Moreover, one can see that there is a unique three-form in the decomposition of the tensor product of a chiral spinor and the symmetrized tensor product of three antichiral spinors: Therefore, equation (3.21) imposes at most one linear equation on the three-forms. Contracting both sides of (3.21) with (γ [a ) αβ (γ bc] ) δ γ in order to saturate the spinor indices, we obtain 7 In terms of irreducible representations the BI decomposes as Hence the BI imposes at most one linear equation on the two-forms and one on the four-forms.
In terms of representations, the BI decomposes as I.e. the BI imposes at most two constraints on the scalars, four on the two-forms and five on the four-forms. Let us examine each representation in turn.

Scalars:
Using the independent structures (S 1 ) αβ γδ := (γ a ) αβ (γ a ) γδ and (S 2 ) αβ γδ := (γ a 1 ...a 5 ) αβ (γ a 1 ...a 5 ) γδ to saturate the spinor indices 8 , we obtain and in the rest of the analysis of the BI's, we have applied many more contractions than the number of independent ones. This is not strictly-speaking necessary, but the 'redundant ' contractions serve as useful consistency checks. 8 The fact that these structures are independent is seen as follows. Contracting the equation with S1, S2, we obtain A = B = 0. Note that there are exactly two independent structures as follows from the representation-theoretic analysis (3.27).
Hence in this case both independent contractions yield the same equation.

Two-forms:
Contracting with the four independent structures (γ Remarkably, all four independent contractions yield only the two constraints above. We also remark that equations (3.29) imply (3.25), which is therefore not independent. Four-forms: I.e. all five independent contractions yield only the two constraints above. Moreover equations (3.30) imply (3.26).
• Case 2.2: (α, β, γ, δ) = (α, β, γ , δ ) The BI decomposes as Therefore the BI will yield at most three constraints on the vectors and four on the three-forms. It can be seen that the vector part of the BI is equivalent to In terms of representations, this is the same as case 2.2 above. The equations which follow from this BI turn out to be identical to (3.32, 3.33).
In terms of representations, this is the same as case 2.1. Proceeding similarly, we obtain the following equations Scalars: Two-forms: Four-forms: In terms of representations, this is the same as case 1.2. Proceeding similarly we obtain the following equations In terms of representations, this is the same as case 1.1. Proceeding similarly we obtain The system of equations above imply in particular and We shall see in the following that in the case g 1 = −4, L = −L ′ , the existence of φ is indeed implied by the higherdimension BI's. But we have seen in section 2.2 that in this case the coefficient g 1 can be shifted by field redefinitions. Hence the case g 1 = −4, L = −L ′ is in fact equivalent to the case g 1 = −4, L = −L ′ . In the following we shall set g 1 = −4, so that we can treat both L = −L ′ and L = −L ′ cases simultaneously. Note that if L = −L ′ , there cannot exist a φ such that λ α = D α φ, µ α = D α φ.

Dimension-3 2 BI
Taking into account our gauge-fixing in section 2.2, the Bianchi identities at canonical mass dimension three-half read and Equation (3.42) can be solved for the dimension three-half supercurvature to give In the following we shall expand the dimension-3 2 torsion into irreducible (gamma-traceless) parts as follows where we have suppressed all spinor indices. Similarly, we expand the spinor derivatives of the various dimension-one fields in irreducible representations, as follows (again suppressing spinor indices) (3.46) Let us now come to equation (3.43). We distinguish the following cases.
• Case 1.1: (α, β, δ) = (α, β, δ) In terms of irreducible representations the BI decomposes as Hence the BI imposes at most three constraints on the spinors, four on the vector-spinors, etc. Let us analyze each representation in turn:

Two-form-spinors
Contracting with the two independent structures (γ As in the previous case, the projections onto the gamma-traceless part of the two-form-spinors are explained in appendix A.

Four-form-spinors
Contracting with (γ abcd e ) αβ δ δ γ , we obtain L (4) hence there will be at most three constraints on the spinors, three on the vector-spinors, etc. Let us examine each irreducible representation in turn:

Two-form-spinors
Contracting with the three independent structures δ [α and projecting onto the gamma-traceless part, we obtain ab µ) . (3.54)

Four-form-spinors
Contracting with the two independent structures (γ abcd e ) αβ δ γ δ and (γ abcd e ) δ(α δ γ β) , we obtain L (4) Hence the BI imposes at most five constraints on the spinors, seven on the vector-spinors, etc. Let us analyze each representation in turn:

Four-form-spinors
Contracting with the independent structures (γ e ) γβ (γ abcd ) δ α and (γ [a eg ) γβ (γ bcd]g ) δ α , we obtain L (4) This is related to the previous case by parity-inversion. The analysis proceeds in a similar fashion.

Two-form-spinors
Contracting with the three independent structures δ [α and projecting onto the gamma-traceless part, we obtain ab λ) . (3.69)

Two-form-spinors
Contracting with the two independent structures (γ [a ) αβ (γ b] e ) δ γ , (γ ab eg 1 g 2 ) αβ (γ g 1 g 2 ) δ γ , we obtain As in the previous case, the projections onto the gamma-traceless part of the two-form-spinors are explained in appendix A.

Four-form-spinors
Contracting with (γ abcd e ) αβ δ δ γ , we obtain L (4) The solution of the (highly overdetermined) system of equations above is given in section 4. Note that the spinor equations for ( L − L ′ ) α and ( L − L ′ ) α (equations (4.14,4.19), respectively) can actually be solved for (L − L ′ ), as we now show. Let us parameterize (L − L ′ ) as follows where m is a massive constant and φ is some real scalar superfield (possibly a constant) of canonical mass dimension zero. Plugging (3.77) in (4.14,4.19) and taking into account the action of the spinor derivative on λ α , µ α , we obtain and Finally, the spinor derivatives of K a are computed by employing the identities The result is given in section 4.

Dimension-2 BI
Taking into account our gauge-fixing for the supertorsion components, the first BI at dimension two reads The second BI at dimension two can be cast in the form This simply determines the spinor derivative of T ab , provided the right-hand-side of the equation above is consistent with the gamma-tracelessness of T ab . In other words, the right-hand-side of (3.82) should be annihilated by (γ a ) γ δ . This condition turns out to be equivalent to the system of all equationsof-motion and Bianchi identities for the bosonic fields! To see this in detail, let us distinguish the following cases.
• Case 1: (α, δ) = (α, δ) For simplicity of presentation, here and in the remainder of this section we shall focus on the bosonic terms, ignoring quadratic and quartic fermionic terms. In this case, the condition for the tracelessness of the right-hand-side of (3.82) can be written as where the coefficients in the expansion above are given by To arrive at these expressions we have Hodge-dualized the γ (7) and γ (9) terms. We have also made use of the identity (3.85) • Case 2: (α, δ) = (α, δ ) In this case the tracelessness condition takes the form where As in the previous case, we have Hodge-dualized the γ (6) , γ (8) , γ (10) contributions. Also, we have taken into account the identity This is similar to the previous case. The gamma-tracelessness condition is of the form The coefficients C b , C bcd and C bcdef can be obtained from B b , B bcd and B bcdef respectively, by making the substitutions L ↔ L ′ , K (3) → −K (3) , L 4 → −L 4 and ε (10) → −ε (10) .
This is related to case 1 by parity inversion. The gamma-tracelessness condition takes the form The coefficients D bc , D bcde and D bcdef g above can be obtained from A bc , A bcde and A bcdef g respectively, by making the substitutions described in the previous case.
In conclusion, the gamma-tracelessness condition is equivalent to It is straightforward to recognize that in the case (b) at the end of the previous subsection (i.e. for L − L ′ = 3 2 (µλ) + me 2φ , L + L ′ = 0 and λ α = D α φ, µ α = D α φ, equations (3.91) reduce to those of Romans supergravity.
In case (a1) (i.e. for L − L ′ = 3 2 (µλ), L + L ′ = 0) we can see, using the results of the preceding analysis of the BI's, that the super-one-form Λ A defined by is closed: It follows that there exists a scalar superfield φ such that Λ = Dφ. In particular, λ α = D α φ, µ α = D α φ and K a = i 2 D a φ. It is now straightforward to see that in this case equations (3.91) reduce to those of massless type IIA supergravity [23]. Note that (3.93) also holds in the case of Romans supergravity.
In the case where spacetime M is not simply connected, it may be that Λ = Dφ + ψ, where ψ is closed but not exact. For example, if M contains an S 1 we may take ψ = mdz, where z is the S 1 coordinate and m is a mass parameter. Upon compactification on S 1 , this would amount to a Scherk-Schwarz reduction. The possibility for this topological modification is the remnant in ten dimensions of the freedom to modify ordinary eleven-dimensional supergravity to MM-theory.

Dimension-5 2 BI
It can be seen that the dimension-5 2 BI does not introduce any new constraints, other than determining the action of the spinor derivative on the dimension-2 component of the supercurvature. Explicitly:

Summary
Here we summarize the result of the analysis of the BI's carried out in the preceding section. At each order in canonical mass dimension we give the solution for the components of the torsion and curvature, as well as the action of the spinor derivative on the various superfields at that dimension 9 . At dimension two we have chosen to include the bosonic part of the equations-of-motion and Bianchi identities for the bosonic fields of the relatively less-known HLW supergravity.
Dimension 0 (4.1) 9 We have checked that our formulae are compatible with overlapping literature. For example, in order to compare with [14], where the supertorsion components were given up to dimension-1 2 , one should make the following identifications: Λα → (µ α , λα) and One also needs to take into account the identity Dimension 1 Torsion: Curvature: Spinor derivatives: where For the scalar fields L, L ′ , we have the following cases 10 Massless IIA: HLW: where in both Romans and massless IIA supergravities we have (modulo the possibility for the topological modification of massless IIA explained at the end of section 3.4) D α φ = λ α , D α φ = µ α . In HLW supergravity, ∄ φ: where for the right-handed spinors T α , T α a we have (suppressing spinor indices) and similarly for the left-handed spinors T α , T aα , No confusion should arise from the slight abuse of notation, as it is immediately clear by the righthand-sides of equations (4.9,4.10) what the chiralities of the spinors T are in each case. Curvature Spinor derivatives b] + γ ab L (2) DL abcd = L (4) abcdα + γ [a L (4) bcd] + γ [ab L (4) cd] + γ abc L (4) d] + γ abc L (4) , (4.12) where 14) and for the right-handed spinors L α , L   where T , T a are given in (4.9, 4.10) and Bosonic equations-of-motion (HLW) where we have introduced the more conventional notation: iK (1) = 3 2 mA, L (2) = 3 16 F , iK (3) = H, L (4) = G. This is exactly the gauge-fixed form of the equations presented in [8] 11 . HLW can also be obtained by a generalized Scherk-Schwarz reduction of ordinary eleven-dimensional supergravity [12] (see also [24]). We have checked that the equations presented here indeed coincide with those in [12] 12 .

Conclusions
In this paper we have employed a systematic procedure in order to search for massive deformations of IIA supergravity. It is amusing to think that had we not known about them, we would have been able to discover in this way both Romans and HLW supergravities in one go. The method used here is quite general; it would therefore be of interest to apply it to other supersymmetric systems. It is quite plausible that new massive supergravities can be discovered in this way.
As already mentioned in the introduction, HLW supergravity arises upon compactification of a topologically modified version of eleven-dimensional supergravity, MM-theory. However, it is not known at present whether MM-theory can be given a microscopic quantum-mechanical description. Given that de Sitter space is an (unstable) vacuum of MM-theory, if the latter can somehow be related to Mtheory it would provide a mechanism for embedding de Sitter in M-theory. This interesting direction deserves to be pursued further, alongside with more recent proposals for the realization of de Sitter space in string/M-theory [25].
In a step towards this direction, it was shown in [11] that HLW supergravity supports (nonsupersymmetric) multi-zero-brane solutions. It was also argued that these states may indeed be associated with a microscopic description of MM-theory 13 and that the latter should represent an unstable phase of M-theory. A better understanding of the dynamics of these zero-branes is important in testing the proposal of [11]. To that end it would be interesting to construct the world-volume theories of 'massive' kappa-symmetric objects propagating in a HLW background (or perhaps directly in MM-theory) 14 , either within the superembedding formalism or by directly imposing kappa symmetry.