Axionic symmetry gaugings in {\cal N}=4 supergravities and their higher-dimensional origin

# Axionic symmetry gaugings in N=4 supergravities and their higher-dimensional origin

Jean-Pierre Derendinger, P. Marios Petropoulos and Nikolaos Prezas
Institut de Physique, Université de Neuchâtel
Breguet 1, 2000 Neuchâtel, Switzerland
Centre de Physique Théorique, Ecole Polytechnique, CNRS1
91128 Palaiseau, France
CERN PH-TH
1211 Genève, Switzerland

E-mail:
11Unité mixte UMR 7644.
###### Abstract:

We study the class of four-dimensional supergravities obtained by gauging the axionic shift and axionic rescaling symmetries. We formulate these theories using the machinery of embedding tensors, characterize the full gauge algebras and discuss several specific features of this family of gauged supergravities. We exhibit in particular a generalized duality between massive vectors and massive two-forms in four dimensions, inherited from the gauging of the shift symmetry. We show that these theories can be deduced from higher dimensions by a Scherk–Schwarz reduction, where a twist with respect to a non-compact symmetry is required. The four-dimensional generalized duality plays a crucial role in identifying the higher-dimensional ascendent.

preprint: NEIP–07–01
CPHT-RR015.0307
CERN-PH-TH 2007/057

## 1 Introduction

The effective field theories describing the low-energy dynamics of typical string and M-theory compactifications are plagued with massless scalar fields that hamper any attempt to contact four-dimensional phenomenology. One way of eliminating some of these unwanted massless scalars is the introduction of fluxes in the internal compactification space (see [1] for a review). From the effective field theory perspective, turning on fluxes corresponds to a “gauging” of the original theory obtained without fluxes. The term gauging refers to the fact that a subgroup of the global duality symmetry of the original theory is promoted to a local gauge symmetry. Simultaneously, part of the original Abelian gauge symmetry is promoted to a non-Abelian one and various fields, including the scalars, acquire minimal couplings to the gauge fields. The connection with the moduli stabilization issue stems from the fact that for extended supergravity theories, the only way to generate a potential is through the gauging. The resulting theories are known as gauged supergravities (see [2] for a concise review).

One can envisage a bottom-up approach to the problem of moduli stabilization where instead of looking for a specific higher-dimensional background whose corresponding low-energy effective theory is phenomenologically viable, one first constructs a gauged supergravity theory with the required phenomenological properties and then attempts to engineer it from higher dimensions. Such a programme was initiated in [3] and models were proposed there exhibiting full moduli stabilization.

Evidently such a programme depends crucially on having a good picture of the “landscape” of possible gauged supergravity theories. Therefore, a problem of paramount importance is to classify and describe all possible gaugings of supergravity in diverse spacetime dimensions and with various amounts of supersymmetry. This problem was tackled in a series of publications [4, 5, 6, 7, 8] where the general method of embedding tensors was developed.

Ultimately, however, one would like to know that the effective theory constructed in the bottom-up approach is consistent, i.e. can be embedded in a certain string or M-theory setup. Although for many classes of gauged supergravities their higher-dimensional realization is known, for the generic gauged supergravity there is no recipe – not even guaranteed existence of a higher-dimensional origin. Presumably such an endeavor would require a better understanding of the classes of string backgrounds dubbed non-geometric, whose significance has recently been investigated in the framework of flux compactifications and supergravity theories.

Here, our aim is to contribute in this direction by analyzing a family of four-dimensional gauged supergravities and explain how they can be obtained from string theory or higher-dimensional supergravity theories. Gauged supergravity with supersymmetry was the arena of [3] since string theories with 16 supercharges are the starting point of a variety of realistic string constructions. Recently the most general theory of this type was explicitly constructed in [9] using the formalism of embedding tensors. We will use this formalism to study in detail the specific class of theories obtained by gauging the axionic symmetries, namely the axionic shifts and rescalings. Consistency requires that several directions are also gauged. The general structure of the gauge algebra is systematically worked out and it exhibits the following characteristic property: it is non-flat contrary to what happens in more conventional gaugings.

In order to uncover the higher-dimensional origin of these gaugings one needs to perform a Scherk–Schwarz reduction of heterotic supergravity (or of the common sector in general) on a torus. The crucial ingredient here is a twist by a non-compact duality symmetry of ten-dimensional supergravity.

The identification of the theory obtained from the reduction with the gauged supergravity under consideration is not at all straightforward. It relies heavily on a duality between massive vectors and massive two-forms in four dimensions. This duality is a necessary extension of the more standard duality between a massless two-form and an axion scalar field in four dimensions : it incorporates the Stückelberg-like terms that are generated by the axionic gauging.

The organization of this paper is as follows. Section 2 is devoted to a reminder of gauged [10, 11, 12] supergravity in [13, 14, 15]. We emphasize the approach of the embedding tensor following Refs. [8, 9, 16]. In Sec. 3 we specialize to the so-called electric gaugings and we analyze a particular class of algebras for which we elaborate on the corresponding gauged supergravity. These theories admit two equivalent formulations related by a duality between massive vectors and massive two-forms in four dimensions. In Sec. 4 we move on to higher dimensions. Our aim is to analyze the ten-dimensional origin of the four-dimensional theory constructed in Sec. 3. We show that it can be obtained using a generalized Scherk–Schwarz reduction of heterotic supergravity. Furthermore, we comment on the higher-dimensional origin of other classes of gaugings. In Sec. 5 we present our conclusions and discuss some open problems.

## 2 Reminder on N=4 gauged supergravities in D=4

In this section we review gauged supergravity following Ref. [9]. After some general remarks on the possible gauge algebras and the constraints on the gauging parameters we present for reference the most general bosonic Lagrangian.

### 2.1 Gauge algebras and the embedding tensor

Four-dimensional supergravity has a very restricted structure. It contains generically the gravity multiplet and vector multiplets. The bosonic sector of the theory consists of the graviton, vectors and scalars. In the ungauged version, the gauge group is Abelian, , and there is no potential for the scalars.

Interactions are induced upon elimination of the auxiliary fields. They affect the scalars whose non-linearities are captured by a universal coset manifold [17, 18]

 M=SL(2,R)U(1)×SO(6,n)SO(6)×SO(n). (1)

All fields are non-minimally coupled to the Abelian vectors. The gauge kinetic terms have scalar-field-dependent coefficients, whereas the action is at most quadratic in the gauge-field strengths with no explicit dependence on the gauge potentials. For this reason, the symmetry of the scalar manifold is globally realized as a U-duality symmetry. Although the scalar manifold survives any deformation of the plain theory triggered by gaugings, the U-duality is broken as a consequence of the introduction of non-Abelian field strengths and minimal couplings, all of which depend explicitly on the gauge potentials.

The duality group acts as a symmetry of the field equations and the Bianchi identities of the gauge fields. In the standard formulation of supergravity only a subgroup of it is realized off-shell as a genuine symmetry of the Lagrangian. This includes the plus a two-dimensional non-semi-simple subalgebra of generated by axionic shifts and axionic rescalings. The third transformation in , corresponding to the truly electric-magnetic duality, is an on-shell symmetry which relates different Lagrangians associated to different choices of symplectic frames.

The only known deformation of supergravity compatible with supersymmetry is the gauging. This consists in transmuting part of the local symmetry into an non-Abelian gauge symmetry, or equivalently in promoting part of the global U-duality symmetry to a local symmetry, using some of the available vectors. This operation should not alter the total number of propagating degrees of freedom, as required e.g. by supersymmetry.

It is possible to parameterize all gaugings of supergravity by using the so-called embedding tensor. The latter describes how the gauge algebra is realized in terms of the U-duality generators. It captures all possible situations, including those where some of the gauge fields are the magnetic duals of the vectors originally present in the Lagrangian, as well as the option of gauging the duality rotation between electric and magnetic vectors, which, as already stressed, appears only as an on-shell symmetry. In this procedure, one doubles the number of vector degrees of freedom, keeping however unchanged the number of propagating ones thanks to appropriate auxiliary fields. One therefore avoids any choice of symplectic frame until a specific gauging is performed. We will not elaborate on these general properties but summarize the structure of the embedding tensor that we will use in the ensuing. We refer the reader to [8, 9, 16] for further information on this subject.

The electric vector fields belong to the fundamental vector representation of . Their magnetic duals form also a vector of , but carry opposite charge with respect to the that generates the axionic rescalings.

The algebra is generated by , which obey the following commutation relations:

 [Sαβ,Sγδ]=−ϵαγSβδ−ϵβδSαγ−ϵαδSβγ−ϵβγSαδ (2)

with . In the vector representation and they explicitly read111Since with the present conventions for , , we can raise and lower -indices unambiguously as follows: and . This leads to and . In particular, , and .

 S++=S−−=(0020),S+−=−S+−=(−1001),S−−=S++=(0−200). (3)

The axion and the dilaton form a complex scalar , which parameterizes the coset. We define

 Mαβ=1Imτ(|τ|2ReτReτ1) (4)

and we denote by its inverse. The action of

 g=(abcd)∈SL(2,R) (5)

is linear on : while it acts on as a Möbius transformation: . Therefore, generates the axionic shifts, the axionic rescalings, whereas the electric-magnetic duality is generated by . In this basis, form a doublet of with diagonal and correspondingly the rescaling charges are and .

The is generated by , obeying

 [TKL,TJM]=ηLJTKM+ηKMTLJ−ηKJTLM−ηLMTKJ, (6)

with being the metric222Indices are lowered and raised with and (inverse matrix).. In the fundamental representation the generators read: .

The scalars coming from the vector multiplets live on the coset and can be parameterized by a symmetric matrix of elements . Introducing vielbeins with and we can write and define the fully antisymmetric tensor

 MMNPQRS=ϵmnpqrsVmMVnNVpPVqQVrRVsS (7)

that appears in the scalar potential. We will denote by the inverse of .

The gauging of supergravity proceeds by selecting a subalgebra of generated by some linear combination of . The coefficients of this combination are the components of the embedding tensor and are subject to various constraints discussed in detail in the aforementioned references. In summary, this tensor belongs a priori to the of . However, there are linear constraints resulting from the requirement that the commutator provides an adjoint action and that supersymmetry is preserved333This is actually the minimal set of constraints that one can consistently impose and they also guarantee that a Lagrangian exists propagating the correct number of degrees of freedom, as one learns from general studies on gaugings of maximal supergravities [4, 5, 6, 7, 8].. These reduce the representation content of the embedding tensor to . Furthermore, there are quadratic constraints guaranteeing the closure of the gauge algebra – Jacobi identity. Putting everything together, one finds that the admissible generators of the gauge algebra are of the form

 ΞαL=12(fαLMNTMN−ηPQξαPTQL+ϵγβξβLSγα) (8)

where are fully antisymmetric in and satisfy

 (i)ηMNξαMξβN=0∀α,β. (9)

These parameters characterize completely the gauging.

The set of consistency conditions is completed as follows:

 (ii) ηMNξ(αMfβ)NIJ=0, (10) (iii) ϵαβ(ξαIξβJ+ηMNξαMfβNIJ)=0 (11)

and

 (iv)ηMNfαMI[JfβKL]N = 12ξα[JfβKL]I+16ϵαβϵγδξγIfδJKL (12) − 12ηMNξαMfβN[JKηL]I−16fαJKLξβI,

where and stand for antisymmetrization and symmetrization with respect to different indices belonging to the same family (e.g. ).

Several comments are in order here. A general gauging is manifestly expressed in terms of parameters subject to four conditions (i,ii,iii,iv). The gauge algebra, as defined by this set of parameters, is characterized by the commutation relations of the subset of independent ’s. These generators are indeed constrained (they satisfy e.g. as a consequence of their definition (8) and Eqs. (9), (10), (11)) as they should since no more that vectors can propagate. The structure constants of the gauge algebra are not directly read off from the ’s, which are not necessarily structure constants of some algebra. They can, however, be expressed in terms of the ’s and ’s and describe a variety of situations which capture simple, semi-simple or even non-semi-simple examples. For all of these, always provides an invariant metric although the Cartan–Killing metric of the corresponding gauge algebra can be degenerate.

The duality phases of de Roo and Wagemans [19, 20, 21] are also captured by the present formalism when the ’s are absent444For vanishing ’s, Eq. (12) is the only constraint; it is an ordinary Jacobi identity when ., as relative orientations of with respect to for each simple component [9]. We will not elaborate any longer on the general aspects of the embedding tensor and the variety of physical possibilities and for further details we will refer the reader to the already quoted literature.

### 2.2 Lagrangian formulation

The bosonic Lagrangian corresponding to the most general gauging was presented in [9]. For the sake of completeness we reproduce this result here and provide several comments. The Lagrangian consists of a kinetic term, a topological term, and a potential for the scalars:

 L=Lkin+Ltop+Lpot. (13)

 e−1Lkin=12R + 116DμMMNDμMMN−14(Imτ)2Dμτ¯¯¯¯¯¯¯¯¯¯Dμτ (14) − 14e−2ϕMMNHMμνHNμν+18aηMNϵμνκλHMμνHNκλ

with the covariant derivatives defined as

 DμMMN = ∂μMMN+2gAPαμΘαP(MQαP(MMN)Q, (15) Dμτ = ∂μτ+gAM−μξ+M+g(AM+μξ+M−AM−μξ−M)τ−igAM+μξ−M¯τ2 (16)

and the generalized gauge-field strengths being

 HMμν=2∂[μAM+ν]−g^fαNPMαNPANα[μAP+ν]+g2Θ−M−MNPCNPμν+g2ξ+M+C++μν+ξ+M−C+−μν. (17)

The combinations and are defined in terms of the gauging parameters and as

 ΘαMNP = fαMNP−ξα[NηP]M, (18) ^fαMNP = fαMNP−ξα[MηP]N−32ξαNηMP. (19)

The tensor gauge fields and are auxiliary and their elimination ensures that the correct number of gauge field degrees of freedom are propagated. For that purpose one needs to introduce a topological term in the Lagrangian

 e−1Ltop = −g2ϵμνρλ(ξ+MηNPAM−μAN+ν∂ρAP+λ−(^f−MNP+2ξ−NηMP)AM−μAN+ν∂ρAP−λ −g4^fαMNR^fβPQRβPQAMαμAN+νAPβρAQ−λ+g16Θ+MNPΘ−M−MQRCNPμνCQRρλ −14(Θ−MNPCNPμν+ξ−MC+−μν+ξ+MC++μν)(2∂ρAM−λ−g^fαQRMαQRAQαρAR−λ)).

Finally, there is a potential for the scalar fields that takes the form

 e−1Lpot = −g216(fαMNPfβQRSMαβ(13MMQMNRMPS+(23ηMQ−MMQ)ηNRηPS) (21) −49fαMNPfβQRSϵαβMMNPQRS+3ξMαξNβMαβMMN).

The basic feature of this Lagrangian is that it depends explicitly on both the electric gauge potentials and their magnetic duals . Therefore it allows the gauging of any subgroup of the full duality group , where in principle both electric and magnetic potentials can participate. The field equations derived from this Lagrangian and the gauge transformations can be found in [9].

## 3 Gauging the N=4 axionic symmetries

Here we present a class of gauged supergravities obtained by making local the axionic shifts and axionic rescalings. This is a subclass of the electric gaugings and we will refer to them as non-unimodular gaugings. Their existence was pointed out in [9] but here we analyze in detail and full generality the properties of the corresponding gauge algebra and discuss some interesting features of their Lagrangian description, such as a duality between massive vectors and massive two-forms.

### 3.1 Electric-magnetic duality and electric gaugings

The most general gauging is described in terms of two tensors and , and , satisfying four quadratic conditions (i, ii, iii, iv) displayed in Eqs. (9)–(12). This general formalism, defined in an arbitrary symplectic frame, captures in particular the gauging of the electric-magnetic duality symmetry generated by (see Sec. 2.1).

Gaugings with pure ’s have been studied extensively in the literature. They correspond to switching on gauge algebras entirely embedded in , as shown by (8). On the other hand, turning on the ’s allows one to gauge both and . This situation has not attracted much attention and only a few examples of the corresponding gauge algebras have been analyzed (see e.g. [9, 35]). Our aim is to study systematically a class of such gaugings and show that they correspond to a specific pattern of higher-dimensional reduction which generalizes the Scherk–Schwarz mechanism.

We will focus here on electric gaugings, namely gaugings that do not involve the generator of . Axionic shifts or axionic rescalings will however be gauged, accompanied by the appropriate generators. Hence, this class of gaugings is defined by setting

 ξ−I=0. (22)

We will also set

 f−LMN=0, (23)

although this is not compulsory for electric gaugings555What is meant by electric gauging is not universally set in the literature. In [9], for example, electric gaugings are defined as those with , which is somewhat too restrictive., while keeping and . Furthermore, we will focus on the case , which is related to pure gravity in ten dimensions, and adopt the off-block-diagonal metric:

 η=(0I6I60). (24)

The quadratic constraint (iii), Eq. (11), is now automatically satisfied while the constraints (i, ii, iv) – Eqs. (9), (10), (12) – reduce to

 ηMNξMξN=0,ηMNξMfNIJ=0, (25) ηMNfMI[JfKL]N=23f[IJKξL]. (26)

### 3.2 Non-unimodular gaugings

The fundamental representation 12 of decomposes into under the diagonal subgroup and correspondingly the -indices decompose into with both and ranging from to . Then, the metric can be written in the following way: , whereas . In this basis the -invariant inner product takes the form and we can write .

#### A specific solution with f’s and ξ’s.

Now, a non-trivial solution to Eqs. (25) and (26) is

 ξi=λiξi′=0for alli′, (27) fi′ij=fiji′=fji′i=−λ[iδj]i′all others vanishing, (28)

with arbitrary real numbers. The existence of the gauging described in Eqs. (27) and (28) was pointed out in Ref. [9]. Actually, there exist a whole class of gaugings of this type with more components of the tensor turned on. As discussed in [9], besides Eqs. (27) and (28) one can turn on . Then, the quadratic constraints reduce to a single equation

 f[ijkλl]=0. (29)

Since the novel feature of this class of gaugings is the presence of a non-zero parameter, we will restrict ourselves to the simplest example, namely the gauging with .

The gauging under consideration will be called “non-unimodular” for reasons that will become clear at the end of Sec. 4.2, or “traceful” since

 fijjij=−52λi. (30)

This is slightly misleading, however, since the gauge algebra is traceless as a consequence of the full antisymmetry of its genuine structure constants. The latter are not but specific combinations of and read off from the commutation relations of generators (8).

#### The gauge algebra.

In the rest of this section, we will characterize the gauge algebra, which will be further studied from a higher-dimensional perspective in the next chapters. Using Eqs. (27) and (28) in expression (8), we obtain:

 Ξ−i = −λi2S−−≡λiΥ, (31) Ξ−i′ = 0, (32) Ξ+i = −λi2(Tjjj+S−+)≡λiΞ, (33) Ξ+i′ = −λjTjji′≡Ξi′. (34)

The gauge algebra at hand has at most 18 non-vanishing generators but only 7 are independent: , plus 5 of the due to the constraint .

Their commutation relations follow from Eqs. (2) and (6):

 [Ξi′,Ξj′] = 0, (35) [Υ,Ξj′] = 0, (36) [Ξi′,Ξ] = Ξi′, (37) [Υ,Ξ] = −Υ. (38)

The set spans an Abelian Lie subalgebra. Furthermore, the algebra generated by is a non-compact subalgebra of .

The above algebra is non-flat666An algebra is called flat when it is generated by a set satisfying the commutation relations with . Then the Levi–Civita connection on the corresponding group manifold has zero curvature, therefore justifying the name flat., in contrast to the algebras obtained by standard Scherk–Schwarz reductions. As we will see in later, the gaugings at hand are related to twisted versions of these reductions, which relax therefore the flatness of the gauge algebras. The particular case appears when compactifying from five to four dimensions (see [35]) with a non-compact twist generated by the five-dimensional rescaling. The full algebra (31)–(38) will also emerge (see Sec. 4) as a ten-dimensional heterotic reduction with twist. Richer non-Abelian extensions are possible in this case, that eventually lead to non-vanishing as already advertised, and which are not possible when compactifying from five dimensions. These issues will be extensively analyzed in Sec. 4.2.

### 3.3 Lagrangian description

It is straightforward to derive the bosonic Lagrangian for the gaugings we have just described by using the general formulas of the previous section. As a first step, we will implement (22) and (23), and later set (27) and (28). Finally, we will dualize a vector, which acquires a Stückelberg-like mass via the gauging at hand, into a massive two-form field potential.

#### Electric gaugings ξ−I=f−KLM=0.

 e−1Lkin=12R + 116DμMMNDμMMN−14e4ϕDμaDμa−DμϕDμϕ (39) − 14e−2ϕMMNHMμνHNμν+18aηMNϵμνκλHMμνHNκλ,

where now

 DμMMN=∂μMMN+2gAPμΘP(MQP(MMN)Q (40)

and

 HMμν=2∂[μAMν]−g^fNPMNPAN[μAPν]+g2ξMCμν (41)

with .

Since and are zero and hence and are zero as well, we have omitted the “+” index from all coefficients. We will use the notation for the gauge potentials in order to avoid cluttering of the formulas with indices. Furthermore, we define the linear combinations

 Xμ≡ξMXMμ,Aμ≡ξMAMμ. (42)

The gauge covariant derivatives of the axion and dilaton take the form777From now on we set the coupling constant equal to 1 for simplicity.

 Dμa = ∂μa+Xμ+Aμa, (43) Dμϕ = ∂μϕ−12Aμ. (44)

By turning on the parameters we have gauged a non-Abelian two-dimensional subgroup of the global axion-dilaton symmetry. The “magnetic” potential corresponds to the gauging of the shift symmetry of the axion and acts as a Stückelberg field, while gauges the dilatation symmetry , . In terms of the notation of the previous subsection, the corresponding gauge algebra is the one spanned by and with commutation relation (38).

The topological term in the Lagrangian for the class of gaugings under consideration becomes

 e−1Ltop=−12ϵμνρλ(ξMηNPXMμANν∂ρAPλ − 14ξMCμνGMμν (45) − 14^fMNR^fPQRPQAMμANνAPρXQλ).

We have defined the following gauge field strengths

 FMμν = 2∂[μAMν]−^fNPMNPAN[μAPν], (46) GMμν = 2∂[κXMλ]−^fQRMQRAQ[κXRλ], (47)

in terms of which one can write .

Finally, the potential terms are

 e−1Lpot = −116M++(3ξMξNMMN (48) +fMNPfQRS(13MMQMNRMPS+(23ηMQ−MMQ)ηNRηPS))

with .

#### Non-unimodular gaugings.

We will now specify the gauge parameters by setting (27) and (28). With this gauging, the indices and corresponding to the and of are treated differently. We first notice that the magnetic field strengths do not appear in the Lagrangian. The gauge field strengths that appear are

 Fmμν = 2∂[μAmν], (49) Fm′μν = 2∂[μAm′ν]+(2λjAm′[μAjν]−λm′Aj[μAjν]), (50) Gmκλ = 2∂[κXmλ]+(λiAi[κXmλ]+λiAm[κXiλ]), (51)

where now

 Xμ≡ξMXMμ=λmXmμ,Aμ≡ξMAMμ=λmAmμ. (52)

One further notices that the Lagrangian actually depends only on the linear combination . This combination can be written in terms of and as

 Gμν=2∂[μXν]+2A[μXν]. (53)

Similarly we introduce .

The natural prescription is to integrate out the auxiliary two-form in order to obtain the final Lagrangian. If we do so, starting from (39), (45) and (48), we obtain

 e−1L=12R + 116DμMMNDμMMN−14e4ϕDμaDμa−DμϕDμϕ (54) − 14e−2ϕMMNFMμνFNμν+14aηMNFMμν~FNμν−14λ2e2ϕZμνZμν − 112ϵμνρλXμωρλν+e−1Lpot.

In this expression , where is the Hodge–Poincaré dual of ; we have also defined and we have introduced the Chern–Simons form:

 ωρλν=AnνFρλn+AnνFnρλ−12AλAmνAmρ+12AνAmλAmρ+cyclic. (55)

The local gauge invariance under axionic shifts enables us to gauge away the axion . Then, the magnetic potential acquires a mass through its Stückelberg coupling to the axion. The final expression for the gauge-fixed Lagrangian is therefore

 e−1L=12R + 116DμMMNDμMMN−DμϕDμϕ−14e−2ϕMMNFMμνFNμν (56) − 14λ2e2ϕ(Gμν+e−2ϕλm′Mm′N~FNμν)(Gμν+e−2ϕλm′Mm′N~FNμν) − 112ϵμνρλXμωρλν+e−1Lpot.

It captures all the relevant information carried by the axionic-symmetry gauging.

#### Stückelberg mass and dualization.

There is a different formulation of the theory, which is actually more suggestive of a higher-dimensional origin. Instead of integrating out the auxiliary antisymmetric tensor , one can promote to a propagating field and integrate out . Using as previously (39), (45) and (48), this procedure yields the following Lagrangian:

 e−1~L=12R + 116DμMMNDμMMN−DμϕDμϕ (57) − 14e−2ϕMMN(FMμν+12ξMCμν)(FNμν+12ξNCμν) − 38e−4ϕ(∂[κCμν]−A[κCμν]−13ωκμν)2+e−1Lpot,

where the two-form acquires now a scalar-field-dependent mass.

The key observation is that the theory described by (57) is related to that described by (56) by an interesting generalized duality. Recall that a massless two-form potential in four dimensions carries one propagating degree of freedom and can be dualized into a scalar. In the case where the two-form comes from the reduction of the ten-dimensional NS-NS two-form in the gravity multiplet, the dual scalar is the axion. If instead, as it happens here, the two-form is massive, it carries three degrees of freedom and the dual potential is a massive vector. This duality is a particular instance of a generalized duality between massive -forms and massive ()-forms in dimensions [22, 23].

For the benefit of the reader we will present schematically how this generalized duality works, leaving as an exercise its precise implementation between (56) and (57). Start from a massive gauge field with Lagrangian

 L=−14g2(∂μXν−∂νXμ)(∂μXν−∂νXμ)+12m2XμXμ. (58)

This Lagrangian can be obtained from

 ~~L=2Cμν∂μXν+g2CμνCμν+12m2XμXμ (59)

by integrating , which is an auxiliary non-propagating antisymmetric tensor. Instead, we can integrate by parts so that becomes non-dynamical, while acquires a dynamics. Integrating out finally yields an action for a massive two-form,

 ~L=−2m2∂μCμν∂λCλν+g2CμνCμν, (60)

which can be brought to a more familiar form for antisymmetric tensors, by Hodge–Poincaré-dualizing .

Following a similar pattern, one can replace the massive vector in (Eq. (56)) by a two-form. This yields precisely (Eq. (57)), therefore demonstrating that the Lagrangians <