Contents

ITP–UU–10/27

SPIN–10/23

NSF-KITP–10/113

New potentials from Scherk-Schwarz reductions

[5mm] Hugo Looyestijn1, Erik Plauschinn1,2, Stefan Vandoren1

1 Institute for Theoretical Physics and Spinoza Institute

Utrecht University, 3508 TD Utrecht, The Netherlands

2 Kavli Institute for Theoretical Physics, Kohn Hall

UCSB, Santa Barbara, CA 93106, USA

H.T.Looijestijn, E.Plauschinn, S.J.G.Vandoren@uu.nl

[3mm]

Abstract

We study compactifications of eleven-dimensional supergravity on Calabi-Yau threefolds times a circle, with a duality twist along the circle a la Scherk-Schwarz. This leads to four-dimensional gauged supergravity with a semi-positive definite potential for the scalar fields, which we derive explicitly. Furthermore, inspired by the orientifold projection in string theory, we define a truncation to supergravity. We determine the D-terms, Kähler- and superpotentials for these models and study the properties of the vacua. Finally, we point out a relation to M-theory compactifications on seven-dimensional manifolds with structure.

## 1 Introduction and motivation

Scherk-Schwarz reductions [1, 2] provide a way to construct gauged supergravities from higher dimensional ungauged ones. They typically lead to semi-positive definite potentials for the scalar fields with local minima that can describe Minkowski or de Sitter vacua. Such models have been studied intensely over recent times in the context of compactifications of string- and M-theory, with and without fluxes. For some background material and earlier references, see e.g. [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].

Two classes of Scherk-Schwarz reductions are usually considered: the case of twisted tori (or twistings of the cohomology of other manifolds), and the case of reductions over a circle with a duality twist along the circle. Sometimes, these two classes are related to each other, and reductions with duality twists can be understood in terms of compactifications on twisted tori. For a discussion on this, see e.g. [12]. This relation will also appear in our investigation, as we will discuss, although we focus primarily on reductions with a duality twist.

In this paper, we present a detailed study of a Scherk-Schwarz reduction of eleven-dimensional supergravity compactified on a Calabi-Yau threefold, denoted by , times a circle, with a duality twist along the circle. Equivalently, this model can be formulated as a compactification on a seven-dimensional manifold, which is a Calabi-Yau fibration over a circle. This yields gauged supergravity in four dimension with a scalar potential for the vector- and hypermultiplet scalars. Moreover, there appear Chern-Simons like terms in four dimensions consistent with supersymmetry, induced from the Chern-Simons terms in five dimensions. These models have also been investigated in [9, 15], which we reproduce and elaborate on, and extend to include also the hypermultiplet sector.

The second part of the paper deals with truncations of our models from to supersymmetry. Inspired by the rules of the orientifold projection in string theory, we define a truncation of eleven-dimensional supergravity on to supergravity in four dimensions. In the absence of the duality twist, our rules are consistent with the results from compactifications of type IIA strings on Calabi-Yau orientifolds [16]. Here, we study the extension of this truncation to the case when the duality twist is non-trivial. On top of the Kähler potential, this yields a class of superpotentials and D-terms which we compute explicitly. It leads to formulas (5.41) and (5.40), which form one of the main new results in this paper. Alternatively, in the picture of the compactification on the seven-dimensional manifold , the supergravity is described by the Kähler potential

 (1.1)

and superpotential

 W=14∫Y(C3+i√8Re(CΩ))∧d(C3+i√8Re(CΩ)) , (1.2)

where denotes the radius of the circle, is the volume of the Calabi-Yau threefold while represents its holomorphic three-form, and is the three-form of eleven-dimensional supergravity. Due to the truncation, loses some degrees of freedom and the remaining ones are contained in , where the compensator will be defined in (5.18). Interestingly, similar formulas for the superpotential have also been obtained in the context of (flux) compactifications of M-theory on -manifolds, see e.g. [17, 18, 13], building on earlier work [19, 20]. This suggests a connection between those models and the ones considered here, which we will discuss in more detail towards the end of this paper.

## 2 M-theory on Calabi-Yau manifolds

In this section, we review aspects of compactifications of eleven-dimensional supergravity on Calabi-Yau threefolds. Almost all material in this section is known, and collected from various places in the literature, which we refer to below. We give this review to recall some of the duality symmetries in five dimensions, and to set our notation for subsequent sections. The reader who is very familiar with five-dimensional matter coupled to supergravity might skip this section and go straight to section 3 where we present the Scherk-Schwarz reduction to four dimensions.

The low-energy limit of M-theory can be described in terms of eleven-dimensional supergravity. In form-notation, the bosonic part of this action reads [21]

 ^S=12∫(^R⋆1−12^F4∧⋆^F4−16^F4∧^F4∧^C3). (2.1)

Here, denotes the eleven-dimensional Ricci scalar and stands for the eleven-dimensional Hodge star operator. Furthermore, is a three-form potential, denotes the corresponding field strength and we have set the eleven-dimensional Planck constant to one.

In the following, we compactify M-theory on a simply-connected Calabi-Yau three-fold , which leads to a supergravity theory in five dimensions with eight supercharges [22].

### 2.1 Calabi-Yau manifolds and dimensional reduction

#### Notation

We begin by establishing some notation for the Calabi-Yau three-fold . Let us denote a basis of harmonic -forms on by

 ωA,A=1,…,h1,1, (2.2)

where here and in the following denote the Hodge numbers of the Calabi-Yau threefold. The triple intersection numbers for are defined by

 KABC=∫XωA∧ωB∧ωC. (2.3)

For the third cohomology group we denote a real basis by

 (2.4)

which is chosen such that

 ∫XαK∧βL=δKL,∫XαK∧αL=0,∫XβK∧βL=0. (2.5)

The Calabi-Yau threefold is endowed with a Kähler form and a holomorphic three-form . In terms of the bases (2.2) and (2.4), these can be decomposed in the following way

 J=vAωA,Ω=ZKαK−GKβK, (2.6)

where the expansion coefficients are real. The functions are the holomorphic sections of special geometry and depend on the complex structure moduli of the Calabi-Yau manifold, where . The volume of can be expressed in terms of the Kähler form as follows

 V=13!∫XJ∧J∧J=13!KABCvAvBvC. (2.7)

#### Ansatz for the compactification

To perform the dimensional reduction of the action (2.1), we make the following ansatz for the eleven-dimensional metric

 ^GMN=(~g~μ~ν00Gmn),~μ,~ν=0,…,4,m,n=1,…,6, (2.8)

where denotes a five-dimensional metric and is the metric of a Calabi-Yau threefold. For the three-form potential, we chose the expansion

 ^C3=~c3+AA∧ωA+C3,C3=√2ξKαK−√2~ξKβK, (2.9)

with a three-form in five dimensions which depends solely on the five-dimensional coordinates . Similarly, are five-dimensional one-forms while and are five-dimensional scalars. Note that since the pure Calabi-Yau part features in the superpotential (1.2), we have separated these terms from and .

#### Five-dimensional supergravity

Performing the dimensional reduction to five dimensions is straight-forward and is briefly reviewed in appendix A. The resulting five-dimensional low-energy-effective action has been presented in equation (A) which we recall for convenience [23, 24]

 S(5)=∫R4,1[ +12R(5)⋆51−14dlogV∧⋆5dlogV+14KABCνCdνA∧⋆5dνB +14(KABCνC−14KACDνCνDKBEFνEνF)dAA∧⋆5dAB −112KABCdAA∧dAB∧AC−Gr¯sdzr∧⋆5d¯¯¯z¯s (2.10) −14V2(da+ξKd~ξK−~ξKdξK)∧⋆5(da+ξLd~ξL−~ξLdξL) +12V(ImM)−1KL(d~ξK−MKNdξN)∧⋆5(d~ξL−¯¯¯¯¯¯¯MLMdξM)].

The first term in this expression is the five-dimensional Ricci scalar, is the volume of the Calabi-Yau manifold and denote the triple intersection numbers defined in (2.3). The matrix as well as the period matrix have been introduced in appendix A.

The scalars are related to the expansion coefficients of the Kähler form by a rescaling with the volume (see equation (A.14)), such that they satisfy

 16KABCνAνBνC=1. (2.11)

Thus, there are scalar degrees of freedom in these fields. Accordingly, the vector fields comprise the graviphoton and additional vector fields to form five-dimensional vector multiplets. The remaining scalar fields form hypermultiplets that parametrize a quaternion-Kähler manifold [22].

### 2.2 Symmetries of the five-dimensional theory

#### 2.2.1 Symmetries in the vector multiplet sector

We begin our discussion on the symmetries of (2.1) with the vector multiplets. Besides the usual gauge invariances acting on the vector potentials, there are additional symmetries in the scalar sector. In particular, the scalars in the vector multiplets parametrize a so-called real special geometry, whose isometries have been studied in [23]. As explained in [25], not all isometries extend to symmetries of the full Lagrangian, but only transformations

 δνA=MABνB,δAA=MABAB, (2.12)

where the constant, real matrix is subject to the constraint

lead to symmetries of the full action, including the Chern-Simons terms.

Generically, the real special manifolds parametrized by the scalars in the vector multiplets need not be homogeneous, and solutions to (2.13) are not known in general. However, for homogeneous spaces a classification can be found in [26, 27]. A special subclass of the latter is given by the manifolds

 SO(1,1)×SO(n+1,1)SO(n+1) , (2.14)

for any integer , with isometry group . This case arises in compactifications in which the Calabi-Yau manifold is a -fibration over a base . In the present context, this situation has been studied in [15].

#### 2.2.2 Symmetries in the hypermultiplet sector

##### Notation

To study the isometries for the hypermultiplets, we first introduce some notation. The hypermultiplet scalars were given by , which parametrize a particular type of quaternionic manifolds called ‘very special’ in [27].

Since we consider M-theory on a Calabi-Yau manifold, the subspace of complex structure deformations is described by special Kähler geometry, for which there exists a prepotential. In the large complex structure limit, it is given by 111We reserve the usual notation and for the special geometry in the vector multiplets.

 G(Z)=−13!drstZrZsZtZ0,r,s,t=1,…,h2,1. (2.15)

Here, is a real symmetric tensor, the appear in the expansion (2.6) of the holomorphic three-form . The connection to the scalars is made by introducing projective coordinates

 zr=ZrZ0,r=1,…,h2,1. (2.16)

 Kcs=−ln(i∫XΩ∧¯¯¯¯Ω)=−ln(43∣∣Z0∣∣2d), (2.17)

where here and in the following we employ the notation

 d=drstxrxsxt,dr=drstxsxt,drs=drstxt, (2.18)

with . From (2.17), we can then compute the Kähler metric as 222The identification of (2.19) with the metric (A.7) can be made by noting that as well as that .

 Gr¯s=∂2∂zr∂¯¯¯z¯sKcs=−32drsd+94drdsd2. (2.19)

With denoting the inverse of (2.19), the curvature for this metric can be computed as follows [27]

 Rrstv=δrsδvt+δrtδvs−43Crvudstu,whereCrst=27641d2Gr¯¯¯uGs¯vGt¯¯¯wduvw. (2.20)
##### Symmetries for zr

Since the scalars appearing in the action (2.1) can be described by a Kähler potential, their kinetic term is invariant provided that (2.17) does not change under the transformations of interest.333Strictly speaking, (2.17) should be invariant up to Kähler transformations, but we will ignore those in the present analysis. We then make the following ansatz for the transformation of the sections appearing in the holomorphic three-form

 δ(ZKGK)=(QKLRKLSKLTKL)(ZLGL), (2.21)

where, , , and are constant, real, square matrices of dimension . Imposing the invariance of the Kähler potential (2.17) under this transformation, i.e.

 δ∫XΩ∧¯¯¯¯Ω=0, (2.22)

we are lead to the constraints

 (2.23)

which means that these isometries have to be contained in the symplectic group . However, because we are considering a Calabi-Yau manifold, we know that the sections are related to through a prepotential as . Therefore, in the ansatz (2.21) the transformation is not independent of , but we have to require

 δGK=∂GK∂ZLδZL. (2.24)

Recalling that is a homogeneous function of degree one in the , that is , we infer from (2.24) that [27]

 0=GTQZ+GTRG−ZTSZ−ZTTG, (2.25)

where matrix multiplication is understood. Furthermore, to leading order in the large -expansion, for Calabi-Yau threefolds the prepotential is given by (2.15). The solution to (2.25) in this case can be found in [27] which we briefly recall. In particular, the matrices and appearing in (2.21) can be parametrized as

 (2.26)

with , , and constant parameters. The matrix is subject to the constraint

 Br(sdtu)v=0, (2.27)

where denotes symmetrization and the constants are constrained by

 (2.28)

With this information, we can compute the transformation of the projective coordinates introduced in (2.16). Employing (2.26) as well as (2.16), we find [27]

 δzr=br−23βzr+Brszs−12Rrstvzsztav, (2.29)

and we note that the condition (2.28) implies that is constant.

##### Symmetries for ξK and ~ξK

To promote the symmetry of the complex structure deformations to a symmetry for the full hypermultiplets, and hence to isometries of the quaternionic space, we follow again [27]. First we note that the period matrix appearing in the action (2.1) (as well as in equations (A.10)) satisfies the relation

 GK=MKLZL. (2.30)

From the transformation of shown in (2.21), we infer that transforms as

 δM=S+TM−MQ−MRM. (2.31)

Requiring the kinetic term of the scalars in (2.1) to be invariant implies their following transformation

 (2.32)

which also leads to the invariance of the terms and agrees with [27]. Hence, just like , the form a symplectic pair.

Finally, we should add that the hypermultiplet space in general possesses more symmetries than the ones described here, for instance the Heisenberg algebra of isometries (which include the Peccei-Quinn shifts on ) that act on the coordinates and only. Furthermore, there are additional isometries that act non-trivially on the volume and the axion – for a complete classification see [27]. Including these in a Scherk-Schwarz reduction would be an interesting extension of our work. We will not consider them in our present discussion.

## 3 Scherk-Schwarz reduction to four dimensions

In this section, we compactify the five-dimensional theory given by (2.1) on a circle of radius . In addition, we impose a non-trivial dependence on the coordinate of the circle. Such a setup was studied first in [9] and, without hypermultiplets, further worked out in [15].

### 3.1 Ansatz for the compactification

To perform the compactification from five to four dimensions, we split the five-dimensional coordinates as

 {~x~μ}⟶{xμ,z},~μ=0,…,4,μ=0,…,3, (3.1)

where denotes the coordinate of the circle normalized as . The dependence of the five-dimensional scalars and the five-dimensional vectors on the coordinate is chosen in the following way

 ∂zνA=MABνB,∂zAA=MABAB, (3.2)

where satisfies (2.13). These expressions can be integrated to obtain

 νA(z)=[exp(Mz)]ABνB(0),AA(z)=[exp(Mz)]ABAB(0), (3.3)

where the exponential of the matrix is understood as a matrix product and where only the -dependence of the fields is shown explicitly.

Clearly, the fields are not periodic around the circle, but are related to each other by the duality transformations (2.12) generated by . These duality transformations form a group , and therefore one should have

 exp(M)∈G . (3.4)

Classically, the group is taken over the real numbers, and hence the entries of can be taken as arbitrary real constants. They determine the masses of the fields in four dimensions, and are treated as continuous parameters which we can take to be arbitrary small – or at least to be smaller than the masses of the Kaluza-Klein (KK) modes that we neglected. In the quantized theory, however, we expect the duality group to be defined over the integers, and hence the masses will be quantized in some units. This no longer guarantees that they are smaller than the masses of the KK modes. In turn, this could lead to complications in the truncation of the theory to the lightest modes, which we will ignore in this paper. For discussions on this issue for toroidal compactifications, see for instance [7, 12]. Essentially, this problem is similar to what one encounters in flux compactifications, where one has to make sure that there is a separation of mass scales, in particular the mass scale induced by the fluxes and the KK mass scale.

After this important side comment, we now turn to the hypermultiplets. For the dependence of the scalars on the coordinate of the circle we take

 ∂z(ξK~ξK) =(QKLRKLSKLTKL)(ξL~ξL), (3.5)

and for the complex structure moduli we choose in a similar fashion

 ∂zzr=br−23βzr+Brszs−12Rrstvavzszt≡Nr. (3.6)

The finite version of these transformations can easily be written down for . For , one first expresses them as transformations for the sections , after which one can integrate. For the scalars and , we choose

 ∂za=0,∂zV=0. (3.7)

Note that, since we have chosen the dependence of the fields on the circle coordinate such that they correspond to Killing vectors of the five-dimensional theory, the full action does not depend on and so we can evaluate the terms at a particular reference point, say .

For the five-dimensional metric, we make the following ansatz for the dimensional reduction

 (3.8)

where is the four-dimensional metric, is the radius of the circle and where the four-vector will become the graviphoton. The factor is chosen such that we end up in Einstein frame. For the five-dimensional gauge fields appearing in the action (2.1), we choose

 AA(5)=AA(4)+bA(dz−A0), (3.9)

where we added subscripts to distinguish between five- and four-dimensional quantities. Using the above ansätze within the action (2.1), one can perform the dimensional reduction, which is outlined in appendix B. Below, we present the results.

### 3.2 The four-dimensional action

The dimensional reduction from five to four dimensions as well as bringing the result into the standard form of supergravity can be found in appendix B. In particular, the four-dimensional action takes the form

where while , and where we have omitted labels on the vector fields indicating four-dimensional quantities. We have furthermore defined

 (3.11)

as well as the complexified Kähler moduli and their derivatives

 tA=bA+iϕA,DtA=dtA−MAB(AB−tBA0), (3.12)

where appeared in (3.9). The Kähler metric is written as

 gA¯¯¯¯B=−14R3(KAB−KAKB4R3), (3.13)

where we have employed the following notation

 KA=KABCϕBϕC,KAB=KABCϕC, (3.14)

with the triple intersection numbers defined in (2.3). Using these as well as (3.11) in the constraint (2.11), we also find

 R3=16KABCϕAϕBϕC. (3.15)

The metric (3.13) is a special Kähler metric and can be derived from the following prepotential

 F=−13!KABCXAXBXCX0,A,B,C=1,…,h1,1, (3.16)

where we employ coordinates with . The corresponding Kähler potential reads

 (3.17)

where due to the symmetries of the theory we can set . The expressions for the period matrix are given in (B.12), and the field strengths appearing in (3.10) are written as

 FΛ=dAΛ+12fΛΣΓAΣ∧AΓ,Λ,Σ,Γ=0,…,h1,1. (3.18)

The structure constants are [9, 15]

 f0AB=0,fCAB=0,fBA0=−MBA, (3.19)

and they define the gauge group which we elaborate on in the next subsection.

We mention here that gauge invariance of the action (3.10) requires the presence of Chern-Simons-like terms, which are inherited from the five-dimensional Chern-Simons term. These arise when the matrix transforms nontrivially under the action of the gauge group, in such a way that it needs to be compensated by an additional term in the action, the last term on the second line in (3.10). The existence of such terms in gauged supergravity was found in [28], and in the present context it was discussed in [15]. Some further applications of these terms in the study of supersymmetric vacua can be found in [29].

Turning to the hypermultiplet sector, we find that it is described by

 huvDμquDμqv=Gr¯sDμzrDμ¯¯¯z¯s+∂μϕ∂μϕ+e4ϕ4(∂μa+ξKDμ~ξK−~ξKDμξK)(∂μa+ξLDμ~ξL−~ξLDμξL)−e2ϕ2(ImM)−1KL(Dμ~ξK−MKPDμξP)(Dμ~ξL−¯¯¯¯¯¯¯MLQDμξQ), (3.20)

where and has been introduced in (2.19). The covariant derivatives appearing here are

 Dμzr=∂μzr−NrA0μ,Dμ(ξ~ξ)=∂μ(ξ~ξ)−N(ξ~ξ)A0μ, (3.21)

where had been defined in (3.6), where appropriate indices for are understood and where the matrix is given by

 N=(QKLRKLSKLTKL). (3.22)

Finally, the scalar potential can be expressed in the following way

 (3.23)

where matrix multiplication with correct contraction of indices is again understood. We will study the properties of this potential in section 6.

### 3.3 Gauged N=2 supergravity formulation

The ungauged part of the Lagrangian (3.10) is already written in the usual form of four-dimensional supergravity. The only changes we have to explain are the modifications due to the gauging, in particular the covariant derivatives for the scalars, and the scalar potential.

#### Covariant derivatives, Killing vectors and isometries

The covariant derivatives are given by

 Dμqu=∂μqu+~kuΛAΛμ,DμtA=∂μtA+kAΛAΛμ, (3.24)

where the quantities and are Killing vectors on the quaternionic and special Kähler spaces, respectively. For the scalars in the vector multiplets, from (3.12) we read off that

 kA0=MABtB,kAB=−MAB,

which means that on the special Kähler space defined by (3.16), the isometries we are gauging are given by

 δtA=−MABaB+MABtBa0 , (3.25)

for some arbitrary parameters and . That these are indeed isometries follows from the analysis of the special Kähler subsector of the hypermultiplets given in (2.29), which is completely analogous. Here, the symmetries (3.25) correspond to the first and third term in (2.29), namely a shift in and a linear transformation with a matrix satisfying (2.13). The gauge group is thus a subgroup of the duality group of isometries on the special Kähler manifold. This duality group contains the one from the five-dimensional theory, but in four dimensions it gets extended to a larger group [27]. The structure constants of the gauge group are given by (3.19), and define a solvable Lie algebra which is the semi-direct product of two Abelian subalgebras of dimension one (graviphoton) and (the other vector potentials) [9, 15].

The isometry group for the hypermultiplets can easily be read off from (3.21). It is a group, realized linearly on the scalars but non-linearly on (see equation (3.6)). The gauge group acts on it only via the graviphoton.

#### Scalar potential

The explicit form of the scalar potential is given in (3.23). It can be written in the standard form of supergravity which reads 444The overall factor compared to the potential of [30] is due to the different normalization in (3.10). When rescaling the four-dimensional metric in (3.10) as , one arrives at the form of [30].

 V=2eKvec(4huv~kuΛ~kvΣ+gA¯¯¯¯BkAΛ¯¯¯kBΣ)¯¯¯¯¯XΛXΣ, (3.26)

where and were defined in (3.17) and (3.13), respectively. In the general expression for the scalar potential, there is an additional term proportional to the quaternionic moment maps (see e.g. [30, 31])

 VP=2(gA¯¯¯¯BfΛAfΣ¯¯¯¯B−3LΛ¯¯¯¯LΣ)PxΛPxΣ. (3.27)

These moment maps in turn are proportional to a covariant derivative on . However, as can be seen from (3.21), the hypermultiplets are only gauged with the graviphoton . Therefore for and their covariant derivative also vanishes, so for . The only term in (3.27) that can contribute is the term with . We then utilize that the vector geometry is specified by (3.16), from which one calculates . Combining these properties, one finds that . This analogue of the no-scale property reduces the full scalar potential to (3.26).

To see that (3.26) reproduces our scalar potential, we use , and as (2.13) implies , with the help of (3.17) we find

 2eKgABkAΛk¯¯¯¯BΣ¯¯¯¯¯XΛXΣ=−14R6KABMACϕCMBDϕD. (3.28)

Employing the expressions for the covariant derivatives of the hyperscalars above, it is then straight-forward to check that (3.26) reproduces (3.23).

## 4 M-theory on twisted seven-manifolds

The Scherk-Schwarz reduction described above, yielding the gauged supergravity Lagrangian (3.10), can also be obtained from a compactification of eleven-dimensional supergravity on a seven-manifold. This point of view had also been taken in [15] for the vector multiplets. We will briefly review and extend this procedure in the present section to also include the hypermultiplet sector.

The seven-dimensional space we are going to compactify on, denoted by in the following, is chosen as a fibration of a Calabi-Yau three-fold over a circle .

 X→Y↓S1 (4.1)

The coordinates of will be denoted by and the coordinate of the circle is again normalized such that . At a particular reference point , we choose a basis of harmonic two- and three-forms of the corresponding Calabi-Yau three-fold as in section 2. We then must indicate how this data changes when moving around the circle.

In words, the difference to the point of view taken in section 3 can be explained as follows: instead of specifying the -dependence in the coefficient functions (i.e. the five-dimensional fields) as we do in the Scherk-Schwarz reduction, we can shift the -dependence from the fields into the basis of two- and three-forms of . This produces a seven-dimensional manifold of the type (4.1), which by construction is equivalent to the Scherk-Schwarz reduction. We now explain this in some more detail.

#### Cohomology

Let us begin our discussion with the cohomology of the compactification space . Analogous to the harmonic -forms on we introduce

 ^ωA(y,z),A=1,…,h1,1(X), (4.2)

with denoting the coordinates on and denoting the coordinate of the circle. The dependence of on is taken as

 ^ωA(y,z)=[exp(zMT)]BAωB, (4.3)

where the exponential of the matrix is understood as a matrix product and is a basis of harmonic -forms on the Calabi-Yau three-fold at a particular reference point . The matrix is not arbitrary but, as explained in [15], has to satisfy the constraint shown in (2.13). Infinitesimally, the relation (4.3) can be written as

 d^ωA=(MT)AB^ωB∧dz,^ωB(y,0)=ωB, (4.4)

so we see that in general the forms are not closed. Their non-closure will be the origin of the gaugings in the resulting four-dimensional action. The triple intersection numbers for the Calabi-Yau three-fold in the present context are given by

 ^KABC≡∫Y^ωA∧^ωB∧^ωC∧dz=∫XωA∧ωB∧ωC=KABC, (4.5)

where the second equality follows by using (2.13).

Analogous to the second cohomology, for the third cohomology we introduce

 {^αK(y,z),^βL(y,z)},K,L=0,…,h2,1(X). (4.6)

Their dependence on the coordinate of the circle is chosen as

 (+^α(y,z)−^β(y,z))=[exp(zNT)](+α−β), (4.7)

where the matrix was defined in (3.22) and proper contraction of indices is understood. Furthermore, denotes the basis of harmonic three-forms on the Calabi-Yau manifold at a particular reference point , and the minus sign has been chosen to match the results from the previous section. Infinitesimally, we can express (4.7) as

 d(−^α−^β)=−NT(−^α−^β)∧dz,(−^α(y,0)−^β(y,0))=(−α−β), (4.8)

where proper contraction of indices is again understood. Finally, using (2.5) and (4.7), one can show that

 ∫X^αK∧^βL=δKL,∫X^αK∧^αL=0,∫X^βK∧^βL=0. (4.9)

#### Dimensional reduction

For the dimensional reduction of the M-theory action (2.1) on the seven-manifold we make the following ansatz for the space-time metric

 ds211=e43ϕR−1gμνdxμdxν+e43ϕR2(dz−A0)2+Gmndymdyn, (4.10)

where is the radius of the circle satisfying (3.15), denotes the graviphoton one-form and is the metric of the Calabi-Yau threefold, whose fluctuations depend on and . For the three-form potential , we consider an ansatz similar to [15] but are more specific about the sector corresponding to the hypermultiplets. In particular, we consider

 ^C3=c3+B∧(dz−A0)+(AA−bAA0)∧^ωA+bA^ωA∧dz+C3,C3=√2ξK^αK−√2~ξK^βK, (4.11)

where is a four-dimensional three-form, denotes a four-dimensional two-form, are one-forms and as well as are scalars in four dimensions. For the corresponding field strength , employing (4.4) as well as (4.8), one finds

 ^F4=dc3+dB∧(dz−A0)−B∧F0+Fa∧^ωA−bAF0∧^ωA+DbA∧^ωA∧(dz−A0)+√2[d(ξ~ξ)T−(ξ)~ξTNTdz]∧(−^α−^β), (4.12)

where and are defined in (3.18). Using the above ansätze in the eleven-dimensional action (2.1), one can perform the dimensional reduction. However, to make contact with (3.10), we have to dualize to a scalar and to a constant , chosen to be zero. A non-zero choice for would correspond to a non-trivial -dependence for the five-dimensional field in the Scherk-Schwarz reduction of section 3, which we did not consider.

Taking into account these points, we then recover the four-dimensional action (3.10), as we have checked explicitly.

## 5 Truncation to N=1 supersymmetry

We now perform a truncation of the theory studied in section 3.2 from to supersymmetry. To motivate this truncation, we note that M-theory compactifications on seven-manifolds of the form can be related to orientifold compactifications of type IIA string theory [32]. In particular, consider M-theory on

 X×S1(¯¯¯σ,−1), (5.1)

where is an anti-holomorphic involution acting on the Calabi-Yau three-fold and where acts on the circle coordinate as . Upon dimensionally reducing on , the resulting theory is type IIA string theory on