Consistent truncation with dilatino condensation on nearly K\"ahler and Calabi-Yau manifolds

We construct a consistent four-scalar truncation of ten-dimensional IIA supergravity on nearly K\"ahler spaces in the presence of dilatino condensates. The truncation is universal, i.e. it does not depend on any detailed features of the compactification manifold other than its nearly K\"ahler property, and admits a smooth limit to a universal four-scalar consistent truncation on Calabi-Yau spaces. The theory admits formal solutions with nonvanishing condensates, of the form $S^{1,3}\times M_6$, where $M_6$ is a six-dimensional nearly K\"ahler or Calabi-Yau manifold, and $S^{1,3}$ can be de Sitter, Minkowski or anti-de Sitter four-dimensional space.


Introduction
Theories with fermionic condensates are interesting in a variety of contexts, not least for cosmological applications. They have been studied in the past mainly in heterotic theory [1,2,3,4,5,6,7,8,9] and eleven-dimensional supergravity [10,11]. More recently the IIA theory has been shown to admit dilatino condensates which, in contrast to the situation in heterotic theory, can lead to a positive cosmological term and (formal) de Sitter solutions [12]. In the present paper we show that theories with fermionic condensates can also admit consistent truncations. To our knowledge this is the first time a consistent truncation (CT) of a higher-dimensional theory has been constructed in the presence of condensates.
A CT of a higher-dimensional theory is a lower-dimensional theory all of whose solutions lift to solutions of the higher-dimensional theory. CT's have proven to be useful in various different contexts, notably in applications of AdS/CFT. Although a CT is not a prerequisite for performing the holographic analysis of a theory, the truncation reduces the higher-dimensional equations of motion to simpler lower-dimensional equations, thus facilitating the construction of a precise holographic dictionary.
In the present paper we will use the latter approach to construct a CT of ten-dimensional (massive) IIA supergravity on nearly Kähler (NK) spaces in the presence of dilatino condensates. Moreover, taking the limit of vanishing scalar curvature, we establish the consistency of a fourscalar truncation of IIA supergravity on Calabi-Yau (CY) manifolds. To our knowledge, this is the first truncation on a CY manifold with massive lower-dimensional fields, whose consistency has been rigorously proven.
There are four homogeneous compact NK six-manifolds: S 6 , S 3 × S 3 , CP 3 and F 1,2 [42]. Until recently these were the only known compact NK six-manifolds. Two nonhomogenous examples were constructed in [43], and it is expected that many more should exist [44]. CT's on homogeneous NK manifolds have been constructed in the past, relying on the coset description of these spaces [37]. 1 The CT we will present in this paper is "universal", meaning it only relies on the 1 A reduction on NK spaces leading to four-dimensional N = 2 gauged supergravity, has been performed in [45], but without a complete proof of consistency. In this approach one postulates the existence of a finite set of forms on the internal manifold satisfying a certain list of constraints [46,47,48,49,50,51]. Plugging the truncation ansatz into the higher-dimensional action can then be shown to result in a lower-dimensional gauged supergravity. The consistency of this procedure, albeit plausible, has not been rigorously proven to date. See [52] for a recent discussion of this approach. NK property of the manifold, but not on any of its detailed features. In particular the CT is valid for any NK manifold whether or not it is homogeneous and admits a coset description.
The outline of the paper is as follows. In section 2 we review the dilatino-condensate action. The solutions with nonvanishing condensates are presented in section 2.2, after reviewing the bosonic AdS 4 solutions in section 2.1. Following a discussion of the case without condensates in section 3, the consistent truncation in the presence of condensates is given in (3.29)-(3.31) of section 3.2. Its CY limit is presented in section 3.3. We conclude with a discussion in section 4.

Dilatonic solutions
In a maximally-invariant vacuum of the theory, all fermion vacuum expectation values (VEV) are assumed to vanish, but quadratic or quartic fermion terms may still develop nonvanishing VEV's. Schematically, where λ collectively denotes the fermions and Φ stands for all fields in the action S[Φ]. The vacuum Φ is obtained by minimizing the effective action S eff with respect to the fields, where, at tree level in the coupling, the effective action coincides with S. Moreover, in the case of the IIA superstring, the two-derivative effective action S eff coincides with the action of tendimensional IIA supergravity to all orders in string perturbation. 2 However, nonperturbatively, S eff may develop nonvanishing VEV's for the quadratic and quartic fermion terms.
We will not examine the mechanism for the generation of fermionic condensates here: we will simply assume their presence and examine the implications. In the following we will look in particular for dilatonic solutions, i.e. for solutions of the dilatino-condensate action of [12]. This is obtained from the IIA supergravity action by setting the Einstein-frame gravitino to zero. Moreover, the quadratic and quartic dilatino terms in the action should be thought of as replaced by their condensate VEV's, and thus become (constant) parameters of the action. The dilatinocondensate action should therefore be regarded as a book-keeping device whose variation with respect to the bosonic fields gives the correct bosonic equations of motion in the presence of dilatino condensates; the fermion equations of motion are trivially satisfied in the maximallyinvariant vacuum, and need not be considered.
In [12] the fermionic terms of IIA supergravity were determined in the ten-dimensional superspace formalism previously developed in [53], resolving an ambiguity in the original literature [54,55,56] concerning the quartic fermions, and finding agreement with [54]. In the conventions of [12], the dilatino-condensate action of (massive) IIA reads, 3 where Λ is the dilatino; S b is the bosonic sector of Romans supergravity [57], where CS denotes the Chern-Simons term. We emphasize that, as mentioned previously, the dilatino terms in (2.3) are not dynamical but should be thought of as parameters of the action.
In particular (ΛΛ) 2 should be thought of as the VEV (ΛΛ) 2 and is therefore a priori independent of (ΛΛ), which should be thought of as the VEV Λ Λ .
The dilaton and Einstein equations following from action (2.3) read, and, The form equations read, Moreover the forms obey the following Bianchi identities, dF = mH ; dH = 0 ; dG = H ∧ F . (2.8)

Bosonic AdS 4 solutions
The equations of motion (2.5)-(2.8) admit bosonic solutions of the form AdS 4 × M 6 , where M 6 is nearly Kähler, cf. section 11.4 of [58]. Let us now review these solutions, before switching on the dilatino condensates in section 2.2.
We take the ten-dimensional spacetime to be of direct product form AdS 4 × M 6 , Let us parameterize, where g µν , g mn are the components of the metric in the external, internal space respectively; Ω is negative for anti-de Sitter space; ω is related to the first torsion class of M 6 through (A.7).
Moreover we set the dilaton to zero, φ = 0, and we parameterize the three-form and RR fluxes as follows, where J is the Kähler form of M 6 , and vol 4 is the volume element of AdS 4 . It is then straightforward to see, using (A.7), that the Bianchi identities (2.8) are satisfied provided, The F -form equation in (2.7) is automatically satisfied, while the H-form equation reduces to, The G-form equation in (2.7) reduces to, af + 6cω = 0 . (2.14) Moreover the dilaton equation reduces to, The mixed (µ, m) components of the Einstein equations are automatically satisfied, while the internal (m, n) components of the Einstein equations reduce to, where we have taken (2.15) into account. Finally the (µ, ν) components of the Einstein equations reduce to, where we have used (2.15) and (2.16).
As noted in [58], these equations admit three general classes of solutions only one of which is supersymmetric and corresponds to the NK solutions first discovered in [59]; it reads, In particular we see that the solution is parameterized by a single parameter (the Romans mass) and reduces to flat space without flux in the massless limit m → 0.

Solutions with dilatino condensates
We will now allow for nonvanishing dilatino bilinear and quadratic condensates. Let Λ ± be the positive-, negative-chirality components of the ten-dimensional dilatino. We decompose, where θ + , θ ′ + are arbitrary anticommuting four-dimensional Weyl spinors of positive chirality, see appendix B for our spinor conventions. Furthermore the reality of Λ ± imposes the conditions, We define the following three complex numbers parameterizing the four-dimensional dilatonic condensate, In terms of these, the ten-dimensional dilaton bilinears decompose as follows, where we have used the formulae of appendix A. For the "kinetic" bilinear terms we will assume that (Λ ± Γ µ ∇ ν Λ ± ) = 0. For a NK internal manifold such that (A.6) holds, we have, (2.24) Let us now substitute the above in the 10d equations of motion, while retaining the same form ansatz (2.11) as in section 2.1. The only difference is that we postulate a 10d metric of the form (2.30) As already mentioned, the last term above should be thought of as the VEV of a quartic fermion term, thus a priori different from the square of the bilinear VEV. Finally the (µ, ν) components of the Einstein equations reduce to, (2.31) In the limit of vanishing condensates one recovers the bosonic AdS 4 × M 6 solutions reviewed in section 2.1. Moreover one can obtain Ω > 0, and thus four-dimensional de Sitter space, e.g. for (ΛΛ) 2 sufficiently large.

Consistent truncation
The solutions of section 2 can be recovered from the equations of motion of a four-dimensional consistent truncation of the ten-dimensional IIA dilatino-condensate action. In the following we will construct the consistent truncation in the case of vanishing condensates. The case of nonvanishing condensates will be considered in section 3.2.
Our ansatz for the ten-dimensional metric includes two new scalars A, B with four-dimensional spacetime dependence, ds 2 (10) = e 2A(x) e 2B(x) g µν dx µ dx ν + g mn dy m dy n . (3.1) From the above we obtain the following formula for the ten-dimensional Laplacian of a scalar S(x) with only four-dimensional spacetime dependence, where the contractions on the right-hand side are taken with respect to the unwarped fourdimensional metric. The Einstein tensor of (3.1) reads, Our ansatz for the forms is such that the Bianchi identities (2.8) are automatically satisfied. It is given in terms of three scalars ϕ, χ, γ which are taken to only carry four-dimensional spacetime dependence. Explicitly, In particular we obtain, where the contractions on the left-hand sides are taken with respect to the ten-dimensional metric while the contractions on the right-hand sides are taken with respect to the unwarped four-and six-dimensional metrics. The following expressions are also useful, where the Hodge star is defined as in [60,12]. Plugging the above ansatz into the equations of motion we obtain the following: the internal (m, n)-components of the Einstein equations (2.6) read, where we have taken (2.10) into account. The external (µ, ν)-components read, (3.9) The F -form equation of motion is automatically satisfied. The H-form equation reduces to, 0 = −∇ µ e −φ+4A+2B ∂ µ χ + 48ω 2 e −φ+4A+4B χ + e 3φ/2+6A+4B m 2 χ + 2me φ/2+2A+4B (mχ 2 + γ)χ − ϕ(mχ 2 + γ) . (3.10) The G-form equation of motion reduces to, together with the following constraint, The latter can be readily integrated to solve for ϕ in terms of the remaining fields, where C is an arbitrary constant.

The Lagrangian
As we can see from (3.1) the scalar B(x) can be reabsorbed in the definition of the 4d metric. We have kept it arbitrary so far with the idea to use the associated freedom in order to obtain a 4d effective theory directly in the Einstein frame. This can be accomplished by choosing, (3.14) With this choice, and taking into account that ϕ is given in eq. (3.13), it is straightforward to check that the ten-dimensional equations given in (3.7)-(3.11) all follow from the 4d effective action, (3.15) where the potential V is given by,

AdS 4 solutions revisited
The consistent truncation (3.15) captures all of the AdS 4 solutions of [58] reviewed in section 2.1. Indeed upon setting the warp factor and the dilaton to zero, A = φ = 0, and the remaining fields γ, χ to constant values, imposing the equations of motion amounts to finding a minimum of the potential V of (3.16). We thus obtain the following three classes of solutions: First class where it is understood that the sign of F is correlated with that of the external part of G, while the sign of H is arbitrary. This can be written equivalently,

Second class
Or, equivalently, where it is understood that the sign of F is correlated with that of the external part of G, while the sign of H is arbitrary. Equivalently, In the above we have noted that Ω is given by the value of V /6 at the minimum, as follows from (3.15), (2.10). These coincide with the three classes of solutions presented in section 11.4 of [58], with the third class being the supersymmetric one, cf. (2.18).

Consistent truncation with condensates
In the presence of condensates, the internal (m, n)-components of the Einstein equations (3.7) get modified as follows, where we have taken (2.10) into account. The external (µ, ν)-components read, while the mixed (µ, m)-components are automatically satisfied. The dilaton equation reads,  together with the following constraint, The latter can be readily integrated to solve for ϕ in terms of the remaining fields, where C is an arbitrary constant.
Upon imposing (3.14) as before, a tedious but straightforward calculation then shows that all the above equations of motion can be obtained from the following four-dimensional action, (3.29) The action has exactly the same kinetic terms as before, cf. (3.15), but the potential now reads, where ϕ is non-dynamical and is given by, It can also be seen that this consistent truncation contains the S 1,3 × M 6 solutions of section 2.2 as special cases.

The Calabi-Yau limit
It can be seen from the equations of motion that the limit ω → 0 can be taken consistently, provided that we first rewrite, where γ 0 is constant while ξ is dynamical. This corresponds to the CY limit, in the sense of the vanishing of all SU (3) torsion classes. 4 More explicitly, in this case our ansatz for the forms becomes, and can be seen to automatically satisfy the BI's (2.8), taking into account that dJ = dΩ = 0. All remaining equations of motion can be obtained from those of section 3.2 by first replacing γ using (3.32) and then taking the ω → 0 limit. Note that this rewriting allows to keep the dynamical field ξ in the limit.
Moreover it can be seen that all equations of motion can be integrated into the following Lagrangian, (3.34) The potential V CY above is given by, where, It can be seen that, in the absence of condensates, unless all flux is zero, the potential is nonnegative and only has runaway minima. However this need no longer be the case in the presence of nonvanishing condensates.
To our knowledge, this is the first truncation on a CY manifold with massive four-dimensional fields, whose consistency has been rigorously proven.

Massless limit
A further truncation to two scalars, the warp factor A and the dilaton φ, can be obtained by taking the massless limit, m = 0, while at the same time setting χ, γ = 0. This amounts to the following flux ansatz: which is of Freund-Rubin type, and automatically satisfies the BI's (2.8). Moreover the remaining form equations reduce to a single constraint, where C is an arbitrary constant. It can then be seen that all equations of motion can be integrated to the following Lagrangian, where the potential V CY 0 is given by,

Discussion
We have constructed a consistent truncation of (massive) ten-dimensional IIA supergravity on NK or CY spaces in the presence of dilatino condensates. It has been argued in [61] that, at least in the case of sphere reductions, the existence of a consistent truncation of a higherdimensional supersymmetric theory to the bosonic sector of a supersymmetric lower-dimensional theory implies the existence of a consistent truncation to the full lower-dimensional theory. In the limit of vanishing condensates, it would be interesting to try to establish a dictionary between the present paper and the bosonic sector of the N = 2 four-dimensional truncation considered in [45]. This would provide strong evidence for the full consistency of the truncation of [45], which has only been shown for supersymmetric solutions.
The solutions of the present paper are formal, in that we have not offered any concrete mechanism for the generation of the dilatino condensate. More importantly perhaps, one would need to provide a controlled setting in which the dilatino condensate is not negligible compared to other quantum corrections. One place where one might be able to carry out this calculation would be in the context of the the uncompactified theory: since the two-derivative low-energy effective action of the ten-dimensional IIA superstring is expected to be exact to all orders in string perturbation, a potential fermion condensate generated nonperturbatively in the string coupling (e.g. via gravitational instantons as in [62]) would not have to "compete" with any string-loop corrections. We hope to return to this in the future.

B. Spinor conventions
Our spinor conventions are identical to those in [60], cf. appendix A therein, except for a slight change of notation with respect to the SU (3)-singlet spinor of the internal manifold M 6 , which is denoted η here and corresponds to η + of [60]. Moreover the η − of that reference corresponds to η c := Cη * here.