Gauged Kaluza-Klein AdS Pseudo-supergravity

We obtain the pseudo-supergravity extension of the D-dimensional Kaluza-Klein theory, which is the circle reduction of pure gravity in D+1 dimensions. The fermionic partners are pseudo-gravitino and pseudo-dilatino. The full Lagrangian is invariant under the pseudo-supersymmetric transformation, up to quadratic order in fermion fields. We find that the theory possesses a U(1) global symmetry that can be gauged so that all the fermions are charged under the Kaluza-Klein vector. The gauging process generates a scalar potential that has a maximum, leading to the AdS vacuum. Whist the highest dimension for gauged AdS supergravity is seven, our gauged AdS pseudo-supergravities can exist in arbitrary dimensions.


Introduction
There are usually two criteria for a successful construction of supergravities. The first is that the degrees of freedom of the bosonic and fermionic fields should match. This condition itself does not provide much restriction on the construction. The second, which is much more non-trivial, is that the bosonic part of the Lagrangian should admit consistent Killing spinor equations, whose projected integrability conditions give rise to the full set of the bosonic equations of motion. Let us consider eleven-dimensional supergravity [1] as an example. The on-shell degrees of freedom of the graviton and 3-form gauge potential match with those of gravitino. However, in addition to the Einstein-Hilbert action and the kinetic term for the 3-form potential, eleven-dimensional supergravity requires a Chern-Simons term for the 3-form with a specific coupling. This term does not affect the total degrees of freedom, but is essential for the consistency of the projected integrability condition for the Killing spinor equations [2,3,4].
Recently, it was discovered that the low-energy effective action of the bosonic string, which is an intrinsically non-supersymmetric theory, admits consistent Killing spinor equations [5]. The results were extended to include Yang-Mills fields and α ′ order corrections [6], as well as the conformal anomaly term [7]. Based on these results, the pseudo-supergravity extension of the bosonic string was constructed in [8]. Pseudo-supersymmetric partners, namely the pseudo-gravitino and pseudo-dilatino, are introduced. The full Lagrangian is invariant under the pseudo-supersymmetric transformation, up to the quadratic fermion order. The pseudo-supersymmetric theory can be extended by coupling it to a Yang-Mills pseudo-supermultiplet. This also allows one to construct "α ′ corrections" involving quadratic curvature terms. An exponential dilaton potential term, associated with the conformal anomaly for a bosonic string outside its critical dimension, can also be pseudosupersymmetrized. Of course, in D = 10, where the degrees of freedom for the bosons and fermions match, the theory may become fully supersymmetric after adding quartic fermion terms. The full ten-dimensional N = 1 supergravity with Yang-Mills supermultiplets were given in [9,10,11]. However, when α ′ correction terms are involved, the supersymmetry was proved only at the quadratic fermion order [10,11].
Killing spinor equations for the D-dimensional Kaluza-Klein theory that is the circle reduction of pure gravity in D + 1 dimensions were obtained in [5]. It was shown in [7] that the Killing spinors can be charged under the Kaluza-Klein vector. This charging process generates a scalar potential, yielding the full bosonic Lagrangian where e = √ −g, F (2) = dA (1) is the 2-form field strength for the Kaluza-Klein vector A (1) , In this paper, we obtain the gauged pseudo-supergravity extension for this bosonic Lagrangian.
The paper is organized as follows. In section 2, we construct ungauged Kaluza-Klein pseudo-supergravity that is pseudo-supersymmetrization of the Kaluza-Klein theory. In section 3, we pseudo-supersymmetrizing the scalar/gravity system as a warmup exercise, since as we explained earlier, the gauging process will generate a scalar potential. In section 4, we combine the results of sections 2 and 3, and obtain ungauged Kaluza-Klein pseudo-supergravity with a scalar potential. We show that the scalar potential has a single exponential term which has no fixed point. The theory possesses U (1) ∼ SO(2) global symmetry that rotates the fermion fields. In section 5, we gauge the theory by letting all the fermions charged under the Kaluza-Klein vector. This effectively turns the U (1) symmetry to become a local one. The gauging process extends the previous scalar potential to involve two exponential terms. The new scalar potential has a maximum, implying that the AdS spacetime is its vacuum solution. We conclude the paper in section 6. We summarize our results in the appendix. Since we consider pseudo-supergravities in all dimensions, we also present the fermion conventions in diverse dimensions in the appendix.

Kaluza-Klein pseudo-supergravity
The D-dimensional Kaluza-Klein theory is the S 1 reduction of pure gravity in D + 1 dimensions. The Lagrangian is given by (1) with g = 0, namely The consistent Killing spinor equations for this were obtained in [5]. Here we examine whether the theory can be pseudo-supersymmetrized. We introduce pseudo-gravitino and dilatino fields (ψ i µ , λ i ), and propose that the fermionic extension is given by where the constant coefficients e 1 , . . . , e 5 are to be determined. Note that this is very different from the previous supergravity construction, which is typically on a specific dimension with specific type of fermion fields. Here we construct the theory in a generic dimension. It is thus necessary to summarize the properties of fermions in general dimensions. We adopt exactly the same convention given in [8], which follows the convention of [12]. We present the convention in Table 1 at the end of the appendix, and refer readers to [8,12] for details.
In addition to the Γ-matrix symmetries and spinor representations in diverse dimensions, we also present the s ij and t ij (and also u ij that will appear in later constructions) in Table   1. The indices i, j takes two values, 1 and 2. From Table 1, we note that s ij and t ij can be the fermions are two copies of Majorana; when s ij = ε ij , they are of a single symplectic Majorana. When s ij = t ij , which occurs only for β = +1, the fermions in (4) can be reduced to either a single copy of Majorana or symplectic Majorana, depending on the dimensions.
Having presented the convention for the fermions in general dimensions, we now give the ansatz for the pseudo-supersymmetric transformation rules where the coefficients c 1 , . . . , c 6 are to be determined. Note that the pseudo-supersymmetric transformation rules for the fermionic fields (ψ i µ , λ i ) are inspired by the Killing spinor equations obtained in [7] for single copy of fermions.
We now require that the full Lagrangian be invariant by the pseudo-supersymmetric transformation (5), up to quadratic order in fermion fields. We find that this fixes the coefficients in the ansatze completely, given by , As discussed in the introduction, our ultimate goal of this paper is to construct gauged Kaluza-Klein pseudo-supergravity. The gauging process will generate a scalar potential.
In this section, as a warmup exercise, we consider the pseudo-supersymmetrization of the scalar/gravity system. The bosonic Lagrangian is given by The potential V can be expressed in terms of a superpotential W , via where a prime denotes a derivative with respect to φ. Killing spinor equations for this system in the context of domain wall solution were given in [13]. We propose the ansatz for the fermionic extension Note that e 1 is given by (7) and e 6 , . . . , e 9 are to be determined. The ansatz for the pseudosupersymmetric transformation rules is given by The coefficients c 2 and c 4 are given by (7) and coefficients b 1 and b 2 are to be determined.
It is worth mentioning that s ij and u ij , which can be found in Table 1 in the appendix, can be the same only for cases with β = −1. We find that the requirement for pseudosupersymmetry implies that 4 Kaluza-Klein pseudo-supergravity with a scalar potential We now combine the results of section 2 and section 3 together. Keep in mind that in section 3, the superpotential W can be an arbitrary function of φ. We shall examine the restriction on W in Kaluza-Klein pseudo-supergravity. The full Lagrangian is given by The pseudo-supersymmetric transformation rules are given by We find that the variation of the Lagrangian, up to quadratic order in fermions, yields where γ = ±1 is given by (34) in the appendix. The vanishing of the above variation requires that where m is a free parameter. Thus we see that in Kaluza-Klein pseudo-supergravity, the scalar potential cannot be arbitrary, but a specific single exponential structure. Note that (s, t, u) cannot be all the same in this case. In dimensions where Majorana spinors are allowed, two copies are needed with ε ij and δ ij bilinear structures. The global symmetry is U (1), which is a subgroup of SU (2). In dimensions where symplectic Majorana is necessary, there is also additional δ ij bilinear structure. The global symmetry is broken down from Sp(2) to U (1). In the next section, we consider gauging the U (1) global symmetry.

Gauged Kaluza-Klein pseudo-supergravity
In the previous sections, we consider Kaluza-Klein pseudo-supergravities where the fermions are all neutral under the Kaluza-Klein vector A (1) . In this section, we gauge the theory by considering that all the fermions are charged under A (1) . As we shall see presently, this turns the global symmetry of the U (1) rotation of the fermions into a local symmetry. The charged covariant derivative on fermions is given by where γ = ±1 is given by (34), and the charge parameter b is a constant to be determined.
Note that here ξ i represents both ψ i µ and λ i . The full Lagrangian for gauged pseudosupergravity is now given by The pseudo-supersymmetric transformation rules also take the same form as (14), but with the covariant derivative D on the spinors replaced by D(A).
We find that the variation of the Lagrangian L KK,gauged leads to the following Thus we have The corresponding scalar potential is Note that the potential has a maximum and we have chosen the parameters so that it occurs Note that in D = 3, appropriate scaling limit should be performed to obtain non-vanishing results. Under the local gauge transformation, the fermions transform as follows or equivalently where Note that the U (1) ∼ SO (2) [14].) The D = 6 example can be embedded [15] in six-dimensional gauged supergravity [16] with a vector multiplet, which has an origin [17,15] in massive type IIA supergravity [18].

Conclusions
In this paper, we construct gauged Kaluza-Klein AdS pseudo-supergravity in diverse dimensions. By pseudo-supergravity, we mean that the full Lagrangian is invariant under pseudo-supersymmetric transformation rules up to quadratic fermion order. We start with pseudo-supersymmetrizing the D-dimensional Kaluza-Klein theory that is the S 1 reduction of pure gravity in D + 1 dimensions. We then consider pseudo-supersymmetrizing the scalar/gravity system. Combining the two results, we obtain the pseudo-supersymmetric Kaluza-Klein theory with a single-exponential scalar potential. By requiring that the fermions are all charged under the Kaluza-Klein vector, we obtain gauged Kaluza-Klein pseudo-supergravity. In dimensions where there can be Majorana spinors, two copies are needed, with bilinear structures of either δ ij or ε ij , which has a U (1) global symmetry.
In dimensions where there can be symplectic-Majorana, the global symmetry is also U (1), which is a subgroup of Sp (2). The effect of the gauging is that the U (1) symmetry becomes a local one, associated with the Kaluza-Klein vector. The scalar potential now involves two exponential terms and it has a maximum, giving rise to the AdS spacetime as its vacuum solution.
The success of our construction, together with the previous example of pseudo-supersymmetrizing of the bosonic string [8], suggests that when a bosonic system admits consistent Killing spinor equations, it can always be pseudo-supersymmetrized. In dimensions when the fermion and boson degrees of freedom happen to match, full supersymmetry may be realized.
The highest dimension in gauged supergravities with AdS vacua is D = 7. In our gauged pseudo-supergravities, it can be arbitrary. Solutions of charged rotating [19] and static black holes [7] were previously obtained. These solutions provide interesting higher dimensional backgrounds to test the AdS/CFT correspondence. The pseudo-supersymmetry that our theories possess makes them superior to an ad hoc concocted AdS theory. The existence of consistent Killing spinor equations implies that we can in principle derive the complete set of solutions that preserves pseudo-supersymmetry. These solutions are as good as BPS solutions in gauged supergravities in testing the AdS/CFT correspondence.
Finally, we would like to emphasize that the possible bosonic theories that can be pseudosupersymmetrized are rather limited. It was shown that Einstein gravity coupled to an n-form field strength cannot in general pseudo-supersymmetrized, unless it happens to be part of a supergravity theory [4].
where F (2) = dA (1) , a 2 = 2(D − 1)/(D − 2) and the charged covariant derivative is given by for any fermion ξ i , where γ = ±1, given by (34). The superpotential W and the potential V are given by (20) and (21) respectively. The quantities (s ij , t ij , u ij ) take either δ ij or ε ij , satisfying the following identities s ij s ik = δ jk , s jk t jl s lm = t km , s jk t jl = t kj s lj , s kj t jl = t kj s jl .
Note that in the above, s and t can interchange, and each can interchange with u and the identities still hold.
The pseudo-supersymmetric transformation rules for all the involved fields are given by To verify that the Lagrangian is indeed invariant under the transformation rules, up to the quadratic fermion order, it is useful to derive first the following projected integrability conditions, given by and Note that for Killing spinors, we have δψ i µ = 0 and δλ i = 0. Substituting these into the above equations and we find that what remain are precisely the full set of bosonic equations of motion attached to various Γ-matrix structures.
It is worth pointing out that since s ij = t ij occurs only for β = +1 and s ij = u ij only for β = −1, the quantities (s, t, u) cannot be all the same in any dimensions. In dimensions where Majorana spinors are allowed, two copies are necessary with both δ ij and ε ij bilinear structures. When symplectic Majorana is available, the symplectic structure is broken down to include δ ij structure as well.
Finally we present the Γ-matrix and fermion conventions. We adopt exactly the same convention given in [8], which follows the convention of [12]. We present the convention in Table 1. In addition to the Γ-matrix symmetries and spinor representations in diverse dimensions, we also present the s ij , t ij and u ij that appear in the construction. From Table   1, we can derive the following important identities s ji t jk s mk t ml = β δ il , s ji u jk s mk u ml = −β δ il , s ji u jk s lk t lm = γβ ε im , where γ = +1 , if t ij = δ ij , D mod 8 CΓ (0) CΓ (1) Spinor β