Type IIB flux compactifications on twistor bundles

We construct a U(1) bundle over N(1,1), usually considered as an SO(3) bundle over CP^2, and show that type IIB supergravity can be consistently compactified over it. With the five form flux turned on, there is a solution for which the metric becomes Einstein. We also turn on 3-form fluxes and show that there is a one parameter family of solutions. In particular, there is a limiting solution of large 3-form fluxes for which two U(1) fiber directions of the metric shrink to zero size. We also discuss compactifications over N(1,1) to AdS_3. All solutions turn out to be non-supersymmetric.


Introduction
Compactifying solutions of supergravity theories provide a natural way of constructing consistent supergravity theories in lower dimensions. Moreover, some of the solutions turn out to be the near horizon geometry of M(D)-branes, and thus are of significance in AdS/CFT duality. Most of such solutions, however, preserve part of the supersymmetry and usually one needs to break it to construct more realistic models. Squashed and stretched solutions with fluxes in the compact direction, on the other hand, provide examples of supergravity solutions in which supersymmetry is spontaneously broken, and therefore, might be of interest in building the phenomenological models in the context of AdS/CFT duality [1].
Recently, we constructed new solutions of eleven-dimensional supergravity compactifying it to AdS 5 and AdS 2 × H 2 . We employed canonical forms on S 7 to write consistent ansätze for the 4-form field strength. The twistor space construction of CP 3 was the key for identifying the new solutions [2], and as we will see in this paper, this construction also proves useful in finding yet more solutions.
In this note, we extend the construction of [2] to the case of compact manifold N(1, 1), and use the twistor space language to describe it as U(1) bundle over a base which itself is an S 2 bundle on CP 2 ; the flag manifold. The first supergravity solutions of this kind were found in [3,4], and then explicit (squashed) metrics were constructed [5]. In section 2, first we consider N(1, 1) as an SO(3) bundle over CP 2 and then rewrite the metric as a U(1) bundle over the flag manifold. We then show that on this 7-dimensional manifold there exists a natural harmonic 2-form, the Kähler form of CP 2 , and use it to construct an 8-dimensional twistor bundle: a U(1) bundle over N(1, 1). Interestingly, as the harmonic 2-form is anti-self-dual, the Ricci tensor of this 8-dimensional metric in a suitable basis is diagonal with constant components. On the other hand, in section 3 we show that on this 8-dimensional manifold there exists a harmonic 3-form which we use to write down an ansatz for the 5-form field strength of type IIB supergravity. In this way, we are able to reduce the field equations to a set of algebraic equations. Among the three solutions we obtain one is Einstein. In section 3.1, we generalize our solution by turning on 3form fluxes, and show that there is a one parameter family of such solutions. In a limit of large 3-form fluxes two U(1) fiber directions of the metric shrink to zero size. In section 3.2, we discuss the supersymmetry of the solutions and show that they break supersymmetry. Section 4 is devoted to a discussion of compactification on N(1, 1) and the supersymmetry of the solution. Conclusions and the discussion are brought in section 5.
2 U (1) bundles over N (1, 1) N(1, 1) can be considered as an SO(3) bundle over CP 2 admitting two Einstein metrics, and hence providing Freund-Rubin type solutions of eleven-dimensional supergravity [6,1]. The bundle structure is very similar to that of S 7 where it is viewed as an SU(2) bundle over S 4 . However, N(1, 1) admits a 2-form, the Kähler form of CP 2 , which, as we will see, is anti-self-dual and harmonic. This allows us to construct a U(1) bundle over N(1, 1) so that the Ricci tensor is diagonal and has constant coefficients. Therefore, with a suitable ansatz for the form fields we are able to reduce the field equations to some algebraic equations. Let us start by taking the following 7-dimensional metric of N(1, 1) written as an SO(3) bundle over CP 2 [5,1]: where λ is the squashing parameter. Here 0 ≤ µ ≤ π/2, and Σ i 's are a set of left-invariant one-forms on SU(2): There is a similar expression for σ i 's: where they now take value on SO(3), i.e., 0 ≤ τ ≤ 2π, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π. They satisfy the SU(2) algebra; As in the case of S 7 , we can see that metric (1) can be rewritten as a U(1) bundle over a base which itself is an S 2 bundle on CP 2 , the flag manifold, [7,8,2]: where, and, A = cos θ dφ + sin θ(cos φA 1 + sin φA 2 ) + cos θA 3 .
In the new form of the metric, (2), we can further rescale the U(1) fibers toλ so that the Ricci tensor, in a basis we introduce shortly, is still diagonal. So, let us take the metric to be and choose the following basis In this basis the Ricci tensor is diagonal and reads For λ 2 =λ 2 = 1/2, and λ 2 =λ 2 = 1/10 the metric becomes Einstein, and thus one can get a solution of the Freund-Rubin type [6]. One can also turn on the 4-form flux in the compact direction to get the Englert type solutions [3,5].
We can now look at an interesting feature of N(1, 1) as a bundle over CP 2 . The base manifold admits a closed 2-form, i.e., the Kähler form: so that dJ = 0. Moreover, we observe that on N(1, 1) with metric (5) J is also co-closed: where we used and, as K and Im Ω are self-dual, whereas J is anti-self-dual on CP 2 . All this indicates that we can use the corresponding U(1) connection of J to construct a U(1) bundle over N(1, 1) so that its Ricci tensor is diagonal with constant coefficients. Therefore, for the metric of this 8-dimensional manifold, M, we take withλ measuring the scale of the new U(1) fiber. Adding to the vielbein basis (6), the 8d Ricci tensor reads We see that as J is harmonic and anti-self-dual, we do not get mixed components and the Ricci tensor remains diagonal.

Type IIB compactifications to AdS 2
We now show that the eight dimensional metric constructed above admits a harmonic 3-form, and then use this 3-form to provide an ansatz for the five form field strength of type IIB supergravity. To begin with, we note that on this manifold there are generally three 4-forms which are closed and self-dual on CP 2 [2]. On the other hand, since de 8 = −4λJ is anti-self-dual we can write down a 5-form which is also closed: * 8 ω 3 = (αω 4 + βK∧e 56 + γe 7 ∧Im Ω)∧e 8 + ξJ∧e 567 , with α , β , γ , and ξ being constant parameters. In fact, using (13), (16), and (17) we can see that d * 8 ω 3 = 0. Taking the Hodge dual (with ǫ 01235678 = 1), we have which we also require to be closed. Using (13) together with we see that ω 3 is closed if Hence, on M there exists a harmonic 3-form; dω 3 = d * 8 ω 3 = 0.
To discuss type IIB supergravity, we take a direct product ansatz for the metric: together with the following ansatz for the self-dual 5-form: which then satisfies the equation of motion, d * F 5 = 0, as ω 3 is harmonic. Next, let us consider the Einstein equations. Taking the dilaton and axion to be constant, in the Einstein frame, they read Using (20) and ansatz (26), the Einstein equations reduce to the following algebraic equations: First we note that there is a solution for which the metric is Einstein. Plugging (24) into the above equations, we get the following solution: with the Ricci tensor along AdS 2 : With the help of Mathematica, we have also found two more solutions of eqs. (28) for which the metric is not Einstein: with R µν ≈ −14.4583 g µν . And, with R µν ≈ −11.5538 g µν .

A one parameter family of solutions
Having found a solution for which the metric is Einstein, we are interested to see whether we can have solutions with H and F 3 fluxes turned on. For this we note that indeed there are two 3-forms which are closed: and, with ζ and η two constants. With the above ansätze for the 3-form fields, let us now turn to the type IIB equations of motion which, in the Einstein frame, read: where, First note that because of (9), (12), and (17) we have H∧F 3 = 0, and hence we can use the same F 5 as in the previous section, namely let so that the last equation of (35) is satisfied. Taking φ to be constant, c = 0, and the form fields as in (33), (34), and (37) we can see that the rest of equations in (35) are also satisfied if This leaves us with four unknown coefficients to be fixed. However, when we plug these into Einstein equations (27) they collapse into 3 equations: with b 2 = e −φ ζ 2 . So, we get a free parameter, b, that is not determined by the equations of motion. Although we have not been able to find the most general solution of (39), by examining the pattern of numerical solutions generated by Mathematica we did derive a particular solution: which can be checked by direct substitution in eqs. (39). Note that for b = 0, we get the solution in the previous section where we had only F 5 turned on. In the extreme limit b → ∞,λ andλ go to zero and thus two U(1) directions of metric (18) and (5) shrink to zero size. Surprisingly, the Ricci tensor of AdS 2 turns out to be independent of b, R µν = −12 g µν .

Supersymmetry
In this section we show that the solution we found in sec. 3, where we had turned on only the five-form flux with constant dilaton, breaks all supersymmetries. When the dilaton and axion are constant and there are no 3-form fluxes, the variation of the dilatino vanishes. However, we need to check whether the supersymmetry variation of the gravitino vanishes too, i.e., To study the Killing equation, (42), on the direct product space AdS 2 × M of sec. 3, let ε = ǫ ⊗ η, with ǫ and η the supersymmetry parameters along AdS 2 and M, respectively. We decompose the 10d Dirac matrices as whereγ µ and γ m are the 2 and 8 dimensional Dirac matrices respectively, witĥ γ 0 = iσ 2 , andγ 1 = σ 1 .
We can see that the supersymmetry is broken by looking at the Killing equation along AdS 2 . First, note that so if we choose Γ 11 ε = (σ 3 ⊗ γ 9 )(ǫ ⊗ η) = ε, with Γ 11 = −Γ 0123456789 , then we have Therefore, to split Killing equation (42) along the AdS 2 and the compact direction, we need to require F mnp01 γ mnp η = kη , for k a constant, so that along AdS 2 we have But, since γ mnp anticommutes with γ 9 and since η has a definite chirality, γ 9 η = η, eq. (45) can only have a zero eigenvalue, i.e., we must have k = 0. On the other hand, if k = 0, then the integrability of Killing spinor equation ∇ µ ǫ = 0 implies that the 2-dimensional Ricci tensor is vanishing which is not consistent with the AdS 2 factor that we obtained from solving the equations of motion. Therefore we conclude that the solution breaks supersymmetry. The above argument also applies to the solutions of sec. 3.1.

Type IIB on N (1, 1)
Now that we have discussed the compactification of type IIB on U(1) bundles over N(1, 1), let us look at the related compactification of type IIB on N(1, 1) itself. We study solutions with only F 5 flux turned on. So, let us take a direct product ansatz for the metric: together with F 5 as which is self-dual and closed because of (15). Using this ansatz and the Ricci components of N(1, 1) in (7), Einstein equations (27) reduce to 6 − 4λ 2 − 2λ 2 = 0 , Note that in this solution the Ricci tensor along the base manifold, i.e., CP 2 , vanishes. The non-compact space is an AdS 3 with Finally, let us discuss the supersymmetry of this solution. For AdS 3 × N(1, 1) compactification, we take the 10d Dirac matrices as whereγ µ and γ m are 2 and 8 dimensional Dirac matrices respectively, withγ 0 = iσ 2 ,γ 1 = σ 1 , andγ 2 = σ 3 . The supersymmetry parameter then decomposes with Γ 11 = 1 ⊗ 1 ⊗ σ 3 . As in the previous section, let us first look at the Killing equation along the AdS 3 factor. Note that so, to split the Killing equation along AdS 3 and N(1, 1) we must have The integrability of the Killing equation along the base manifold CP 2 , on the other hand, requires (−2γ 01 + γ 23 − γ23 +cγ1(γ 03 − γ 12 )) η = 0 , withc a constant. The above equations imply which is in conflict with eq. (53). Therefore, we conclude that the solution breaks supersymmetry.

Conclusions
We constructed a U(1) bundle over N(1, 1), and showed that type IIB supergravity can be consistently compactified over it. The twistor space formalism was crucial in deriving the solutions, specially when there were 3-form fluxes turned on. This approach has earlier been used in deriving new eleven-dimensional supergravity solutions [2], and also in [9] to study new solutions of massive IIA supergravity. We noticed that N(1, 1) admits a harmonic 2-form and used it to write the twistor bundle eight dimensional metric. With this choice of the connection, the Ricci tensor turned out to be diagonal with constant components. Furthermore, we saw that this eight dimensional manifold allows a harmonic 3-form, which was then employed to write a consistent ansatz for the 5-form field strength of type IIB supergravity. In this way, we showed that the field equations could be reduced to a set of algebraic equations. Among the three solutions we found one was Einstein. The discussion became more interesting when we turned on 3-form fluxes and obtained a one parameter family of solutions. Amusingly, we observed that there was a limiting solution for which two fiber directions of the metric were shrinking to zero size, whereas the 2-dimensional cosmological constant turned out to be independent of the free parameter. At the end, we further studied the related compactification over N(1, 1) to AdS 3 .
Since all the solutions we found in this paper break supersymmetry it is interesting to see whether they are associated with some brane configurations. This would then allow us to study AdS/CFT in a non-supersymmetric set up.