A massive class of N = 2 AdS 4 IIA solutions

We initiate a classification of N = 2 supersymmetric AdS 4 solutions of (massive) type IIA super-gravity. The internal space is locally equipped with either an SU(2) or an identity structure. We focus on the SU(2) structure and determine the conditions it satisfies, dictated by supersymmetry. Imposing as an Ansatz that the internal space is complex, we reduce the problem of finding solutions to a Riccati ODE, which we solve analytically. We obtain in this fashion a large number of new families of solutions, both regular as well as with localized O8-planes and conical Calabi–Yau singularities. We also recover many solutions already discussed in the literature.


Introduction
The study of four-dimensional anti-de Sitter solutions of string/M-theory is of considerable interest both in the context of flux compactifications and of the AdS/CFT correspondence.The prototypical class of such solutions is the Freund-Rubin class [1] in M-theory, where the flux is along the anti-de Sitter spacetime and the internal manifold is Einstein.This class allows for various amounts of supersymmetry, which further constrain the geometry of the internal manifold M 7 .Maximal supersymmetry imposes M 7 ≃ S 7 , while with N = 2 supersymmetry, which is the focus of this paper, M 7 is Sasaki-Einstein.Outside of this class very few solutions are known in M-theory [2][3][4].
From a holographic perspective, AdS 4 solutions are dual to Chern-Simons-matter field theories in three dimensions.A good control of the correspondence typically requires extended supersymmetry, and N = 2 provides a nice balance between control and variety.When in M-theory (and its type IIA reduction), the sum of the Chern-Simons levels of the gauge groups that characterize the field theory is zero.A non-zero sum corresponds to a non-zero Romans mass in type IIA string theory [5,6].So far, almost all known N = 2 solutions of massive type IIA supergravity are numerical [7][8][9][10], with one notable exception being the Guarino-Jafferis-Varela solution [11].In this paper, we overturn this status, finding a vast number of analytic solutions.
We will analyze the constraints imposed by supersymmetry by employing the "pure spinor" method, originally devised for N = 1 solutions [12,13].Adapting this method to N = 2 is not entirely straightforward.Until recently, in most of the literature one has resorted to imposing, on top of N = 1 supersymmetry, the presence of a vector field that leaves all fields but the supersymmetry parameters invariant, thus representing the R-symmetry action.This has proven useful, but has to be supplemented by an inspired Ansatz, and so far has resulted in the aforementioned numerical solutions.
A different approach has been put forward in [14], based on the work of [15].In the latter reference, the conditions for supersymmetry were expressed in terms of differential forms in the spirit of generalized geometry, without further assumptions on the form of the solution.In [14], these were adapted to the specific case of N = 2 AdS 4 × M 6 solutions of type IIB supergravity, obtaining a set of N = 2 pure spinor equations.After some work this resulted in a system of partial differential equations which characterize all possible solutions.In this paper we apply the same idea to type IIA supergravity.
The set of N = 2 pure spinor equations we obtain superficially resembles that of [14] in IIB; as is the case for N = 1, it is obtained by exchanging odd with even pure spinors.However, the geometric constraints that follow differ early on in the analysis.While in IIB the structure group on M 6 is exclusively the identity, in IIA it can be either the identity or SU (2). 1 In this paper we will focus on the SU(2) structure case, leaving the identity structure for future work. 2 The set of constraints we obtain from supersymmetry on the SU(2) structure also imply the Bianchi identities for the form fields, and all equations of motion.
Within the SU(2) structure case, we find two classes which we call "class K" and "class HK", because the internal manifold M 6 contains a four-dimensional subspace M 4 equipped with either a Kähler or a hyper-Kähler metric.We work out the supersymmetry constraints in full detail for both classes.Class HK leads to one local metric.On the other hand, class K leads to a very rich structure of solutions.
In particular, a simple and natural Ansatz (inspired by [18]) is that M 6 admits a complex structure.After imposing this, the problem of finding solutions reduces to a single ordinary differential equation (ODE) of Riccati type.The analysis is further subdivided according to whether M 4 is Kähler-Einstein, or is a product of two Riemann surfaces.In both cases we find the most general solution to the ODE analytically.
Beyond this maximal deformation, the solutions still exist, but develop singularities that have a physical interpretation as corresponding to the presence of various orientifold planes.In most cases these are smeared in some directions and localized in others; some limits of the parameters however produce solutions with fully localized O8-planes.
The rest of the paper is structured as follows.In section 2 we specialize the system of tendimensional equations obtained in [15] to N = 2 AdS 4 IIA solutions.As we mentioned, the analysis is similar to the one in IIB [14,Sec. 2,3], but some differences begin to emerge already here, and in particular we see that the SU(2) structure case is admissible, which we then focus on.In section 3 we analyze the system for this case by eliminating redundancies and obtaining the geometrical consequences of the system; the two classes K and HK are analyzed in turn.Being class HK rather limited, we devote section 4 to class K, under the assumption that M 6 admits a complex structure.As anticipated, we obtain two main families of analytic solutions, depending on several parameters.We summarize our findings in section 4.3.

Reduction of the ten-dimensional supersymmetry equations
In this section we will specialize the system of equations obtained in [15] as a set of necessary and sufficient conditions for any ten-dimensional solution of type II supergravity to preserve superymmetry, to the case of an AdS 4 background of type IIA supergravity preserving N = 2 supersymmetry.The process is similar to the one followed for type IIB supergravity in [14] and we refer the reader there for more details, especially on conventions.
We begin by reviewing the system of equations of [15], which are summarized in section 3.1 of that paper, focusing on the following subset of equations: Here ϕ is the dilaton, H is the NS-NS three-form field strength, d H ≡ d − H∧, and F (10d) is an even form obtained as a formal sum of all the R-R field strengths.λ(F p ) ≡ (−1) ⌊p/2⌋ F p , for F p a p-form.Moreover, Φ in (2.1) is an even form that corresponds via the Clifford map γ where ϵ 1,2 are the parameters of the supersymmetry transformations (which we take to be two Majorana-Weyl spinors of positive and negative chirality respectively).The vector K and the one-form K are defined by As we see in the second equation of (2.1b), K is a Killing vector and is in fact a symmetry of all fields in a solution.We will apply (2.1) to AdS 4 solutions, meaning that we will take all fields to preserve its SO(3, 2) isometry group.In particular we will take the metric to be of the warped product form: 3) The symmetry requirement also implies that the H field will only be a form on M 6 , while the R-R field strengths will be decomposed as (2.4) The supersymmetry parameters ϵ 1,2 will also be sums of tensor products of spinors on AdS 4 and M 6 .For N = 2 supersymmetry, the decomposition reads ) (2.5b) The χ's are a basis of AdS 4 Killing spinors: where µ = 0, . . ., 3. We will assume χ 1 + and χ 2 + to be linearly independent, since otherwise we would only have N = 1 supersymmetry.The gamma matrices decompose according to with m = 1, 2, . . .6. γ 5 and γ 7 are the external and internal chirality operators.
Let us now reduce (2.1) for this case of N = 2 AdS 4 solutions.We start with (2.1b): K, K decompose as ) ) (2.9) We thus find that (2.1b) gives where c IJ are constants, and Hence ξ is a Killing vector and in fact realizes the R-symmetry, acting on the I index in (2.5).
Notice that there are subtle sign differences in these formulas with respect to similar ones in IIB [14].
To reduce (2.1a), we write Φ as a wedge product of external and internal forms: (2.12) Again we use bispinors to denote the corresponding polyform under the Clifford map.Using (2.6) we can derive the exterior differential of the four-dimensional spacetime polyforms: where k is the form degree.Using this in (2.1a) and factoring spacetime forms, we get purely internal equations: ) ) and ) ) where and ξ is the complex conjugate of ξ.
As was the case for the corresponding system of equations in type IIB supergravity [14], the system (2.14), although alarmingly large, has a high degree of redundancy.For instance, we will see soon that c IJ can be set proportional to the identity; after that one can see that the equations that involve the R-R fields are redundant, except for (2.14c).Also, the I ̸ = J components are redundant, since the I = J ones furnish two copies of the pure spinor equations [12,13] for N = 1 AdS 4 solutions.Finally, the remaining "pairing equations" [15, (3.1c,d)] are redundant as for [14].
In spite of this redundancy, (2.14) will be more convenient for our analysis than a repeated application of the N = 1 equations [13].

Parametrization of the pure spinors
In this section we will parametrize the pure spinors ϕ IJ ± in terms of a set of differential forms.Before introducing the parametrization, we will fix the constants c IJ of (2.10) as where δ IJ is the Kronecker delta.This is permitted as the decomposition Ansatz (2.5) sets the internal spinors only up to a GL(2, R) transformation that leaves invariant the norms ∥η I i+ ∥ (which are equal to e A , by (2.10)).The details of this transformation can be found in [14].Since (2.17) Furthermore, instead of η I i+ we will work with which have charge ±1 under the U(1) ≃ SO(2) R-symmetry.From (2.10) and (2.16) we then have The internal spinors η ± i+ can be parametrized in terms of a chiral spinor η + of positive chirality (and its complex conjugate η − ≡ (η + ) c ) as follows: where the w i are one-forms, a is a function taking value in C and b, c are real functions.The latter satisfy where • denotes the inner product.
The chiral spinor η + defines an SU(3) structure, characterized by a real two-form J and a holomorphic three-form Ω, as with J, Ω satisfying J ∧ Ω = 0 and J ∧ J ∧ J = 3 4 iΩ ∧ Ω.When they are not all linearly dependent, the one-forms w i parametrize an identity structure and are holomorphic with respect to the almost complex structure J defined by η + .We will leave this generic case to future work; in this paper, we will limit ourselves to analyzing the case of an SU(2) structure, for which the w i are all linearly dependent.Such a case is not guaranteed to be compatible with the supersymmetry equations a priori, and indeed it is not allowed in type IIB supergravity [14].However, as we will see, in type IIA supergravity solutions with N = 2 supersymmetry and an SU(2) structure do exist.
We will thus take and set w 1 ≡ w, with normalized norm ||w|| 2 = 2.Note that (2) structure is defined by the one-form w, a real two-form j and a holomorphic two-form ω, with and w, j and ω satisfying ι w j = 0 = ι w ω, j ∧ ω = 0 and j ∧ j = 1 2 ω ∧ ω.We can now express the pure spinors in terms of forms: ) We also have where (ξ) ♭ is the one-form dual to the vector ξ.

System of equations
In terms of the pure spinors ϕ ±± ± introduced in (2.25), the system of supersymmetry equations (2.14) reads ) ) ) ) ) and ) ) We also have which were obtained from (2.1b) and the condition that the ten-dimensional vector K is Killing.We are showing (2.29) separately because they are in fact implied by (2.28).Even if they are redundant, (2.29a) and (2.29b) are useful to show that the equations of motion and the Bianchi identities of the R-R fields are automatically satisfied; see also our comment after (2.15).
Acting with d H on (2.29a), and using the imaginary part of (2.28b) it follows that which are the equations of motion.Acting with d H on (2.28f), using (2.30b), and subtracting the real part of (2.29b), it follows that which are the Bianchi identities of the R-R fields.This holds under the assumption that the Bianchi identity for H, dH = 0, is satisfied.Although it is not immediately obvious, we shall see that the NS-NS Bianchi identity is in fact implied by the supersymmetry equations.

Analysis of the supersymmetry equations
In this section we analyze the supersymmetry equations obtained in section 2. As we anticipated, not all the equations are independent, and we will be able to reduce them to a significantly smaller set which characterizes the SU(2) structure on the internal manifold.We will distinguish two cases.This is because certain equations, such as the zero-form component of (2.28c), have an overall factor of b, and can thus be solved either by setting b = 0 or by keeping b ̸ = 0 and setting to zero the remaining factor.It turns out that these two cases are qualitatively different, and we will consider them in separate subsections.
We will refer to the first case as "Class K" and the second one as "Class HK", because in these two cases M 6 will turn out to contain respectively a Kähler and a hyper-Kähler four-dimensional submanifold.

Class K
In this section we look at the case b = 0.The condition (2.30a), Im(ξ) = 0, fixes We define y ≡ e A f , which we will use as a coordinate, so that where now We will also introduce a coordinate ψ, adapted to the Killing vector as ξ = 4∂ ψ .
where ρ is a one-form on the four-dimensional subspace orthogonal to w.
The zero-form component of (2.28f) yields while the one-form part of (2.28a)-(2.28e)give We can thus write Note that from the above expression and (2.21), which for b = 0 yields |a| 2 = 1, it follows that F 0 and ℓ cannot be simultaneously zero.
Given the above we find that the two-form part of (2.28a)-(2.28e) is automatically satisfied, while the three-form part yields: We can combine the first two of the above equations so as to obtain one which does not involve the NS-NS field strength H: As pointed out earlier, F 0 and ℓ cannot be simultaneously zero, and hence when either of the two is, it follows from (3.8a) and (3.8b) that H = 0. We will proceed by making a 2 + 4 split of the internal manifold, with coordinates {y, ψ} on the two-dimensional subspace.The differential operator is decomposed as and the metric takes the form with x i , i = 1, 2, 3, 4 coordinates on the four-dimensional subspace, M 4 .
We can now decompose the three-form equations (3.9) and (3.8c): To further analyze the above equations it is convenient to rescale the data of the fourdimensional base as follows: where θ ≡ arg(â).Then (3.12) becomes where The last condition, d 4 ω = i P ∧ ω, suggests that the almost complex structure defined by ω is independent of ψ and y and integrable on the four-dimensional subspace M 4 , i.e. the latter is a complex manifold.In addition, d 4 ȷ = 0, and thus ĝ( 4) is a family of Kähler metrics parametrized by y.Furthermore, P is the canonical Ricci form connection defined by the Kähler metric with the Ricci form R = d 4 P .It is worth noting the similarity of the SU(2) structure we are studying here with the one that characterizes N = 1 supersymmetric AdS 5 solutions of M-theory, studied in [18]. 3This close resemblance allows us to draw upon certain results of the latter reference.
There are certain identities and conditions that derive from the system (3.15), to which we now turn.The equation for ∂ y ω determines the dependence of the volume of M 4 on y: Given that the complex structure is independent of y the following identity holds: where a plus superscript denotes the self-dual part of a two-form on M 4 .Combining with the second equation of (3.15) we arrive at Finally, the restrictions below hold as consequences of (3.15): The four-five-and six-form parts of (2.28a)-(2.28e)are automatically satisfied given the conditions we have derived so far.
Let us now look at the rest of the fields.The dilaton is determined by (3.7) and the condition |a| 2 = 1 descending from (2.21): (3.21) The NS-NS field strength H is given by either (3.8a) or (3.8b), and its Bianchi identity dH = 0 is manifestly satisfied.The R-R fields are determined by (2.28f) and are given by the expressions where we have introduced the auxiliary two-form and vol 6 = Rew ∧ Imw ∧ 1 2 j ∧ j.For future reference, let us also note the following B-twisted fluxes F ≡ e −B F , which will play a role when we examine flux quantization.This necessitates differentiating between the cases with the constants ℓ, F 0 either generic or vanishing.We will consider the twisted fluxes only for the generic case with F 0 ̸ = 0, ℓ ̸ = 0. Local expressions for the NS-NS potential B are easily read off from (3.8a) or (3.8b), While B 2 leads to shorter expressions for the remaining potentials, it has a singularity at y = 0; we will thus work with B 1 .We thus find the following expressions where explicitly We will come back to these expressions in section 4.

Class HK
In this section we look at the case b ̸ = 0. We find that in contrast to Class K this class of solutions is rather restricted and determined up to constant parameters.
The zero-form component of (2.28f) yields Rea = −e −A+ϕ yF 0 while the one-form component of (2.28b) gives d(e 3A−ϕ Ima) = 0. We can thus write So far things are akin to section 3.1.
From now on, however, analysis of the rest of the equations (2.28) puts strong constraints on the SU(2) structure and the functions that determine the solution.We find: where φ is defined via z 2 = e i(φ+ψ) and satisfies ∂ y φ = 0 = ∂ ψ φ.We thus conclude that the four-dimensional base of the internal manifold is hyper-Kähler5 and its metric is independent of y.Also, the connection of the fibration of the U(1) isometry generated by ξ over the base is flat.Furthermore, for the warp factor and the dilaton we find where L and g s are constants.b is also constant and is fixed by the relation The metric on the internal manifold, after a coordinate transformation y = L cos 1/2 (α), reads

.31)
Here Dψ ≡ dψ + ρ and ds 2 HK (x) is the line element on the hyper-Kähler base, with x denoting its coordinates.
Turning to the rest of the fields, the NS-NS field strength H is zero, while the R-R fields can be read from (2.28f).Their expressions are:6 where vol 6 = Rew ∧ Imw ∧ 1 2 j ∧ j.

Class K: complex Ansatz
In this section we explore an Ansatz for the Class K of solutions consisting of This Ansatz is equivalent to requiring that the holomorphic three-form Ω = w∧ω that characterizes the SU(3) structure on the internal space M 6 satisfies dΩ = V ∧ Ω for a one-form V , which in turn is equivalent to requiring that M 6 is a complex manifold.From From the above it can be inferred [18] that the Ricci tensor on M 4 , at fixed y, has two pairs of constant eigenvalues.For compact M 4 , which is the case of interest, we can invoke [26] stating (under the assumption that the Goldberg conjecture is true) that a compact Kähler four-manifold whose Ricci tensor has two distinct pairs of constant eigenvalues is locally the product of two Riemann surfaces of constant curvature.If the two pairs of eigenvalues are the same, then by definition the manifold is Kähler-Einstein.There are thus two classes to consider: either M 4 is Kähler-Einstein or is the product of two Riemann surfaces.

Kähler-Einstein base
with κ = 0 or κ = ±1.The case κ = 0, corresponds to M 4 being hyper-Kähler and turns out to be the b = 0 limit of the Class HK of solutions we examined in the previous section.We will thus restrict to κ = ±1.The dependence of the metric of M 4 on y is given by ĝ(4) (y, where g KE 4 is a Kähler-Einstein metric of constant curvature R = 4κ.When combined with (4.2a), and the fact that ∂ y R = −d 4 ∂ y ρ = 0 which is part of the Ansatz (4.1), the condition (4.3) fixes this becomes a Riccati: We were able to solve this Riccati equation analytically: where µ is a constant parameter.Note that in this parametrization the limit ℓ → 0 is not well-defined since the solution becomes trivial.The ℓ → 0 limit is well-defined after shifting µ → 12ν 2 /(F 2 0 ℓ 2 ) + µ.

Regularity and boundary conditions
We now turn to the analysis of the geometry of the solutions, which we will carry out in terms of a rescaled coordinate x ∝ y.We will specify the constant rescaling factor later on, for the cases (i) F 0 ̸ = 0 and ℓ ̸ = 0 (generic), (ii) ℓ = 0, and (iii) F 0 = 0, separately.The metric (3.11) on the internal manifold takes the form: where q = q(x) is a polynomial (of degree 6 if F 0 ̸ = 0), and a prime denotes differentiation.The warp factor is given by The dilaton is given by Table 1: Various boundary conditions for the polynomial q at an endpoint x 0 , and their interpretation.Empty entries are meant to be non-zero.
L and g s are two integration constants which we will specify in terms of the constants appearing in p later on.Positivity of the metric and the dilaton requires These conditions will only be realized on an interval of x.What happens to q at an endpoint x 0 of this interval dictates the physical interpretation of the solution around that point.We summarize our conclusions in Table 1.For example, we see from there that if q has a simple zero at a point x 0 ̸ = 0, the S 1 parametrized by ψ shrinks in such a way as to make the geometry regular, provided that the periodicity ∆ψ is chosen to be 2π.If this happens at both endpoints of the interval, the solution is fully regular.
Here are some details about each of these cases.
Simple zero: regular endpoint.Near a simple zero x = x 0 of q, the warp factor and dilaton go to constants, while the internal metric behaves as where q ′ 0 ≡ q ′ (x 0 ), q ′′ 0 ≡ q ′′ (x 0 ) (x 0 will never be zero).Positivity of the metric requires that if x 0 > 0 ⇒ x < x 0 , and if x 0 < 0 ⇒ x > x 0 .Choosing for definiteness the first case, and introducing r = √ x 0 − x we see that the parenthesis in (4.11) becomes dr 2 + r 2 Dψ 2 , which is the metric of R 2 (fibred over the Kähler-Einstein base), with the condition that the periodicity of ψ is taken to be ∆ψ = 2π.
Extremum: O4-plane.At a point x 0 ̸ = 0 where q ′ (x 0 ) = 0, the ten-dimensional metric and dilaton behave as ) where q 0 ≡ q(x 0 ).Positivity of the metric and the dilaton requires that if x 0 q ′′ 0 > 0 ⇒ x > x 0 , and if q ′′ 0 x 0 < 0 ⇒ x < x 0 (in the above equation we have recorded the first case).It also requires κq ′′ 0 < 0. One recognizes the usual structure H −1/2 ds 2 ∥ + H 1/2 ds 2 ⊥ for extended objects, with H ∼ x − x 0 .Since there are five parallel directions, this signals the presence of a four-dimensional object; the fact that the function is linear matches with the behavior of an O4-plane near the point where its harmonic function goes to zero.The dilaton matches the behavior e 2ϕ ∝ H (3−p)/2 of an Op-plane again for p = 4. Thus we conclude that this singularity corresponds to the presence of an O4-plane extended along AdS 4 .
We should also point out, however, that the local structure of the singularity does not clarify if the orientifold is smeared over KE 4 .Suppose one places a fully localized O4-plane at the tip of a cone C(Y 4 ) of metric dx 2 + x 2 ds 2 Y 4 .Near the tip, the backreacted metric is then of the form On the other hand, an O4-plane that is partially smeared along a four-dimensional manifold Y 4 would have a metric since H smO4 is now a harmonic function of one dimension only, it is piecewise linear.
The metric (4.14) ceases to make sense at x = x 0 .Expanding around this point, H O4 ∼ (x − x 0 ), and we would obtain (4.12) (up to constants that can be reabsorbed).On the other hand, (4.15) for a = 0 also gives (4.12), upon identifying x − x 0 = |x 9 |.In this sense, it is not entirely clear if (4.12) should be considered as smeared over KE 4 or not.(Such considerations also apply to Op-planes for p ̸ = 4.) For Y 4 = S 4 , (4.14) is the simplest interpretation; for Y 4 a Kähler-Einstein manifold, the singularity C(Y 4 ) would be bad (in that for example it would not be Ricci-flat, as one would expect before placing an object on it), and (4.15) seems the simplest interpretation.We thus conclude (4.12) is an O4-plane that is smeared over KE 4 .
Of course smeared orientifolds have rather limited physical validity; nevertheless, for completeness, we will include them in our analysis.
Triple zero: conical Calabi-Yau singularity.Near a triple zero x = x 0 of q, the warp factor and dilaton go to constants, while the internal metric behaves as The metric in parenthesis is a regular Sasaki-Einstein metric, built as U(1) bundle over the Kähler-Einstein base (KE 4 ).Thus (4.17) represents a conical Calabi-Yau singularity.In the particular case that KE 4 is CP 2 , this is in fact an orbifold singularity.
We can be a little more precise.d(Dψ) is the Ricci form of KE 4 , which in de Rham cohomology represents the first Chern class c 1 .The integral of the latter over the two-cycles C i of KE 4 is 2πn i , n i ∈ Z.If the periodicity of the S 1 coordinate is ∆ψ = 2π, the total space is the U(1) bundle associated to the canonical bundle over KE 4 .In that case the conical singularity (4.17) is the complex cone over KE 4 .If gcd(n i ) ≥ 1, there is also the possibility of taking the periodicity to be 2π × gcd(n i ).
For example, if KE 4 = CP 2 , there is only one cycle C with n = 3; taking ∆ψ = 2π gives the orbifold singularity but one can also consider ∆ψ = 6π, for which (4.17) in fact becomes the fully regular space C 3 .This possibility is not available if at the other endpoint one has a single zero, where ∆ψ is necessarily 2π.However, as we will see, in one case there is a triple zero is present at both endpoints, and in that case ∆ψ = 6π is possible.This will correspond to the Guarino-Jafferis-Varela (GJV) solution [11].
If KE 4 = CP 1 × CP 1 , there are two C i with n i = 2.With ∆ψ = 2π, (4.17) becomes a Z 2 quotient of the conifold singularity; one also has the possibility of taking ∆ψ = 4π, for which one obtains the original conifold singularity.Again this option is only available if a triple zero appears at both endpoints.We will see later, when considering the product base class in section 4.2, that this case has in fact a richer array of possibilities.

Generic case
We now turn to examining the parameter space of solutions for the generic case, by which we mean F 0 ̸ = 0 and ℓ ̸ = 0.The rescaling of the coordinate y that we mentioned at the beginning of 4.1.1 is We will also rescale the constant parameters that appeared in (4.8) and introduce The polynomial q that determines the solution then reads while the constants L, g s introduced in the previous section are: For later use, we note that the Riccati ODE (4.7) implies an ODE directly on the polynomial q: 1 24 It is also useful to notice that in this case We now have to study for which values of the parameters β, γ the positivity conditions (4.10) are satisfied.First of all, in order for q < 0 we need to require β > 0. We then have to subdivide this region according to the nature of the zeros of q, and the presence of maxima or minima.To identify these subregions, it is useful to look at the discriminant of q: For every β, ∆(q) = 0 has two solutions β = β ± , β − ≤ β + .(This can be seen by considering the discriminant of ∆(q) with respect to β, which is always negative.)Notice that β∆(q ′ ) 2 ∝ ∆(q), and that res(q, q ′′ ) divides ∆(q ′ ); this implies a double zero is in fact also a triple zero.This also follows from (4.25).
There are six different cases, which we discuss in turn and describe in figure 1.For definiteness, we will consider γ ≥ 0; the discussion for γ ≤ 0 is similar.
• For β < β − , q has two simples zeros x ± ; in the interval [x − , x + ], the conditions (4.10) are met with κ = +1.According to our discussion in section 4.1.1,at both simple zeros the S 1 circle shrinks in such a way that the solution is regular.Thus the internal space is smooth and the solution is fully regular.
The solutions previously found numerically fall in this region.The first to be found were the ones in [7], which should correspond to γ = 0, β ≤ 1, with KE 4 = CP 2 .In [8] it was later suggested that the CP 2 could be replaced by an arbitrary regular KE 4 .This was worked out explicitly in [10] for KE 4 = CP 1 × CP 1 ; our regular solutions corresponds to the q 1 = q1 • At β = β − , the discriminant ∆(q) = 0, and as we remarked also ∆(q ′ ) = 0; so the simple zero x − becomes a triple zero.As we discussed in section 4.1.1,this means that one of the regular points becomes a conical Calabi-Yau singularity; if KE 4 = CP 2 , this is a Z 3 singularity.
• For β − < β < β + , the triple zero x − splits into a single zero x − , a local minimum x 1 and a local maximum x 2 (x − < x 1 < x 2 ).Now the positivity conditions (4.10) are met for κ = +1 between the maximum and the zero: x ∈ [x 2 , x + ].However, a new possibility also appears: for κ = −1, the positivity conditions are met in the interval x ∈ [x − , x 1 ].Both of these correspond to solutions with a single O4-plane singularity.
• At β = β + , the other simple zero x + becomes a triple zero.Thus the solution with κ = +1 develops a Calabi-Yau singularity, besides the O4-plane singularity it already had; the solution with κ = −1 still has a single O4-plane singularity.
• For β > β + , the new triple zero splits into a local maximum x 3 , a local minimum x 4 and a simple zero x + .Now the κ = +1 interval is x ∈ [x 2 , x 3 ], and corresponds to a solution with two O4-plane singularities.Moreover there are two intervals that are allowed for κ = −1: x ∈ [x − , x 1 ] and x ∈ [x 4 , x + ].Both correspond to a solution with a single O4-plane singularity.
• Finally, at β = 1, γ = 0 we have that β − = β + .In this case the two triple zeros appear together; the allowed interval is between them and works for κ = +1.The solution has two Calabi-Yau singularities.When KE 4 = CP 2 , with ∆ψ = 2π these are two orbifold singularities (4.18).(This is the periodicity originally considered in [7].)As we discussed there, in this case one can also take the periodicity of the S 1 to be ∆ψ = 6π, in which case the space becomes fully smooth again; this is the GJV solution [11]. 7For more general KE 4 , this solution was discussed in [27].

Flux quantization for the generic case
Let us now discuss flux quantization for the generic solutions of section 4.1.2.
First we need to return to the B-twisted fluxes (3.25) and introduce potentials C k−1 such that dC k−1 = Fk .Explicitly, where ) ) The fluxes Fk are closed; they have been defined using a particular choice B 1 for the B field.In fact it is also possible to add to it a closed two-form b, so that B = B 1 + b; this defines new fluxes which are also closed.Explicitly, Flux quantization now imposes that the periods of these should be quantized, as well as that 2πF 0 ≡ n 0 ∈ Z (working in string units l s = 1).It constrains the parameters ℓ, β, γ of the solution, as well as the two-form b.
We will now work out more precisely what this implies for regular generic solutions.In particular, we will assume ∆ψ = 2π and β < β − , in the language of section 4.1.2;topologically, M 6 is an S 2 -bundle over KE 4 .
The second homology of M 6 is given by the fiber C 0 ≡ S 2 ψ,x spanned by ψ and x, and by the two-cycles C i , i = 1, . . ., h 2 (KE 4 ).More precisely, a lift of these two-cycles is given by a section of the fibration obtained by setting x to one of the endpoints, say x + .(x − would also work, but a random value would not define a cycle in M 6 , because of the topological non-triviality of the fibration of the ψ coordinate.)A basis for the cohomology H 2 can be taken to be ω I , I = 0, . . ., h 2 (KE 4 ); ω 0 ≡ d(s(x)Dψ), where s(x) is a function which at the two simple zeros x ± has second-order expansion s ∼ ±(1 + (x − x ± ) 2 ) + . .., and ω i are the elements of a basis for 7 The coordinate transformation that brings the solution to the form of [11] is x = cos α and Dψ = 3η.The parameters e ϕ 0 and L of [11] are identified as e ϕ 0 = 3 −3/8 2 1/4 ℓ −1/4 F −3/4 0 and L 2 = 3 −1/16 2 −5/8 ℓ 5/8 F −1/8 0 .Note also that the solution in [11] is in the Einstein frame, whereas we work in the string frame.
H 2 (KE 4 ).We will expand the closed two-form b in this basis: b = b I ω I .Similarly, a basis of four-cycles can be obtained by the Ci ≡ S 2 ψ,x × C i and by C0 ≡ KE 4 .Finally, the triple intersection form d IJK of M 6 will have non-zero entries d 0JK = c JK , the intersection form of KE 4 .
We can now define the periods The periods at b = 0, n 2I ≡ n b=0 2I , n I 4 ≡ n I b=0

4
, n 6 ≡ n b=0 6 , are computed more directly as integrals of the Fk .The two are related by the b-transform (4.30): this gives From (4.28)), (4.29) we can now compute the relevant integrals: where f k± ≡ f k (x ± ) the K i are the Chern class integers of the canonical bundle; 2πK i are the integrals of the Ricci form over the two-cycles C i of the KE 4 .We also defined .32) can be further evaluated using the expressions for the f k in (4.29).In doing so, it is useful to note that (4.25) implies that at a single zero x 0 of q: So in particular The n 2I now determine b I = 1 n 0 (n b 2I − n 2I ); one can then eliminate them from the remaining quantization conditions.A practical way of doing this is to introduce some combinations of the flux quanta that are invariant under b-transform F → F b , generalizing slightly results in [9]: (These come from the expansion in form basis of F 2 2 − 2F 0 F4 and F 3 2 + 3F 2 0 F6 − 3F 0 F2 F4 .)Indeed one can check that the I I 4 and I 6 remain the same if one replaces For us these invariants evaluate to Once a set of flux quanta is specified, solving these equations will specify the parameters ℓ, β, γ of the solution.

ℓ = 0
We will now examine the solutions with ℓ = 0.The rescaling of the coordinate y, appropriate for this case, is and we will also introduce The solution is then in the form (4.9) with which gives The constants that appear in the warp factor and the dilaton are L 2 = |y 0 | and g 2 s = 72/(|y 0 |F 2 0 ).In this case, the analysis is easier, because there is only one parameter, σ, to vary.The discriminant of q is 2(σ 3 + 9 3 ) 2 ; thus there always two simple zeros, except for σ = −9, when one of the two becomes a triple zero.q ′ always has a double zero at the origin, but the discriminant of q ′ /x 2 is −192(σ 3 + 9 3 ) (so again we have ∆(q) ∝ ∆(q ′ ) 2 ), and the sign shows that there is a single extremum x 1 for σ > −9, and three extrema x 1 , x 2 , x 3 for σ < −9.In addition, there is a inflection point at x = 0, which from section 4.1.1we know to correspond to an O8-plane.
We divide the analysis in three cases: • For σ > −9, between the two zeros x − < 0 and x + > 0, q has a minimum at x = x 1 < 0 and the inflection point at x = 0.The intervals where (4.10) are realized are x ∈ [x − , x 1 ] with κ = −1, and x ∈ [0, x + ] with κ = +1.The former corresponds to a solution with a single O4-plane singularity; the latter to a solution with a single O8-plane singularity.
• For σ = −9, the zero x + becomes triple; the allowed intervals remain the same as in the previous case, but the κ = +1 case now develops a Calabi-Yau conical singularity at x = x + .
• For σ < −9, the zero x + splits in a maximum x 2 , a minimum x 3 and a simple zero x + (all three greater than zero).There are now three allowed intervals: the old one x ∈ [x − , x 1 ], still for κ = −1, an interval x ∈ [0, x 2 ] for κ = +1, and a new one x ∈ [x 2 , x + ] for κ = −1.These two new possibilites correspond to a solution with an O8-plane and an O4-plane singularity, and to a solution with a single O4-plane singularity, respectively.(4.42) The massless limit is now obtained by taking β → 0 with s constant.We obtain (4.9) with These solutions uplift to M-theory AdS 4 × Y p,k solutions, where Y p,k are the well-known Sasaki-Einstein seven-manifolds of [19].To see this, one has to perform the further change of coordinate and set the constants in [19,Sec. 2] as {Λ, κ, λ} = {8, 12s − 1, 2ℓ/ν}.In the limit s → 0, the interval of definition for x shrinks to zero.However, one can define S ≡ s/ν 2 and take S → 0; in this limit the solution remains well-defined.After introducing the coordinate θ by cos θ ≡ 2/s x, it becomes (for KE 4 = CP 2 ) the IIA reduction of M 3,2 [22,23], as worked out in [7, (2.10)].

Regularity and boundary conditions
We now turn to the analysis of the geometry of the solutions, in a manner similar to the one in the previous section.
The metric (3.11) on the internal manifold takes the form: where q, u 1 , u 2 are polynomials.The warp factor is given by The dilaton is given by The above is valid for κ 1 , κ 2 ̸ = 0.When one of the two Riemann surfaces is flat, a slight modification is required and we will treat the case κ 2 = 0 separately.Notice that 3q ′ − xq ′′ = x 2 (u 1 + u 2 ) ; (4.53) so we see that for u 1 = u 2 we recover (4.9).Positivity of the metric and the dilaton now requires either Given the similarity between (4.52) and (4.9), most of the analysis leading to Table 1 is the same.There is the additional possibility of the occurrence of a double zero.Moreover, the case of an inflection point now ramifies into three different branches.See Table 2. Double zero: orbifold singularity.This did not occur in section 4.1, because a multiple zero was always a triple zero.This is no longer the case in the present section: a double zero which is not also triple can occur, and we must analyze it separately.A crucial fact is that ∆(q) ∝ res(q, u 1 )res(q, u 2 ).Thus when a double zero occurs either u 1 or u 2 has a zero.Choosing the latter, one finds the local expression for the metric while the warp factor and dilaton are constant.Here u 10 ≡ u 1 (x 0 ) and u ′ 20 ≡ u ′ 2 (x 0 ).From (4.53) we see that −2q ′′ 0 /u ′ 10 = 1.Positivity of the metric requires that x 0 (x 0 − x) > 0, and κ 1 = 1, which selects Σ 1 = S 2 .With the choice r = √ x 0 − x, the parenthesis becomes proportional to dr 2 + 1 4 r 2 (Dψ 2 + ds 2 S 2 ).If the S 1 periodicity is ∆ψ = 2π, this is the local metric for an R 4 /Z 2 singularity; if ∆ψ = 4π, it is R 4 , and we have a regular point.
Inflection point at the origin: O4-, O6-or O8-plane.In section 4.1, at an inflection point at the origin the denominator of the coefficient of ds 2 KE 4 has a double zero, canceling the double zero of the numerator, so that the full coefficient remains constant.In (4.52a), however, the functions u a in the denominator are independent of q.If neither of u a vanishes, the local metric and dilaton read According to the discussion underneath (4.14), this locus describes an O4-plane smeared over Σ 1 × Σ 2 .

.57)
This locus corresponds to an O6-plane singularity.Adapting our discussion below (4.14), we conclude that it is localized if Finally, if both u a vanish at the inflection point, there is an O8-plane singularity as in (4.19), with the KE 4 base replaced by Σ 1 × Σ 2 .
Quartic maximum at the origin: O4/O8-plane.When q ∼ q 0 + 1 4! q 4 x 4 , then both u a have a single zero.We have which is the appropriate behavior for an O4/O8-plane singularity.This case occurs only if κ a have opposite signs, so the O4-plane is smeared over one of the Σ a .

Generic case
Here we define new parameters and the constants L, g s determined by (4.24).We again note that the Riccati ODE (4.50) implies an ODE directly on q, u 1 , u 2 : similar to (4.25)In this case we will not give an exhaustive discussion as in section 4.1.2for the Kähler-Einstein base.There are many different possibilities, and a full discussion would be rather tedious.Let us instead give here a summary of the main features of the parameter space.
As in the Kähler-Einstein base case, the interpretation of the solution depends on the properties of the polynomial q, and less importantly on u 1 , u 2 .The most important features are the presence of extrema, and the presence of zeros.These can be decided again by looking at the discriminants of q and q ′ .Unlike in section 4.1, these are unrelated, and vanish on different loci of parameter space.It is also useful to notice that 2 26 3 10 ∆(q) = β res(q ′ , u 1 )res(q ′ , u 2 ) ∝ ∆(q); thus, the sign of u 1 , u 2 at an extremum of q (which has to do with the sign of κ 1 , κ 2 ) changes on the discriminant of q.The expressions of the resultants are res(q ′ , u 1 ) = −2 18 Depending on the signs of κ a , there are three possibilities to consider: (i) For κ a = −1 there are solutions with a regular endpoint and an endpoint with an O4-plane singularity.(ii) For κ a = +1 there are regular solutions and ones with orbifold singularities which were found numerically in [10].There are also solutions which for ∆ψ = 4π have CP 3 topology and were found numerically in [9].There are also solutions with one or two O4-plane singularities.(iii) For κ 1 = +1, κ 2 = −1 (or viceversa) there are solutions with one or two O4-plane singularities, as well as solutions with an orbifold singularity.
The flux potentials are again defined by (3.25), now with ȷ = Q 1 j 1 + Q 2 j 2 .Using ρ = ρ 1 + ρ 2 , with d 4 ρ a = −κ a j a (no summation), we find where ) 2 . (4.64c) The integration constant in the indefinite integral for the g a is chosen such that In this case we introduce

κ 2 = 0
In this case we need to adjust the form of the solution as presented in 4.2.1, by replacing the factor κ 2 in front of the line element of Σ 2 by a constant m, to be specified below.The functions that determine the solution read: where the coordinate x = y/L 2 , with L 2 related to the constant parameters appearing in (4.51) as: For n, l and m we have the following relations: Finally, g 2 s = 2/( √ 3F 2 0 L 8 ).In this case: • κ 1 = +1: there is a solution with an O6-plane smeared along the T 2 , for l ̸ = 0.This becomes an O8-plane for l = 0.For n > 1 it also has an O4-plane singularity; for n = 1 it has an orbifold singularity; for n < 1, n ̸ = 0 no other singularity.

F 0 = 0
Similarly to section 4.1.5,we define the coordinate x = ν 2 ℓ 2 y; this differs from the x defined in section 4.1.2by a factor γ 2 , recalling the definitions in (4.59).Let us define t i ≡ γ i √ β.Taking the limit β → 0 with t i kept constant yields now the solution ( Moreover L 2 = ℓ 2 κ 1 t 2 /(ν 1 t 1 ), g s = 2L 3 /ℓ.Regular solutions exist for κ a = +1.The parameter space is obtained by considering the equations for the resultants in (4.62), after taking β → 0 with t i kept constant; this gives two cubics, and the allowed parameter space is enclosed by them.The result is shown in figure 2. (Points related by inversion t 1 ↔ t 2 correspond to the same solution.)These correspond to the solutions found in [20,Sec. 4.5] and [21], more or less in the coordinates of the first reference.They were studied in more detail in [10,Sec. 3] where they were referred to as A p,q,r solutions.
• The green boundary (obtained by replacing the inequality with equality in (4.73)) corresponds to solutions with a conical Calabi-Yau singularity.For KE 4 = CP 2 , this is an orbifold singularity C 3 /Z 3 .
• The point at β = 1, s = 0 has two conical Calabi-Yau singularities.For KE 4 = CP 2 , these are both orbifold singularities C 3 /Z 3 .In this case, however, it is also possible to take ∆ψ = 6π, in which case the space becomes fully regular; this is the Guarino-Jafferis-Varela solution [11], whose generalization for arbitrary KE 4 was considered in [27].Solutions with product base and without orientifolds.There are regular solutions with κ a = +1.The parameter space is shown in figure 4, which is defined by certain branches of res(q ′ , u a ) = 0 (see (4.62)).This time we use the original parameters γ 1 , γ 2 , β.These solutions were discussed in [10,Sec. 5] in detail, although they were only known numerically in that paper; thus we will be brief.
• The region in figure 4 between the plotted surface and β = 0 corresponds to regular solutions with the topology of an S 2 -bundle over CP 1 × CP 1 .
• The limit β → 0, with t a = γ a √ β kept constant, reproduces the A p,q,r massless solutions whose parameter space was shown in figure 2.
• The boundary of figure 4 corresponds to solutions with a Z 2 orbifold singularity.
• The intersection of the boundary with the locus {γ 1 = −γ 2 } (visible as the ridge in figure 2) corresponds to solutions with topology CP 3 /Z 2 .In this case one also has the option of taking ∆ψ = 4π, thus making the topology directly CP 3 .These are the solutions studied in [9].
Figure 4: The allowed moduli space for regular solutions in the generic case with product base is between the plotted surface and the plane β = 0.
Solutions with O8-planes.There are many solutions which are regular except for a single O8-plane singularity.They occur for ℓ = 0, both for KE 4 and for product base.Here is a list of possibilities: • KE 4 base with κ = +1, and σ > −9 in (4.39).

Figure 1 :
Figure 1: Plots of q(x) corresponding to various regions of parameter space.

4. 1 . 5 F
0 = 0 In this limit, the rescaling of the coordinate y to the coordinate x is parameter µ is rescaled to s as µ = 3(ℓ 4 /ν 2 )s.From (4.22) we see

1 βFigure 3 :
Figure 3: The allowed parameter space for regular solutions in the generic case with KE 4 base is the interior of the purple region.

Table 2 :
Additional singularities that occur in the product base case.The O8-plane one, which already appeared in Table1, is repeated for comparison.